Mathematical Physics I: Dynamical Systems and Classical Mechanics : Lecture Notes [1 ed.] 9783832587413, 9783832535698

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NOTES LECTURE

Mathematical Physics I

Dynamical Systems and Classical Mechanics

Matteo Petrera

λογος

Mathematical Physics I. Dynamical Systems and Classical Mechanics. Lecture Notes

Matteo Petrera ¨ Mathematik, MA 7-2 Institut fur Technische Universit¨at Berlin Strasse des 17. Juni 136

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de .

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Copyright Logos Verlag Berlin GmbH 2013 All rights reserved. ISBN 978-3-8325-3569-8

Logos Verlag Berlin GmbH Comeniushof, Gubener Str. 47, 10243 Berlin Tel.: +49 (0)30 42 85 10 90 Fax: +49 (0)30 42 85 10 92 INTERNET: http://www.logos-verlag.de

Motivations These Lecture Notes are based on a one-semester course taught in the Winter semester of 2011/2012 at the Technical University of Berlin, to bachelor undergraduate mathematics and physics students. The exercises at the end of each chapter have been either solved in class, during Tutorial’s hours, or assigned as weekly homework. These Lecture Notes are based on the references listed on the next pages. It is worthwhile to warn the reader that there are several excellent and exhaustive books and monographs on the topics that were covered in this course. A practical drawback of some of these books is that they are not really suited for a 4-hours per week, one-semester course. On the contrary, these notes contain only those topics that were actually explained in class. They are a kind of one-to-one copy of blackboard lectures. Some topics, some aspects of the theory and some proofs were left out because of time constraints. A characteristic feature of these notes is that they present subjects in a synthetic and schematic way, thus following exactly the same pedagogical strategy used in class. Notions, concepts, statements and proofs are intentionally written and organized in a way that I found well suited for a systematic and effective understanding/learning process. The aim is to provide students with practical tools that allow them to prepare themselves for their exams and not to substitute the role of an exhaustive book. This purpose has, of course, drawbacks and benefits at the same time. As a matter of fact, many students wish to have a “product” which is readable, compact and selfcontained. In other words, something that is necessary and sufficient to get a good mark with a reasonable effort. This is - at least ideally - the positive side of good Lecture Notes. On the other hand, the risk is that their understanding might not be fluid and therefore too confined. Indeed, I always encourage my students to also consult more “standard” books like the ones quoted on the next pages. Aknowledgments The DFG (Deutsche Forschungsgemeinschaft) collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” is acknowledged. I am grateful to Yuri Suris and to the BMS (Berlin Mathematical School) for having given me the opportunity to teach this course. Matteo Petrera ¨ Mathematik, MA 7-2, Technische Universit¨at Berlin Institut fur Strasse des 17. Juni 136 [email protected] November 14, 2013

Books and references used during the preparation of these Lecture Notes Ch1 Initial Value Problems

X [Ch] C. Chicone, Ordinary Differential Equations with Applications, Springer, 2006. X [Ge] G. Gentile, Meccanica Lagrangiana e Hamiltoniana, Lecture Notes (in Italian) available at http://www.mat.uniroma3.it/users/gentile/2011/ testo/testo.html. X [Te] G. Teschl, Ordinary Differential Equations and Dynamical Systems, preliminary version of the book available at http://www.mat.univie.ac.at/ ∼gerald/ftp/book-ode/ode.pdf. Ch2 Continuous Dynamical Systems

X [Be] N. Berglund, Geometric Theory of Dynamical Systems, Lecture Notes available at http://arxiv.org/abs/math/0111177. X [BlKu] G.W. Bluman, S. Kumei, Symmetries and Integration Methods for Differential Equations, Springer, 1989. X [BuNe] P. Butt`a, P. Negrini, Sistemi Dinamici, Lecture Notes (in Italian) available at http://www1.mat.uniroma1.it/people/butta/didattica/sisdin. pdf. X [Ch] C. Chicone, Ordinary Differential Equations with Applications, Springer, 2006. X [FaMa] A. Fasano, S. Marmi, Meccanica Analitica: una Introduzione, Bollati Boringhieri, 2002. X [Ge] G. Gentile, Meccanica Lagrangiana e Hamiltoniana, Lecture Notes (in Italian) available at http://www.mat.uniroma3.it/users/gentile/2011/ testo/testo.html. X M.W. Hirsch, S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974. X [Ku] Yu. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, 1995, and Dynamical Systems Notes, Lecture Notes available at http://www.staff. science.uu.nl/∼kouzn101/NLDV/index.html. X [Ol1] P.J. Olver, Applied Mathematics Lecture Notes, Lecture Notes available at http://www.math.umn.edu/∼olver/appl.html. X [McMe] P.D. McSwiggen, K.R. Meyer, Conjugate Phase Portraits of Linear Systems, American Mathematical Monthly, Vol. 115, No. 7, 2008. X [MaRaAb] J.E. Marsden, T. Ratiu, R. Abraham, Manifolds, Tensor Analysis and Applications, Springer, 2001.

