Contributions to the Theory of Nonlinear Oscillations (AM-41), Volume IV 9781400881758

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ANNALS OF MATHEMATICS STUDIES

Number 41

ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers, by H

ermann

W

eyl

3. Consistency of the Continuum Hypothesis, by K u r t G o d e l 11. Introduction to Nonlinear Mechanics, by N. K r y l o f f and N. B o g o l i u b o f f 16. Transcendental Numbers, by C a r l L u d w i g Siegel 19. Fourier Transforms, by S. B o c h n e r and K. C h a n d r a s e k h a r a n 20. Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. L e f s c h e t z 21. Functional Operators, Vol. I, by Jo h n

von

Neumann

24. Contributions to the Theory of Games, Vol. I, edited by H. W. K u h n and A. W. T u c k e r 25. Contributions to Fourier Analysis, edited by A. Z y g m u n d , W. T r a n s u e , M . M o r s e , A. P. C a l d e r o n , and S. B o c h n e r

A.W. T u c k e r

28. Contributions to the Theory of Games, Vol. II, edited by H. W. K u h n and 29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S.

L efsc h et z

30. Contributions to the Theory of Riemann Surfaces, edited by L. A h l f o r s et al. 33. Contributions to the Theory of Partial Differential Equations, edited by n e r , and F. Jo h n 34. Automata Studies, edited by C. E. Sh a n n o n and J. M

L. B e r s , S. B o c h ­

cC arthy

36. Contributions to the Theory of Nonlinear Oscillations, Vol. Ill, edited by S. L e f s c h e t z 38. Linear Inequalities and Related Systems, edited by H. W. K u h n and

A. W.

T uck er

39. Contributions to the Theory of Games, Vol. Ill, edited by M. D r e s h e r , A. W. T u c k e r

and P. W

olfe

40. Contributions to the Theory of Games, Vol. IV, edited by R. D u n c a n L u c e and A. W. T ucker

41. Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S. L e f s c h e t z 42. Lectures on Fourier Integrals, by S. B o c h n e r . In preparation 43. Ramification Theoretic Methods in Algebraic Geometry, by S. A b h y a n k a r 44. Stationary Processes and Prediction Theory, by H. F u r s t e n b e r g 45. Contributions to the Theory of Nonlinear Oscillations, Vol. V, C e s a r i , L a S a l l e , and L e f s c h e tl

46. Seminar on Transformation Groups, by A. B o r e l et al. 47. Theory of Formal Systems, by R. Sm u l l y a n 48. Lectures on Modular Forms, by R. C. G u n n i n g 49.

Composition Methods in Homotopy Groups of Spheres, by H.

50.

Cohomology Operations, lectures by N. E. Steenrod, written and revised by D. B. A.

51.

Lectures on Morse Theory, by J . W . M il n o r

52.

Advances in Game Theory, edited by M. D r e s h e r , L. Sh a p l e y ,

Toda

E p s t e in

and A.W. T u c k e r

CONTRIBUTIONS TO THE THEORY OF NONLINEAR OSCILLATIONS V O L U M E

H. ANTOSIEWICZ

I V

W. T . KYNER

R. BASS

S. LEFSCHETZ

D. BUSHAW

L. MARKUS

R. De VOGELAERE

P. MENDELSON

S. KAKUTANI

G. SEIFER T

D. SLOTNICK

EDITED BY S. LEFSCHETZ

PRINCETON PRINCETON UNIVERSITY PRESS

1958

Copyright

0 , A and B are real numbers. This delay-differential equa­ tion also occurs in the theory of certain servo-mechanisms [3]. If B = 0, the resulting differential equation is elementary and we shall henceforth assume B 4 °* We simplify equation (1 ) by writing z(t) = BTy(Tt). Then (2)

z !(t) = [a - z(t - l)]z(t)

,

where a = A t . We shall investigate the functional equation (2) and the results can easily be reinterpreted for (1 ). DEFINITION. A solution z(t) of (2) Is a real con­ tinuous function defined on 0 < t < 1 + e, e > 0 , where z(t) € 1 ^ on 1 < t < 1 + e, and there satisfies the functional equation (2). Clearly the constants z = 0 and z = a are solutions. It Is interesting to note that these are the only solutions of period one since 1

KAKUTANI AND MARKUS

2 z ’(t) = [a - z(t)3z(t)

has no other periodic solutions.

Quite general existence theorems are available [1, 2] for differ­ ence-differential equations. However we obtain precise knowledge of the behavior of the solutions of (2) and in our detailed investigations the general theory is not directly applicable. 2. GENERAL PROPERTIES OP THE SOLUTIONS THEOREM 1. Let cp(t), 0 < t < 1 , be a continuous, real-valued function prescribed as an initial con­ dition. Then there exists a solution z(t) of (2), defined on 0 < t < °o, for which z(t) = cp(t ) on 0 < t < 1. Moreover z(t ) is unique in that each solution of (2), agreeing with cp(t ) on 0 < t < 1, also agrees with z(t) on their common domain of definition. PROOF.

Let

z(t) = cp(t ) on

0 < t < 1.

Define

t-1 z (t ) = cp(i ) exp

on 1 < t < 2. then define

If

z(t)

a(t - 1 ) -

J '

is well-defined on

cp(s )ds

0 < t < n, n =

2,

3,

O

.••,

t-1 z(t ) = z(n ) exp

a(t - n) -

J

z(s )ds

n-1 on n < t < n + 1. Clearly z(t) over z(t) € C^1 ^ on 1 < t < °°, At

is continuous on 0 < t < °o and moreexcept possibly at t = n.

t = n, z !(n + 0) = z(n)(a - z(n - 1)]

and n-1

z !(n - 0) = z(n - Ota - z(n - OJexp

J

n-2 = z(n)[a - z(n - 1)J• Thus

z(t) € C^1 ^ for

1 < t < 00 and there satisfies

z(s)ds

A DIFFERENCE-DIFPT1RENTIAL EQUATION

3

z 1(t) = [a - z(t - 1 )]z(t). If w(t) is another solution, corresponding to the initial func­ tion cp(t ), then let tQ > 1 be the l.u.b. {t | w(t) = z(t)). But on t0 < t < tQ + 1, t-1 z(t ) = z(t0 ) exp

a(t - tQ )

J

z(s)ds

V1 and t-1

w(t) = ¥(t ) exp

Since ¥(tQ ) = z ( t Q ) ¥e have ¥(t) = z(t) of the finite number definition. on

J ¥(s)ds V1

a(t - tQ )

and furthermore ¥(s) = z(s) on tQ - 1 < s < tQ, on t < t < tQ + 1 ¥hich contradicts the existence tQ . Thus ¥(t) = z(t) on their common domain of ).E.D.

Hereafter, by a solution of (2), 0 < t < 00.

ve

shall mean a solution defined

COROLLARY. The solution z(t) € C ^1 ^ on 0 < t < 00 if and only If cp(t ) e C ^1 ^ on 0 < t < 1 and also cp1(1 ) = [a - cp(0 )]cp(1 ). PROOF. If z(t) e C ^1 ^ on 0 < t < 00 then cp(t ) € C ^1 ^ on 0 < t< 1 . Furthermore z(t) satisfies the functional equation (2) at t = 1+ €, e > 0, and thus at t = 1. But at t = 1, z( 1 ) = cp(1 ), z 1(1 )= cp1(1 ) and cp1(1 ) = [a - cp(o)]cp(l). Conversely if cp(t) € C^1 ^ on 0 < t < 1, then z(t) € C (1 ) on 0 < t < 00, except possibly at t = 1. But both z 1(1 - 0) = cp1(1 ) and z f(l + 0) = [a - cp(0) ]cp(1 ) exist and are equal. Thus z(t) e C'(1 ) on 0 < t < °o. Q.E.D. Clearly, if ¥e require cp(t ) e C^°°^ on 0 < t < 1 and also that the derivatives of cp(t ) at t = 1 be related to those at t = 0 by the functional equation (2) and by those equations obtained from (2) by differ­ entiation, then the solution z(t) e C on 0 < t < 00. It is not apparent ¥hether or not there are analytic solutions of (2). The next theorem sho¥s that there are no (entire) analytic solutions of (2) (other than z = a, z = 0) ¥hich can be expressed in terms of elementary functions.

KAKUTANI AND MARKUS

THEOREM 2. Let z(t) be an entire function of the complex variable t and let z(t) satisfy z!(t) = [a - z(t - 1)]z(t). Then, unless z = 0 or z = a, for each integer k, max |z(t)| = M(r) > exp exp ... expr

111 =r

(k repetitions) for all sufficiently large r. PROOF. If z(tQ) = 0, then z!(t0) = o andclearly z^n^(tQ) so that z = 0. Thus assume that z(t) has no zeros. If z(t) is of finite order, then by Hadamard’s theorem, z(t) = exp P(t) where P(t) a polynomial. But then P’(t) = [a - exp P(t - 1)] which is impossible unless P(t) Is a constant. But the only constant solutions are z = o and z = a. Now allow z(t) to be of Infinite order. Let Log z(t) be an entire function such that exp Log z(t) = z(t). For any entire function f(t) one knows1 max It | = r

|f'(t)| < —

(a (r ) + f(o))

,

“ ( R - r )2

where A(R) = max Re{f(t)) 111=R provided A(R) >0, C is a positive constant, and o < r < R. Bet R = 2r, f(t) = Log z(t) and we obtain max — ■Log z(t) < ~ |t|=r " p

max Log |z(t)| + C2 =2r

111

where C]P C2 are positive constants and the above result applies since max 111= 2r

Log |z(t)| = log

4

max

|z(t)| >■ > 0

I 111= 2r

Set M(u) = max |z(t)| 11 |=u Dr. J. Wermer suggested this argument.

J

A DIFFERENCE-DIFFERENTIAL EQUATION

5

Then z\

(t) z(t)"

max 111 =r

for a positive constant

< -yf log M( 2r]

• But |z(t)|
a for all large t, then a - z(t - 1 ) < 0 and z f(t) < 0. Thus z(t) decreases monotonely to a limit which is easily seen to be a. Furthermore z"(t) = {[a - z(t - 1 )]2 - z'(t - l )} z(t) > o

KAKUTMI AND MARKUS

8 so that zero.

z f(t)

increases monotonely to a limit which is easily seen to be

If z(t) < a for all large t, then a - z(t - 1) > o and z 1(t) > 0. Thus z(t)increases to the limita. Also z 1(t) > [a - z(t)] z(t) > [a - z(t)]2 for large t so that z"(t) < 0. Thus z'(t) decreases monotonely to the limit zero.

z(t)

If neither z(t) > a nor z(t) < a oscillates about the line z = a.

for all large

t,

then Q.E.D.

As we shall later see, each of these alternatives is possible for certain values of the parameter a. However, before determining which values of the parameter a produce which behaviors, we shall Investigate the qualitative form of the oscillatory solutions. THEOREM 7 • Let a > o, cp(1 ) > 0 and z(t), the solu­ tion of (2) corresponding to cp(t ), oscillate about z = a.Assume the zeros of z(t) - a are a discrete set on o < t < co. Then, for sufficientlylarge t, each zero of z(t) ~ a is simple and there is exactly one zero of z !(t) between consecutive zeros of z (t ) - a . PROOF. Let the number of zeros of z(t) - a on n < t < n + 1, n = 1, 2, 3 , •••> be k(n). Then define the number k(n) of potential zeros to be k(n) = k(n) If the last zero = n + 1 or if |z (t ) - a | > o on We

max(n, t^ ) < t < n + 1 and define k(n) = k(n) + 1 inall other cases. show that k(n) is a non-increasing function of n.

Suppose k(n) = k(n) and t^ = n + 1. Then thero are at most k(n) - 1 bend points of z(t) - a in n + 1 < t < n + 2. Thus k(n + 1) < k(n) and for equality one must have a situation in which k(n + 1) = k(n + 1). Therefore in this case k(n + 1 ) < k(n). Next suppose

k(n) = k(n) ^

and

|z(t) - a| > o

on max (n, t^) < t < n + 1. Then on n + 1 < t < n + 2 there are at most k(n) bend points, at least one of which must occur before the first zero of z(t) - a. Thus k ( n + i ) < k ( n ) . If k ( n + l ) = k ( n ) , then no zero of z f(t) occurs following the last zero of z(t) - a on n + 1 < t < n + 2,

A DIFFERENCE-DIFFERENTIAL EQUATION and hence

k(n + 1 ) = k(n + 1).

Therefore

9

k(n + 1) < k(n)

in this case.

Finally suppose k(n) = k(n) + 1. There are at most k(n) bend points of z(t) - a on n + 1 < t < n + 2 and thus k(n + 1) < k(n) + 1. But if k(n + 1) = k(n) + 1 the situation is such that k(n + 1) = k(n + 1). Therefore k(n + 1) < k(n) in all cases. Now let lim k(n) = k n=oo and say, for n > N, k(n) = k. Suppose on N + r < t < N + r + 1, r = 1, 2, 3 ,..., there are at least two zeros of z 1(t) on the open In­ terval between two successive zeros of z(t) - a. Then there are at least k(N + r) zeros of z 1(t) on N +p r < t < N + r + 1. But then there are at least k(N + r) zeros of z(t) - a on N + r - l < t < N + r , that is, k(N + r - 1)>k (N + r). Furthermore equality holds only when k(N + r - 1) = k(N + r - 1) + 1. Thus k(N + r - 1 ) > k(N + r) which con­ tradicts the property that k(n) = k for n > N. A similar argument shows that a double root of results in a decrease In k(n) and so can not occur for

z(t) - a n > N.

also Q E.D.

COROLLARY. Let a > 0, cp(1 ) > o and let the solution z(t) oscillate about z = a. Let cp(t ) - a e C ^1 ^ on 0 < t < 1 have at most one potential zero (in particular if cp(t ) - a 4 0), then the interval from each zero of z(t) - a, t > 1, to the following extremum is of length one. The interval from an extremum of z(t) - a, of amplitude e > o, to the following zero is of length d€ > ™ log (1 + e/a) If

a > 1 and if

a

e

.

is sufficiently small, then - c/2a

PROOF. If cp(t ) - a has no zeros on ° < t < 1, then z 1(t) / C on 1 < t < 2 and k(l ) = 1. If cp(t ) - a has one zero followed by a zero of cpl(t) on o < t < 1, then again z'(t) has one zero on 1 < t < 2 and k (1) = 1. Now consider an extremum where z 1(tQ ) = oon n < tQ < n + 1, n = 1, 2, ... . Then on n - 1 < t < n there is one, and thus only one, zero of z(t) - a, since k(n - 1) = k(n - 1) = 1. If there were another extremum on tQ - 1 < t < tQ, then k(n) > 2 which is impossible. Therefore

KAKUTANI M D MARKUS z(t) - a

is strictly monotone on Let

tQ - 1 < t < tQ .

t1 be an extremum where

z(t1) = a + e.

Then

-e(t-t- ) z(t) > (a + e )z for

t1 < t < t1 +

But for t - t 1 = — log (1 + e/a) , -e(t-t1 ) (a + e )c

Thus

- (a + e) ( a f r )

= a

d 6 > “ log (1 + e/a). For a minimum where

z("t2 ) = a -

g,

e(t-t2 ) z(t ) < (a “

g

< a

)'

when (t - t ) < ~ log Thus J

d G > min If 1

g

J

log (1 + e/a),

is small, say

g

< a,

a log a- g

log (1 + c/a)

then

E/a - ~ (c/a)2 + j (c/a

E/a

- ™

( g / u )c

Thus

Now take a > 1. Suppose that following a maximum z(t1 ) = a + z(t) >a on t1 < t < t 1 +2. But then on t1 + 1 < t < t1 + 2^ - z ’(t) > g 1z(t) and -

z(t) < (a +

g

(t-t--1 )

g. j)e

where g1 = z(t1 + 1 ) - a. Then z(t 1 + 2) < (a + which is a contradiction. In this case d £ < 2. z(t) and

Similarly, after a minimum where €2z(t )

-

A DIFFERENCE-DIP^ERENTIAL EQUATION where e2 = a - z(t2 + 1). here also d£ < 2.

Then

z(t2 + 2) > (a - e2 )(1 + e2 ) > a.

Thus Q.E.D.

The most interesting oscillations are those which are strictly monotone from each zero of z(t) - a, for a unit length, until the following extremum. Such an oscillation is concave up on t 1 + 1 < t < t2 + 1, where t1 and t2 are successive maxima and minima respectively. Thus on t2

€

log (l + € / a ) < t < t 2 + i

or on

where € < a is the amplitude of the maximum, the solution cave upwards.

z(t) is con­

THEOREM 8. Let 0 < a < 1 and let z(t) - a oscillate with discrete zeros. Then the oscillations are damped and lim z(t) = a t — 00

PROOF. Let €1, €2, ... and 5 1, 8 , ... be the amplitudes of successive maxima and minima respectively, with 5 ^ occurring before and we consider only large t so that the oscillations have the form described in Theorem 7 * Suppose e^^ is the maximum which follows by one unit the first zero of z(t) - a after 5 1 . Then clearly e ^ is the maximum following by one unit the first zero of z(t) - a after &n , n = 1, 2, ... . Also 5n+k+1 after e .

rr]iriim' un] following by one unit the first zero of

A trivial estimate yields n and a

“

8n+ k+ 1 >

ae

n

But then €n < a

|e x p

a

en €n < ^ e.n-2k-1 • Consider the function

We show that ldjL

1 -

e

n

1

-

1 j-

z(t) - a

12

KAKUTANI AND MARKUS f*(€ ) = a jexp a

Now

f(o) = o

€ -- > f(e)

itself.

1 - e G - e/a

exp a

e > 0, f r(e ) < a

Thus

- ij*

and f 1(€ ) = a

Thus, for

1 - e“€J

Since

exp a

< a

1 - e - e

o

onto

¥e have

llm enl+(2k+1)n = 0 n=co where

n . = 1, 2, .. ., 2k.

But then lim €n = 0 n=oo

Since 5n+k+l < a

1 - e

n

lim n=oo

5n

= o

Therefore lim z(t) = a t=co

Q.E.D.

Thus for 0 < a < 1, cp(1 ) > 0, every solution asymptotic to z = a or is a damped oscillation and thus

z (t) is either

lira z(t) = a t— It seems likely that for a > 1, some of the oscillations are not damped (there are only oscillations, see Theorem 9 ) and do not approach a limit value. THEOREM 9. Let a > 1/e and cp(1 ) > 0. Then no solution z(t) is asymptotic to z = a (except z(t) s a). PROOF. If z*(t) = [a - z(t - 1 )]z(t), let y(t) = ~ z(t) and then y(t) is a solution of y !(t) = a[1 - y(t - l)]y(t) whichis asymptotic to y = 1if and only if z(t) is asymptotic to z = a. Let ?(t) = 1 - y(t) and then £(t) ^ o with ?’(t) ? 0 in casez(t) ^a or y(t) ? 1 as t — — > Suppose

z(t)

is asymptotic to

z = a

from below.

Now

A DIFFERENCE-DIFFERENTIAL EQUATION

13

H-LHU , a i i E l l i [, _ {( t) ] t(t) £ o. Write

a[l - £(t)] = ay(t) > a* = a - e > ^ >

R(t) = ~

for

> 0

e(t) and then t R(t) > a 1 exp

R(s)ds

J

t-1 Say on n - 1 < s < n, R(s) > M > 0. Then on n < s < n + 1, R(n) > a ’eM > M and (since there is no point at which R(s) = a ’eM ) M R(s) > a ’e . By induction on n + k < s < n + k + l we have R(s) > a ’exp R(n + k - 1). M M Consider the number sequence C1 = a ’e , C2 = a ’ exp a ’e , ..., Cn+1 = a ’ exp Cn, ... . By observing that the graph of a ’ex lies above that of x, we note that Cn+1 > Cn and lim C = + oo n=°° n Thus whenever a > 1/e,

-- > + oo

~ 5 (t)

as

t -- > oo. Let

T

be so large that ~ -S-'.(. t) > k 5 (t)

whenever t > T - 1 where k is a largeconstant specified later. on T < t< T + 1, R(t) > k, then

R (t + I + where

o
a'ek/2 = a,

. Also

R (t + |- + t2)

> a' exp (j + -q- a'e1^ 2)

= a2

Since

KAKUTANI AND MARKUS where

R

where

(

0




1 °°n

2

log 100 + 2

an+1 = ane,

so that

100.

>

an/2n+1

...

.

Then one easily calculates

so lim aR = + °° n=oo

But this implies that R(t) isnot bounded on contradicts the assumption that£(t) 4 0and thus from below is impossible.

T < t < T + 1.However this the asymptoticapproach

Next we show that there are no asymptotic solutions to y = 1 from above when a > i/e. Here let |(t) = y(t) - 1 and suppose |(t) IV 11(t) t o , and -

4 1(t)

r1

I

r

ft ) ]

TTtT

Also take

t

so large that

0,

^0-1 ) ITET

•

a [1 + |(t)] > a > 1/eand let

0 '

p(t) - + ! + > Again t P(t) = a exp

J"

P(s)ds

t-1 and P(t) -- > + 00 as t -- > 00. Take P(t) > k whenever t > T - 1 and as above P(t) is not bounded on T < t < T + 1 which contradicts the assumption of an asymptotic solution with |(t) 4 °* Q.E.D. Theorems 8 and 9 show that if 1 /e < a < 1 and cp(1 ) > 0, every solution is a damped oscillation tending to the limit value of The next result is the complement of Theorem istence of asymptotic solutions for 0 < a < 1/e.

9

then a.

and proves the ex­

A DIFFERENCE-DIFFERENTIAL EQUATION

15

THEOREM 10. Let 0 < a < 1/e and cp(l ) > 0. Let the Interval between the zeros of z(t) - a be at least one, for large t . Then the solution z(t) Is asymptotic to z = a. PROOF. Suppose z(t) has an oscillation, necessarily damped, about z = a. Then y(t) = ~ z(t) is a solution of y 1(t) = a [1 -y(t - 1 )]y(t), and has a damped oscillation about y = 1. More­ over the interval fromeach zero of y(t) -1 (for large t > T) to the following extremum is of length one. Consider the family of curves w = 1 - Be""^ for B > 0. We shall show that there exists a curve w(t) of this family which is tangent to y(t) between the minimum at tQ > T and the following zero t * of y(t) - 1. Along y(t), define B(t), tQ < t < t f, as the parameter value of the curve of the family through (t, y(t)). Then near tQ, B(t) is monotone increasing. Also lim B(t) = 0 t=t!

.

Define B =

max B(t) t p

we have several possibilities.

Ib. q > p + 2. This requires m- = p + 1, q, Taking first n = p + 1, y = xp+1y l we find

and hence

r = q.

dy, (xy -a)(y.-bx^'P-1) + (p+i )y (y -cxq-p~1 ) . - l i ---- 3----- 1----------- H -;rT 1 1--------dx - x(y1-cxq P )

(3.1,)

The critical equation is (3.5)

(p + Oy^ - ay1 = 0

Its only root 4 0

is

y 1 = p-f--f = “ • Setting

(,.6 )

H-v dx

Hence thispoint For (3-7)

. y 1 - a = y*

= Xx,+ ay* + ... _- ax rw-v + .. .

is a saddle-point and so it is ruled m-=

q

and

there follows

out.

y = x % 1 we have

dy1 > (y1xq_p-a)(y1-b) + q y ^ ^ P -1(y^c) - xq"P(y1-c)

dx

The only critical value is

y 1 = b.

(3.8)

Setting

, _ - a y *

dx

y1 - b = y

+

there follows

.

- xq~P(y +b-c )

a Bendixson critical point and as before this rules out an s. n. o. hence

(3.9 )

Ic. q = p + i. Hence i rP+1ir1. This time y = xp+1y

r = p + 1 and we only have

iy, (y,x-a)(y -b) + (p+1)y (y -c) Li- ! ------ ! ------------! -- ! --dx - x(y1-c)

The critical point equation is f ( y !) =

(p +

1 )y,(y,

- c ) - a(y1 - b ) =

o

n = p + 1,

2k

LEFSCHETZ

Suppose first b/ c. Since r| is aboverv necessarily b > c. Since some TO-curves must be above and some below r b must be between the roots of f(y1 ) if there is to be an s. n. o. This requires that f(b) < o. However f(b) = (p + 1)b(b - c) > 0 and so there is no s. n. o. here. Suppose now

b = c.