X [Te] G. Teschl, Ordinary Differential Equations and Dynamical Systems, preliminary version of the book available at http://www.mat.univie.ac.at/ ∼gerald/ftp/book-ode/ode.pdf. Ch3 Lagrangian and Hamiltonian Mechanics on Euclidean Spaces

X [Ar] V. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989. X [FaMa] A. Fasano, S. Marmi, Meccanica Analitica: una Introduzione, Bollati Boringhieri, 2002. X [Ge] G. Gentile, Meccanica Lagrangiana e Hamiltoniana, Lecture Notes (in Italian) available at http://www.mat.uniroma3.it/users/gentile/2011/ testo/testo.html. Ch4 Introduction to Hamiltonian Mechanics on Poisson Manifolds

X [AdMoVa] M. Adler, P. van Moerbeke, P. Vanhaecke, Algebraic Integrability, Painlev´e Geometry and Lie Algebras, Springer, 2004. X [Ar] V. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989. X [FaMa] A. Fasano, S. Marmi, Meccanica Analitica: una Introduzione, Bollati Boringhieri, 2002. X [Ho] D.D. Holm, Geometric Mechanics, Imperial College Press, 2008. X [MaRa] J.E. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, Springer, 1999. X [Ol2] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, 1998.

Contents

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3

Initial Value Problems 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definition of an initial value problem (IVP) . . . . . . . . . 1.3 Existence and uniqueness of solutions of IVPs . . . . . . . 1.4 Dependence of solutions on initial values and parameters 1.5 Prolongation of solutions . . . . . . . . . . . . . . . . . . . 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Continuous Dynamical Systems 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition of a dynamical system . . . . . . . . . . . . . . . . . 2.2.1 Orbits, invariant sets, invariant functions and stability 2.3 Autonomous IVPs as continuous dynamical systems . . . . . 2.3.1 Flows, vector fields and invariant functions . . . . . . 2.3.2 Evolution of phase space volume . . . . . . . . . . . . 2.3.3 Stability of fixed points and Lyapunov functions . . . 2.3.4 Stability properties of linear homogeneous IVPs . . . . 2.3.5 Stability properties of linearized IVPs . . . . . . . . . . 2.4 Topological equivalence of dynamical systems . . . . . . . . . 2.5 Stability properties of nonlinear IVPs . . . . . . . . . . . . . . 2.5.1 Existence of invariant stable and unstable manifolds . 2.5.2 Existence of invariant center manifolds . . . . . . . . . 2.6 Basic facts on non-autonomous linear IVPs . . . . . . . . . . . 2.6.1 Periodic linear IVPs . . . . . . . . . . . . . . . . . . . . 2.7 Basic facts on local bifurcation theory . . . . . . . . . . . . . . 2.7.1 One-parameter local bifurcations . . . . . . . . . . . . 2.7.2 Saddle-node bifurcations . . . . . . . . . . . . . . . . . 2.7.3 Hopf bifurcations . . . . . . . . . . . . . . . . . . . . . 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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23 23 24 27 30 35 48 53 61 71 76 81 82 85 88 92 96 99 100 103 107

Lagrangian and Hamiltonian Mechanics on Euclidean Spaces 125 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.2 Definition of a mechanical system and Newton equations . . . . . . . 126

Contents 3.3

3.4

3.5 4

Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Lagrangians for conservative systems . . . . . . . . . . . . . . . 3.3.3 Symmetries of Lagrangians and Noether Theorem . . . . . . . Canonical Hamiltonian mechanics . . . . . . . . . . . . . . . . . . . . . 3.4.1 Hamilton equations and Hamiltonian flows . . . . . . . . . . . 3.4.2 Symplectic structure of the canonical Hamiltonian phase space 3.4.3 Canonical Poisson brackets . . . . . . . . . . . . . . . . . . . . . 3.4.4 Canonical and symplectic transformations . . . . . . . . . . . . 3.4.5 The Lie condition and the canonical symplectic 2-form . . . . . 3.4.6 Generating functions of canonical transformations . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction to Hamiltonian Mechanics on Poisson Manifolds 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basic facts on smooth manifolds . . . . . . . . . . . . . . . . 4.2.1 Definition of a smooth manifold . . . . . . . . . . . 4.2.2 1-forms and vector fields . . . . . . . . . . . . . . . 4.2.3 Maps between manifolds . . . . . . . . . . . . . . . 4.2.4 Distributions . . . . . . . . . . . . . . . . . . . . . . 4.2.5 k-forms and k-vector fields . . . . . . . . . . . . . . 4.2.6 Lie derivatives . . . . . . . . . . . . . . . . . . . . . 4.2.7 Matrix Lie groups and matrix Lie algebras . . . . . 4.3 Hamiltonian mechanics on Poisson manifolds . . . . . . . 4.4 Hamiltonian mechanics on symplectic manifolds . . . . . . 4.5 Foliation of a Poisson manifold . . . . . . . . . . . . . . . . 4.6 Completely integrable systems on symplectic manifolds . 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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131 132 137 140 146 146 150 155 160 170 180 185 199 199 202 203 205 208 212 214 218 220 224 232 237 241 250

1 Initial Value Problems 1.1

Introduction

I In classical mechanics, the position x ∈ R3 of a point particle of mass m > 0 is described by a continuous function of time, a real variable t ∈ I ⊆ R, i.e., x : I → R3 .

Fig. 1.1. Position and velocity vectors of a particle in R2 ([Te]).

• The first and the second derivatives of this function w.r.t. time define respectively the velocity, v := dx/dt, and the acceleration, d2 x/dt2 , of the particle. • Assume that the particle is moving under the influence of an external vector field f : R3 → R3 , called force. Then the Newton law of motion states that at each point x the force acting on the particle is d2 x dt2

f (x) = m

∀ t ∈ I.

(1.1)

• Formula (1.1) is a relation between x and its second derivative: it defines a system of three second-order ordinary differential equations. The variable x is called dependent variable and the variable t is called independent variable. Given a vector field f the problem is to find the general solution x = x (t) satisfying (1.1).

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• We can increase the number of dependent variables by considering the pair ( x, v) ∈ R6 . The advantage is that we get an equivalent first-order system of six ordinary differential equations:  dx   = v,  dt    dv = f ( x ) . dt m

I Informally, we can think of a dynamical system as the time evolution (continuous or discrete) of some physical system, such as the motion of two planets under the 1

2

1 Initial Value Problems

influence of the gravitational field, or the motion of a spinning top. One is often interested in knowing the fate of the system for long times. The theory of dynamical systems tries to predict the future of physical systems and understand the stability and limitations of these predictions. The first basic results on dynamical systems were found by I. Newton (1643-1727) but H. Poincar´e (1854-1912) can be considered as the founder of the modern theory of dynamical systems.

I Dynamical systems naturally split into two classes, according to nature of the time variable t: • If t ∈ I ⊆ R we have a continuous dynamical system, which is typically described by the flow of a system of first-order ordinary differential equations with prescribed initial conditions (initial value problem). • If t ∈ I ⊆ Z we have a discrete dynamical system, which is typically described by the iteration of an invertible map.

I We here present two examples of dynamical systems, the first being with continuous time, the second with discrete time. Example 1.1 (Particle in R3 in a gravitational field) • In the vicinity of the surface of the earth, the gravitational force acting on a point particle of mass m, whose position is x := ( x1 , x2 , x3 ) ∈ R3 , can be approximated by the vector field f ( x ) : = − m g e3 , where e3 := (0, 0, 1) ∈ equations (1.1),

R3

and g > 0 is the constant gravitational acceleration. Newton

d2 x = − m g e3 , dt2 can be easily integrated to get the position vector of the particle m

x ( t ) = x (0) + v (0) t −

1 g e3 t 2 , 2

where t = 0 has be chosen as the initial time. The velocity vector is x˙ (t) = v(t) = v(0) − g e3 t. In this case we have a dynamical system defined in terms of a one-parameter (t is the parameter) family of maps (called flow) Φt : R × R6 → R6 , Φt : (0, ( x (0), v(0)) 7→ ( x (t, ( x (0), v(0)), v(t, ( x (0), v(0))) . • Solutions of systems of ordinary differential equations of the type (1.1) cannot always be found by a straightforward integration. Indeed, a refinement of the model, which is valid not only in the vicinity of the surface of the earth, takes the real gravitational force f ( x ) := − G m M which is a central vector field.

x , k x k3

1.1 Introduction

3

Fig. 1.2. The gravitational force field ([MaRaAb]). Here M is the mass of the earth, G > 0 is the gravitational constant and k x k := ( x12 + x22 + x32 )1/2 . Newton equations (1.1) now read d2 x x = −G m M , dt2 k x k3

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and it is no longer evident how to solve them. Moreover it is even unclear whether solutions exist at all.

Example 1.2 (Logistic map)

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A discrete model for the evolution of a population Pn : Z+ → [0, ∞) of animals is

where α > 0 is the natality rate and β > 0 is a parameter which limits the growth. Note that if β = 0 then Pn = αn P0 (“unrealistic” exponential growth). The rescaled variable xn := β Pn obeys the recurrence equation x n +1 = Φ ( x n , α ) : = α x n (1 − x n ). For α ∈ [0, 4] then Φ maps [0, 1] into itself. The dynamics of the sequence ( xn )n∈Z+ , defined in terms of iterations of the parametric map Φ, depends drastically on the parameter α. • α ∈ [0, 1]. The population will eventually die, irrespective of the initial population. • α ∈ (1, 3]. The population approaches a stable equilibrium. √ • α ∈ (3, 1 + 6). From almost all initial conditions the population will approach permanent oscillations between two values, which depend on α. √ • α ∈ (1 + 6, 3.54). From almost all initial conditions the population will approach permanent oscillations among four values, which depend on α. • α > 3.54. From almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, . . . (period-doubling cascade). At α approximately 3.57 is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions we can no longer see any oscillations of finite period. Slight variations in the initial population yield dramatically different results over time.