: y = a(x),

More generally we may have representations r| : y = p(x),

ry : y =

where p , y are of order p + 1, P - 7 = 8 if 6 = dxs + ..., then d > 0. Thus here

y(x)

is of order

, s > p + 1,

and

dy ^ (y-a)(y-p) dx -( y - 7 ) The regular transformation x = x,

y = z + p(x)

replaces topologically the right of the y axis and its s. n. o.'s by the right of the z axis and its s. n. o .1s. Thus it will be sufficient to eliminate the latter. Now dz ^ (z-(a-ft))z + ft} « (z+ 5 ) dx - (z+5 ) ( 3 *1 0 )

= zg--.f.(.?.)2 -t.P .L g . - (z+5 ) (z-5 1 (x) )(z-5 2 (x)) - (z+5 )

Notice that since ftf 5 = (p + 1 )bdx^+c^ + ...

,

ft! 5> 0 for x small. Hence if s ^ s2 are the orders of £ , £ then S1+ s2 = p + s and £2 have the same signs for x small. Since 5 is then also positive, (3.10) is like our initial system but with rv be~ 1 2 low Ox. If £■]> £2 are > °; rn an(^ rH are a^ove and an el0“ mentary sketch excludes at once an s. n. o. Suppose now replaces (3•10 ) by

£2

both

< 0.

The change of variable z = - z1

ON CRITICAL POINTS dz

25

(2,+e,)(z1+?2 )

dx

- (zr O

where now - -j, s > o ^ or x s m a P P a n & their orders are still s1 , s2, s with s1 + s2 = p + s. Suppose s1 g s2 . The only possibility for an s. n. o. is s2 = s1 + 1 = s. This yields s = p + 1 which contradicts s > p + 1 and eliminates the s. n. o. in the present instance. To sum up then no s. n. o. may arise under case I. b . Passing now to case

x = u2

II of No.

3,

a first change of variables:

will yield dy ^ 2u(y-¥(u)-au2p;f1 )(y-¥(u)+au2cp*~1 du - (y-cu2r) ¥ = cm2p + ... + Pu2q-;

q > p;

a,

a > 0

From Figure 1 we know that an s. n. o . requires TO-curves between the two branches and so of order 2p, and also that c > 0. We apply then the transformation u = u,

y = u2py

and obtain u2 (y1-a )2 + py^y-j-cu2^1* p ^ du

- u(y1-cu2^r“p ^)

There are now two possibilities: (a) r > p. The only critical point is at the origin. This means that possible TO-curves are of order > p and it excludes an s. n. o. (b) r = p and hence c 4 0. There are critical points at the origin and at y 1 = c. The origin has characteristic roots c, - 2pc so that it is a saddle point. The point A is of Bendixson type. Therefore, as under case (a), there is no s. n. o. This completes the proof that thesystem s. n. o. to the right of the y axis.

(1.1)possesses

no

5 . It is much easier to show that(1.1) possesses no s. n. o. to the left of the y axis. To begin with the change of variables x > - x, y -- > y reduces (1.1) to the type

(5.1 )

LEFSCHETZ

26

for which we must show that it has no s. n. o. to the right of the

y

axis.

Upon drawing the possible sketches for (5.1 ) it is readily found that Figure 2 is the only one consistent with an s. n. o. The two branches are separated by the x axis but the branch may run above

FIGURE or below

Ox I.

2

or even coincide with it.We have rH : y ^ rH : J

’

possible cases:

a >0

" + 13x^

r . y = cxp, V *

II.

again two

h
p or else and hence c 4

r = p, 0

rH1 : y =. axP + 2 P + LH * tr : y = cxr,

c > o,

r > p +

ON CRITICAL POINTS

27

The justification of the representations for the two Their general form is

is as follows.

y = A(x) + >/a In order that they he separated by Ox, must be of lower order than A(x) and this leads to the indicated representations. 1 to ■)£ As a matter of fact the change of variables x ' > x re­ duces II to the type I, so that we only need to consider the latter. Thus t (5 *1 ) takes the form (5.2)

dy ■ (y-a3cP)(y+taq ) dx (y-cxr )

If q < p, the transformation y -- > - y will replace (5 *2 ) by a similar equation with q > p. Observe also that r > p. Thus we may assume q > p. The Newton polygon method gives rise to the following special cases:(a) r = p. The possible orders are ia = p, q + 1 . The order q + 1 is out since an s. n. o. requiresTO-curves of orders p and g q.Thus pis the only possible order. The adequate transformation is y = x^)y 1 . It yields dy 1 # x(yl-a)(y1-bxq“p ) + p y ^ y ^ c ) x(y1-c)

dx

The only admissible critical point is y 1 = c. It is found to be a Bendixson point and hence excluded as before (see 3 , la). (b) r > p; q = p + 1 . The only possible order Is Since one order must be g p this case is ruled out. (c) r > p, q = p. and is ruled out as under (b).

The only admissible orderIs

(d) q > p + 1 .Here we find and sothis case is ruled out also. Thus no s. n. o. can 6. There remains to crossed by the y axis.

|i= q, p +

1

= p +

1.

|i= p +

which excludes |i< p

occur to one side of the yaxis. be shown that at most ones.

n. o.can be

Prom (3 -1 ) follows that the polar coordinate equation for Is SI - * y(y - ex")

de/dt

LEFSCHETZ

28

Since r > 1 the only possible tangent of the paths at the origin is y = o. Upon drawing one s. n. o. tangent to Ox, it is found to absorb one Iy branch (Figure 3). A second set would have to cause the existence of a second such branch. Since there is only one, there can exist at most one set of nested ovals crossed by the y axis. This completes the proof of our theorem.

FIGURE 3 That a set of nested ovals corresponding say to Figure 3 can exist is proved by the system $2 .

dx

Universidad Nacional de Mexico

=

--SL

y - x

2

.

III.

OPTIMAL DISCONTINUOUS FORCING TERMS D. Bushaw § 1 . INTRODUCTION

This paper is devoted to a problem which arose originally in the theory of automatic control system design (see [2 ] or [7]). The problem may be stated in the following form: If g(x, y ) is a given function, find a function the properties:

(1

)

assumes only the values

-

(A)

cp(x, y)

(B)

For any point (xQ, yQ ) in some plane domain R containing the origin, a solution x(t) of the differential system

1

and

cp(x, y ) with

+ 1.

x + g(x, x) - q>(x, x), x( 0 ) = xQ, x(0 ) - yQ exists, and there exists a (least) positive value of t, say t , such that for this solution x(tQ ) = *(tQ ) » 0. (C) For all points in R, tQ is a minimal with respect to the class of functions cp satisfy­ ing (A) and (B).

Despite the essentially variational character of this problem, all attempts to apply standard variational methods to it have proved fruitless, and it has been found necessary to develop special approaches which might be described as elaborate combinations of fairly elementary techniques. The present paper is divided into three principal parts. In first part (Sections 2 and 3) the problem is recast in a form in which geometric point of view comes to the fore; in the second part (Section certain general results are obtained which lead to a simplification of 29

the the k) the

BUSHAW

30

problem; and in the last and longest part (Sections 5 -7 ) I give the .solu­ tion of the problem for the case in which the given function g is linear. §2.

TERMINOLOGY

Instead of the differential equation (1 ), it will be convenient to consider the corresponding ,!phaseM system (2)

x = y, y = o and y(t) — — > 0 as t -- > t , the path being parametrized in the obvious way in terms of the parametrization of the constituent arcs. *0

5

) No two of the arcs intersect.

In order to avoid a conflict between 3 ) and 5), each arc is to be regarded as containing its initial point but not its terminal point; this convention will not restrict the generality of what follows. A path from p can therefore almost be described as a curve which could occur as that part of a solution from p (for some 9) which connects p with the origin. "Almost, n because 5) need not hold for every solution of (2); but since our object is to find solution curves of shortest possible time length,nothing will be lost if we leave self-intersecting solutions out of consideration. A path from, p whose time length t is not longer than that of any other path from p will be called a minimal path from p. In order to solve the problem, stated in Section 1 it is sufficient to prove that there exists a unique mini­ mal path from each point p in some domain R con­ taining the origin. Namely, one needs occur In the minimal paths, the minimal paths. Such a as solution curves, and the the solution curve of least origin.

only to define 9(x, y) = + 1 on P-arcs which and 9(x, y) = - 1 on N-arcs which occur in 9(x, y) automatically yields the minimal paths minimal path from, a point p is, by definition, possible time length connecting p with the

This method defines 9(x, y) uniquely at every point of R ex­ cept the origin, where the value of 9 doesn't matter. To verify this, observe first that every point p of R must lie on at least one minimal path, the minimal path which begins at p. Thus 9 is given a value at every point of R. This value is unique, for if there were a point p at which it failed to be unique, this point would necessarily lie both on an N-arc belonging to one minimal path A 1 (say from the point p 1 ) and on a P-arc belonging to another minimal path A2 (from p^). Denote those parts of A^ and Ag which lie between p and the origin by A 1 and A2 respectively. Their time lengths t (A^ ) and t (a 2 ) stand In some relation to each other, say t (a 1 ) < t (a 2 ); then t (a 2 - A2 + a 1 ) < t (a 2 ).

OPTIMAL DISCONTINUOUS FORCING TERMS

33

The curve a 2 - A2 + A 1 may not be a true path (for it may cross Itself), but a true path may be obtained from it by cutting out whatever closed loops or retracings It may contain; and the resulting path is clearly short­ er in time length than A2, but this contradicts the assumption that A2 was the unique minimal path from p2 . §4 . CANONICAL PATHS A path will be called canonical If it contains no NP-comers above the x-axis and no PN-corners below it. In saying that a corner lies above or below the x-axis, I mean that nearby parts of the arcs meeting at the corner are above or below it; the corner itself, regarded as a point, may lie on the axis. THEOREM 1. Given any path A from p which is not canonical, one can find a canonical path from p whose time length is less than that of A. The idea of the proof is simple: given, say, a path with the NP-corner p above the x-axis, one denotes by p l either the last corner of the path preceding p or the last intersection preceding p of the path with the x-axis (whichever is nearer to p), and denotes by p 11 the corresponding point following p on the path. One then draws the P-curve forward from p 1 and the N-curve backward from p 11, thereby obtaining a four-sided figure as shown (Figure 1). If one now modifies the given path

FIGURE

1

by replacing the section p lppM by the section p ,p ,f!p M , the NP-corner is removed, no other such corner is introduced, and the time length of the path Is reduced. To see this last fact, note that by (2) t(p'pp")=

f

y _ 1cix,

P 1pp 11

t(p'p"'p")=

J

y _1d x

P 1P 1*’P 1*

,

BUSHAW where the integrals are extended over the indicated curves. However, for any given value of x the value of y is greater on p !p l,Tp M than on p !ppM ; therefore the second integral must be smaller than the first, as claimed. If one applies this process to every NP-corner above the x-axis, and the corresponding process to every PN-corner below the axis, the re­ sult is a canonical path shorter than the given path. To complete the justification of this conclusion, one must verify two things: (A) that it is always possible to construct the ,fquadrilateral" shown in Figure 1; and (B) that the process described introduces no self-crossings, so that a true path is in fact obtained. The assertion (B) may be established by a straightforward survey of the possibilities, the fact that P- and N-curves can cross only in certain ways being taken into account. The verification of (A) follows. Let p, p 1, and p M be as describedabove; if the initial point of a path is regarded as a corner, p ! always exists, and the exist­ ence of p !1 follows from the fact that the path goes to the origin. It will be shown firstthat the P-semicurve n beginning at p 1 passes over p ’ppM and crosses the vertical line through p M . That n moves to the right as long as itremains above the x-axis follows from the first equa­ tion in (2). If H is given by thefunctions x(t), y (t), where p ! = (5 (0 ), y (0)), then 11has one of thefollowing threeproperties: (i) For

some

t > 0, y(tQ ) = 0.

(ii) lim inf y(t) = 0. t — > 00 (iii) x (t ) —

and avalue this gives

> 00 as

t --- > 00.

For if both(i) and (ii) are false, there exists a number e > 0 T > 0 of t such that for t > T, y(t) > e. Since x = y, t

x(t) - x(T) =

J

y(t! )dt! > e (t - T) -- > 00 as

t

> 00,

T so that (iii) holds. Thus, as t increases n must either approach the x-axis as closely as one wishes or move off infinitely far to the right. n starts off from p* above p'pp,!, for in the upper half-plane the P-curve through a point always has a greater slope there than the N-curve through that point, and even if p ! lies on the x-axis (so that both curves have vertical tangents at p 1), the radius of curvature of the P-curve at p ! is greater than that of the N curve. This all follows from (2). n cannot cross the N-arc p*p, by what was just said about slopes;

OPTIMAL DISCONTINUOUS FORCING TERMS

35

it cannot cross the P-arp pp f1, since P-curves do not intersect one an­ other; and it cannot tend as t ---> °° to either of the points p or p ,! (which may be on the x-axis), for this would imply that the point con­ cerned would be a singular point of the P~system, and this in turn would be inconsistent with the fact that the time length of ppf1 is finite. Thus all that was claimed for n is true. The corresponding argument can be given for N, the N-semicurve obtained by following the N-curve through p ! 1 backwards from p ,!: N also lies above p *ppM , and crosses the vertical line through p 1. Thus n and N must intersect at least once; but they can intersect only once, for if there were two consecutive intersections one of them would nec­ essarily involve an impossible inequality between the slopes. This gives the unique point p !,f and the "quadrilateral" sought. This completes the proof of Theorem

1.

COROLLARY. In seeking a minimal path from a pointit is sufficient to consider only canonical paths from that point. For it follows immediately from Theorem 1 that a path which is minimal with respect to the class of all canonical paths is also minimal with respect to the class of all paths whatever. From this point on all paths considered are therefore assumed to be canonical. If g(x, y) happens to have the property g(- x, - y) s - g(x, y), discussions can be simplified; for if on this hypothesis we make in the equations (2) the substitutions x = - X, y = - Y, cp(x, y) = - $(X, Y), we obtain

X = Y,

Y = ®(X, Y) - g(X, Y)

,

and these equations have exactly the same form as (2). This means that if p and q are two points symmetrical with respect to the origin, then whatever can be said about the P- and N-curves at p can be said about the N- and P-curves at q: e.g., if it can be proved that any minimal path from p must begin with a P-arc, it follows at once that a minimal path from q must begin with an N-arc. Since g(x, y) always has the property named when it is linear (homogeneous), and g(x, y) will be linear in sub­ sequent sections, this observation will be extensively used. Any result obtained from another by an appeal to it will be said to have been obtain­ ed by symmetry.

BUSHAW

36

§5 - THE LINEAR CASE WITH COMPLEX CHARACTERISTIC ROOTS In the rest of this paper the fundamental problem of Section 1 will be restricted by supposing that g(x, y) = Ax + By, where A and B are (real) constants. It is well known that the qualitative behavior of the P- and N-curves will then depend on the nature of the characteristic roots of the matrix

The first case considered here is that in which these characteristic roots are complex. In this case (after a suitable redefinition of the units in which x and t are measured, if necessary) the equations (2) become

x = y,

(3 ) where

|b|
0.

When b > 0 the P-curves for (3) are clockwise-oriented spirals for which the point (1,0) is the (stable) focus. If b = 0 the P-curves are ellipses with (1,0) as the center. The N-system may be obtained in either case by translating the P-system two units to the left along the x-axis. Let

and consider the transformation X = x + by,

(O

Y = ay

The use of X and Y may be regarded as the use of oblique coordinates in the (x,y)-plane. Under (*0 , the system of equations (3) becomes

(5 )

X = - bX + aY + bcp,

Y = - aX - bY + acp

The P- and N-curves are given explicitly by

(6 )

X(t) = e“bt(Aeiat + Be-iat) +

1

Y(t) = ie-bt(Aelat - Be-±at) , where

(7)

A =

~

[X(o) - iY(0 ) +

1 ],

B = A

OPTIMAL DISCONTINUOUS FORCING TERMS

37

(Here and throughout what follows the upper sign pertains to the P-system., the lower tothe N-system. The functions (6) represent ordinary logarith­ mic spirals or, if b =0, circles.) LEMMA 1. The time length of any arc cut off from a P- or N-curve by two successive intersections with the X-axis is */a. This may be easily verified by considering (6) and (7 ). A useful sequence of numbers in Is defined as follows: First, put iQ = o. The P-curve starting at U Q, o) = (o, o) reaches, after the lapse of - jc/a time units, a point (| 1, o); the N-curve starting at (l-j, o) reaches, after the lapse of n/ a time units, a point (i2, o)j and so on. LEMMA 2.

The values of the numbers n-i

!n = (_i)n |enW a

+ i

+

2

ln

Y

are given by (n = ^

This formula may be derived from equations (6) and (7 ) by com­ plete induction. The letter r will be used in this section to denote the P-arc joining the point (|1, 0) with the origin. LEMMA 3 • If p e R, where R is the set which consists of the interval 0 < X < on the X-axis and the in­ terior of the region bounded by this interval and r, and if A is that path from p which is obtained by following the N-curve through p to r and then following r into the origin, t (a ) < it/a.

only.

PROOF. It will suffice to prove this for points on the X-axis The equations of r are X(t) = 1 - •le-bt(elat + e~lat) ,

(8 )

2

Y(t) = lie-bt(e-lat - elat)

,

where - it/a < t < o. The point q = (X , Y ) is given by these equa­ tions when t assumes some value - \ ( o < X < */a). If qIs regarded as the initial point of the N-arc pq, the time necessary to reach p being -n(o one may obtain as parametric equations of En Xn (cr) =

e nlDn/a (

1 _ ebocos

aa)

+ U nl

>

(1 3 ) Yn (a) = enb*/a (- ebosin aa)

,

where 0 < a < it/a. EQ is merely r; but from (13) it appears that En is the curve obtained by magnifying r by the factor enbT('a and then translating the result \i \ units to the right. For each n, the curves E^ and En+1 meet in a cusp at the point ( U n+1 l> °)# From these con­ siderations it is easy to see that the paths asserted to be minimal in the theorem exist and are unique; moreover, their corners are all on the curves En and those obtained from the curves En by reflection through the origin. We are now prepared to embark on the proof proper of Theorem 4 . As in previous cases, one begins by deciding how a path from p (p = (Xp, 0), |£n | < X 1 < |Sn+1|) which might be minimal can behave. It begins, let us say, with an N-arc of time length X, where X > 0 . We shall assume for the present that X also satisfies the condition X
3 , A begins with a curve pqrs as described above, and s is followed by a curve sq’r !s 1 of the same sort* If v and v f are the points at which the arcs qr and q fr 1 intersect the X-axis, then one can use methods already developed to show that each of the four pieces pqv, vrs, sq’v 1, and v !r !s f has a time length greater than x / 2a ; and since the timelength of that part of A following s 1 is, by the inductive assumption and (1 9 )> greater than (n - 1 )it/a, the time length of A itself must be greaterthan (n + 1) % / a , which, by (19 )> is in turn greater than t (a ^ ). The remaining cases, N(a ) = 2 and N(a ) = 1, may be settled individually by use of the method of path contraction already used so ex­ tensively above; the details, which arenovel only in minorrespects, will therefore be omitted from this account. This completes the proof of Theorem

4,

COROLLARY. If p lies in the closed region bounded by one of the curves En and the X-axis, the unique minimal path from p is that obtained by following the path described in Theorem t which passes through P* PROOF. Suppose that A^ were a path from p such that t (Ap) < T(2Sp), where A^ is that path from p which the corollary asserts to be minimal. Let p 1 be that point on the X-axis which is first reach­ ed by following the N-curve through p backwards; clearly, |ln | < X p , < |ln+1|. The curve composed of the N-arc p fp and A^ will be (after the

BUSHAW elimination of any loops, retracings, etc., that it may happen to contain) a certain path from p *, say A^,. But

T(Ap,

) < t(p'p) +

T(Ap )
q> MnA = A,

MqA =

and MnMqA = Pn-qA = A Prom a similar argument we obtain

(6)

^n-qj

C aMn

Also, as a corollary of (6),

and because of (^) and (5)>

^ n , q C^ q

and

P n-q

and so,

(7)

V r ^ n , q

We remind ourselves also that (8)

‘ pn

C

P,kn

’

of

E

PERIODIC SOLUTIONS OP CONSERVATIVE SYSTEMS (9)

Also, I f

^n,q k - k' + i

(1 0)

4

P (n,q)

57

*

o

^ n , q C e/^k(n-q)+n, k'(n-q)+q

for, because of (5 ) and (8 ), if

and because of (2 !) and

A €

n,q-

MnA = A,

similarly we have

the twd indices in the second set of (10) must be different, hence the re­ striction on k and k f. In the special case

q = o, k = -

1

and

k1 = -

1,

^ n , o C ^o,-n and because of the symmetry of this relation (11 ) v J

cM

n,o

cM

o,-n

Finally corresponding to (8 ) and (9) we have (12) (121 )

®^n, 1 C ^kfn-i )+i,1

(1 3 ) (1 3 ') For the applications we have in mind, we introduce the following 9 a terminology: c4i n will be called a set of symmetric points and set of doubly symmetric points, so that the relations (5) and (7 ) are a more precise form of the following theorems:

58

DEVOGELAERE THEOREM 1. Every doubly symmetric point is period­ ic under T. THEOREM 2* Every symmetric periodic point is doubly symmetric«

3. Transformations of the Invariant Sets. All the sets of sym­ metric points may be deduced, from cMo and cM^ by means of powers of the transformation T because cM

(1k) K J ( 1M)

^

2n

= p cM n o

2 n + 1 = Pn ^ l

for, in general (1 5 )

p c4{ = cM „ q n n+2q

As corollary, ( l 6)

V * n , q = ^ n +2 r ,q +2 r

Also because of ju-n = P-ncMn = MoM ncMn = MocMn

, '

we have (1 7 )

Ai

'

Up to a transformation Pn all doubly symmetric sets cM n, q may be reduced to the sets ^ 2 k o ¥^en n and 0. ape even cM cM 2k+

,

Q when n is odd and q is even - . when n and q are odd 1? 1

,

cM 2k_2 ~1 ¥^en n is even and q is odd for instance, because of (16),

,

;

PERIODIC SOLUTIONS OP CONSERVATIVE SYSTEMS

59

The last set may be reduced to the second one because of (16) and (11); (l8)

p-k+i^2k-2,-i =e/^o,-2k+i =e/^2k-i,o

;

but no further reduction is possiblek. Classification of Symmetric Periodic Points. Because of the Theorems 1 and 2, symmetric periodic points are doubly symmetric and con­ versely, so that we have to classify in a unique manner the points of the sets But one of the iterates of these points is contained in n, q

c/^ 2k,o’

‘^ k — 1 ,0 ’ e/^ 2k+i,i

and conversely, iterates of points of these sets are points in c M n ^ is thus sufficient to classify points of {< ? l ).

It

Points are in different sets (*o for different values of the indices, but because of (12) to (131) each point is contained in only one of the sets c °

m

(k > 0)

defined by Z ? = cM

k

®

2

~

2k,o

,0

c i

u cM

n, o ok-1 &

And we have (1 9 )

c \ . = R C £ " J" 1

(1 9 ')

K

(1 9 " )

v

£-J = 5*

Other Properties. Prom a similar theorem on the indices

follows: THEOREM sets Z

3-

The sets q Table Igives the partition ofc M n = cM ^ n corresponding to some values of nand q: |n| < k , |q| < k , n f q. If the sets c M Li, p are obtained in the order of increasing n and for a fixed n in the order of decreasing p, parenthesis indicate the first time the sets C ^ ^ ^ ^ jjj. are obtained for a fixed k, what­ ever be j• brackets indicate the first time the sets are obtained for

61

PERIODIC SOLUTIONS OF CONSERVATIVE SYSTEMS TABLE I The Partitions of the Sets

cM

n, q are Given Below (For the conventions see Section 5.)

-fc,-3 2,0

-It,-2

-3,-2

-3,-1

-it,-i

2 ° t° Z°

-2 , 0 0. 7 .Introduction

of Symmetry. We shall now make the further hy­ pothesis U(x, - y, a) = U(x, y, a), which means that the problem is symmetrical with respect to y = o. y = o is a solution of (21) and the solution on the axis of symmetry is reduced to a problem of one degree of freedom (20). When a is fixed, a solution of the problem which crosses y = 0, is determined up to a symmetry by the values of x and x at the time of crossing because y is determined up to the sign by (22* )• The two di­ mensionalsubspace V of the plane x, x 2U(x,

o, a) - x2 > 0

,

is ageneralization of a surface of section: this surface is made upof analytic pieces, but some discussion will have to be made when there are double points on the boundary and for regular boundedness [I, II, p. 70]; we do not ask that all the trajectories cross y = 0. If we are looking for periodic solutions of (20) and (21 ), sym­ metry is very important, because then symmetric periodic solutions may exist and these are much more easy to find than non-symmetric periodic orbits as will be indicated in Section 10. We shall consider here the sym­ metry defined above, but what follows may be generalized to other cases. The two subspaces and f f will mostly be used. Each point A of V not on the boundary gives the initial conditions oftwo sym­ metric curves in £ crossing the line of symmetry y = 0 at A and followed in a definite sense and conversely. The points on the boundary of V correspond to the solution y = 0, x = 2U.