4 1.2

1 Initial Value Problems Definition of an initial value problem (IVP)

I We start defining an ordinary differential equation. Definition 1.1 1. A continuous n-th order ordinary differential equation (ODE) with independent variable t ∈ I ⊆ R and dependent (unknown) R-valued variable x : t 7→ x (t) is a relation of the form   F t, x, x (1) , . . . , x (n) = 0,

x (i ) ≡

di x , i = 1, . . . , n, dti

(1.2)

where F is some continuous function of its arguments. 2. A solution of (1.2) is a function φ : t 7→ φ(t), φ ∈ C n ( I0 , R), I0 ⊆ I, such that   F t, φ(t), φ(1) (t), . . . , φ(n) (t) = 0 ∀ t ∈ I0 .

I We assume that (1.2) is solvable w.r.t. x (n) (ODE in normal form), that is   x (n) = f t, x, x (1) , . . . , x (n−1) ,

(1.3)

for some continuous function f . Thanks to the “Implicit  function Theorem”, this can (1)

(n)

be done at least locally in a neighborhood of some point t0 , x0 , x0 , . . . , x0 there holds ∂F
6= 0. ∂x (n) t0 ,x0 ,x0(1) ,...,x0(n)

where

I Notational remarks: • We introduce the “dot” notation for time-derivatives of the dependent variable x:   dx d2 x (2) (1) x˙ ≡ x ≡ x¨ ≡ x ≡ 2 , . . . . dt dt From now on our dependent variable x is defined in M, which is an open set of the Euclidean space Rn . • Let M, N be open subsets, not necessarily of the same dimension, of the Euclidean space. We denote by C k ( M, N ), k > 0, the set of functions from M to N having continuous derivatives up to order k. We set C ( M, N ) ≡ C0 ( M, N ), the set of continuous functions and F ( M, N ) ≡ C ∞ ( M, N ), the set of smooth functions.

1.2 Definition of an initial value problem (IVP)

5

Definition 1.2 1. A continuous system of n first-order ODEs, with independent variable t ∈ I ⊆ R and dependent (unknown) Rn -valued variable x : t 7→ x (t), is a set of n ODEs in normal form: x˙ = f (t, x ) , (1.4) where f ∈ C ( I × M, Rn ). System (1.4) is autonomous if f does not depend explicitly on t. 2. An initial value problem (IVP) (or Cauchy problem) is the system (1.4) together with an initial value x (t0 ) = x0 , (t0 , x0 ) ∈ I × M: ( x˙ = f (t, x ), (1.5) x ( t0 ) = x0 . 3. A solution of (1.5) is a function φ : t → φ(t), φ ∈ C1 ( I0 , Rn ), I0 ⊆ I, such that φ˙ (t) = f (t, φ(t)) ∀ t ∈ I0 , satisfying φ(t0 ) = x0 .

I Notational remarks: • From now on the solution of the IVP (1.5) will be denoted by φ(t, t0 , x0 ) instead of φ(t). This is to emphasize explicitly the dependencies of the solution of (1.5) on its arguments. Note that t 7→ φ(t, t0 , x0 ) is a Rn -valued function of t. The arguments (t0 , x0 ) are fixed, and we may think of them as parameters. The initial condition reads φ(t0 , t0 , x0 ) = x0 . If t0 = 0 we slightly simplify the notation omitting the second argument and thus denoting the solution by φ(t, x0 ) with φ(0, x0 ) = x0 . • If f depends on some parameters α ∈ A ⊆ R p , p > 1, i.e., f = f (t, x, α), so does the solution of (1.5). In such a case the solution will be denoted by φ(t, t0 , x0 , α). • In what follows, we will be interested in regularities conditions of the solution of (1.5) w.r.t. the whole set of its arguments t, t0 , x0 , α. So we shall write φ(t, t0 , x0 , α) ∈ C k ( I0 × I00 × M0 × A0 , Rn ) to say that the solution is of class C k w.r.t. t ∈ I0 , t0 ∈ I00 ⊆ I0 , x0 ∈ M0 ⊆ M and α ∈ A0 ⊆ A. Example 1.3 (Linear IVPs in Rn ) 1. A linear IVP in Rn takes the form

(

x˙ = A x, x (0) = x0 ,

6

1 Initial Value Problems where A is a constant n × n matrix. Its solution is given by φ(t, x0 ) = e A t x0 , where

eA t

φ(0, x0 ) = x0 ,

is the matrix exponential: ∞

e A t :=

∑

k =0

tk k A . k!

2. If n = 1, so that A is just a coefficient a ∈ R, we simply have φ(t, x0 ) = ea t x0 ,

φ(0, x0 ) = x0 .

I Any n-th order ODE (1.3) may be reduced to a system of nfirst-order ODEs (1.4)  by introducing n new dependent variables (y1 , y2 , . . . , yn ) := x, x (1) , . . . , x (n−1) . • Indeed, the variables (y1 , y2 , . . . , yn ) obeys the system of ODEs    y˙ 1 = y2 ,    .. . y˙ n−1 = yn ,      y˙ n = f (t, y1 , . . . , yn ). • We can even define the set of variables z := (t, y1 , y2 , . . . , yn ) in such a way that (z1 , . . . , zn+1 ) satisfy the system of ODEs  z˙ 1 = 1,      z˙ 2 = y˙ 1 = y2 = z3 ,    .. .     z˙ n = y˙ n−1 = yn = zn+1 ,     z˙ n+1 = f (z). • This shows that, at least formally, it suffices to consider the case of autonomous first-order systems of ODEs. 1.3

Existence and uniqueness of solutions of IVPs

I Le us recall some facts from Analysis. • Let ( X, k · k X ) be a Banach space, i.e., a complete vector space X with norm k · k X : X → [0, ∞). Every Cauchy sequence in ( X, k · k X ) is convergent. • Let D be a (nonempty) closed subset of X and consider a map K : D → D.