PERIODIC SOLUTIONS OP CONSERVATIVE SYSTEMS

65

8 . Transformations of the Set Zf .Let us consider a trajectory corresponding to a point A of Z f ; if this trajectory again crosses y = 0 , then to this point A there corresponds a pointB of Zf and the differential equations define a transformation T of into itself, T is obtained by following the same trajectory in the other sense giving, if the trajectory crosses y = 0 at D: T ~ 1 A = D. Let us now define the involution R of Zf asthe reflexion aboutx = 0 ; this corresponds in ^ to the trajectory starting at thesame point on y = 0 in the opposite direction. If RA = C, we have IRC= B and TRB = C and so TR is also an involution.

If T is a one to one mapping of Zf onto itself, i.e., if to each A corresponds a B and a D, then the application of Part I is immediateand E = Z f ; if not, Ewill be the largest subset of Zf for which the transformation T deduced from T is one to one onto E. An a priori knowledge of E is then, in general, difficult to obtain, but is not necessary. 9 . Periodic Orbits and Symmetric Periodic Orbits. Solutions of (20) and (21) will be called here indifferently, trajectories or orbits. It is immediate that the sets P n of E correspond to the periodic orbits t crossing y = 0 at least once and conversely. We have now to find the meaning of the sets n • is the set of points invariant under R, i.e., the set of points on x = 0 in E. These points corre­ spond to trajectories perpendicular to y = 0 in £ , i.e., to tra­ jectories symmetric with respect to the x axis or trajectories symmetric with respect to the plane y = 0 , x = 0 In the phase space. 0 is the dH transform of c M Q -under T (1*0 ; these points corresponds to the n crossing of trajectories symmetric with respect to y = 0. If A Is a point of cM .j, TRA = A; to A corresponds in the phase space a trajectory in. with initial conditions X(x O . y^O = 0, O x O. y ), to RA corresponds the trajectory with initial conditions X ( x Q, 0 , - x Q, - f Q ) and

x(- t, After a time TRA = A

2 t 1,

crosses

*(2 t,,

X)

= - x(t,

y = o

Y)

=

at

Y) X,

x(0, X)

. and we have, because of

;

but because of the uniqueness in the phase space, this was true at any pre­ ceding time, for instance - t1, and so i(t,,

Y) = *(- t,, X) = - x(t^, Y) = 0 .

DEVOGELAERE

66

For the same reason y ^ , Y) = 0 . The orbit will have one point of zero velocity or in £ a point on the boundary 2U = 0 , and A is the next point of intersection with y = o. Because of (1U 1 ) ^ 2n+1 the set in E of the orbits starting from the zero velocity line and crossing y = 0 for the n ^ time. These orbits are symmetric in the phase space about x = 0 , y = o. Symmetric points correspond to symmetric orbits in the preceding sense. The sets c M n are defined in E, but we extend their definition in & in the obvious manner; they will be, in general, lines which inter­ sect in doubly symmetric points c M ii,q , which correspond (Theorem 1 ) to symmetric periodic orbits; the converse is also true because of Theorem 2 . 10.

On the Advantage of Symmetry. Actually the sets c M ^ in & will be obtained by integrating the differential equations (20) and (21) with initial conditions (x, y= 0 ) € S, x = 0 , y given by (22!) or with the initial conditions (x, y) satisfyingU(x, y) =0, x = y = 0. The sets do not give all the periodic solutions, namely those which do n, q not cross y = 0, and those which cross y = 0 and are not symmetric. The last ones may in general only be obtained with much more work, because we have then to integrate all the solutions starting withany value of x, x in f f up to the second crossing with y = 0 in £ ; by using the symmetry argument, the integration with the indicated initial conditions up to the first crossing with y = 0 is sufficient. Periodic orbits which do not cross y = 0 may be obtained in the case where there exists another plane of symmetry in the phase space, which could be for instance the x = 0, y = 0 plane. But to deduce from this, that all non-symmetric periodic solu­ tions are obtained, we must have more knowledge on the surfaces of section and prove that every trajectory crosses in the phase space either x = 0, y = 0 or y = 0 , x = 0 . 11. Classification of Periodic Orbits. Because of the corre­ spondence of the symmetric periodic points and the symmetric periodic orbits, Section A gives us a classification of symmetric periodic orbits. With the notation introduced at the end of Section 7, if A we have with

2k, o

B = T^A, MQA - A,

MqB = B

(Section

k,

ii)

PERIODIC SOLUTIONS OP CONSERVATIVE SYSTEMS

z\,

3 A± = T^A = T1RA = RT- 1 A = R (T k _ i B )

i

= 1,

2,

67

k -1

all the points being different because of the definition of corresponding trajectory ^ crosses y = 0 perpendicularly at B and has exactly k - 1 other points of intersection A^ with such a trajectory will be noted

The A and y = o,

Similarly to A e corresponds a trajectory 7^ with ex­ actly # k points of intersection with y = 0 and with two points of zero velocity. Also to A € corresponds a trajectory 6^ with exactly k points of intersection with y = 0 at one of which (A) the crossing is perpendicular and with two points of zero velocity symmetric with respect to y = 0. Using Table 1, one sees that from the computation of and 71 may be obtained, i.e., the periodic orbits symmetric or not with respect to y = 0 and having two points of zero velocity. The additional computation of cM^ gives every orbit , e2 and &2, i.e., the symmetric periodic orbits with respect to y = 0 with two points where the crossing is perpendicular and without or with an additional crossing point and the symmetric periodic orbits with respect to y = 0 with two symmetric points of zero velocity and two points on y = 0, one at which the crossing is perpendicular. And so on. 5j

Theorem.

5,

12. E1 =

( ff

n

Non Existence of Symmetric Periodic Solutions. If in the x > 0) and E2 = (JT n x < 0) one has

THEOREM 7 *If all trajectories starting from y = 0 with y > 0 and x > 0 and which cross y = 0 are such that at the first crossing point x > 0, there is no symmetric periodic orbit of type 6 or e. Moreover, if

E^ = E 1 and

E^ = E2

we derive from Theorem 6 bis:

THEOREM 8. If the hypothesis of Theorem 7 are verified and if all trajectories starting with zero velocity and y > 0 and which cross y = 0 are such that at the first crossing point x > 0, there is no symmetric periodic orbit. The additional hypothesis means that cA/i

1 —

YicAt^

cM^

C E^

C RE^ — E^

hence

DEVOGELAERE

68

REMARK. As such the two preceding theorems are not very useful, hut will become so when T is a topo­ logical transformation. 13.

Properties of the Transformation

T.

THEOREM 9. Let S ?1 C Q be a closed bounded domain in which U(x, y, a) has continuous bounded first partial derivatives in x and y and in which these derivatives satisfy Lipschitz!s condition. Let I(x0 ) be a closed set of intervals in xQ such that

and

p x

such that

then to every point PQ € p x corresponds a unique so­ lution of the differential equations (20) and (21 ); if this solution crosses again y = 0 after a finite time and before leaving f*1, the corresponding point P 1 is in a set p x; and the original point is in the set the correspondence between p xQ and , p x is one to one. PROOF. The system of first order differential equation (20) and (21) satisfies the condition for existence and uniqueness of a solution with initial conditions at t = tQ, (xQ, yQ ) e g x and xQ, yQ satisfy­ ing (22); moreover those solutions are continuous functions of xQ, y , x , yQ and (t - t ) in [28, II, iAi]j hence the correspondence be­ tween some subsets of P is one to one and it is easy to see that p ^ and p x are such subsets. It is not so that those solutions are neces­ sarily continuous functions of the initial conditions alone, for, t may increase indefinitely along a solution without these solutions leaving g x, for instance when the solution is asymptotic to a periodic solution. THEOREM 10. If besides the hypothesis of Theorem 9 , U admits in continuous bounded second partial derivatives which satisfy a Lipschitz condition, the correspondence between p x and p x is continuous and thus topological.

69

PERIODIC SOLUTIONS OP CONSERVATIVE SYSTEMS We shall restrict ourselves at first to the open domains of

p 'Q

and P\, a solution corresponding to a point in ;P x has then continu­ ous first partial derivatives in xQ, yQ, x Q, f Q and t - tQ . In par­ ticular, if yQ = 0 and y Q is determined by (22), y is a continuous function of xQ, xQ, and t - tQ, since yQ is by (3) a continuous func­ tion of xQ and xQ . Now y becomes zero for t1 =t - tQ, is differ­ entiable andits derivative with respect to t does not become zero at P 1 (otherwise the trajectory would be on y = 0 and P 1 on the boundary of P l), nor in the vicinity of P 1 (since y^ is bounded by (21)). Hence there exists a unique differentiable function t1= ^(Xq , x q ) which satisfies identically y(xQ, xQ, t1 ) = 0 [28, I, 1A1— 1A-2]. Ifwe replace 11 by 9 in the continuous and differentiable functions x and x, we obtain a result even stronger than the one stated. We have now to discuss points on the boundaryof p x . If such a point is not on the boundary of P , it is immediatethat x and x are continuous and differentiable for any variation of xQ, xQ which points to the interior of P x. If the point is on the boundary of P 9 we need to consider more closely the solutions of the differential equa­ tions near y = 0. If x(t), y = 0 Is a solution, any solution in the vicinity x + j*, r], satisfies the differential equations. (2 3 )

dx2

(210 ^ (2 5) with errors of the type if the time is bounded.

a 2u

I =

S>2U V Sy2

ix = x ^

e(|£ | + |r)|),

(' — ) 8x 8y ^

€

tending to zero with

t

and

t)

= 0

because of the symmetry of U. The first equation is equivalent to the third, and (25) shows that the variation | is of the form Ax. For any initial condition t] = 0, ^ / 0, the solution of the second equation at any finite time cannot be zero at the same time as rj, and so at P 1, tj — 0 can be solved explicitly in t and continuity and differentiability follow easily for points on the boundary of p , for directions interior to P x. The proof of the theorem is completed if the points on the boundary are considered as limit points of corresponding sequences In the interior of P . This asks for the condition of finite time entering in

DEVOGELAERE

70

the definition of The following interpretation must then be given to points L on the boundary of V' : if L is the limit of a sequence of points interior to f , the time t^ after which the correspond­ ing trajectories cross y = o tends to a finite time t as tends to L. Theorems 9 and 10 may serve to prove under certain conditions, con­ tinuity of the curves e M 2n as iterates of eM 0 * It Is also possible to write similar conditions which will serve to prove continuity for 0. The end points of the curves are on the boundary of f , and correspond to trajectories Infinitely close to the line of symmetry. Prop­ erties of the Immediate vicinity of y = 0 may hence furnish some proper­ ties on symmetric periodic orbits; for instance. THEOREM 12. Let us consider two trajectoriesp and q infinitely near the line of symmetry, such that y = 0 (hence x = 0) at P and Q respectively and limited to a section corresponding to the segment PQ; if their total number of intersections with the open segment PQ is m, under the conditions of Theorem 11, there exists an odd number of periodic solutions of type

PERIODIC SOLUTIONS OF CONSERVATIVE SYSTEMS

71

5 if m is different from one and an even number if m equals one. The case where p or q crosses PQ at P or Q requires a special investigation. It is also necessary for the hypothesis of Theorem 11 to be fulfilled that the periodic orbit on y = 0 has a nonreal characteristic exponent (k < 1 ).

The solutions near the axis of symmetry are given by Hill*s equa­ tion (see, for instance, [5]) (26)

where the partial derivative is computed on the periodic solution. The Sturm-Liouville Theorem applies and the roots of two solutions of (26) separate each other, so that if one of the sections of the trajectories considered has at least two points of intersection with PQ, the other will have at least one. However, the first point of intersection of both sections gives oneend point of c M ^, hence if m > 2 both end points are in regions of JT separated by y = 0; if m = 2 both sections must have one intersection with PQ and cM^ has also an odd number of intersections with PQ; if m = 1 both end points are not separated by PQ and the number of intersections with PQ is even; if m = 0, we must use Korteweg1s Theorem [29, p. kok ] which states that if t)^, are the values of t) for t = pT, (p + 1 )T, (p + 2 )T where T is the period of the orbit on y = 0, we have (27)

+ r)p = 2k T]p+1,

k = constant = cosh &T,

a

being called the characteristic exponent. Hence the trajectories p, q, when extended, will be.such that at P and Q or Q and P their ordinate r) is proportional to (28)

1, t)1, d 1, c ^, tjjdg,

cn,

j] 1dn+1,

...

(29) with

c.n

cosh n2T d. u

2 cosh OT - d,n-1 - d. *n-2

d_-j = d 1 = cosh ~ If the orbit on

or

dn = cosh (n + 1/2 )a T

y = 0 has an odd instability, [1 9]> [5]> i.e.,

DEVOGELAERE

72

k = °1 ^ - 1,

then the second and third members of "both series (28) and

(29) alternate in sign, for q ^ and q2d2 are Positive so the end points of q^ are separated by x = 0.

(m = 0)

and

If k > - 1, we can always choose ft so that d 1 is positive, hence q1 and q2 are positive. If now two consecutive terms of the series (28) are of opposite sign, it will be so for the corresponding terms of (29). The first alternation in sign must occur for corresponding couples in the two series and once again the end points of cM^ are separated by x = 0. •

The exceptional case corresponds to a series for which cn and dn are all positive, implying k = cosh ftT > 1. In the case k = 1, there exists at least one periodic solution for q. That p is such a solution Is impossible, for if we take the family of orbits with their in­ itial points on U = 0 tending to P, the time of the first crossing would become infinite, as indicated by the behavior of p; in all other cases the solution for q is of the form t0(t ) + Y(t ) where 0 and I are periodic in t with the same period as the orbit on y = 0. This Indicates that this orbit is unstable, and the same Is true if k > 1. In these cases the time of crossing of p or q may be made as large as we want as the initial q becomes small. This however is ex­ cluded by hypothesis. (See end of proof of Theorem 10.) In the case k = - 1, at least one of the orbits, p or q, crosses PQ at Q, or P and the variational equations of first order (26)is notsufficient to derive the behavior of eM near its end point. As a corollary we mention that the number of Intersections of the curves eM ^ and c M 2 is of the same parity because every intersection with PQ of cM ^ and not c M 1 corresponds to a curve e which has two points on PQ. Other topological properties of the invariant curves may be deduced from Section 1; for instance, c M ^ and- cM ^ (p, q > 0) intersect only at known points or at symmetric points of known points. It is then possible, if all c M ^ ,g are known for n < p, to determine a domain, some­ times much smaller than (? which contains cM , since most of its inter­ sections with c M Q and all of its intersections with cM n (n < p) are known. This in turn will furnish an solutions to which the knowledge of c M

approximationof leads.

thenew periodic

19« Continuous Variation of the Parameter a . A continuous vari­ ation of the parameter a will now be considered. In this case, If U is a continuous function of a, the invariant curves will vary continuously,

PERIODIC SOLUTIONS OF CONSERVATIVE SYSTEMS

73

this indicates that the periodic points appear and disappear in pairs [20] except if the point of disappearance is on the boundary of $ (and on oM q ); in this case, an orbit disappears by flattening on y = 0, this may be given the interpretation that two symmetric orbits disappear on y = 0, they coincide if they are of the type 5 or €. If two pointsin £ k appear, they give only one orbit and an interpretation must be given to the theorem of Poincare referred to. At first sight one would think that in general the contact of and c M 0 will be of first order but just as often the contact may be of second order. To make thisclear let us consider for instance a family of periodic solutions of type8 1, the invariant k = cosh dT ^1^ is a continuous function of the parameter and may takevalues + 1, or - 1, in this case orbits near & 1 appear or disappear. This has been studied in [9]*The results are as follows: When k = - 1, either

When

a)

cM^

b)

cM ^

k = 1,

is orthogonal to c M 0 orbits of type 71, or

and near

81

exist

has an inflection point and near 81 exist orbits of type €1. Inflection point in this con­ text means that the order of contact is at least two.

either

c)

there exist near 81 non symmetric periodic orbits which cross 81 near the points of zero velocity and have a double point on y = 0, or

d)

cM

1 and cM 2 have .an inflection point and near 81 exist orbits of type 81 distinct from the original family.

Conversely, if at a point where i)

ii)

iii)

is orthogonal to a) and orbits

crosses e M Q,

Q

we have

k = - 1,

case

cM ^

has an inflection point with cM Q, it will be so for c M 2 and k = + 1, case d) and orbits 8 1,

cM cM

iv)

cM ^

2 has an inflection point with c M 0 , but not 1, we have k = - 1, case b) and orbits €1,

cM 2 is orthogonal to c M Q, k 4 ± 1> there exists an orbit in the vicinity of 8 1 which starts orthogonally to y = 0, crosses 8 1 after t = T

^ ^ T i s here the half period T/2 i.e., the time between two consecutive crossings of y = 0 by 8 , this is the natural definition as follows from [9 J•

DEVOGELAERE on y = 0 and will be again orthogonal to y = o after t = 2 T, this means that cosh ftT = - 1 or cosh ftT = 0 and that theorbit is of type g2 * cM 2

is tangent to cM but no otherpreceding property is satisfied; k = + 1, there exists orbits in the vicinity of 8 1 which starts with zero velocity crosses y = 0 the second time orthogonally; in this case after t = 3T again the velocity is zero and cosh ft3T = 1, hence cosh ftT = - — and the orbits are of type &2« The points of zero velocity being on one side of 5 1, we may expect for values of y 1 nearby two orbits of type &2, one with the points of zero velocity on one side of 8 ^ and one with points of zero velocity on the other side* Case c) can not be discovered from the behavior of c M 1 and c M 2.

v)

y = o

at

The similar results for periodic solutions X and Y as follows: When k = - 1, a 1 ) , b ’)

When

k = +

1,

of type

g

crossing

cM ^

is orthogonal to c M Q either at X or at Y = TX and near e1 exist orbits of type £2»

either

c !) there exists near € 1 a non symmetric orbit cross­ ing e 1 at X and T, or d 1)

vi)

vii)

eM

has an inflection point with Q and near g 1 exist orbits of the same type g 1 distinct from the original family* Conversely^ if at a point where c M 2 crosses cM Q but not cM 1 2

cM ^ is

orthogonal to Q, we have case a) or b) and orbits g 2^

k

= - 1,

cM ^ has an inflection point with o M Q, we have k = + 1 , case d) and orbits g ^; case c f) can not be discovered from the behavior of c M 2 *

Another interesting property is that if for a certain value of a = aQ one orbit of type g1 appears, corresponding to I on x = o, c M 2 must have a contactor order at least two with x = 0 at I* Let us note the order of the contact at I by k 1 for c M y and k 2 for c M 2 , with the convention k = 0 when the curve cM crosses x = o at I but is not tangent at x = o and k = - 1 when cM does not pass through I*

PERIODIC SOLUTIONS OP CONSERVATIVE SYSTEMS

75

We know that k2 ^ 1, we must disprove k2 = 1. If k2 would be one, 2 would cross x = 0 for a let us say smaller than aQ at two points I1 and I2, these points are of period two under the transformation T. If I2 = T(I1), and a increases to aQ, I2 and I1 tend to I and the orbit at I is of type 5 1 and not €1 unless k2 > 1. If J 1 = T(I1) and J2 = T(I2 ) when a Increases to aQ, 1^ and 1^ tend to a point J Iterate of I, distinct from I, this means that for a = aQ two orbits of type €1 appeared and not one, this is not an ordinary situation.

cM

CHAPTER III.

APPLICATION TO 3T0RMERfS PROBLEM

15. GeneralIties. As application of the preceding results, we will choose a typical conservative problem of two degrees of freedom which is of interest in physics, namely the study of the motion of an electrical particle in the magnetic field ojf an elementary dipole; the Lagrangian of hhft T Y m h l ftm

1a

->

->

L = T + e v *A

where T is the kinetic energy, e the charge of the particle, v its velocity and A the potential vector of a magnetic dipole of moment M situated at the origin in the direction of the z axis; this vector in polar coordinates may be taken as tangent to a parallel, positive in the West direction and of length M cos x,/r2; this gives the Hamiltonian

hence the Hamiltonian and the momentum p^ are constant [27] . When p^ is not positive, Stormer has proven [21, p. 23] that no periodic solution exists; when p^ is positive, with as unities m, v and Me and with the change of variable Pjpi* = ex,

da = e~2xp 3 dt

the equations deduced from the new Hamiltonians are: x = ae2x - e~x + e~2xcos2A = X = dU/dx

(30) (31 ) (32)

x

x2 +

= ^e-2xcos2\ + 1 + tan2 x.^tan x2

X

= A = c)U/3 x.

= ae2x - 1 - tan2A. + 2e_x - e-2xcos2A = 2U

DEVOGELAERE

76

/ here a = is used, r 1 = - 7 = P^ / 2 ape equivalent parameters. To every solution in the meridian plane r, x correspond an infinity of tra­ jectory in the space obtained by integrating the equation for the longitude (3 3 )

9 = cos~2\ - e“x

.

This integration is a trivial problem and the properties of the trajectories in the three dimensional space are deduced easily from those in the meridian plane, for instance to a periodic solution of period z in this plane correspond trajectories on the surface of revolution around the z axis having the periodic solution as directrix. If

$ =

J

^cos 2 X - e"~x ^) da

is commensurable with *, all corresponding trajectories are periodic in space, if 0 is incommensurable each trajectory recurs infinitely often to the neighborhood of an initial state on the surface of revolution and may be classified as almost periodic; hence the general problem is easily solved if the problem in the x, X plane, is; for an example see [11]. Let us now consider some general properties of the equations (30, 31). 16. Properties. 'Ehe equations are of the type discussed in Part II, X playing the role of the variable y, the x axis is the axis of symmetry; the portion of the x, X plane, U > 0, has been described as follows (see for instance [27] ) } the boundary U = 0 is made up of a branch asymptotic for x = ~ ° o to X = it/2 cutting the x axis orthogo­ nally at P with x = g and when 0 < a < 1/16 of two similar branches one asymptotic for x = - 00 to X = it /2 cutting x = 0 -at Q with g 1 > g, the other asymptotic for x = +«> to x = it/2 cutting x = 0 at > S* £ ps then formed of two disconnected regions; when a > 1/16 the two last branches become connected and do not cross 1 = 0 , Q is then formed of one connected region. When a = 1/16 the boundary has a double point at x = log 2, X = 0 and a special discussion is needed. When a = 0, g reduces to a line (3 ^)

ex = cos2\

called the thalweg, all points of this line are equilibrium points and this case will be excluded In the following. For every finite point X and a are analytic; by a change of variable one finds that the points at Infinity are regular with one

PERIODIC SOLUTIONS OF CONSERVATIVE SYSTEMS

77

exception. The point x = X = */2 is an essential singularity and has been studied by Stormer [22, p. 1k^>, 235], [23], [26, 19^7, 19^9]> the proof that there exists a solution through that point has been furnished by Malmquist [18], but no proof of uniqueness is available although this seems highly probable. The general results on periodic solutions are as follows: every periodic solution is in the domain x < f = log (27^^ ) < g2 [25, p. 61]; this shows that no periodic solution extends to + 00 and exists in the region which extends to + 00 in the case a < 1/16. Every periodic solu­ tion crosses the x axis [9], [7]. There is only one equilibrium point, which is the double point on theboundary of (? when a = 1/16. Also any trajectory asymptotic to the equilibrium point or a periodic solution which does not extend to in­ finity crosses the x axis after a finite time. It has also been proven that no periodic solutions exist for a greater than k [61 because then f < g. A lot of specific results on periodic solutions have been obtained for the value 7 = 0.97 by Stormer [ 2k ] , [26 , 1950] and on a certain number of families of periodic solutions, the principal one [ 2k ] , [13], [1^], the oval family [3], [ k ] , and the horseshoe family [9] and the family on the equator [5]. It will be seen now how the method of Part II leads very naturally to those results. 17 * Surface of Section. The conceptof surface of section, be­ ing especially useful to find periodic solutions, we will not consider case 7 1 g ° where no such solutions exist; when 7 -j > 0 we have to dis­ tinguishthree cases, we will treat first the most straightforward one.

i) 7 1 > 1 or a < 0.0625* It was proven by Graef that for these values of the parameter every trajectory crosses the line X = 0 or is asymptotic to this line [12, p. 30, Th. VII and convention p. 28 Th. V]. Furthermore if a trajectory is asymptotic to X = 0, the time between successive crossings is finite [ 7 , Section 7 ] , hence the segment PQ de­ fined in the preceding section is a line of section (see also [7, Section 8 ] ) for every trajectory of the region 6^ of g not extending to + 00. To € 1 corresponds a connected bounded domain V inside the strip g < x < gi with boundary (3 5 )

x2

=

2U(x,

0, a)

.

Unfortunately the Theorem 11 can not be used, because (? 1 is not bounded and because the point at infinity is an essential singularity of U. We have to restrict 6^, let us therefore use the coordinates of LemaltreBossy [16], obtained as follows: we draw from the point (x, X ) a

the

DEVOGELAERE

78

perpendicular to the thalweg (3*0> the distance to the thalweg is u, the latitude of thepoint on the perpendicular and the thalweg is y; the Jacobian of thetransformation is positive and when 1 = 0 we have y = 0, x = u and x = u. Let us follow the trajectories in the space u, u, y; the trajectories which are extremum, (y = o) for a given valueof y = yQ are represented in the plane y = yQ by points on a closed curve cQ approximated by [i6]: •2

u

/ v-2 2 A + (cos y cos v) u = a cos y

with tan v = 2tan y . A point in the interior ofcQ corresponds to a trajectory having an extremum in yQ greater than yQ or passing through the origin. These trajectories will be represented on another plane y = J-] < y0 by curve c* interior to c1 because every trajectory has only one extremum in y between each crossing of y = o [7]. For instance when y = 0, there exists a set of closed curves c£ each one corresponds to trajectorieshaving their extremum at y = y^ > 0 and cj

is interior to

cl

if

yi > yj • Let us now consider

Q\ = ^

n ( |y| % y±) •

! is a closed bounded domain in which U is analytic, if we exclude from f f ^ I the domain inside cj and call thisp 1, and define P Q = RS^.,, to every point P 0 € P 0 corresponds a unique solution of the differential equations (30) to (32) and this solution crosses again 1 = 0 after a finite time at a corresponding point P 1 e p ^ hence by Theorem 10 the correspondance is topological. Moreover, the part of the invariant curves and c M 2 contained in P ^ is continuous. If it is true, as con­ jectured by Stormer [22, p. 1^5], that there is only one trajectory through the origin^1 ^ and if for this trajectory the return is by convention the same path backwards, this will mean that the curves c| converge to the point 0 corresponding to the trajectory through the origin, when y^ tends to The transformation T should then be topological in P and the cM ^ continuous. We will suppose this to be true in the following; if the conjecture is disproved, the results here stated will need an interpretation. As no trajectory through the origin crosses the first time X - 0 orthogonally, cM ^ does not pass through 0 and is analytic, cM^ is made up of two branches corresponding to the two branches of the boundary of £* which converge towards 0. ii) 7 = 1 or a = 0.0625. This case is similar tothe pre­ cedingone except for the fact that the boundary of g and P has a %T

^ In the following trajectory through the origin is by convention a tra­ jectory through the origin in the r, X coordinates, i.e., a trajectory passing through the essential singularity at infinity in the negative direction of the x axis.

PERIODIC SOLUTIONS OF CONSERVATIVE SYSTEMS

79

double point, this does not alter any conclusion of i. iii) o < 71 < 1 or a > 0.0625. Here the situation is not as nice as in the preceding cases; in this case a trajectory starting from X = 0 either crosses again X = 0 or is unbounded, hence there does not exist a nice closed section of the x, x for which the transformation T transforms this section into itself. But the ideas of the second part of this paper may still be used because every periodic trajectory is bounded by the line x = f. Let us bound the domain inside of (3 5 ) by the line x = f and call this S f ; if A e JT, TA is not necessarily in or may even not exist if the trajectory goes to infinity before crossing X = 0, but if A is periodic all the iterates of A are in Z f ; the invariant lines c M n are not entirely in but their intersections e M „ ^ are necessarily in Jf, hence to find the periodic solutions it n, p is not necessary to know which part of V is transformed into itself but only to find the part of the iterates of c4i 0 and c M ^ which is inside & . This can be done in succession and we know that the iterate of a part of c M ^ not in f f will not be in V . We will from here on summarize the content of the Report bearing the same title as this paper. 18.

Data on the Computations. A sufficient number of points on the curves cM 1 and c M have been determined for a sufficient number of values of the parameter a or y ^ . The choice has been made with great care and was dependent on preceding computation and on extensive compu­ tations done with a desk computer. The computations were done on a high­ speed computer (the SEAC of the National Bureau of Standards). In the Report we have indicated clearly how the initial conditions were deter­ mined, which method of integration was used and which method of checking was involved. A total of 1288 trajectories were integrated for some 21 values of the parameter. The trajectories near y = 0 have also been Integrated with another method to give the boundary points of oM ^ and It may be interesting to note that the time for the integration of one step was 0.35 sec. for the high-speed computer versus 25 min. for the desk computer. For a precision of five to six decimal places, from 4 o to 70 steps were necessary, in the mean. 19. Presentation of the Results. The results have been pre­ sented in the form, of tables which are available on microfilm and in the form of diagrams, one for each value of the parameter y ^; for y = c.97, detailed study has been made which leads to two diagrams, one of which is enclosed (Figure 1). Some data about Figure 1 will now be given. The subspace

5^

is bounded by the curve

KJP

and its symmetric.

80

DEVOGELAERE

PERIODIC SOLUTIONS OP CONSERVATIVE SYSTEMS

81

The curves eM^ and c M 2 have been determined by computation; each point corresponds to a trajectory with special initial conditions (Section 9 ). and e M _2 have been deduced by symmetry. The two branches of eM^ spiral towards a point 0 1, which corresponds to a trajectory a tending to the singularity x = - «>, \ = £ . The curves cM^ have been deformed in the neighborhood of 0 1 to simplify the drawing (of cM ^ ). The inter­ sections of _ 2 , cM_ pp. 87-91.

[ 153

LEMAITRE, G., "Applications de le m^canique celeste au probleme de Stormer, f Ann. Soc. Sci. Bruxelles, Vol. 63, 19 ^9 , pp. 83 -9 7 > Vol. 6/ 1950, pp. 76-82.

[16]

LEMAITRE. G. et BOSSY, L., MSur un cas limite du problbme de Stormer, 1 Ac. Roy. Belg., Vol. 31, 19^5 ; pp. 3 5 7 -361+.

[17 ] LIPSHITZ, J., n0n the stability of the principal periodic orbits in the theory of primary cosmic rays," J. Math, and Phys., Vol. 21, 191+2, pp. 28A-292. [18]

MALMQUIST, J., "Sur les systemes d !Equations differentielles," Ark. Math. Astr. och Eysik, Vol. 3 0A, 19I+1+, No. 5 ; pp. 1-8 .

[1 9 ] MOULTON. P. R., "On the stability of direct and retrograde satellite orbits, 1 Monthly Not. of the R.A.S., Vol. 7 5 > 191^ pp. 1+0-57. [20]

POINCARE, Les m.6 thodes nouvelles de la mecanique cdleste," Paris, Gauthier-Vlllars, 1899, Tome I and III.

[21]

STORMER, C., "Sur le mouvement d lun point matdriel portant une charge d !6 lectricite sous 1 !action d !un aimant Elementaire," Vidensk. Selsk. Skriffer Christiana, 190I+, No. 3 , pp. 1-3 2 .

[22]

STORMER, C., "Sur les trajectoires des corpuscules blectrises dans l !espace sous 1 !action du magnetisme terrestre avec application aux aurores bordales," Arch. Sc. Phys. Nat., Vol. 2 t, 190/ pp. 113 - 198 , 221-2I+7. STORMER, C ., "Resultats des calculs numeriques des trajectoires des corpuscules electriques dans le champ d !un aimant elementaire, I. Trajectoires par 1 !origine, " Vidensk. Skriffer, 1913 , No. 1+, pp. 1-7 3 .

[23]

[21+]

STORMER, C., "Periodische Elecktronenbahnen im Pelder eines Elementarmagneteny Astroph. Vol. 1, 1930, pp. 237-271+.

[29]

STORMER, C., "On the trajectories of electric particles in the field of a magnetic dipole," Astroph. Norveg., Vol. 2, 1936, pp. 1-121.

[26]

STORMER, C., "Resultats des calculs numeriques des trajectoires des corpuscules electriques dans le champ d'un aimant elementaire," Skriffer Norske Vidensk Ak., 19^7 , No. 1, pp. 5-80; 19I+9, No. 2, pp. 5 -7 9 ; 1950, No. 1, pp. 5 -7 3 .

[27]

VALLARTA, M. S., "An outline of the theory of the allowed cone of cosmic radiation," Univ. of Toronto Press, 1938, pp. 1-56.

[28]

de la VATiTEE POUSSIN, C., "Cours d !analyse infinit6 simale, " Louvain, Librairie Univ., 193 7 , I-II.

[29]

WHITTAKER, E., "Analytical dynamics," Cambr. Univ. Press, 1917.

V.

ASYMPTOTIC BEHAVIOR OP SOLUTIONS OF CANONICAL SYSTEMS NEAR A CLOSED, UNSTABLE ORBIT1 D. L. Slotnick* §1 .

INTRODUCTION

We investigate in this paper the behavior of the solutions of a certain class of Hamiltonian systems with one degree of freedom, i.e., systems of the form * - By,

y - - Hx

where the Hamiltonian H(x, y, t ) is a real analytic function of x and y, periodic in t, and vanishing at the origin x = y = o. In the neigh­ borhood of a closed orbit, autonomous Hamiltonian systems with two degrees of freedom can be reduced to this form, where the periodic orbit now corre­ sponds to the zero solution. The historical interest in these systems rests largely on this ground. We will deal with a class of these equations which have the feature of being unstable despite the stability of the linearized systems. For systems of this class we will establish the existence of an interesting family of unstable solutions and examine the analytic behavior of the sur­ face on which these solutions escape from the origin. This behavior Is marked by a rather unexpected phenomenon; namely while it would appear that this family should be analytic in some neighborhood of the origin it turns out to have an essential singularity at the origin. This has the immediate consequence of making power series methods Inapplicable for obtaining these solutions. In order to discuss the results in greater detail we give the This paper was submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at New York University. This work was supported by the Office of Naval Research under contract No. N onr-285(06). 85

86

SLOTNICK

explicit form of the Hamiltonians under consideration:

H(x, y, t) =

(x2 + y2 ) + H 3 (x, y, t) + H^x, y, t) + ...

is a convergent power series in some neighborhood of the origin. The terms are homogeneous polynomials of degree a in x and y whose coefficients are continuous functions of t with period 2 *. cd is a positive constant. H (x, y, t ), o = 3, k, 5, ...,

The question of stability for these systems is still unsolved. Certainly the linear terms are stable but the answer for the full system is locked in the higher order terms. We denote by s the Index for which H_o (x, y, t) t 0 but H u (x, y, t ) = 0 for a < s. Then we can write H(x, j ,

t) = -

(x2 + y2 ) + Hs(x,

j,

t) + H g+1(x, y, t ) + ...

An example of the dependency of the stability question on the higher order terms is furnished in [3]* where it is proved that under certain conditions on cd and on Hg — related to Its definiteness — a strong growth con­ dition holds for the solutions near the origin. Solutions satisfying this growth condition are called, by the author, almost-stable. Since we want to deal with unstable systems we have made a corresponding assumption of indefiniteness from which instability easily follows. This assumption is formulated In Section 3 after the Hamiltonian is brought into a simple normal form. In Sections 2 and 3 this normal form is obtained by exploit­ ing the elegant algebraic properties of Hamiltonian systems. In particu­ lar the Invariance of their form under canonical transformations. This allows one to concentrate completely on simplifying the Hamiltonian which, of course, then characterizes the corresponding system of differential equations. In Section k we introduce, following Poincare, the analytic mapping corresponding to the differential equations. This mapping assigns to a point (xQ, yQ ) the new point x 1 = x(2*), y = y(2 it) where x(t), y(t) is the solution of the system with initial values x , y . Under the indefiniteness assumptions mentioned above we prove that this mapping has the form. x i = s (xo )xo + x(xo>

Yo )

(1. 1 ) =

+

Y ^ 0>

y0 }

Numbers in the square brackets [ ] refer to the bibliography at the end.

INSTABILITY IN CANONICAL SYSTEMS

87

where s(xQ ) > 1 > t(x ) for xQ positive and s(o) = t(o) = 1. X and Y denote terns which in a suitable region are small compared with the leading terms. This mapping is analogous in appearance with u.j = su + ...

( 1. 2 ) V1 = tv + ...

,

where s and t are constants which obey s > 1 > t and the dots denote convergent power series in u, v which start with quadratic terms. For the mapping (1.2) Poincare proved, by power series methods, the existence of an analytic curve which is invariant under the mapping [k]. Such a curve corresponds to a one-parameter family of solutions of the differential equations which move asymptotically away from the origin. 2 2 Hadamard, assuming just the estimate o(u + v ) and continuity for the higher order terms In (1.2) proved in [2], with a geometric argument, the existence of a continuous Invariant curve. Recently, Sternberg [6] proved that the invariant curve for (1.2) is just as regular as the higherterms. Birkhoff [1] treated the mapping (1.1 ) for the case s = 3 and forthis case, which is included In our result, proved the same result obtained here, by methods which are, however, completely different from, those we employ and which are perhaps more complicated. We prove in Section 5 that the mapping (1.1) possesses an in­ variant curvewhich is analytic for xQ > 0 and is infinitely differ­ entiable at xQ = 0• We conclude by giving, in Section 6, an example adapted from [1] of such a cruve which has an essential singularityat x0

= °-

The author would like to thank his thesis adviser, Professor Jurgen Moser, for proposing the problem and for his generous help and en­ couragement throughout the time this paper was in preparation. Our thanks are due also to Professor K. 0 . Friedrichs for his kind cooperation. §2.

NORMALIZATION OF A HAMILTONIAN WITH A DEFINITE QUADRATIC TERM

This section and the next are devoted to showing that the period­ ic, analytic Hamiltonians we consider can be canonically transformed to an advantageous normal form. In this section we deal with those simplifica­ tions of the Hamiltonian that are a consequence of the assumed form of the quadratic term. In the next section we restrict our consideration to those Hamiltonians which are the subject of this paper by imposing conditions on

88

SLOTNICK

the next highest term and performing the further simplifications made possible by these additional assumptions. The reduction made in this sec­ tion is specialized from one employed in [3]® We assume the Hamiltonian to be a real power-series, convergent in some neighborhood of x = y = 0,viz.: (2.1 )

H(x, y, t) =

—

■ (x2

y2 ) +Hg(x, y, t)+ H g+1 (x, y, t) + ...

+

where the subscript o in Ha(x, y, t), here and throughout, denotes the degree of a homogeneous polynomial in x and y whose coefficients are continuous functions of t for all t and have period 2 *. a> is a posi­ tive constant. By introducing appropriate new variables |, t\, we will show that as many terms of H(x, y, t ) as we desire, say Hg, Hg+1, ..., Hs+n, can be transformed into a simple normal form. We want the system that re­ sults from this transformation to beagain canonical and periodic in t, moreover since the quadratic term of H(x, y, t) is already in a con­ venient form we will want the transformation to leave it unaltered. These ends can be accomplished by defining the transformation with a generating function of the form (2.2)

u(x, t), t) = xt] + us(x, t],t) + u s+1 (x, ri, t) + ... + ug+n(x, ti, t)

whereby the equations of the transformation are ^ = U^X, y = ux (x,

r,, t) = Ti,

t) =

T,

X

+ U ST1 + Us+1^

+ ...

+

us+n ^

+ ugx + us+lx + ... + u.s+n x

and the Hamiltonian of the new system is readily seen to have the form (2.3)

n(i, ti, t) = ~

( t ,2

+ n2 ) + ns(S,

n,

t) + «S+1 U, i, t) + ...

.

The coefficients of each u (x, r\, t), a = s, s+1, ..., s+n, will be de­ termined so as to have period 2 *, thus insuring that ft(|, t), t) also has this property, and so as to make fto a, fta s+,-, 1 •••, ft^^ s+n as simple as possible. We carry this out first for n = 0 , i.e., for the term H_(x, y, t ). o The Hamiltonian a ( z , n, t) Is, as Is well known, related to H(x, y, t) by G(u , T|, t) = ut + H(x, u^., t)

89

INSTABILITY IN CANONICAL SYSTEMS which gives + Tl2 ^ +

F' "t) + •••

= ut - ~ (x2 + u2 ) + Ha(x, ux, t ) + ... Comparing the terms of degree

Ust + a>^;xu3 r1 ~

^2,i|)

s

1USX}

.

in this last equation gives

~

Q3 ^X ’

^

= “

E s^x ’

^

'

a partial differential equation for u (x, rj, t). It is convenient in dis­ cussing this equation to introduce the complex conjugate variables 5

= i + in,

The system, in the variables

1

£ = i - iii

•

is canonical and its Hamiltonian

L t) = - 2ifl (I -p -t = io)?f + ®3(?,

t)

1,

t) + 4>a+1(£, S, t) + ...

If we denote

«. by

v (£,

t ) and replace V st +

where

(h x

- ?Vs p

*. ^

£ in (2.1 ) then (2 .1 ) becomes

by -

t)

i

4>3 (£,

I , t)

= J s (5,

5,

t)

Js(£> f, t ) is the known function

k —z Farther, denoting the term in £ £ , k + & = s, in va, 0. J by k-i k-i k~£ respectively then we see upon comparing the coefficients of in the last equation that

SLOTNICK

90

We would like to determine the so as to have period 2jr and so as to make as many of the as possible zero. We claim this can he done for each k, i for which (k - I )cd is not an integer. Thus if co were irrational, for example, this could be done for all indices k 4 This can be proved as follows: set = o, then (2.5) becomes (

d dt and integrating from

ico(k-i)t ^ VW

V 0

to

_ ia)(k-i)tT e Jki

-

2 jc gives

e 2 '1“ (‘'-''vk ( (2 , ) - v k J (0 ) .

t)dt

f '

.

o Thus since v^^ (t) has period must choose

2ojt if and only if

vk({0) -

v ^ (°)= v-^ (2 *)we

f

- ,) ’’

o

which can be done since the quantity in parentheses is not zero if co(k - &) is not an integer. This proves that we can take = 0 ^ ca(k -i) is not an integer. Observe that has period 2jr. We now treat the remaining case: co(k - z ) can occur only when k = & or when cd is rational. this case we can take (2 .6 )

where

V

cp^

(t).

V

an integer, which We assert that in

-1“ ^

of

s

of the

system.

since we will

systems we ma y

to f o r m u l a t e

into

A_(z, z)

case

a .

ia>(; £

(2 . 7 )

by

Aa s

z

of

N ow,

H

INDEFINITE

and

If f u r t h e r m o r e the

a

.

assume

that

Ag

is

special

Since w e will,

a defi­ of the

in this p a p e r

a (z, z)

is

indefinite.

introduce polar

co-

for the moment,

we omit the superscript

letting

the

stability

concern ourselves

this more preci s e l y w e

= pe

examine

being a time-independent

k n o w n theorem, o f D i r i c h l e t

of u n s t a b l e

this normalization

of the m o t i o n therefore w ill be d e t e r ­

the H a m i l t o n i a n in the f o r m

s

efficients

that

eliminated.

H a m i l t o n i a n is a n i n t e g r a l

ordinates

A

remark

FURTHER NORMALIZATION FOR THE CASE

A = a (z, z).

nite form

section we

for the new Hamiltonian

but

§3-

case

series

as w i l l b e this

n — > oo

which would be

order. In concluding this

shows

to t h e l i m i t

form.

s

on the co-

INSTABILITY' IN CANONICAL SYSTEMS

an easy

calcu l a t i o n shows

2ipS 1

As =

where we have denoted has

a real

zero The

the

filled by

the following

,

For

in this

case

odd m u l t i p l e average zero.

(k -

of

value

q.

Notice

the

be

conditions terms,

i)a>

s

on

can be

s

and s

($).

by

is

e a s i l y s e e n to b e f u l ­

q

contains which

even there

no

equals

are

the

if

or

s

z).

a rational

odd wit h

only if

constant

implies

Now

($)

of

co: co

and

>

a_ s A(z,

an integer if and

2 it,

to

2ip3

= 2 i PS a s (l J )

indefinite

and both

cr (tf)

o

that for

+ Tk.e

indefiniteness

that

Thus

zero on

^

sum multiplying

requirement

in lowest

s ln

ensure

number

£

that

I'P k . J

it w i l l

93

s > q.

k

-

i

is

an

term and hence has

existence

of a real

is a t e r m s

2 2 in

A

3

not have

giving rise a real

to a constant

In this

without

o

loss

assume

the prop e r t y

moreover

can be made

q>

_

cr3 (tf)

in

3 3

w h i c h thus need

2 2

We henceforth has

term

z e ro.

that

that

that is a

of this

has

simple

section we will

on the basis

the H a m i l t o n i a n

crs (tf)

the

further

assumption.

of g e n e r a l i t y

that

is

to r e i n troduce

H(x, zero

y,

dQ

t) and

zero.

show what last

a real

simplifications Note

that we

of

A

can assume

coordinates have been rotated

so t h a t

0. It

convenient

z = u + iv,

The H a m i l t o n i a n of the

transformed

real variables

z = u

to

( 2 .7) b y

- iv

system

with

Q cr(u,

v)

= acru a + b cru0- 1v + ... + e crv a ;

a = s,✓ s+1 s+n ✓, .... ✓ ✓

,

9^

SLOTNICK

where, in particular, aa = -

I

Im 'Pki

and

ba = -

I

b

(k -

* )Re K l

h

Now since we have assumed that o is a simple zero of crs(^) at is clear that Qa(u, v) m.ust contain v as a simple factor, i.e., that aS0 = 0 b while b b0 ^ o. The role of b S Is particularly important and for simplicity we will henceforth denote it by b. We have thus shown that as a consequence of our last assumption Q q (u , v ) must have v as a factor, i.e., that aa In fact, we now b S = 0. prove that In appropriate new variables U, V, we can make V a factor of as many terms of the Hamiltonian as is desired. To this end we will determine a polynomial

with constant coefficients such that the canonical transformation U = u,

V = v + Pn+1 (u)

of Q(u, v t) yields a new Hamiltonian whose terms of degree s + k + 1 (k = o, i, •.•, n- 1 ) contain V as a factor, i.e., the value correspond­ ing to a s+k+1 (k = o, 1, ..., n-1 ) in the new Hamiltonian is zero. PROOF. Q

The transformed Hamiltonian is u, V - Pn+ 1 (U)

u, V - Pn+ 1 (U) + Q,'s+k+i

One etelly sees that the coefficient of

u, V - Pn+ 1 (U) us+k+i

“ bpk+2 + as+k+i + gs+k where

depends only on

p2, p^, ..., p-^+1

Q-s' %+■) ’ %+k* Thus lf we considep induction, we can choose

pk+2 since

b ^ o,

V 2>

and the coefficients of P3>

as+k+i + gs+k ET

which concludes the proof.

•••>

Pk+1

as known by

INSTABILITY IN CANONICAL SYSTEMS

95

To summarize, we have shown that in appropriate coordinates a Hamiltonian whose quadratic term is definite and whose next highest term is indefinite in the fashion described above can be written in the form. s+n (3-2)

Q(u, v,

t)

=

Qa(u, v) + o- s

Q

t

(u ,

v,

t)

T=s+n+i

where Qa(u, v) = b 0u0~1v + ... + e^v0; ticular, b_a = b 4 0.

a

= s, s+1 , ..., s+n,

and in par­

Here again as in the previous sectiop we would like to go to the limit n — > «> as then we would have v as a factor of Q. This would mean that if u(t) were a solution of u = Q^u, 0 , t) then u = u(t), v = 0 would be a family of unstable solutions of the system derived from Q. This family would be of the type whose existence we seek to establish, with the curve v = 0 being the curve in question. However, we cannot 2 3 let n — > 00 as the series p2u + + ... thus obtained will in gen­ eral diverge as will be seen from the counter-example of Section 6 . §4 . EXISTENCE OP A CURVE INVARIANT UNDER THE ASSOCIATED MAPPING In this section and the next we will utilize the normal form (3.2) to establish the behavior of the solutions of the differential equations arising from (3*2) in the vicinity of the solution u = v = 0. In this section we will actually need only the fact that Qa - can be s and CDs+1 normalized as in (3*2). For the next section however we will need the full normalization to arbitrary order n. k 9 We define the weight of the term u v as k + 2i. The result of the last sectionthen yields that (4.1 )

Q(u, v, t) =bus~1v + terms of weightexceeding

s

+ 1, b 4 0

This is so because ag = aQ+1 =0. It is precisely this consequence of the form (3.2) of Q, that is important for this section in that if we confine u andv to the region R given by (4 .2 ) where is the Q in of the

0< u
0 , u and v are analytic functions of uQ, vQ for |uQ |, |vQ | < p1 and 0 < t < 2 *. Thus if we represent u and v as power series in uQ, vQ with time-dependent coefficients, then substitution into the equa­ tion and comparison of coefficients, denoting u(uQ, vQ, 2 n), v(uQ, vQ, 2 *) by u 1 = f(uQ, vQ ), v 1 = g(uQ, vQ ), and setting t = 2 * gives u 1 = f(u, v) = u(l + bu3_1) + F(u, v) (4 .3 ) v 1 = g(u, v) = v(l

- b(s -

1 )u3"2 )

+ G(u, v)

where wehave for simplicity denoted u , v , 2 *b by u, v, b and where the power series F(u, v), G(u, v) begin with terms of weight exceeding s - i: s respectively. If we denote 1 + bus~1, 1 - b(s - 1 )u3” 2 by s(u), t(u) respectively and make the assumption that b > o, which we will later re­ move, then (1*-•3) has a form analogous to the mapping (studied by Poincare): (4 .4 )

u 1 = su +

...,

v 1 = tv +

...

where s and t are constants obeying s > 1 > t and the dots denote quadratic and higher terms in u and v. The essential difference between (4.3) and (4 .4 ) is that in (4 .3 ) s(u)and t(u) while obeying s(u) > 1 > t(u) for u > 0 however become equal to one for u = 0. This produces a striking difference in the properties of the two mappings name­ ly that while It is proved in [4 ] that if (4 .4 ) is analytic then it possesses an invariant curve passing through u = v= 0 which is itself analytic for u > 0, the invariant curve we shall prove exists for the analytic mapping given by (4 .3 ) will in general have a singularity at the origin. The analogy between (4 .3 ) and (4 .4 ) does hold insofar as in the region Rthe terms s(u)u, t(u)v arethe dominating terms in u 1, v 1 despite the fact that F and G will in general contain terms of the same degree as s(u)u and t(u)v respectively. This is precisely ex­ pressed In the following estimates: obtained in the usual way.