1.3 Existence and uniqueness of solutions of IVPs

7

• A fixed point of K is an element x ∈ D such that K ( x ) = x. • The map K is a contraction if there exists η ∈ [0, 1) such that

kK ( x ) − K (y)k X 6 η k x − yk X

∀ x, y ∈ D.

• The following claim holds true: If K is a contraction then K has a unique fixed point x ∈ D (“Contraction Principle” or “Banach fixed point Theorem”). Example 1.4 (Banach spaces) 1. X = Rn is a Banach space with norm given by the Euclidean norm q q k x k X = k x k := h x, x i := x12 + · · · + xn2 . 2. X = C ( I, Rn ), where I is a compact interval of R, is a Banach space with norm given by

k x k X = supk x (t)k. t∈ I

• Let (t, x ) ∈ I × M. A function f ∈ C ( I × M, Rn ) is locally Lipschitz continuous in x, uniformly w.r.t. t, and we write f ∈ Lip ( I × M, Rn ), if, for every (t0 , x0 ) ∈ I × M, there exists a compact neighborhood V of (t0 , x0 ) and L > 0 (Lipschitz constant) such that

k f (t, x ) − f (t, y)k 6 Lk x − yk

∀ (t, x ), (t, y) ∈ V.

(1.6)

If (1.6) holds true for all (t, x ), (t, y) ∈ I × M then f is globally Lipschitz continuous in x, uniformly w.r.t. t. • The following claim holds true: If f ∈ C1 ( M, Rn ) then f ∈ Lip ( M, Rn ). Example 1.5 (One-variable real-valued Lipschitz continuous functions) Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is no greater than a definite real number, the Lipschitz constant. 1. f ( x ) := sin x is C ∞ (R, [−1, 1]). It is globally Lipschitz continuous because its derivative cos x is bounded above by 1 in absolute value. 2. f ( x ) := | x | is not C1 (R, R). It is globally Lipschitz continuous by the reverse triangle inequality || x | − |y|| 6 | x − y| for all x, y ∈ R. 3. f ( x ) := x2 is C ∞ (R, R). It is not globally Lipschitz continuous because it becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.

I The following fundamental Theorem provides sufficient conditions for the existence and uniqueness of solutions of the IVP (1.5).

8

1 Initial Value Problems

¨ ) Theorem 1.1 (Picard-Lindelof Consider the IVP (1.5). If f ∈ Lip ( I × M, Rn ), then there exists, around each point (t0 , x0 ) ∈ I × M, an open set I0 × M0 ⊂ I × M, such that the IVP admits a unique local solution φ(t, t0 , x0 ) ∈ C1 ( I0 , Rn ). Proof. We proceed by steps. • Integrating both sides of the system of ODEs x˙ = f (t, x ) w.r.t. t we get an integral equation which formally defines the solution of (1.5): x (t) = K ( x )(t) := x0 +

Z t t0

f (s, x (s)) ds.

(1.7)

Note that x0 (t) := x0 is an approximating solution of (1.7) at least for small t. • Plugging x0 (t) := x0 into (1.7) we get another approximating solution x1 (t) := K ( x0 )(t) = x0 +

Z t t0

f (s, x0 (s)) ds,

and so on, x2 (t) := K2 ( x0 )(t) = (K ◦ K )( x0 )(t) = x0 +

Z t t0

f (s, x1 (s)) ds.

• The iteration of such a procedure, called Picard iteration, produces a a sequence of approximating solutions xm (t) := K m ( x0 )(t),

m ∈ N.

• This observation allows us to apply the “Contraction Principle” to the fixed point equation (1.7), which we compactly write x = K ( x ). We set t0 = 0 for notational simplicity and consider only the case t > 0 to avoid absolute values in the estimates which follow. • We need a Banach space X and a closed subset D ⊆ X such that K : D → D is a contraction. The natural choice is X := C ([0, T ], Rn ) for some T > 0. Since I × M is open and (0, x0 ) ∈ I × M we can choose V := [0, T ] × M0 ⊂ I × M, where M0 := { x ∈ Rn : k x − x0 k < δ}, and define P := max k f (t, x )k, (t,x )∈V

where the maximum exists by continuity of f and compactness of V. • Then we have:

kK ( x )(t) − x0 k 6 whenever {(t, x ) : t ∈ [0, T ]} ⊂ V.