INSTABILITY IN CANONICAL SYSTEMS If

(u, v)

is in

R

then for a suitable constant

with

M |G(u, v)| < Mus+1

|F(u, v )| < Mus,

(^•5 )

97

In addition, we list here for later use the following estimates: (I+.6 ) where

|FJ < Mus" \ M

|Fy | < Mus~2,

|Gj < Mus,

|Gy | < Mus_1

is taken large enough. We now assert that THEOREM 1 . There exists a unique function v = cp(u) lying in R with cp(o) = 0, invariant under the mapping (k .3 )•

mapping

Note that the image cp1 (u) of a function cp0 (u) is determined by the functional equation

under the

(k.3)

g(u, cpG (u)) = cp1 LEMMA 1. If under (^.3) then for

|cp0 (u)| < u2,

0 < u < Min

From

and

1

b W >

i.e., the image of a function in PROOF.

f(u, cp0(u))

B'(s-i'7 R

u,

|cp1 (u1 )| < u 2 < u 2 LEMMA 2 . Let Le as an Lemma 1 . In addition assume 9 0 ^u ^ is differentiable with |cp^(u)| < a, where a is a positive constant with a < b(s - 1 )/2M. Then cp1 (u), which is also differentiable, also obeys |cp!(u)| < a for u < Mint a/s-1, i/b(s. - 1 )], where

PROOF. Different iat ing (h .7 ) give s

gu + ? P u )gv =

1

+ b(s -

1

+ u3~2 [(b(s -

=

1

+ b(s -

1 )u3~2

1 )us 2

- Mu 3" 1)

1

+ Fu + cp^(u)Fv

- aMu3" 2

~ aM) - Mu]

|cp* (u1 )|,

The numerator in the expression for

|gu + $2; •••>j Tn; Tn have these properties. We will prove that

The idea of this

$Q (u) be curves which have the ITqI < a > then their iterates will, by Lemmas 1 and 2 also for sufficiently small u

lim |cpn (u) - $n (u) | = 0 n— > 00 from which Theorem 1 will easily follow.

.

1 00

SLOTNICK We begin by proving that

(^•8 )

_cpn+i^u i)l - a^u) lcPn ^u) "

where a(u) =

1

-

PROOF OF

2

us

and

cr(u) >

0

for

u


+ Mus ~ 1 + aMus

- 1) - aM) + Mus 1

2

*n+l ( U 1

}|

this last quantity is 9 n (u))

- f(u, cpn (u))|

|cpn (u) - cpn (u)| |q>n (u) - ?n (u)|

which for

is k n (u) - " $ n (u)| = a(u) |cpn (u) - '$n (u)|

,

where cr (u) =

1- us

-2

Now, if we denote by %_-} (u, cpn (u)) and in general denote the pre-image of ^n-k-1 ^u-k-i ))

and

we can

c o n ^ rm e

P^e-image of the point ^n -k ^ -k ^

(^*8 ) as follows:

In the brackets { } v must be replaced by a mean value between Tn (u ) before applying (4 .6 ).

cPn (u )

i01

INSTABILITY IN CANONICAL SYSTEMS

• " W

V

-

*

ff(u)a(u_1 )

< a(u)cr(u^1 )a(u^2 ) . . .

Now, since know that

!„_! ( U _ , )

-

Tn _

cr(a _n )

1( u - 1 }

- ? 0 (u _n )l

cpn was obtained as the nth iterate of a curve (u_n , cpQ (u_n )) lies in R; thus we know that

l ‘P0 ( u - n ) " L

2»i

(u- n } l
u

we obtain

log Pl + i ^ = log p + itfo + log Equating imaginary parts gives + Im log =

=

$Q

+ Im

+ b |u| s ~2 sin(s - 2 )oQ + terms containing

|u| S~1

SLOTNICK

1Ok We can thus determine is

K(iQ ) such that if

> 00 + | |u| 3-2 Note that

that if

3in(s

-

2 )* 0

K(iQ ) goes to zero with

.

*0

+ f

To conclude the proof of the first part and M

then

>

|u| < K(lQ ) this last quantity

|u1| > |u|

|u| 3 " 2 > O0

.

of (5*1) we easily see

< -jf

since |u,| > |u| |1 + bu3"2 | - M|u|s (1 + Lfrf.:. 2.l)

> |u|

1

- M|u|

+ |us_2| (| - M|u|

>

u

Thus choosing

o

t(s-2 )

and p 1 < m in

1»

m>

K(V

insures that the first part of (5 * 0 holds. to insure the second part of (5.1 ) for

We claim that this also serves

b (s’-1") PROOF. by (k .7 ) we have

Denote by

cp1(u)

*

the image under (t.3) of

cp1 (u1 ) = 90 (u)(l - b(s - 1 )u3” 2 ) + G(u, cpQ (n) ) u

and taking moduli we get (u1 2 U

>

-0 O

Dividing by

< U


U we have

U

IS

< 1

sufficiently small to insure that

IV U !}l o

- P-N-1

which concludes the proof of (b). We can now remark that the entire argument of Sections k and 5 can be carried out for any simple real zero Notice also that with Ao. + * is a root of a (tf) , thus the invariant curves o s come in pairs which are tangent to the same line at the origin but these pairs are not the analytic continuation of each other because of the singu­ larity at the origin. Also, notice that the transformation yielding (2.7) has the in t, where a> = £ Thus in the original coordinates the period invariant curve has the following significance. A solution x(t), y(t) with initial values on the curve returns to the curve after the time 2qit at a point closer to the origin. Thus after a time t = 2q* the entire curve cp(u) will return to its position at t = 0. For values of t be­ tween 0 and 2qjc the curve will sweep out some surface whose plane sections perpendicular to the t-axis are indicated in Figure 2.

FIGURE 2 Finally we can remove the restriction that the constant b > 0 by remarking that if we consider the inverse mapping of (1-3) then the role of b is played by - b, so that we can reduce the case b < 0 to the one we have treated by working In this case with the inverse mapping.

INSTABILITY IN CANONICAL SYSTEMS §6 . COUNTEREXAMPLE TO ANALYTICITY AT

u =

0

We now give the counterexample to analyticity of cp(u) at u = o mentioned at the beginning of Section 5. We will first have to recall some properties of the function Y(z) = where

r(z)

log r(Z)

,

is. the gamma function.

r(z) obeys r(z + 1) = zr(z); entiating k times gives (6. 1)

taking logarithms and differ­

¥ , M ( z + l ) = - - ^ + ' F ,tf(z) z

It Is this functional equation for

.

¥ 11 1(z) that we will have need for.

We take as our mapping u

-p- = f (u)

d+us _ 2 )s " 2 (6.2) v, = (1 +

3-1 S-2 \S-2

(v - 6u33"7 ) = g(u, v)

which an easy calculation shows to be area preserving and of the form (^.3) when considered expanded into its power series. First, we make the useful observation that

< s - 3 )

•

Now we claim that the curve given by

proposed a perturbation theory where the objects studied are periodic k-surfaces. (A surface generated by solutions to a system of differential equations is called a periodic k-surface if it is the homeomorphic image of the product of k circles. ) I made a study of a class of transforma­ tions that are used in this theory and obtained sufficient conditions for the existence of an invariant surface. My result was published in Volume Three of this series [ k ] . The purpose of this paper is to apply this theorem to a perturba­ tion problem closely related to that of Kryloff and Bogoliuboff. Their This report represents results obtained at the Institute of Mathematical Sciences, New York University, under the auspices of the U. S. Army Office of Ordnance Research, Contract No. DA-30-069 -0RD- 1258. K--& Now with Shell Development Company, Emeryville, California. 111

KYNER

torus of solutions is a periodic 2-surface. In the problem studied here, a set of periodic (k + 1 )-surfaces is shown to exist. The members of this set are generated by solutions starting on periodic k-surfaces. SECTION 1 Using vector notation, a system of can be written (1)

n

differential equations

dx dt = X(x)

*

We assume that the vector valued function X has continuous third partial derivatives. We also assume that the system, of differential equations has a periodic solution x = u(t) of period a>. In the (x, t) closed curve in the t = (1 ) whose initial values These solutions generate ness theorem, a solution all t.

space, the range of this periodic solution is a o plane. Let us call it p . The solutions to lie on are periodic solutions, x = u(t + t ). a cylinder parallel to the t-axis. By the unique­ once touching this surface must remain on it for

Now consider a perturbed system (2)

{jjf =

X(x, t, 7)

.

We assume that the third partial derivatives of X with respect to x and t are continuous In x, t, and 7, and are periodic in t with period T. 7 is a real parameter. At 7 = 0 , the system (2) becomes the autono­ mous system (1 ). In the t = 0 plane, let p be a smooth simple closed curve near the curve defined by the periodic solution to (1). If 7 is small, we can use the points of the curve p as sets of initial values for solu­ tions to the perturbed system. A surface will be generated whose inter­ section with the plane t = T will be a simple closed curve. In this way a transformation is defined on a class of curves. If it is possible to find a curve that is fixed under this transformation, then the solutions starting on its image in the t = T plane will trace out the same curve in the t = mT plane, for any integer m. This follows from the period­ icity of the perturbed system. Therefore if, for some 7, an invariant curve exists, the solutions to (2) that start on it will generate a period­ ic surface in the (x, t) space. By identifying cross-sections at t = 0 and t = T, this surface becomes a two dimensional torus in n space. The generating curves are solutions to a differential equation on this torus.

PERIODIC PERTURBATION OF AN AUTONOMOUS SYSTEM It exist.

is

It is

easy to

equations

x

was

= u(t)

such invariant

curves usually will

t h e r e f o r e n e c e s s a r y to p l a c e f u r t h e r

perturbed

first

see that

(1).

strongly

[2]

Levinson

stable.

assumed

This means

113

restrictions

that

that

on the u n ­

the periodic

the linear

not

solution

e q u a t i o n of

variation n

/*>)

dv „ V SX(u(t)) „

(3)

at -

L

—

vj

j=i has

n

-

1

characteristic

exponents with negative

proved

that

for each

7

curve

p

depending

continuously on

as

tends

7

periodic

,

to

zero,

s o l u t i o n to

the p e r iodic

sufficiently

(1).

surfaces

cylinder defined by

7, pQ ,

In the

t )

defined by

about

(x, these

teristic bility G.

conjectured exponents

the

[5] Let

Take

a

system

the

He

a unique

then

invariant

the p r o p e r t y

that

curve determined by

space,

invariant

as

7

tends

curves

characteristic of the

it w o u l d b e

have non-zero

of a n o p e r a t o r

Hufford

that

is

to

collapse

the

zero, onto

the

solution.

the n o n - v a n i s h i n g J a c o b i a n hypothesis Diliberto

real parts.

there

and having

approaches

the periodic

The hypothesis

small,

sufficient

real p a r t s .

corresponding,

exponents

implicit

in a

if the

This

sense,

n

insures

to

is

similar to

f u n c t i o n theorem.. -

1

the

charac­ inverti-

the Jacobian.

1953 ,

In

p r o v e d D i l i b e r t o 1s c o n j e c t u r e . us

consider

the f o l l o w i n g g e n e r a l i z a t i o n of

this problem.

of vec t o r equations dx.

cFET*

(^)

where

for

each

e q u a t i o n is

i,

assumed

m a y b e u n e q u a l ). products, to

these

“ ^ i ^ x i^

we have

an

to h a v e

a periodic

In the

kn

solutions

(4 ) s t a r t i n g

1,

i =

n

dimensional

a

k

k

,

vector equation.

solution with period

dimensional

define

...,

space formed b y

dimensional

Each vector (the

taking Cartesian

torus.

The

solutions

this k-torus generate a cylinder parallel to the

on

The perturbed

t-axis.

system.

dXi (5)

= X ± ( x 1, x 2 ,

introduces

coupling

disappears

as

dimensional

7

(and p e r i o d i c

tends

to

surface near

then defined by following

zero.

...,

x k , t,

7)

time dependence,

with period

We

of

take

the k - t o r u s .

For

solutions until

as

a

set

small they

7,

initial

T ) points

that a

k

a t r a n s f o r m a t i o n is

intersect

the

t = T

plane.

KYNER

The generalized problem is to find sufficient conditions so that for each 7 sufficiently small,there is a unique invariant surface. This invariant surface will define a periodic surface in the (x,t) space that can be consideredas a (k + 1) dimensional torus in a (kn+ 1) dimension­ al space. As 7 tends to zero, this must collapse onto theoriginal surface. The first results that apply to the generalized problem were ob­ tained by M. Marcus [6] in 1953. His theorem, when specialized to k = 1, includes Levinson1s but not Huffordfs. By using a fixed point theorem that is stated in Section 3 (and proved in [1] ), I have shown that an invariant k-torus exists, if, for each i, the unperturbed vector equation (5 ) satisfies Hufford’s hypothesis. SECTION 2 In order to prove the existence of an invariant surface, we make use of a coordinate transformation introduced by Levinson for the k = 1 problem. For each i, we can transform the corresponding n dimensional ri"~1, e^) by space to coordinates (y^1, y^2, ..., y^ x± = ujL(e1) + 3±(Q±^± •

(6)

Si(0i) Is an n x (n-1) matrix whose entries are periodic functions of the single variable 0 ^ with period u^ is the periodic solution of the unperturbed i-th vector equation. This transformation is valid near the curve defined by the range of u^. It induces a transformation of the product space. In this coordinate system, the periodic solution u^ is given by y± = 0 , 0 ^ = t. Hie system of vector equations (1) can be written

9b (7 )

where

de. dt “ Y^

and have period in e^. The perturbed system becomes dyi

cTF- =YiL'

(8 )

Yi' 0i^ Y^(o,

7k' V

V

ep

=

o, 0 ^ ( 0 , ep

^

de. = ©

i L '

• ••>

e i?

• ••>

yS>

•

= 1.

PERIODIC PERTURBATION OF AN AUTONOMOUS SYSTEM and have period zero, (8 ) becomes (7 ).

cjd^

In

and period

T

In

t.

115 As

7 becomes

Consider a solution to the perturbed system that starts from a point in the t = 0 plane with coordinates (y, 0 ) = (y1, ..., y^, e1, ..., 0^)• (Recall that (y^, 0 ^) is a point in the n dimensional space of Xj_. ) This solution can be written y = F(t,

j,

y)

e,

(9 ) y

e = G(t, y, e, Since y = we can write (9 ) as

0, 0

= t +

0

)

.

is the solution of

y

=

0,

y =

0,

0 = 0,

y = N(e )y + R(y, e, 7 )

( 1 0) e = e + T + H(y, e, 7)

*

where the solution is evaluated at t = t. N(e) is the k(n-l) x k(n-l) matrix formed by the partial derivatives of the vector function F with respect to y evaluated at 7 = 0 , y = 0 , 0 = 0• R is a k(n-l) 2 vector, H a k vector. Both are C functions of y and 0, with period in 0^. At

y = 0, 7 = 0,

we have R (0 , 0, 0) - 0

(11 ^

,

dR(o,9,Q) _ 0

Sy H( 0,

0,

0) E

0

.

Equation (10) defines a mapping of the point (y, 0) in the t= 0 plane onto the point(y, 0) in the t = T plane. Thisinduces a map of surfaces onto surfaces by taking y = p(01, ..., 0^), where p has period in 0 ^ and is of class C2 . The image of p is a surface n in the t = T plane. n is given by jt(0)

= N(o)p(0) + R(P (0 ), 0, 7)

(12 ) 9

=

0

+ T + H(p (0 ),

0,

7)

KYNER

To compute the value of tc at 0 , it is necessary to solve the second equation of (1 2 ) for e as a function of p, 0 , and 7. Geometrically, we are finding the 9 coordinate of the point (rc(e), 0) in the t = T plane by going back along a solution curve to the point (0(9 ), 0 ) in the t = 0 plane. This can be done if the norms of p (see below) and of 7 are small enough. Let be the transformation such that n = Q^(p). This trans­ formation of surfaces into surfaces can be considered as a non-linear transformation defined on a subset of a space of multiply periodic func­ tions. Our problem is to show that this transformation has a fixed point. SECTION

3

If we let m. = k(n - 1), our function space is the space of real multiply periodic continuously differentiable m-dimensional vector valued functions of k real variables. If p = (p^ . » . , p_) is such a func­ tion, then must be periodic with period in We use

where max max |p .(0 )| i

0

J

For fixed 7, a transformation n = Q (P) is given by equations (12). To insure that 0 can be solved for 0, we restrict j|p|| and 17 1 so that the matrix

is non-singular. If ||N|| 1,

7

then

(if

||f| | < r . 17 1 < 7 ),

hence we have

J“1 has the characteristic root

(1 A y ) < 1,

and -1 (3 6 )

||G^|| < 1

1 - V

Jv

V 1) V 1

J -

+ q S q ! small.

We again have a uniform bound. —1 Retracing our steps, we see that as a transformation on a space of complex valued functions is uniformly bounded. The transformation induced by it on the space of real valued function has the same bound. This completes the proof of the proposition. CONCLUDING RPMARKS My fixed point theorem was applied to a special situation where the existence of a periodic coordinate transformation and the existence of L“1 depended on their existence in the single variable case. If the un­ perturbed periodic surface of initial values is not the product of periodic solutions, it is not yet possible to show that the differential equations (5) can be transformed into the system (3 ) so that the original periodic surface is given by y = 0, 0 = 0. In the general k variable problem the concept of characteristic exponents is not applicable. A method of characterizing the unperturbed system of differential equations is needed so that one could easily tell if L~1 exists and is uniformly bounded. Diliberto has recently obtained such a result for the special case of m = 1 [page 5*3 of [7]]. APPENDIX PROPOSITION.

R j p p = P(e1 + T)J P"1(el ).

PROOF. Dropping subscripts, we have that 0 = G(t, y, e ) is the solution of

g -

Y(y, e)

(1 ) § | = 0 (y,

0 )

y = F(t, y, 0 ),

KYNER

122

that starts at (y, 0) in the t = 0 plane. Since y = o , 0 = T + 0, is a solution, Y(o, e) = 0, ©(0, 0) = 1. The equation of first variation is

I SY(0 ,t+9 ) \ ^y dw dt

(2)

0, there exist < rQ, and |r| < 70> then ||Gp7H < 1 + qt

rQ,

yQ

so that if

PROOF. (e ) = it( G ^ (e ) )

(8 )

Since

e = G^^(e) = 0 + T + H(p(e ), e,

'o

is one to one,

max max \ n .(e ) = i e I 1 »

i\ p7

i

y)

m

(0 )) de. r

^

z

8±. + de.

^

i=i

1

5H.(p(e ),e,7 ) Se4

j=i

( 10) &H.(p (e ),e,y

)

^TT

is -uniformly bounded, and H(o, e, o ) = o. |7 | small enough to Insure that

(11

)

drt (Gp^, (e ))

Hence we can take

||p||

Sit

N

det

[1

+ q !]

3et

Therefore

(1 2 )

llGp7 ic|| < ||*||

[1

+ q']

Having this, we can show that the inverse map,

(13)

l | Vpr *||


tq, n = 1, 2, ..., whence ,|)||> e /2 on Jn = [Tn - e/2M, tn + e/2M]; clearly, we may assume = o, n = 1, 2, ..., and t 1 > tq + e/2M. There exist constants v (e) > 0 such that V(t, x )> p, V ’(t, x) < - v, for t > T, < pl. Hence

n < V ^ T n + e/2M,

P ( T n + e/2M,

tq ,

which is absurd for sufficiently large

£ ))< V(

tq ,

|) -

v ne/M

n.

COROLLARY. If f(t, x) is independent of t or periodic In t on I x N, and if there exists, on I x N, a Lyapunov function V(t, x ) such that

,

MTOSIEWICZ

150

V1(t, x) Is negative definite on I x N, x = 0 is uniform-asymptotically stable. The assumptions of Theorem 7; in general, do not even imply the equiasymptotic stability of x = o, [23], [2 k]. Theorem 1 , which is due to Marachkov [23], generalizes Lyapunov!s classical theorem [13] on asymptotic stability. Aside from more stringent regularity assumptions on the function V(t, x), Lyapunov assumes the hy­ potheses of Theorem 6 and the further condition that V(t, x) — -> 0 with x uniformly on I. These assumptions where shown by Persidskii [35] to be sufficient (cf. Theorem 13) and by Malkin [22] to be necessary (cf. Theo­ rem 15) for the uniform asymptotic stability of x = 0. Zubov, [^2], [^3], [^ ], proved, among others, criteria weaker than Lyapunov’s theorem which are sufficient as well as necessary for the asymptotic stability for x = 0. These criteria are based on considerations of functions which are closely related to Lyapunov functions. The requirement that V(t, x)— >0 with x uniformly on I Is too restrictive to be implied by the equiasymptotic stability, even when the hypotheses on V’(t, x) are weakened similarly as mentioned above, [2 k]. Yet, as Theorem 8, [2 k], shows, whenever f(t, x) is linear in x on I x N, the existence of a Lyapunov function satisfying a weak­ er additional requirement does follow from the asymptotic stability of x = 0, which in thiscase is equivalent with equiasymptotic stability. THEOREM 8. If f(t, x) is linear in x on I x N and, for every x € N, of class on I for some integer k > 1, and if x = 0 is asymptotically, hence equiasymptotically, stable, there exists a Lyapunov function V(t, x), defined and of class on I x N, such that V'(t, x) Is negative definite on I x N. Moreover, V(t, x) has the property that, given any £, 0 < | < p, and any a > T, there are constants t)(£), ° t, V(t, x) < V(s, y) imply ||x|) < a.

PROOF. Since f(t, x) Is linear in x on I x N, given any (tQ, xQ) e I x N, we have F(t, tQ, xQ) = X(t)X“1(tQ)xQ where X(t) is a non-singular (matrix) solution of the associated matrix equation. Thus, for t € I, ||x|| S ||X(t)X 1(T )P(T, t, x)|| < ||X(t )X-1 (T) || ||P(T, t, x )II

A SURVEY OP LYAPUNOV1S SECOND METHOD

151

Let g(t) = ||X(t)X~1(T)||; by hypothesis., g(t) is defined on I and g(t) --> 0 as t --> oo* It follows that, given any e > 0, there exists a n(e) > 0 such that ||F(T, t, x)|| > m- for t > T, ||x|| > e; moreover, ||F(T, t, x)|| --> | for every £ > o. Let G(t) be the function associated with g(t) and h(t) s 1 by Lemma 3, let 0(t, x) = ||F(T, t, x)||, and define 00

V(t, x) =

J

00

G^g(s)0(t, x)^ds +

T

J

G^g(s)0(t, x)^ds .

t

According to Lemma 3, V(t, x) is defined and of class t > T, ||x|| > e, then 0(t, x) > \x and hence 00

V(t, x) >

on I x N. If

00

f G^g(s)0(t, xyds > y G^g(s)^iyds T T

so that V(t, x) is positive definite on I x N. Given any (t , xQ) € I x N, OO v(t,

F(t, t0, x0 )^ =

J

CO G^g(s)0(to,

xQ))ds +

G^g(s)0(to, xQ )) .

J

whence v(t, F(t, tQ, xQ ))= - G(g(t)0(to, x0 )^

.