Z t 0

k f (s, x (s))k ds 6 t P,

1.3 Existence and uniqueness of solutions of IVPs

9

• Hence for



δ t 6 T0 := min T, P

 ,

we have T0 P 6 δ and the graph of K ( x ) restricted to I0 := [0, T0 ] is again in V. Note that since [0, T0 ] ⊆ [0, T ] the same constant P bounds k f k on V0 := [0, T0 ] × M0 ⊂ I × M. • Therefore, if we choose X := C ([0, T0 ], Rn ) as our Banach space with norm

k x k X := max k x (t)k, t∈[0,T0 ]

and D := { x ∈ X : k x − x0 k X 6 δ} as our closed subset, then K is a map from D to D. • The contraction property of K follows from the Lipschitz continuity property (1.6) of f . If y ∈ D we have:

kK ( x )(t) − K (y)(t)k 6

Z t 0

k f (s, x (s)) − f (s, y(s))k ds

Z t

k x (s) − y(s)k ds

6

L

6

L t sup k x (s) − y(s)k,

0

s∈[0,t]

provided that the graphs of both x and y lie in V0 . In other words:

kK ( x ) − K (y)k X 6 η k x − yk X ,

∀ x, y ∈ D,

with η := L T0 . Choosing T0 < L−1 we see that K is a contraction. • The existence and the convergence of lim xm (t) = K m ( x0 )(t) =: φ(t, x0 )

m→∞

follows from the fact that X is a Banach space and its uniqueness follows from the “Contraction Principle”. • The fact that φ is differentiable w.r.t. t follows from the fact that φ˙ (t, x0 ) = f (t, φ(t, x0 )), where f ∈ C ( I × M, Rn ). The Theorem is proved.



I Remarks: • Reformulation of Theorem 1.1: Let V := [t0 , t0 + T ] × M0 ⊂ I × M and P be the maximum of k f k on V. Then the solution of the IVP (1.5) exists at least for t ∈ [t0 , t0 + T0 ], where T0 := min( T, δ/P), and remains in M0 . The same holds for the interval [t0 − T, t0 ]. In these intervals the solution is of class C1 w.r.t. t.

10

1 Initial Value Problems • Theorem 1.1 can be interpreted as a principle of determinism: if we know the initial conditions of a system, then we can predict its future states. Although such a principle is mathematically validated by Theorem 1.1, its physical interpretation is not as clear as it might seem. The main reasons are: 1. To find the explicit solution of an IVP (1.5) can be very complicated (if not impossible). 2. To know the initial state exactly may be very difficult (if not impossible).

Corollary 1.1 Consider the IVP (1.5). If f ∈ C k ( I × M, Rn ), k > 1, then there exists, around each point (t0 , x0 ) ∈ I × M, an open set I0 × M0 ⊂ I × M, such that the IVP admits a unique local solution φ(t, t0 , x0 ) ∈ C k+1 ( I0 , Rn ). Proof. It is a consequence of the fact that φ˙ (t, t0 , x0 ) = f (t, φ(t, t0 , x0 )), where f ∈ C k ( I × M, Rn ). 

I The following result states that if f is continuous but not locally Lipschitz continuous in x, uniformly w.r.t. t, we still have existence of solutions, but me may lose their uniqueness. In other words, it provides sufficient conditions for existence of solutions of the IVP (1.5). Theorem 1.2 (Peano) Consider the IVP (1.5). If f ∈ C ( I × M, Rn ), then there exists, around each point (t0 , x0 ) ∈ I × M, an open set I0 × M0 ⊂ I × M, such that the IVP admits a local solution φ(t, t0 , x0 ) ∈ C1 ( I0 , Rn ). No Proof. 1.4

Dependence of solutions on initial values and parameters

I An IVP (1.5) is well-posed if: 1. There exists a unique local solution φ(t, t0 , x0 ). 2. The solution admits a continuous (or even C k , k > 1) dependence on the initial values (t0 , x0 ). If the IVP (1.5) is well-posed, one expects that small changes in the initial values will result in small changes of the solution.

I We start with the following technical Lemma.

1.4 Dependence of solutions on initial values and parameters

11

Lemma 1.1 (Gronwall inequality) Let T > 0. Let ζ : [0, T ] → R be a non-negative function such that ζ (t) 6 α +

Z t 0

α > 0, t ∈ [0, T ],

β(s) ζ (s) ds,

where β(s) > 0, s ∈ [0, T ]. Then ζ (t) 6 α exp

t

Z 0

 β(s) ds ,

t ∈ [0, T ].