Thus, on I x N, V’(t, x) = -G(g(t)0(t, x)) < - G(||x||) sothat V!(t, x) is negative definite on I x N. Given any £, 0 < £ < p, and any a> T, let r\= £ and choose t(cj, I)> or solarge that if u € [T, a3 andy In ||y|| < £, then t > t and 00

2

J T

CO

G^g(s)0(u, y )^)ds >

f

G^g(s)0(t,

x)^}s

T

Imply ||x|| < 5. Clearly, this is always possible. Hence, if u e [T, a] and y in ||y|| < e,j then the inequalities t > t, V(t, x) < V(u, y) imply IIxl| < |, for oo J

' G^g(s)0(t, x)^)ds < V(t, x) < V(u, y) < 2

oo J

G^g(s)0(u, y)^)ds •

ANTOSIEWIGZ

152

The next theorem due to Massera [25] shows that the hypothesis on f (t, x) in Theorem 7 can he omitted at the expense of restricting V f(t, x) further. THEOREM 9 * If thereexists, on I x N, a real scalar function V(t, x), locally lipschitzian and positive definite, and if there exists a real scalar function W(t), defined, continuous and increasing for t > 0, W(o) = 0 , such that V !(t, x) < - W(V(t, x )) on I x N, x = 0 is asymptotically stable. PROOF. By Theorem 1, x = 0 isstable. Hence, given any p 1, 0 < p ' < p, and any tQ € I, there exists a 5 (tQ, p ?) > 0 such that ||x0|| < 5 implies ||P(t, tQ, xQ )|| < p' for t > tQ . Since

V'(t, F(t,t Q, xQ )) < - ¥(V(t,

F(t,

t Q,

xQ )))

for

t > tQ,

/ W W T ^ ' (t - tQ ) V(t0 ,xQ ) and hence V(t, F(t, tQ, xQ )) — — > 0 as t -- ■> 00. Thus, F(t, tQ, xQ ) — > as t --> «>. 3. The next two theorems concern sufficient conditions for the equiasymptotic stability of x = 0. Theorem 10 is due to Massera [2 t]. Theorem 11 wasoriginally stated by Malkin [18] who merely proved sta­ bility; in this formulation, Theorem 11 follows as a corollary from theo­ rems on stability given earlier by Halikoff [8 ], [9]. Massera [2k ] ex­ tended Malkin!s proof to yield equiasymptotic stability. THEOREM 1 0 . If there exists, onI x N, a Lyapunov function V(t, x ) such thatV 1(t, x) is negative definite on I x N, and if V(t, x ) has the further property that, given any I,0 < | < p,and any cr > T,there are constants p (I ), 0 < t] < £, and t ( a , £ ) > cr such that, for any s e [T, a] and any y in ||y|| < t], the inequalities t > t , V(t, x ) < V(s, y ) imply ||x|| < I, then x = 0 is equiasymptotically stable. PROOF. By Theorem 6 , x = 0 is stable and, given any pf, 0 < p 1< p, and anytQ e I, there exists a 8 (tQ, p !) > 0 and, given any xQ in ||x|| < 5 and any i], 0 < -q < p1, there exist constants cr( 1 0, t ]) > 0 and t 1 e [tQ, tQ + a ) such that ||F(t1, tQ, x q )|| < r\. Let

A SURVEY OF LYAPUNOV1S SECOND METHOD

153

e be given, 0 < e < p f, and let r\ = t)(e) < €, t (tQ, e) > tQ + cr be the constants corresponding to € and t + a according to the hypothesis. Then V(t, F(t, tQ, xQ )) < V(t1, F(t!, tQ, xQ )) for t > t 1 and, a fortiori, for t > t whence ||F(t, t , x )|| < e for t > t . Theorem 8 states that, if f(t, x ) is linear in x on I x N, the assumptions of Theorem 10 are also necessary for the equiasymptotic stability of x = 0. Theorem 11 below shows that the hypothesis on V 1(t, x ) in Theorem 1o can be weakened at the expense of requiring that V(t, x )--- ■> 0 with x uniformly on I. THEOREM 11. If there exist, on I x N, real scalar positive definite functions U(t, x ), V(t, x) such that V(t, x ) is continuous on I x N, V(t, x) — —> o with x uniformly on I and V 1(t, x) + U(t, x ) ~— > 0 as t -- > oo uniformly on p 1 < ||x|| < p 2 for every positive P-j, P2 < p> x = o is equiasymptotically stable. PROOF. Given any non-increasing null sequence (7n ), 0 < 7n < p> letnn (7n ) = inf (V(t, x) |t > T, ||x|| = 7n ) and \ ( 7n ) = sup {V(t, x) | t > T, ||xIi < 7n )j clearly, nn > 0, 0 < in < », and V(t, x) = pn im­ plies t > T, Pn < ||x|| < rn for some Pn (7n ) > There exist constants an^7n^ > ° ’ vn^7n^ 51 0 suc]l that < ^n for t 2vn for t > T, an < ||x|| < p; clearly, an < Pn < 7 . More­ over, there existsa divergent sequence [rn }, 'rri(7n ) > T, such that V (t, x) + U(t, x)< vn for t > Tn, an < ||x|| < i.e., V'(t, x) < - vn for t > t ,an < ||x|| < 71; without loss of generality we may assume Tn+2 > Tn+1 + V

vn+l' n = 1, 2, ...

.

By the hypothesis concerning the continuous dependence of F(t, t ,x ) upon (tQ, xQ ), given any tQ > T, there exists a 5 (7 ,, tQ ) = 5 * ( t 1 , tQ ) > 0 such that ||x0| | < 5 implies ||F(t, tQ,xQ )|| < on [tQ, tQ + t1]. Then ||F(t, tQ, xQ )|| < 7, for t > tQ + t,; for, otherwise, ||F(t, t , x )|| = 7, at some time t = t' > tQ + t, which im­ plies V(t', F(t', t , xQ )) > p,. Since by construction V(tQ + t , , F(t + t,, t , xQ )) < p,, there exists a t" e (tQ + t ,, t'] such that V(t", F(t", tQ, xQ )) = n1 whence p, < ||F(t", tQ, xQ )|| < 7, and V'(t", F(t", t , xQ )) < - v,, which is absurd. This proves the stability of X = 0. Now, either ||F(tQ + t2, tQ, xQ )|| < a2 or not. In the first case, we can conclude by an argument similar to the one above that l|F(t, t , x )|| < 72 for t > t + t2 . If, on the other hand,

MTOSIEWICZ IIF(tQ + t2, tQ, xo )|| > a2, there exists a t2 e = [tQ + t2, tQ + t2 + X-j/vg] such that ||F(t2, tQ, xQ )|| < a2; for, otherwise, c*2 < ||P(t, tQ, xQ )|| < 7-, throughout J2 whence V 1(t, F(t, tQ, xQ )) < - v2 throughout J2 and V(tQ+T2+7 1/v2, P(tQ+T2+X1/v2,tQ,xQ )) < V(tQ+T2,F(t0+T2,tQ,xQ )) \^ < 0, which is absurd. Again, by an argument similar to the one above, ||p(t, t , xQ )|| < 72 for t > t2 and hence, a fortiori, for t > tQ + t2 + X-j/vg. Thus, in any case, ||P(t, tQ, xQ )|| < r2 for t > tQ + t3 . Continuing this way, we conclude that ||F(t, tQ, xQ )|| < rn for t > tQ + Tn+1, n = 1, 2, ... . REMARK. Note that {Tn ) is independent of tQ; solely upon (rn ) and the functions U(t, x ), V(t, x).

it depends

COROLLARY. If f(t, x) is lipschitzian for some constant k > o on I x N, and if the hypotheses of Theorem 11 are satisfied, x = 0 is uniformasymptotically stable. The proof parallels that of Theorem. 11. Observe that by the hy­ pothesis on f(t, x ), given any (tQ, xQ ) e I x N, ||P(t, t , xQ )|| < IIx0IIexp k(t - tQ ) for t ;> tQ . In view of the corollary, the assumptions of Theorem 11 are not necessary for the equiasymptotic stability of x = o, even when f(t, x) is linear in x on I x N. The next theorem may, In certain cases, be easier to apply than the foregoing Theorems 1o and 11. THEOREM 12. If x = o is uniformly stable and if there exists, on I x N, a Lyapunov function V(t, x) such that V 1(t, x) is negative definite on I x N, x = 0 is equiasymptotically stable. PROOF. Clearly, the quasl-equiasymptotic stability of needs only to be proved.

x = 0

Let a positive constant p T < p be so chosen that given any e, 0 < €< p1, there exists a 5 (e) > 0 such that tQ e I, ||xQ|| < 6 imply ||F(t, tQ,xQ )|| < € for t > tQ, and let 6q = 5 (p' ). By Theorem 6, given any tQ e I and any xQ in ||x|| < 8 , thereexists a t (tQ, e )> 0 and a t 'e [tQ, tQ + t ] such that ||F(t1, tQ, xQ )|| < 6(e). Hence ||F(t, tQ, xQ )|| < e for t > t f and, a fortiori, for t > tQ + t . k

bilityof

. The theorems that follow concern the uniform-asymptotic sta­ x = o. Theorem 13 is Lyapunov’s classical theorem on asymptotic

A SURVEY OF LYAPUNOV fS SECOND METHOD

155

stability, [131 > as foxmilated by Persidskii [27], [3 13> £32], [353 • THEOREM 13. If there exists, on I x N, a Lyapunov function V(t, x ) such that V(t, x ) -- > 0 with x uniformly on I and V*(t, x) is negative definite on I x N, x = 0 is uniform-asyraptotically stable. PROOF. By Theorem 3 > x = 0 is uniformly stable. Thus, given any p 1, 0 < p 1 < p, there exists a 8(pr) = 8q > 0 such that tQ € I, HXqII < 60 llP(t, tQ, xQ )|| < p f for t > tQ . Given any e, 0 < € < p !, thereexists a 5 (e) > 0 and, by Corollary 2 to Theorem 6, given any xQ in ||x|| < §Q there exist constants t(p 1, 5 (e)) = t*(€ ) > 0 and t 1 € [tQ, tQ +t *) such that||F(t1, tQ, xQ )|| < 8(e). Therefore, ||F(t, tQ, xQ )||< e for t > t 1and, a fortiori, for t > tQ + t *. The severity of the assumptions in Theorem 13 is illustrated by the following theorem. THEOREM 1k . If V(t, x) is a Lyapunov function on I x N such that V(t, x )— ~> 0 with xuniformly on I and V f(t, x) is negative definite on I x N, then given any constants p 1, p,f, 0 < p" < p1 < p, there exist constants A,(p!, pn) > 0, v (pH ) > 0 such that W(t, x ) = exp(xt)V(t, x ) satisfies W f(t, x ) < - v for t > T, p " < ||x|| < p ’. The proof is trivial and therefore is omitted. The next theorem due to Malkin [22] is a partial converse to Theo­ rem 13. In Malkin!s original formulation the first order partial deriva­ tives of f(t, x) with respect to x are required to be bounded on I x N. THEOREM 1 5 . If f(t, x)is, for some integer k > 1, v of class C with respect to x on I x N and, for every x € N, of class on I, and if x = 0 is uniform-asymptotically stable, there exists on some set I x M, M C N, a Lyapunov function V(t, x ) of class such that V(t, x ) -- > 0 with x uni­ formly on I and V 1(t, x) is negative definite on I x M. PROOF. By Lemma 2, we may assume without loss of generality that IcUL (t, x)/dxj| < 1 on I x N, 1 < i, j < n. Let a positive constant

p* < p be so chosen that, given any

e,

MTOsmricz

156 0

< € < p*, there exists a 5 (e) > 0 such that t e l , ||x|| < 5 imply ||F(t + s, t, x)|| < e for s >o. By hypothesis, there exists a 5 q > o and, given any tj > o, however small, a t (t)) > T such that t e l , ||x|| < bQ imply ||F(t + s, t, x)|| < r\ for s > t . Let p1 = min (6q, 6(p*)) and let Mbe the open sphere ||x|| < p 1. Clearly, for any s > o, F(t + s, t,x) is of class onI x M. The boundedness, on I x M, of the first partial derivatives of f(t, x) with respect to x implies that, for any s > o, the first partial derivatives of ||F(t + s, t, x)|| with respect to t and x are bounded, on I x M, by h(s) = kexp(Xs) where k = k(p*) > 0 and X >o are certain constants independent of (t,x ). More specifically, k is an upper bound of ||f(t, x)|| for t e l , ||x|| < p*. Given any non-increasing null sequence (7n ) >° < rn < p 1>there exists an increasing divergent sequence{tn), tn (7n ) > o, such that (t, x) e I x M implies ||F(t + s, t, x)||< ? n for s > t . Let g(s) be a real scalar function, defined and continuous, positive, non-increasing for s> o, g(s) --- > 0 as s -— -> », such that given any (t, x ) e I x M, IIF(t +s, t, x )|| < g(s) on [0, t2] and g(tn+1 ) = 7n >n = 1, 2, .. . . Then g(tn+1 ) < g(s) < g(tn ) on [tn, tn+1] and hence ||F(t + s, t, x)|| < 7n < g(s) on ttn+1, tn+2J, n = 1, 2, therefore ||F(t + s, t, g(s) for s > o.

tions

Let G(s) be the function associated by Lemma g(s ), h(s) and define

3

with the func­

00 V(t, x) =

J

G(||P(t + s, t, x)||)ds

.

o Clearly, V(t, x) is defined on I x M and, by Lemma 3, is of class Ck . Moreover, the first partial derivatives of V(t, x) with respect to x are bounded on I x M so that V(t, x) -- > o with x uniformly on I. Given any

(t, x) e I x M,

we have for

s > 0

t+s ||F(t + s, t, x ) - x||
(1/2 )||x||. Therefore,

V(t, x) >

J

J|x||/2k

o

G (||F(t + s, t, x )||)ds > ||x||/(2k)G(||x||/2 )

x)||
o, there exists, on I x N, a real scalar function V(t, x ) of class C1 which is a positive definite form in x of degree m such that V(t, x) ---> o with x uniformly on I and V 1(t, x) = - W(t, x) on I x N. For the proof we refer to [20], and to [1] if quadratic form in x independent of t .

W(t, x)

is a

Theorem 16 generalizes an analogous result of Lyapunov [13] for linear systems with constant coefficients. It also extends an earlier theorem of Malkin [16] on the necessity of the existence of a Lyapunov function of class C1 on I x N, satisfying the hypotheses of Theorem 13 , in order that, given any (tQ, xQ ) e I x N, every solution F(t, t , xQ ) satisfy Persidskii1s condition [31]: ||F(t, tQ,xQ )|| < M||xo|| for t > t + t (M, x ) where t > 0 and M is any given constant,0 < M < 1. For the relation of these results to work by Perron [29] > [30] on the boundedness of every solution of linear non-homogeneous systemswith bound­ ed forcing term, we refer to [1 ], [16], [19]. The following theorem of Massera [25] proves a stronger con­ tention than Theorem 15 under considerably weaker hypotheses.

158

AOTOSIEWTCZ

THEOREM 17* If f(t,x) is locally lipschitzian on I x N and if x =0is uniform-asymptotically stable, there exists on some set I x M, M C N, a Lyapunov function V(t, x),possessing partial derivatives with respect to t andxofany order, such that V(t, x) — > o with x uniformly on I and V’(t, x) is negative definite on I x M. If f(t, x) is lipschitzian on I x N, the partial derivatives of V(t, x) are bound­ ed on I x M; if, on I x N, f(t, x)is independent of t or periodic in t, V(t, x) isindependent of t or periodic in t on I x M. Por the proof we refer to [25]• Theorem 17 contains earlier results of Massera [25] and Barbasin [2] on the existence of a Lyapunov function V(t, x) on I x M, M C N, such that V!(t, x) is negativedefinite on I xM for the case of asymptotic (and hence uniform-asymptotic) stability of x= 0 when f(t, x) Is independent of t or periodic in t on Ix N. These re­ sults are converse theorems to the corollary toTheorem 7• The next theorems, though essentially elementary, may be useful in some applications. Theorem 18 is related to Theorem 12 on the equi­ asymptotic stability of x = 0; it generalizes an analogous theorem of Massera [25] based on the assumptions of Theorem 9* If f(t, x) is locally lipschitzian on I x N, the converse to Theorem 18 Is obviously true as a consequence of Theorem17* Theorems 19and 20 due to Massera [25] show that the hypotheses ofTheorem 9 can beImplemented so as to yield the uniform-asymptotic stability of x = 0. THEOREM 18. If x = 0 Is uniformly stable and if there exists, on I x N, a bounded Lyapunov function V(t, x) such that Vf(t, x) is negative definite on I x N, x = 0 is uniform-asymptotically stable. The proof is very similar to that of Theorem 12 and therefore is omitted. THEOREM 19. If f(t, x), is lipschitzian on I x N and’if there exists a real scalar function V(t, x) defined, locally lipschitzian and positive definite on I x N, and a real scalar function W(t), de­ fined, continuous and increasing for t > 0, W(o) = 0, such that V!(t, x) < - W(V(t, x)) on I x N, then x = 0 is uniform-asymptotically stable.

A SURVEY OF LYAPUNOV'S SECOND METHOD

159

PROOF. By Theorem 18, the uniform stability of only to be proved.

and let 0 < e < e < ||x|| tQ e I, t > tQ +

x =

0

needs

Let f (t, x) be lipschitzian for the constant k > o on I x N, K > 0 be an upper bound of V(t, x) on I x N. Given any €, p, there is a ^(e) > 0 such that V(t, x ) > \i for t e l , < p. Let t(€)= /^(€ )dV/W(V) and 5 (e) = e exp(- kx). Then ||x0 1| < 5 imply ||F(t, tQ, xQ )|| < e for t e [tQ, tQ + t ] ; for t, it follows that V

and thus

\x,

V(t, F(t, tQ, xQ ))
o, W( 0 ) = o, g(t) positive on I and /Qdt/g(t) = oo, such that V !(t, x) < - W(V(t, x)) and (x, f (t, x)) < ||x||g( ||x||) on I x N, then x = 0 is uniform-asymptotically stable. PROOF. By Theorem 18, the uniform stability of only to be proved. Given any €, 0 < € < proof of Theorem 19 , and choose Since

on I x N, given any for t e [tQ, tQ + t]

tQ € I

p,

let

§(e)

and any

> 0,

M-(e)

> 0

t(

such that

xQ in

||x||
t(€ ).

e) > 0

5,

it follows that

e g(l|x||)

~

Jb

g(||x||)

whence ||F(t, t , xQ )|| < e; and, by the same argument as in the proof of Theorem 19, ||F(t, tQ, xQ )|| < e for t > tQ + t.

ANTOSIEWICZ

160

IV. BOUNDEDNESS AND ASYMPTOTIC STABILITY IN THE LARGE 1.

We shall now suppose that the basic differential equation x = f (t, x)

is defined on I x Rn and that f (t, x) has the same regularity prop­ erties on I x Rn as it was assumed in (I.) to have on I x S, except that for boundedness results we shall not require f (t, o) = o on I. NC will denote the set ||x|| > p > 0 in Rn and I, as before, some interval [T, «>), T > o. DEFINITION

3•

Every solution is said to be

(3-1)

bounded: if, given any tQ e I and any rQ > 0, there exists an r(tQ, rQ ) > 0 such that ||xQ|| < rQ implies i|F(t, tQ, xo )|| < r for t > tQ .

(3-2)

uniformly bounded: if, given any rQ > 0, there exists an r(rQ )> o such that tQ € I, ||xQ||< rQ imply ||F(t, tQ, xQ )|| < r for t > tQ .

(3*3)

ultimately bounded: if, given any rQ,r 1, rQ > r1 > 0, there exist an r(r1 ) > 0 and a t (rQ, r1)> 0 such that tQ € I, ||xQ|| < rQ imply ||F(t, t, x)|| tQ + T.

DEFINITION

k

. The solution

x = 0

is said to be

(t.1)

asymptotically stable in the large [3]: if it is stable and if (tQ, xQ ) e I x Rn implies F(t, tQ, xQ ) — ~> 0 as t — —>

0.2 )

uniform-asymptotically stable in the large [ b ] : if every solution is uniformly bounded and if, given any positive rQ, r 1, there exists a T(r0, r1) > 0 such that tQ e I, ||x0|| < rQ imply ||P(t, tQ, xQ )|| < r1 for t > t + t .

It can be shown similarly as in the case of asymptoticstability that if f(t, x) Is independent of t or periodic in t on I x Rn, then asymptotic stability in the large of x = 0 implies uniform-asymptotic stability In the large of x = 0, (cf., o»g* > [25]). 2. We begin with theorems on boundedness which are related to some results of Yoshizawa [39], O o ] , O1 ].

1 61

A SURVEY OF LYAPUNOV’S SECOND METHOD THEOREM 2 1 . If there exists, on I x Nc, a real scalar function ¥(t, x), defined, locally lipschitzian and positive definite, such that ¥(t, x ) -- > °o with x uniformly on I and ¥ ’(t, x ) < 0 on I x Nc, every solution is bounded. PROOF. Given sup{¥(t0, x) | p < ||x|| < ¥(t, x )>o) for t > T, ||F(t, tQ, xQ )|| < r for some time t = t 1 > tQ which is absurd.

any tQ e I and any rQ > p, let o>(t0, rQ ) = rQ) and let r(tQ, rQ ) > p be such that ||x|| > r. Then ||xQ || < rQimplies t > t Q; for, otherwise, ||F(t, tQ, xQ )|| = r at so that a> < ¥(t1, F(tf, tQ, xQ )) < W(t , xQ ) < a>,

THEOREM 22. If there exists, on I x Nc, a realscalar function ¥(t, x), defined, locally lipschitzian and positive definite, such that, for any open sphere S )N, ¥(t, x ) is bounded on I x Nc H Sc, ¥(t, x )— -— > » with x uniformly on I, and ¥ ’(t, x) < 0 on I x Nc, then every solution is uniformly bounded. The proof is very similar to that of Theorem

21

and therefore is

omitted. THEOREM 23. If there exists, on I x Nc, a real scalar function ¥(t, x ), defined, locally lipschitzian and positive definite, such that for any open sphere S } N, ¥(t, x ) is bounded on I x Nc fl S°, ¥(t, x) — > 00 with x uniformly on I, and ¥* (t, x ) is negative definite on I x Nc,then every solution is ultimate­ ly bounded. PROOF. Given any rQ, r1, rQ > r., > p, let o>(rQ ) = sup (¥(t, x) | t € I, p < ||x|| < rQ), let v(r1 ) > 0 be such that ¥ !(t, x) < - v for t e l , ||x|| > r 1 , and put x*1 ) = ^/v* Given any tQ e I and any xQ in ||x|| < rQ, either ||xQ|| < r 1 or r 1 < ||xQ || < rQ . In the first case, there exists an r(r1 ) > p such that ||F(t, tQ, xQ )|| < r for t > tQ . In the second case, there exists a t f e [tQ, tQ +t] such that ||F(t 1, tQ, xQ )||< r1; for, otherwise, ||F(t, tQ, xQ )||>r 1 throughout [tQ, tQ + t ] and hence ¥(tQ + t, F(tQ+ t, tQ, xQ )) < ¥(tQ, xQ ) - v t < 0, which Is absurd. Thus, in either case, ||F(t, tQ, xQ )|| < r for t > t ’ and hence,a fortiori, for t > tQ + t .

ANTOSIEWICZ In a series of papers [36], [37], [38] concerning second order non-linear differential equations of the Cartwright-Littlewood type, Reuter used gauge functions closely related to those of Theorem 23 to prove the ultimate boundedness of every solution and the existence of forced almost periodic solutions. He appears to have introduced the term ultimate bound­ edness. 3.

The theorems that follow concern the uniform-asymptotic sta­ bility in the large of x = 0. Theorem 2k due to Massera [25] generalizes a similar theorem of Barbasin and Krasovskii [3] for the case when f(t, x) is independent of t on I x Rn . THEOREM 2k . If there exists, on I x Rn, a Lyapunov function V(t, x) such that V(t, x) — -> 0 and 00 as ||x|| -— > 0 and 00uniformly on Iand V 1 (t, x) is negative definite on I x Rn, x = 0 is uniformasymptotically stable in the large. The proof resembles that of Theorem

12

and therefore is omitted.

The next theorem, also due to Massera [25], is a generalization of an analogous result of Barbasin and Krasovskii I k ]. Their proof hinges on Lemma 3 and requires f(t, x) to be of class C 1on I x Rn and to have boundedfirst partial derivatives with respect to t and x on I x Rn . This result is itself an extension of an earlier theorem of theirs [3] for the case when f(t, x ) is independent of t on I x Rn and allsolutions exist in the past; theproof of this latter theorem uses Barbasin1s method of sections [2]. THEOREM 25. If f(t, x ) is locally lipschitzian on I x Rn and if x = 0 is uniform-asymptotically stable in the large, there exists, on I x Rn, a Lyapunov function V(t, x ), possessing partial derivatives with respect to t and x of any order, such that V(t, x) > 0 and °° as ||x|| — > 0 and 00uni­ formly on-- I and V 1 (t, x) isnegative definite on I x Rn . If f(t, x) is lipschitzian on I x Rn, the partial derivatives of V(t, x) are bounded on I x H for every bounded set H C Rn; if f(t, x ) is independent of t or periodic in t on I x Rn, so is V(t, x ). For the proof we refer to [25].

A SURVEY OP LYAPUNOV'S SECOND METHOD

163

ADDED IN PROOF (September, l9 5 7 )• Since this survey was pre­ pared for publication, a number of papers have appeared dealing with the theory of Lyapunov's second method and, in particular, with the inversion of the basic theorems on asymptotic stability and instability. While it has been impossible to Incorporate these new results, it is nevertheless felt that a few remarks about some of them are in order here. Krasovskii [4 6 ], [4 8 ] examines several types of asymptotic sta­ bility and obtains necessary and sufficient conditions for them in terms of Lyapunov functions satisfying various additional requirements; for example, [4 6 ], he states without proof necessary and sufficient conditions for the asymptotic stability of x = 0 as well as for the uniform (simple) and equiasymptotic stability of x = 0. In [45] he gives new proofs for the converse of Theorem 13 in the case of autonomous equations and for the converse of Cetaev's theorem on instability, which was proved by Vrkoc [52]. In [4 7 ] he introduces the concept of Mnon-critical and uniform" be­ havior of a solution in an €-neighborhood of x = 0 in terms of which he formulates necessary and sufficient criteria for the existence of Lyapunov functions in the case of asymptotic stability and instability. Kurzweil [4 9 ] proves a converse to Theorem 15 under considerably weaker hypotheses, similar to Theorem 17 of Massera who himself extends his original method of proof, [2 4 ], to infinite dimensional spaces in [50]. Persidskii [51] and Zubov [ 5 3 ]consider various types of asymptotic stability, their interrelationships [53], and criteria for them involving Lyapunov functions, [51 ], and functions closely related to them, [53].

National Bureau of Standards BIBLIOGRAPHY [1 ]

ANT0 SIEW1 CZ, H. A., and DAVIS, P., "Some implications of Liapunov's conditions for stability," J . Rational Mech. Anal., Vol. 3 (1 9 5 4 ), 4 4 7 -4 5 7 .

[2 ]

BARBASIN. E. A., "The method of sections in the theory of dynamical systems, (Russian), Mat. Sb., Vol. 29 (1 9 5 1 )> 233-280.

[3]

BARBASIN, E. A., and KRASOVSKII, N. N., "On stability of motion in the large," (Russian), Doklady Akad. Nauk SSSR, Vol. 86 (1952), 4 5 4 -456 .

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[5]

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1 6k

ANTOSIEWICZ

[6 ]

GORSIN, S., ,TOn stability of motion with constantly acting dis­ turbances, f (Russian), Izvestiya Akad. Nauk Kazah. SSR 56, Ser. Mat. Meh., Vol. 2 (19^8 ), *+6 -7 3 -

[7]

--- ------ , "Critical cases," (Russian), Ibid., Vol. 2 (191+8), 7I+-101.

[8]

HALIKOFF, H. S., "Sur la stabilite des integrates des equations differentielles ’Im Grosser1," (Russian), Izvestiya Fiz.-Mat. Obschchestvo Kazan. Univ., Vol. 9 (3 ) (1 93 7 ), 31 -5 9 *

[9]

----------, "Une note sur un theoreme de stabilite du mouvement," (Russian), Ibid., Vol. 11(3) (1 9 3 8 ), 199-200.

[10 ] KRASOVSKII, N. N., "On stability of motion in the large for constant­ ly acting disturbances," (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., Vol. 18 (195 *0 , 95-102. [1 1 ]

■, "On a converse of a theorem of K. P. Persidskii on uni­ formstability," (Russian), Ibid., Vol. 19 (1 9 5 5 ), 273-278.

[12]

KURZWEIL, J., "On the reversibility of the first theorem of Lyapunov concerning the stability of motion," (Russian), Czech. Math. J., Vol. 5 (8 0 ) (1 9 5 5 ), 382-398.

[13]

LIAPOUNOFF, A., "Probleme general de la stabilite du mouvement," Annals of Math. Studies No. 17 , Princeton Univ. Press, 19*+9 , Princeton, N. J.

[11+]

MAKAROFF, S., "Quelques generalisations des theoremes fondamentaux de Liapounoff sur la stabilite du mouvement," (Russian), Izvestiya Fiz.-Mat. Obschchestvo Kazan. Univ., Vol. 10(3) (1938), 139-159.

[15]

MALKIN, I. G., "On stability in the first approximation," (Russian), Sbornik Naucnyh Trudov Kazan. Avlac. Inst., No. 3, 1 9 3 5 .

[16]

--------- , "Certain questions in the theory of stability of motion in the sense of Liapounoff," (Russian), Ibid., No. 7 , 1937 , Amer. Math. Soc. Translation No. 20.

[17]

---- -----, "On the stability of motion in the sense of Lyapunov," (Russian), Mat. Sb., Vol. 3 (*+5 ) (1938 ), 1+7 - 100 , Amer. Math. Soc. Translation No. 1+1 . MALKIN, I. G., "Verallgemeinerung des Fundamentalsatzes von Liapunoff uber die Stabilitat der Bewegungen," C. R. (Doklady) Acad. Sci. URSS, Vol. 18 (1938), 162-16I+.

[18]

[19]

----- — — , "On stability under constantly acting disturbances," (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., Vol. 8 (19kk ), 2 A1-2A5 , Amer. Math. Soc. Translation No. 8 .

[20]

-------- —, "On the construction of Lyapunov functions for systems oflinear equations," (Russian), Ibid., Vol. 16 (1952), 239-2I+2.

[21]

--------- , "Theory of stability of motion," (Russian), Izdat. Tehn.-teor. Lit., Moscow-Leningrad, 19 5 2 .

[22]

----- — — , "On a question of a converse theorem of Lyapunov on asymptotic stability," (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., Vol. 18 (195I+), 129-138.

[23]

MARACHKOV, M., "On a theorem of stability," (Uber einen Liapounoffschen Satz), (Russian), Izvestiya Fiz.-Mat. Obschchestvo Kazan. Univ., Vol. 12(3) (19^0 ), 171-171+.

Gosudarstv.

A SURVEY OP LYAPUNOV’S SECOND METHOD [214-]

165

MASSERA, J. L., "On Liapounoff’s conditions of stability/1 Annals Math., Vol. 50 (19^9 ), 705 --7 2 1 .

[2 5 ] --------- , "Contributions to stability theory, " Annals Math., Vol. 6*4 (1956), 182-206. [26]

MOISSEJEV, N., "Uber den wesentlichen Character einer der Beschrankungen, welche den topographischen Systemen in der Liapounoffschen Stabilitatstheorie auferlegt werden," C . R. (Doklady) Akad. Nauk SSSR., Vol. 10 (1936), 165-166.

[2 7 ] NEMITSKII, V. V., and STEPANOV, V. V., "Qualitative theory of differ­ ential equations," (Russian), 2nd ed., Gosudarstv. Izdat. Tehn.-teor. Lit., Moscow-Leningrad, 19^9 • [28]

NEMITSKII, V. V., "Some problems of the qualitative theory of differ­ ential equations," (Russian), Uspehi Matem. Nauk (N.S.), Vol. 9(61 ) (195 *0 , 39 -5 6 .

[2 9 ] PERRON, 0 ., "Uber Stabilitat und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, f Math. Zeit., Vol. 29 (1928), 129-160. ----— , "Die Stabilitatsfrage bei Differentialgleichungen," Math. [3 0 ] Zeit., Vol. 32 (1 9 3 0 ), 703-728. [3 1 ] PERSIDSKY, K., "Uber die Stabilitat einer Bewegung nach der ersten Naherung, (Russian), Mat. Sb., Vol. *40 (1 933 ), 28*4-293. [3 2 ] — — — — — , "Sur la theorie de stabilite des integrales du systeme des equations differentielles," (Russian), Izvestiya Piz.-Mat. Obschchestvo Kazan. Univ., Vol. 11(3) (1936-37 ), 0 -8 5 . [3 3 ]

— -— — , "On the theory of stability of systems of differential equations, 1 (Russian), Ibid., Vol. 11(3) (1 9 3 8 ), 2 9 - O .

[3 ^ ] --------- , "Uber einen Satz von Liapounoff," C. R. (Doklady) Acad. Sci. URSS, Vol. 1*4 (1 937 ), 5 O - 5 O . »"On the theory of stability of solutions of differential [3 5 ] —— ---equations, f (Summary of dissertation), (Russian), Uspehi Mat. Nauk, Vol. 1 (19^6), 2 5 0 -2 5 5 . [3 6 ] REUTER, G. E. H., "On certain non-linear differential equations with almost periodic solutions," J. London Math. Soc., Vol. 26 (1951), 215-221. [3 7 ] --------- , "A boundedness theorem for non-linear differential equa­ tions of the second order," Proc. Cambridge Phil. Soc., Vol. *47 (1 951 ), * 49-5*4 . [3 8 ] ■ --------- , "Boundedness theorems for non-linear differential equa­ tions of the second order, II," J. London Math. Soc., Vol. 27 (1 952 ), *48-58. [3 9 ] YOSHIZAWA, T., "On the non-linear differential equation," Mem. Univ. of Kyoto, Ser. A., Vol. 28 (195*4), 133-1*41 . [ho]

----- , "Note on the existence theorem of a periodic solution of the non-linear differential equation," Ibid., Vol. 28 (195 *0 , 153 -1 5 9 .

[in ] —-------- , "Note on the boundedness of solutions of a system of differential equations," Ibid., Vol. 28 (195*4), 293-298.

166

ANTOSLEWICZ

[*42]

Z U B O V , V. I., D o k l a d y Akad.

nO n t h e t h e o r y o f L y a p u n o v ’s s e c o n d m e t h o d , " N a u k S S S R . , V o l . 99 ( 195 *0 , 3^-1 - 3 ^*+ •

[A3]

--- --- — — , " P r o b l e m s i n t h e t h e o r y o f L y a p u n o v ’s s e c o n d m e t h o d ; c o n s t r u c t i o n of g ene r a l so l u t i o n in the r e g i o n of a s y mptotic s t a ­ b i l i t y , " ( R u s s i a n ) , A k a d . N a u k S S S R . P r i k l . M a t . M e h . , V o l . 19 ( 1 9 5 5 ), 1 7 9 - 2 1 0 .

[*4*4]

-----------~, " O n t h e

t h e o r y o f A.

M.

(Russian),

L y a p u n o v ’s s e c o n d m e t h o d , "

(Russian), Doklady Akad. Nauk SSSR., Vol. 100 (1 9 5 5 ),

8 5 7 -8 5 9 -

[*45]

K R A S O V S K I I , N. N . , " O n t h e i n v e r s i o n o f t h e o r e m s i n t h e s e c o n d m e t h o d o f A.. M. L y a p u n o v f o r i n v e s t i g a t i o n o f s t a b i l i t y o f m o t i o n , " ( R u s s i a n ) . U s p e h i M a t e m . N a u k (N.S. ), V o l . 11 (69) (1958), 159-1 6*4.

[*46]

— , "On the theory of Lyapunov’s second method in studying stability of motion," (Russian). Doklady Akad. Nauk SSSR, Vol. 109 (1958), *460-A463 •

[*4 7 ]----— — —-, "On the theory of the second method of A. M. Lyapunov the investigation (1 9 5 8 ), 57-6*4•

for

[*48]

of

stability,"

(Russian).

Mat.

S b . , Vol.

— - — ----— , " i n v e r s i o n o f t h e o r e m s o n L y a p u n o v ’s s e c o n d m e t h o d q u e s t i o n s of s t a b i l i t y of m o t i o n i n the f i r s t a p p r o x i m a t i o n , "

(Russian).

*40

and

Akad. Nauk SSSR. Prikl. Mat. Meh., Vol. 20 (1956), 255-265.

[*49]

K U R Z W E I L , J., " O n t h e i n v e r s i o n o f t h e s e c o n d t h e o r e m o f L y a p u n o v on s t a bil ity of m o t i o n , " (Russian). C z e c h . M a t h . J., V o l . 6 (81) (1956), 217-259, a n d *455-48*4.

[50]

MASSERA, J. L., "Sobre la estabilidad en espacios de dimension infinita," Rev. Un. Mat. Argentina, Vol. 17 (1 9 5 5 ), 135 -1^7 *

[51]

P E R S I D S K I I , K . , "On the s e c o n d m e t h o d of L y a p u n o v , " (Russian). I z v e s t i y a A k a d . N a u k K a z a h . S SR. Ser. M a t . M e h . , V o l . *4 (8 )

(1 956

[52]

VRKOC,

), *43-^7 * I.,

"On the

inverse

t h eorem of C e t a e v , " (Russian). .

Czech.

Math. J., Vol. 5 (80) (1 9 5 5 ), *451 -*461 [53]

Z U B O V , V. I., " C o n d i t i o n s f o r a s y m p t o t i c s t a b i l i t y i n c a s e o f n o n s t a t i o n a r y m o t i o n a nd e s t i m a t e of the ra t e of d e c r e a s e of t h e g e n e r a l solution, f (Russian). V e s t n i k L e n i n g r a d U n i v . , V o l . 12 (1 9 5 7 ),

110-129, 208.

IX.

ON PHASE PORTRAITS OP CRITICAL POINTS IN n-SPACE* Pinchas Mendelson 1.

INTRODUCTION

In this paper we study the geometric behavior of the solution curves of the (vector) differential system

-yr = Ax

+

Q(x),

x = (xQ, x 1?

xn )

near the isolated critical point x = 0. Here A is a real, constant, (n+1) x (n+1) matrix (n = 1, 2, 3 , . • • ) , with one zero characteristic root. All the n remaining characteristic roots have real parts of equal sign. The analytic vector function Q(x) consists of higher order (non­ linear) terms, subject to a certain mild restriction. Such differential systems occur in many important stability theo­ ries (cf. Liapounov). In the two-dimensional case, n = 1, a rather com­ plete analysis of the local phase portrait in the neighborhood of the origin has been carried out by several authors. Poincard investigated the case when the matrix A is non-singular. Bendixson, Forster and Lefschetz have completed the analysis for a general matrix A in the particular case n = 1. In higher dimensions Siegel has obtained significant results, but only under the restriction that A be non-singular, a restriction which excludes many of the most important applications. This paper seems to contain the first investigation of the geometrical aspect of the criti­ cal point at the origin when A is permitted to have a zero characteristic root. The notions of "fan-cone" and "saddle-cone" are defined and the critical points of the given system are analyzed in terms of them. It is shown that in the particular case of E^(n =2), some major aspects of This work was carried out under Office of Naval Research Contract N 60 RI-105, Mathematics Dept., Princeton University, Princeton, N. J. 167

MENDELSON

168

these new types of singularities are natural generalizations of the classi­ cal Bendixson singularities in the plane. 2.

NOTATION AND TERMINOLOGY

In the following discussion x with the components (xQ, x 1, ..., xn ).

will denote a real (n+1)-vector

We shall use the following notations: [x]^> (h = 1, 2,... ), to of which, x ^ ..., (i in (x , x 1, xn ), convergent and beginning with terms of degree

denote an (n+1)-vector, thecomponents - o, 1, ..., n)), are real power series in some region containing the origin, at least h.

*[x , x 1, ..., x ] , (h = 1, 2, ...),will denote a real power o i n h series in (xQ, x^, ...,xn ) beginning with terms of degree at least h and having no terms in xQ alone. E(x q, x 1, ...,xn ) will denote a power series in (xQ, x 1, ..., xn ), convergent in some region containing the origin and satisfying E(o, 0, ..., o) = 1.We shall refer to E(x q, x 1, ..., xn ) as a unit. An analytic transformation P : En+1 -— >E^+1 of the form xjf = f^(x^, x 1, ..., x ^ )j f^ (o, o, ..., o) = o, (i = o,1, ..., n), is said to be regular atthe origin if the Jacobian

I af i

■5x7

J

(0,0, . . . , o )

^ 0

’

Two systems of differential equations, which can be transformed into one another by an analytic transformation regular at the origin, are said to have >Tthe same behaviorn in the neighborhood of the origin (cf. Lefschetz [1] p. 118). The following notations and definitions will be used extensively in Section 6: (i) (ii) (iii)

The double cone {x | denoted by C(m).

= m.2x2 } will be

The solid cone {x | S^=1x| < m2x2 } will be denoted by S(m). The sets C(m.) fl (x | 0 < xQ < e} and C(m) n {x | - e < xQ < 0}, e > 0, will be denoted by C+ (m, e) and C“(m, €), respectively. These sets will sometimes be

CRITICAL POINTS IN n-SPACE

169

referred to as "the lateral surfaces." (iv) The sets S(m) fl{x | 0 < xQ < e) and S(m) fl {x | - € < x Q < 0}, e > 0, will be denoted by S+ (m, €) and S~(m, €), respectively. These sets will sometimes be referred to as "the solid cones." (v) The sets S(m) fl fx | xQ = e) and S(m) D (x | xQ = - e), e > 0, will be denoted by D+ (m, €) and D“(m, e ), respectively. ¥e shall sometimes refer to them as "the upper and lower discs" of the cones S (m, € ) and S“ (m, €) • ¥e

shall also find it usefulto employ the following definitions:

DEFINITION 1. Let $ be an autonomous system ofdifferential equations defined in a region ft C En and satisfying the uniqueness of solutions there. Let x e ft, be the (unique) trajectory of $ pass­ ing through x, and let A be any point set contained in ft. Then Ta = Crx | x € A)

t„ A

DEFINITION 2. is a time such that

.

Let r be as defined above, and suppose that x rA (tA ) = x. Then rA = {r A(t ) It > t A ) and

rx = frx (t) I t < V * NOTE. ¥e shall occasionally denote a point x € ft by Roman capitals. The symbols rA , At ,A r* and r~ will then be written as A Tp, tp, Tp and Tp, respectively. DEFINITION 3 • Let m be an open set contained in ft and let B(m) be the boundary of Then, we say that a point P € ft fl B(cd) is a point of egress from m (with respect to the given system $ and the set a), if there exists a positive number € such that £p(t) e a> for tp - € < t < tp. If, moreover, there exists a positivenumber tj such that Tp(t) € ft - ai for tp < t < tp + t], the point P Is called a point ofstrict egress from cd. (A. PliS, p. M 5). DEFINITION b. Let o1 be the system which is obtained from $ by the transformation t----> - t . Then P e ft fl B(ao) is a point of strict access into a> (with respect to the given system $ and the set ft) if P is a point of strict egress from cd with respect to the system and the set ft. DEFINITION 5 • Let 0 be as defined above, and suppose that 0 has an isolated singularity at 0 e ft. The cone S+ (m, e ) is said to be

MENDELSON a stable [unstable] fan-cone of

(i)

(ii)

*

if:

P e C+ (m, e) 0 D+(m, e) = = > Ip C S+ (m, e)[r"C S+ (m, e)]. That is, the vector field of $ on the lateral surface and upper disc of the solid cone never points out of [into] the cone. 0

is asymptotically stable with respect to

T as S (m, e )

t *-- > + oo [t

DEFINITION

6.

If in Definition

DEFINITION the first kind) if:

7*

S+ (m,

5

> - oo]

f € T has a S (m, e ) definite tangent at the origin, and if furthermore these tangents are the same for all r e T , then S+ (m, €) will be called a simple fan-cone. ^ (ni>e)

(i)

(ii) (iii)

(iv)

e)

every

is said to be a saddle-cone of

0

(of

If x € C+ (m, e ), then x is a point of strict access into the interior of S+ (m, € ). x e C+(m, e) = = > r* t S+ (m, e ) . A If x is a point of the interior of D+ (m, € ), then x is a point of strict egress from the interior of S+ (m., €). That is, all trajec­ tories passing through the upper disc leave the solid cone (upon continuation in the positive direction). Let A denote the set of all points x € D+(m, €) such that r~ C S+ (m, e ). A Then o is negatively asymptotically stable with respect to T^.

DEFINITION 8 . S+ (m, e) is said to be a saddle-cone of $ (of the second kind) if the transformation t — -> - t reduces it to asaddlecone of the first kind. DEFINITION 9 * S+ (m, e ) is said to be a simple saddle-cone of o (of thefirstor second kind)if the set A of Definition 7 , con­ dition (iv),consists of exactly one point. NOTE. The above definitions were given in terms of S+ (m, e and Cf (m, e), but apply, in the obvious way, also to S”(m, e) and C‘(m, e). -X+

CRITICAL POINTS IN n-SPACE 3.

3 •1 •

(1

171

A CANONICAL FORM

We begin with the system

)

dx = Ax + [x] 2 ^

where (i)

(ii)

(iii)

A = (a^j) is a real, constant, singular matrix of rank n.

(n+1 ) x (n+l ),

The non-zero characteristic roots of have negative (positive) real parts,

A

all

Z^=0x| > 0 in S - 0 , where S is a suit­ able spherical neighborhood of the origin,

A real, non-singular, constant matrix B can be found such that the transformation x = By will transform the given system (1 ) into a system in y of the form: ( i )'

y = Py + [ y ] 2

where P is in real Jordan canonical form. Furthermore, system (1 ) 1 still satisfies conditions (i), (ii), (iii). We may assume, therefore, that in the given system (1 ), A is already in canonical form, namely

A

where

A1

Ai

is an (nxn) real, non-singular matrix in Jordan canonical form,

3 .2 We observe that the components [xQ, x 1, ..., xn ]^, (i = o, 1 , .•., n), of [x] 2 are uniformly and absolutely convergent in some spherical neighborhood of the origin. Hence in that neighborhood these series may be rearranged. We can write, therefore:

[ x o>

(2 )

X 1> •••' xn ] 2 = P2 (xo ) + *[xo> (i = o,

1,

where P 2 (xQ ) are power series in ing with terms of degree at least

2.

xQ

..., n)

xnl2 ’

,

(with constant coefficients) start­

MENDELSON

172 3 .3

Let x ± + [xQ, xn ]| (i = 1, ..., n), and consider the system of equations: (3:

q ( x0' •••> xi

= °>

C =

1,

n)

•,

*' xn }’

.

We have f\(0 ,

0,

0

0,

)=

(i =

1,

. . n)

OO aq HA1 1| 4 o

.

Hence, by the implicit function theorem, it follows that system (3 ) has a solution in the neighborhood of the origin, of the form: (5)

where

x 1 =

uj_(x 0 h

^ ( x ^ ,

(i =

1,

* 2

^

•••, n), UpCo) =

u

2 (x q ),

xn

xQ = xQ,

x±

The transformation

Zk ( x o^

+

u n ^ x o^

are analytic functions of 0,

(i =

1,

. . n)

We now return to the given system formation U defined by (cf. Liapounov, pp. (6 )

=

= u± + z±, U

xQ

satisfying:

.

(1

) and apply to It the trans­ 30 2 -3 ):

(i »

1,

..., n)

.

reduces system (1 ) to the form:

^x o ’

Z 1'

zn ] 2

’

Zk(xo ) + *[xo> z 1 ' •••’ zn ] 2 (7 :

A,

,ZB xc0 where

Z^(x ), (i = 0, 1, ..., n),

+

Cxo> z 1’ •••■’ zn ] 2

are power series In x^ beginning with 0 0 ^ o terms of degree at least k, and Z^(x ) = P2 ^xo^ + ^xo* u 19 ^n^' where the right side is as defined in 3 *2 . Furthermore

CRITICAL POINTS IN n-SPACE

173

(8)

3

• Suppose first that z§(xQ ) = 0 . Then, by virtue of (7 ) and (8 ) - if we write (x1, ..., xn ) for (z1, ..., z ) - U transforms the given system (1 ) into: xo

(9 )

The transformation U is analytic and regular at the origin. Hence, in the neighborhood of the origin the behavior of the trajectories of system (9 ) is the same as that of system (1 ). We note, however, that x 1 = x2 = ... = xn = o, -Xq = ° is a solution of (9) for any real constant c. Thus in (9 )> and hence in (1), the origin is not an isolated singu­ larity, contradicting condition (iii). It follows therefore that

^ 0.

3 *5 * The above cons iderat ions have transformed the given system (1) into one of the form:

(10)

where (i) A is an (n x n), real, non-singular matrix in real Jordan canonical form. (ii) (iii)

REMARK. of 3 .1 .

x£(xQ ) = 2j=kCjX^,

°k

4

k ^ 2.