No Proof. Theorem 1.3 Consider the following IVPs: ( x˙ = f (t, x ),

(

x ( t0 ) = x0 ,

y˙ = g(t, y), y ( t0 ) = y0 ,

(1.8)

where f , g ∈ Lip ( I × M, Rn ) (with Lipschitz constant L > 0) and (t0 , x0 ), (t0 , y0 ) ∈ I × M. Let φ(t, t0 , x0 ) and ψ(t, t0 , y0 ) be the unique local solutions of the IVPs (1.8). Then there holds

kφ(t, t0 , x0 ) − ψ(t, t0 , y0 )k 6 k x0 − y0 ke L|t−t0 | +

 P  L | t − t0 | e −1 , L

(1.9)

where P := sup k f (t, x ) − g(t, x )k, (t,x )∈V

with V ⊂ I × M being a set containing the graphs of φ(t, t0 , x0 ) and ψ(t, t0 , y0 ). Proof. Without any loss of generality we set t0 = 0. We proceed by steps. • We have f ∈ Lip ( I × M, Rn ), so that

k f (s, φ(s, x0 )) − g(s, ψ(s, y0 ))k 6 k f (s, φ(s, x0 )) − f (s, ψ(s, y0 ))k + k f (s, ψ(s, y0 )) − g(s, ψ(s, y0 ))k 6 Lkφ(s, x0 ) − ψ(s, y0 )k + P =: L ζ (s) > 0. • We also have:

kφ(t, x0 ) − ψ(t, y0 )k 6 k x0 − y0 k +

Z t 0

k f (s, φ(s, x0 )) − g(s, ψ(s, y0 ))k ds,

12

1 Initial Value Problems that is

kφ(t, x0 ) − ψ(t, y0 )k = ζ (t) −

P 6 k x0 − y0 k + L

Z t 0

L ζ (s) ds,

or, equivalently, ζ ( t ) 6 k x0 − y0 k +

P +L L

Z t 0

ζ (s) ds.

• Use Lemma 1.1 with α := k x0 − y0 k + P/L > 0: ζ (t ) 6 α e L|t| , which is

kφ(t, x0 ) − ψ(t, y0 )k +

P 6 L



k x0 − y0 k +

P L



e L|t| ,

which coincides with (1.9) for t0 = 0. The Theorem is proved.



Corollary 1.2 Consider the IVP (1.5). If f ∈ Lip ( I × M, Rn ) (with Lipschitz constant L > 0) and (t0 , x0 ), (t0 , y0 ) ∈ I × M, then

kφ(t, t0 , x0 ) − φ(t, t0 , y0 )k 6 k x0 − y0 k e L|t−t0 | .

Proof. It is a consequence of Theorem 1.3 with f = g (i.e., P = 0).



I We now denote by φ(t, s, ξ ) the local solution of the IVP (1.5) to emphasize its dependence on the initial value (s, ξ ) ∈ I × M which is now free to be varied within a neighborhood of (t0 , x0 ). Note that with this notation we mean that ξ is the initial point and s is the initial time. Theorem 1.4 Consider the IVP (1.5). If f ∈ Lip ( I × M, Rn ), then there exists, around each point (t0 , x0 ) ∈ I × M a compact set I0 × M0 ⊂ I × M such that the IVP admits a unique local solution φ(t, s, ξ ) ∈ C ( I0 × I0 × M0 , Rn ). Proof. Using the same notation as in the Proof of Theorem 1.1 we can find a compact set V := [t0 − ε, t0 + ε] × B0 , where B0 := { x ∈ Rn : k x − x0 k < δ}, such that φ(t, t0 , x0 ) exists for |t − t0 | 6 ε for ε > 0. Then we can find V 0 := [t00 − ε/2, t00 + ε/2] × C0 , where C0 := { x ∈ Rn : k x − x00 k < δ/2}, where φ(t, t00 , x00 ) exists for

1.4 Dependence of solutions on initial values and parameters

13

|t − t00 | 6 ε/2 whenever |t00 − t0 | 6 ε/2 and k x00 − x0 k 6 δ/2. Hence we can choose I0 := (t0 − ε/4, t0 + ε/4) and M0 := C0 . The claim follows.  I In some cases the continuity w.r.t. (s, ξ ) of the solution φ(t, s, ξ ) assured by Theorem 1.4 is not good enough and we need differentiability w.r.t. the initial point ξ. • Consider the IVP (1.5) with initial point ξ ∈ M. Suppose that φ(t, t0 , ξ ) is differentiable w.r.t. ξ. Then the same is true for φ˙ (t, t0 , ξ ). Let us write the IVP (1.5) as ∂ φ(t, t0 , ξ ) = f (t, φ(t, t0 , ξ )). ∂t • Differentiating w.r.t. ξ and using the chain rule we get:     ∂ ∂ ∂ ∂ φ(t, t0 , ξ ) = f (t, φ(t, t0 , ξ )) φ(t, t0 , ξ ) ∂ξ ∂t ∂φ ∂ξ   ∂ ∂ = φ(t, t0 , ξ ) , ∂t ∂ξ where in the last step we interchanged the t and ξ partial derivatives. • Defining the n × n matrix variable Θ(t, t0 , ξ ) :=

∂ φ(t, t0 , ξ ), ∂ξ

(1.10)

and the n × n matrix A(t, t0 , ξ ) :=

∂ f (t, φ(t, t0 , ξ )), ∂φ

we see that Θ obeys the matrix differential equation ˙ (t, t0 , ξ ) = A(t, t0 , ξ ) Θ(t, t0 , ξ ), Θ

(1.11)

which is called first variational equation. Note that the matrix A is, formally, nothing but the Jacobian matrix (w.r.t. x) of the function f = f (t, x ). • Integration of (1.11) w.r.t. t yields the integral equation Θ(t, t0 , ξ ) = Θ(t0 , t0 , ξ ) +

Z t t0

A(s, t0 , ξ ) Θ(s, t0 , ξ ) ds,

where Θ(t0 , t0 , ξ ) = ∂φ(t0 , t0 , ξ )/∂ξ = 1n (here 1n is the identity n × n matrix) due to the fact that φ(t0 , t0 , ξ ) = ξ.