X^(x ), (i = 1, n), are real power series in xQ beginning with terms of degree at least k. The matrix

A

appearing in (10) is the matrix

A^

MENBELSON

17^

3.6. Writing X^(xQ ) = apply the transformation t -- > t.

), where defined by:

E(xQ ) is a unit, we

dt1 = ckE(x0 )dt thereby transforming (10) into dxQ dt1

ftx0, x,,

xn ]° c"1E 1(x0 )

Xi (xo} + *[V

A i ck V * o A (n)

= A

+

\ dv where

**•' xn]2

Xk (xo } + *[xo' *•*' xn ]

xn°k W

E 1(xQ ) = E(xQ )~1

is also a unit.

NOTE. Since E(x ) is a unity there exists a small enough neighborhood S of the origin such that E(xQ )> ^ throughout S. Let now P 1 he the vector field defined by system (1o) and let E2 be the vector fielddefined by system(11 ). Then and F2 differ through­ out S only by a continuous scalar factor (namely,c^.E(x0 )) which is positive in case c^ > 0 and negative in case c-^ < o. The fields de­ fine two systems of trajectories throughout S which are identical, ex­ cept for the fact that in the case c^ < ° the sense of the trajectories is reversed. Rearranging terms, system (11) becomes: 1o

=

l x l\

l* '\

(12 )

r + * LX_ o',

o

' xn J? 2

/pk (x0 } + *[xo> *•*' xn ]A +

= A \ ± J

.

W

ip£(x0 ) + *[x0,

where A is as in (1 o ) and (I in xQ beginning with terms of degree at least

n),

xn ] * J

are power series

k.

3 .7 * It follows from the above discussion that system (12 ) is a canonical form for real analytic systems of differential equations having an isolated singularity with one zero characteristic root at the origin. We shall assume henceforth that the given system has already been reduced to that form.

CRITICAL POINTS IN n-SPACE

175

3 .8 . The study of the most general systems of the type (1 ) is rather complicated. We shall therefore restrict our attention to systems (1 ) whose canonical form (1 2 ) satisfies:

xn = x * + o ~ *0 Irr T i.e., .e., xQ xQ

*[x . Xn 1k °+1 o ’ *'•' is the terrr term of least degree in the expression for x o

(iv)

Geometrically this added condition implies that the surfaces xQ = + *[xQ, •••, xn ] ^ +1 = 0 , when they exist, are tangent to the xQ = o plane at the origin. k.

space

THE TRANSFORMATION

T

U.1 . Let x ! denote pointsin a second (n + he defined by: E^+1. Let T : En+1 -- > E ^

(1 3 )

x±

xQ - x£,

x j x Q,

=

(i =

1, 2 ,

1 )dimensional

n)

.

Then T can be thought to map theorigin o of En+1 onto the x^ = o plane, and all other points of the xQ = o plane Into the point at Infinity of the plane x^ = 0 . Outside xQ = o the map is one-to-one and analytic at every point. ^.2 . For

xQ

i

o we have:

xo = xo'

xi = xixo1>

(i = 1’ •••■’ n)

and hence ±i

Xq = Xq.

= x±x ”

1

- x±x-2 i0,

The transform of (1 2 ) by X-

=

(1M

(X-)X +

[*«,

T

(1 =

1, 2 ,

..., n)

.

yields: X AX - ] ° + 1 ,

k >

2

n Xj_ =

^

a ijx i + clx o + x i [xi> X 1*'

x n ]t

^

= 1’

j=i The term c^x^ can only appear in the case k = 2 , but in that case we apply a suitable linear transformation which reduces system (1h ) further to:

MENDELSON

176

+

(*i)k

XA

/x.\

/±{ \

xAx-]°+i;

[X',

I

xo lx o ’

k > 2

XA]!^

•••'

= A

(1 5 ) X'

\

\ Xni

n/

h .3 .

Since xAx - ] ° +1 = (x.)1- 1* ^ , X],

*[x>, x'x^

xA ]°

it follows that (16)

xA = (xA )k + (xA )k+ 1*[xA, x»,

n

(xA )*E(xA,

1

The transformation t — — > t 1 defined by reduces system (15) further to the form: k >

*0 = (xo )k '

- A

(17)

\*il U .1+. Let ed in xQ > 0 [x < relations:

R

x Q > 0 tx0 < 0],

Let

dt 1 = Etx^, ..., x^Jdt

y I1J1^ xn

+ XI

\ txA> •••>

\XA /

0 ].

.

2

iL f x-v;qI ,

/+.\

xA )

X ! ]n 1

n

1/

be a ray emanating from the origin 0 and contain­ Then R can be represented by the set of (n + 1 )

X, = m 1x 0 ,

x 2 = m 2x oJ

R ! = T(R). Then clearly

x* > ° [x’ < 0],

x] -

R1

xn = n y ^

is the half-line:

x^ = m2,

i.e., the half line perpendicularto the x^ = that plane at the point (0, m 1, m2, ..., m^).

x^ = mn

.

0 plane and intersecting

Furthermore, if C+(m, e) is the half cone defined above, then T(C+(m, e)) is the half cylinder {x1 | 0 < x^ < e, z ^ 1xj2 = m2). Similarly, T(C“(m, e)) = {x1 | ~ € < x^ < 0 , z ^ x ^ 2 = m2}. in

Let now r(t) be a trajectory of (12) which is contained C+(m, e) [C“(m, e)] and let r f(t) be the image of r(t) under T.

CRITICAL POINTS IN n-SPACE

177

Then r!(t) is contained in the cylinder T(C+ (m, e)) [T(C~(m, e))]; and if r(t) tends to o in E n + 1 for t ---- > + °o(t ---- > - oo), then r f(t) tends to x£ = o in E^+1 (i.e., r* (t) tends to the base of the cylinder). Moreover, if r(t) tends to o with a definite tangent R : x 1 = m 1xQ, •••> xn - mnx0' t h e n r»(t) tends to the point (o, m 1, m2, ..., mn ). In particular, if r(t) tends to the origin tangent to the xQ-axis, then r1(t) tends to the origin o*. Conversely, if rf(t) is contained in a half cylinder, as above, then r(t) is contained in the corresponding cone. If r!(t) tends to the = 0 Plane (inside the cylinder), then x^ = xjx^ < mx^ — > o as x^ -- > o, and hence r(t) tends to the origin. Finally, if pt(t) > (o, ri^, ..., mn ) as t -- > + «>, then r(t) has a definite tangent at the origin, this tangent being the line: x 1 = m -|x0'

xn = V o 1 We shall find it convenient to study system (1 7 ) in detail and obtain thereby, in view of the above considerations, information about the system (12). 5.

5 .1 . [ ^

Replacing ^

q j

PHASE PORTRAIT OF SYSTEM (1 7 )

=

1 '

(xp . .•,> An o' •. * ^ K i n), by X± = X± (x0 b y

xn ) and xn ) we may rewrite

(17) as: k > 2

(18)

+ xo

A

The matrix A appearing In (18) Is in real Jordan canonical form. We may assume, without loss of generality, that

i-1 where

i-1

i-1

R^(x) = e^(x)X^. Therefore, n

(26)

pp < -

Xq 2 +

Y

p |xQ |

>

!%!

i-1 or, n (27)

p < - Xp + |x0 | £

|R± |

•

1=1

Let S be the common domain of convergence of R 1, ..., Rn and let C 1 be aclosed cylinder of height 2h f and radius t) which is con­ tained in S (Figure 1 ). Furthermore, let

M - Max X€C

: ( I ' i -1

Let

C

be another cylinder of radius

h < M

’

r\

h < h'

^

)

•

' and height

(Figure

1)

2h,

where

181

CRITICAL POINTS IN n-SPACE

n C'

■{£

x| < i)2 }

- h' < xQ < h

x? < I2;

- h < x

< h

) 4Si (£ 1=1

IRI 1

1=1

j

n

■{£ FIGURE Obviously n Max. xeC

( 1=1 I

'

M

'

Therefore, for points on the lateral surface of

C,

we have from (27):

MENDELSON (28)

p < -

Xr\ +

hM < - Xrj + —jJ- • M - -

< 0

Hence, on the lateral surface of C, the vector field defined by (18)points into the cylinder C. Notethat in the casebeingdiscussed k = 2p and therefore x^ > 0 for xQ / 0. Hence, on the bottom disc the vector field points into C, whereas on the top disc the vector field points out of C. 5.3.4.

Consider the set

C 1 = C n|(xQ,

(29)

xn )I xQ < oj-

and let n

(30) C> = jx I - h < xQ < 0,

Y,

n

x| = T]2J U |x I xQ = -

1=1

h,

Y

X1
tp . Since xQ = x^p > 0 throughout C1, rp must rise monotonically towards the (x1, ...,xn ) plane and cannot have any point Q e C 1 in its positive limit set A+ (rp ). Thus, the positive limit set of rp is contained in the disc n

D:

=

0,

Y

4

< f

'

1=1

However, if

A+(rp )

3

P € D,

where

P 1 = (0, £ 1, ..., (; ) with

n

I

A = e> 0 >

1=1

an argument identical with the one used previously will show that there exists an h ( 0 such that for |x | < h(£)> (i.e., for t large enough) p < 0 in the neighborhood of P 1 . Thus P 1 ^ A+(rp ). Hence A+(rp ) con­ sists of the origin only. Thus all trajectories entering tend to the origin in the direction of increasing time. It will be shown later that they do so

183

CRITICAL POINTS IN n-SPACE

tangent to the xQ-axis. The phase-portrait is that ofa ,!fan,? tangent to the xQ-axis as indicated in Figure 3 * 5.3.5. Consider next the set C2 = C n P(C0, £1> •••> £n ) he a point on its open lateral the trajectory passing through P at time tp. We fp must enter C2 atP. Since xQ = x ^ > 0 is for all points of Tp at time t> tp. Hence Tp finite time Tp, and must do so through the upper first, point of departure by P .

(x |xQ > 0) . Let surface and let fp he have seen before that monotonic, xQ > > leaves C2 after a disc. We denote the

Let f he the mapping of the lateral surface of upper disc defined by: f : P -- > P

Let P( t), 0 , ...,0), (m = 0, 1, ... ), points on the lateral surface of C2 such that:

-- >

0

as

m

Tp

which is

be a sequence of

> “ »

be the (n-1 ) sphere defined

and letJ , (m = 0, 1,...,),

into the

.

Since Tp traverses its path from P to P in a time finite, it follows that f is a homeomorphism.

£0 = I '

C2

by

n £

=

T)2 ,

x0 =

(Figure

2

)

.

i=i Then f (Jm ) = is contained in the upper disc of C2* Let A^, (m = o, 1, 2 , . . .)> denote and itsinterior. Then A^ is a closed bounded set lying in the upper disc and A^ ^ •\1+1 for m = 0 , 1, 2 , Hence CO

A = n

.

m=o Let Q€ A and consider the trajectorypassingthroughQ at time tQ. We shall prove thatTq stays in C2 for all time t < tQ. For suppose Tq, when continued in the negative direction, leaves C2 at some point P. Then P( t)q , r)1, ..., r\n ) must be on the openlateral surface of C2, and thus r\Q > 0 .Hence t)0 > for some ra implying that Q, i Arr]. It follows that Q, 4 A, contradicting our original

18if

MENDELSON

FIGURE 2 assumption. t < tQ .

Therefore we must conclude that

remains in

C2

for all

Using an argument similar to the one used with respect to tra­ jectories approaching the origin from below the (x19 . xn ) hyperplane^ we can show that the negative limit set of consists solely of the singular point at the origin. We have thus proved: LEMMA 1 . System (18), case 1A, has at least one trajectory tending to the origin from above the (x1, xn )plane in the negative direction. We now proceed to prove: LEMMA 2 . System (18), case 1A, has only one tra­ jectory tending to the origin from above the (x1, . xn ) plane in the negative direction. PROOF.

Assume there are two such trajectories and let them be

CRITICAL POINTS IN n-SPACE denoted by

(xl x],

x? - x{, (1 =

(31

1,

•> x^) and (x^, x®, ., n), be denoted by

2,

185

x^), respectively. and define,

Let

' I

)

i=1 Then replacing (i = above

1 , ...,

XqX^x^

n),

and

( 5 *3 .3 ), w e

pp < - Xp

••

X ± (x 0 >

*n ) in (18) by

following

a calculation

similar

•••> xn )> to o n e

carried

out

get:

+

I ui[xi ( v 1=1 n

xi )

•••'

-

xi ( v

x3

•••» xA)_

n

bx ±

,2 r c - ~ Xp + L u i L 1=1 j=i (32)

u j + ei

n
En be a map f = f (x, y, t ). Consider the vector sys­ tem of ordinary differential equations

f

(S)

x" = f(x, x !, t),

(f = d/dt)

where the scalar product (R)

x*f(x, y, t ) > o,

unless

x = o

The condition (R) implies that f(o, y, t) = o and that x = o Is the only equilibrium solution (xl = x" = o) of the system (S) - (R). Numbered footnotes appear at end of paper. 201

2 02

BASS

In the autonomous case f = f (x, x 1 ), f e C2, suppose also that the Jacohian matrix f„(o, 0 ) issymmetric. Then in E2n replace (S) by (Sa ) : x' = y, y* = f(x, y), and at the origin (x, y) = (o, o) the system (S ) has, by (R), exactly n negative characteristic roots and exactly n positive characteristic roots. (In fact, (R) implies that fTr(o, o) = o and that f° (o, o) is positive definite.3 ) Hence well-known / theorems (cf. e.g., the recent hooks of Coddington-Levinson or Lefschetz) imply that the geometrical description given above holds, at least in some sufficiently small neighborhood of the origin. The extension of this 2n description to all of E (and in particular, the assertion that the "stable11 manifold intersects every hyperplane x = const.) can be effected if and only if the growth of f (x, y) as a function of y is further restricted. For a simple example, consider a particle sliding under the in­ fluence of gravity on a vertical circle (and free to leave the circle at its horizontal extremities). Measuring arc length x from the unstable rest point, and letting g = 1, we have (E)

x,f = f(x),

xf(x) > o

for

x

±

o ,

where here f(x) = sin x for |x| < jc/2 and f(x) = sgn x for |x | > it/2 . By explicit solution of (E), it can be seen that to each initial position xQ there corresponds (I) an initial velocity x_(xQ ) for which the corre­ sponding solution x(t) exists on I and satisfies |x(t)| < |x |; and (II) an initial velocity x+ (xQ ) for which the corresponding solution x(t) satisfies |x(t)| -- > + oo as t — > t^ (for some ^

< ->•

For the example (E), the planar phase-portrait can be readily sketched. The origin is a saddle point. In case (I), (x(t), x !(t)) — — > (o, o) as t -— > oo. The assertions (I), (II) are simply that at least one "exceptional integral curve" (i.e., "one through the origin") and also at least one "divergent" integral curve cuts each line x = xQ. For general scalar systems of the type of (E), and more generally, x" = f(x, t), the assertions (I) - (II) are implied by a theorem of A. Kneser [8 ]. Hartman and Wintner [ k ] extended Kneser!s theorem to include velocity dependent forces, i.e., they dealt with (S) in the case n = 1 . (Subsequently [5] they generalized this result to first order systems in such a way as to include [8 ] and [A].) However, Hartman and Wintner

NON-LINEAR REPULSIVE FORCES require that

f

satisfy not only (R) but also for all

y

and each

R > o

If (x, y, t)| < cpR (|y| ), (G) ° < t < R, 0 < |x| < R where

cp = cpR

is such that 00

(Q) The* ("quadratic”) Nagumo condition (G) - (Q,) permits e.g., x ,f = (1 + (x1 ) 2 )sin x, but not x ,! = (1 + (x* )2+€ )sin x for any € > 0. An example of Nagumo shows that (G) - (Q) is the best possible general restriction (for n = 1)• In the scalar autonomous case, there is a simple geometrical explanation of the meaning of condition (G) - (Q). In fact, (S) can be re­ placed by the first order equation dy/dx = F(x, y), where F(x, y ) = f (x, y )/y. Then (Q) is just Wintner!s general criterion 00

for the unrestricted existence of solutions y = y(x); cf. [1t] or Ex. 5 , p. 61 of Coddington-Levinson1s Ordinary Differential Equations (1955). In other words, the exceptional integral curve y = y(x) has less than ex­ ponential growth; hence it cuts every vertical line x = const. Because of the distinctions between ordinary and scalar products the arguments used for n = 1 cannot be generalized to the case n > 1. However, for linear systems Wintner [15] generalized Kneser1s theorem as follows. Let xn = F(t)x, where the matrix F(t) is continuous on I+ and non-negative definite; then (I) - (II) holds and there are n linearly independent solutions of each type. Moreover, the same is true ([3]; cf. also [6 ]) for systems x" = G(t)x’ + F(t)x, where F, G are continuous matrices and the symmetric part F° of F satisfies F° > GG* ( > 0). The object of this note is to generalize Kneser1s theorem to the system (S) - (R), for all n > 1 , in such a way as to include the linear result of Harman and Wintner ([3] ) just quoted. However, we shall replace (G) - (Q) by the less general condition (L)

l|f(x, y, t )II < CR + Kp|jy ||

BASS for all vectors y and all R > o, and for all o < t < R and ||x|| < R, where the "constants” C-^ and depend only on R. For the system (S) - (R) - (L) the conclusions (I) and (II) hold. The proof is similar to that of Kneser except that a certain boundary value problem arising in a lemma is more difficult. This problem is reduced to a non-linear integral equation, for whose solutions (L) pro­ vides an a priori bound. Then in the manner of Birkhoff-Kellogg, Schauder, 1+ and Leray, the Lefschetz fixed-point theorem for compact HLC spaces can be used to solve this integral equation. Despite considerable effort, I have been unable to relax (L) to (G) - (Q) in the case n > 1, or to provide a counter-example; this inter­ esting problem remains open. The remainder of this note is similar to Part Two of my 1955 dissertation, which was written at The Johns Hopkins University under the direction of Professor Wintner. I wish to thank Professor Wintner for suggesting this topic, and to thank both him and Professor Hartman for their assistance and advice. I also Wish to thank Professors Lefschetz, Markus, and Barocio for the illuminating geometrical discussion of this result which was reproduced above; this discussion was developed following my presentation of the result to Professor Lefschetz1s Princeton seminar. 2. To give a more precise formulation of the results to be ob­ tained, let x be the coordinate vector (x1, ..., xn ) and let f(x, x !, t) be a real vector function (f^, ..., fn ) which is continuous on the halfspace

(1 )

0

t < CO,

— oo < X_^ < + 00,

~ 0 0 < x | < + 00,

(i =

1,

..., n)

As a generalization of the non-negative definiteness of the coefficient matrix

F°

in the simple linear case

(2) for all (3 )

G s o,

suppose that

x •f(x, x !, t) > o x, x f, t

on (1 ).

We may then say that the differential equations x" = f(x, x *, t)

represent a mechanical system, in which, by virtue of (2 ), the forces have no "centrally attractive" component. p Now put r = ||x|| = x*x . In addition to (2), we shall assume that for every rQ > 0 and T > 0 there exist positive numbers C = C(rQ, T ) and K = K(rQ, T ) such that

NON-LINEAR REPULSIVE FORCES OO for

l|f(x, x», t)|| < t, x

0

+ K||x11|

satisfying

(5 a)

(5b)

o ^ t g T,

o g r ^ rQ

,

and for all x'. Since C(rQ, T ) and K(rQ, T) can always be taken to be monotone increasing and continuous in T, we shall assume for later convenience that this has been done.

(2

) and

(3

Next, note that ) we have

(6 )

r !(t) =

rn > o,

2x*x!,

where

r,f(t) =

2x«x"

r = ||x||2

.

+

2 x*

*x’,

and by

This fundamental inequality implies that the squared length r(t) of any solution vector of (3 ) - (2) is convex toward the t-axis. The results which follow actually depend on (6 ) rather than. (2). In the linear case, F° > GG* is sufficient for ( 6 ) ; cf. [31 • Also it is clear that in the linear case conditions (* 0 - (5) are satisfied. By means of (* 0 and (6 )we shall show (cf. Section 3 ) that all local solutions of (-3 ) at t = 0 can be continued onto a ("maximal11) interval (7a)

0

but not onto

0

t g t°,

(7b)

where

t < t°

,

t° < °° if and only if

r(t) -- > 00

as

t — -> t°

Consequently, all solutions of (3) are unrestricted except (7b) holds for some t° < °o; in particular, solutions monotone non-increasing exist on (7a) with t° = °o.

thosefor which

forwhich r(t)

is

Let us define a divergent solution of (3 ) as onewhich exists on (7a) and satisfies (7b) for some t° ^ ». On the other hand, we shall say that a solution is asymptotic if r(t) satisfies (8 )

r(t) > 0,

r !(t) ^ 0,

r,!(t) > 0

for

0 g t < 00 ,

so that, in particular, lim r(t) t — .>00

exists

(< 00)

and

lim r !(t) = 0 t — *>00

BASS Finally, let x(t; x , x^) denote any continuation onto its maximal in­ terval (7a) of any solution of (3) which is defined at t = o and satisfies there (9)

x(0; xQ, x p = xQ,

x 1(0; xQ, x p = x^

.

(This solution is not in general uniquely determined by the initial con­ ditions (9). ) Using the above terminology, we now state our result as follows* THEOREM. Let f be continuous on (1) and satisfy (if) - (5). Then corresponding to every initial position x , there exist two vectors xj(xQ ) and x ’(xQ ) and corresponding solutions of (3) which are such that ( 10)

every x(t; xQ, x|(xQ )) is divergent (0 g t < t° g 00),

(1 1 )

some

x(t; x , x^(xQ )) is asymptotic

(0 S t
0; then r !(c>) > 0, and (by the con­ vexity of r(t)) (7b) must hold for some t° g ) We now establish this alternative. 3.

a local solution t° < 00 such that

According to a result of Wintner [1A], all continuations of x(t; x , x^) are unrestricted unless there exists a

limjr(t) + ||x!(t)||2j- = »,

as

t -- > t°

Thus, if r(t) is bounded on 0 g t < t° (so that (5b) holds there for some rQ ), then lim||x1(t)|| = as---- t -> t°. From this we derive a contradiction with (A), by means of the following lemma. LEMMA 1. Let x(t) be a solution of (3 ), defined on (5a), where f(x, x !, t) is continuous on (1) and

NON-LINEAR REPULSIVE FORCES has the property (U) - (5) (hut not necessarily (2) - (6 ))• Then if x(t)satisfies (5b), there exists a constant D= D(rQ, T) such that x T(t) satisfies (12)

0 T)

•

This a priori bound for x !(t) follows from modification of a device of Hartman and Wintner [3]• Since x f(t) is continuous on (5a) there exists some t* in (5 a) such that |jx1 (t*) || = max||x!(t)|| on (5a). Now let (1 3 )

M

= l/(mK(r0, T))

,

where m = m(r , T) > n is so large that |t | g T/2 , and choose sgn t such that t = t* + t lies in (5a). Then by the mean valuetheorem there exist numbers 0 ^ (i = 1, ..., n) lying in (5a) and such that if we put x M(0) = (x" ( 0 1 ), ..., x^(0n ))> we have (1*0

x(t* +

But by (3 ) and ( b ) , by (5) and (-\b),

t)

= x(t*) + TX !(t*) + 1 r2x ”(e)

.

||xn (0 )|| g nC + riElIx* (t*)|| g nC + mK||xf(t*)||,

sothat

M||x«(t*)|| g 2 (r0 )l/2 + l m K T 2 ||x'(t# )|| + 1 nCt2

i.e., |t ((1 - I mK| t| )||x’(t^)|| g 2 (rQ ) 1 ^2 + ^-hGt2,which from (13 ) gives This proves (12 ) with D(r0, T) = Am(ro ) 1/2K(rQ, T)+ nC(rQ, T)/mK(r , T). the lemma and so the alternative (7a) -(?b). A. It remains only to select x^(xQ ) and a corresponding solu­ tion (which is not unique) in such a manner as to satisfy (11 ).Inour construction of an asymptotic solution of (3 ) , the following property (J ) proves to be fundamental: (J). To every vector xQ and every positive number T there corresponds a solution x^(t) of (3 ) , de­ fined on (5 a), and satisfying (15)

XT (0 )

= X Q,

Xrp(T)

= 0

Hereafter we shall refer to (J ) as the joinability property for (5 a). Because the successive approximations by means of which Picard [9] established local joinability (under certain smoothness assumptions for a

BASS system similar to (3 ) do not converge for large T, it will be necessary to prove in some other manner that (6) implies joinability in the large. To this end we shall employ the following lemma (which is independent of (2), (10). LEMMA 2. Suppose that f (x, x 1, t) and bounded, say |f| g M, on (16 )

0g t

g T,

< + oo,

- oo