I The next Theorem shows, in particular, that if f ∈ C1 ( I × M, Rn ) then the unique solution of the matrix differential equation (1.11), with Θ(t0 , t0 , ξ ) = 1n , is exactly given by (1.10).

14

1 Initial Value Problems

Theorem 1.5 Consider the IVP (1.5). If f ∈ C k ( I × M, Rn ), k > 1, then there exists, around each point (t0 , x0 ) ∈ I × M, an open set I0 × M0 ⊂ I × M such that the IVP admits a unique local solution φ(t, s, ξ ) ∈ C k ( I0 × I0 × M0 , Rn ). No Proof.

I Finally, we assume that f depends on a set of p parameters α ∈ A ⊆ R p and consider the IVP ( x˙ = f (t, x, α), (1.12) x ( t0 ) = x0 . We are now interested in varying the initial condition (t0 , x0 ) and α. Therefore, we denote by φ(t, s, ξ, α) the local solution of the IVP (1.12) to emphasize these dependencies.

I The following result, which generalizes Theorem 1.5, establishes a C k -dependence on (t0 , x0 ) and α for the IVP (1.12). As for the case of initial conditions, it is natural to expect that small changes in the parameters will result in small changes of the solution. Theorem 1.6 Consider the IVP (1.12). If f ∈ C k ( I × M × A, Rn ), k > 1, then there exists, around each point (t0 , x0 , α) ∈ I × M × A, an open set I0 × M0 × A0 ⊂ I × M × A such that the IVP admits a unique local solution φ(t, s, ξ, α) ∈ C k ( I0 × I0 × M0 × A 0 , Rn ). No Proof. 1.5

Prolongation of solutions

I It is now clear that the local solution of an IVP (1.5) is locally defined on I0 ⊂ I, where I is the definition domain of t for the function f . In particular, even though the IVP is defined for t ∈ I = R, solutions might not exist for all t ∈ R. This raises the question of existence of a maximal interval on which a maximal solution of (1.5) can be defined. In other words, we are now interested in understanding how much we can prolong the interval of existence I0 . Definition 1.3 Let φ(t, t0 , x0 ) ∈ C1 ( I0 , Rn ) be the unique local solution of the IVP (1.5) defined in I0 . e(t, t0 , x0 ) ∈ C1 ( e 1. A solution φ I0 , Rn ) of (1.5) is a prolongation of φ(t, t0 , x0 ) if e(t, t0 , x0 ) = φ(t, t0 , x0 ) for all t ∈ I0 . I0 ⊂ e I0 and φ

1.5 Prolongation of solutions

15

2. A solution φ(t, t0 , x0 ) is called maximal solution if for any prolongation e(t, t0 , x0 ) of φ(t, t0 , x0 ) we have I0 = e φ I0 . In this case I0 is called maximal interval of existence. Theorem 1.7 Consider the IVP (1.5). If f ∈ Lip ( I × M, Rn ) then there exists a unique maximal solution defined on some open maximal interval (t− , t+ ), where t± depend on (t0 , x0 ) ∈ I × M. No Proof.

I We know from Theorem 1.2 that if f is continuous but not locally Lipschitz continuous in x we still have existence of solutions, but we may lose uniqueness. Even without uniqueness, two given solutions of the IVP (1.5) can be glued together at some point. Example 1.6 (Non-uniqueness of the solution) Consider the following IVP in R: (

x˙ = 3 x2/3 , x (0) = 0.

The function f ( x ) := family of solutions

3 x2/3

is not Lipschitz continuous at x = 0. The IVP admits the a two-parameter  ( t − t1 )3    0 φ(t, 0) =    ( t − t2 )3

t < t1 , t1 6 t 6 t2 , t > t2 ,

for all t1 < 0 < t2 .

Fig. 1.3. Plot of the function x (t) = φ(t, 0) ([Ge]).

I The following claim holds.

16

1 Initial Value Problems

Theorem 1.8 Consider the IVP (1.5). Assume that f ∈ Lip ( I × M, Rn ) and let (t− , t+ ), with t± depending on (t0 , x0 ) ∈ I × M, be the maximal interval of existence. If t+ < ∞ then the solution must eventually leave every compact set D with [t0 , t+ ] × D ⊂ I × M as t approaches t+ . In particular, for I × M = R × Rn , the solution must tend to infinity as t approaches t+ . No Proof.

I As a consequence of Theorem 1.8, it is possible to show that solutions exist for all t ∈ R if f (t, x ) grows at most linearly w.r.t. x. Theorem 1.9 Let I × M = R × Rn and assume that for every T > 0 there exist S1 ( T ), S2 ( T ) > 0 such that

(t, x ) ∈ [− T, T ] × Rn .

k f (t, x )k 6 S1 ( T ) + S2 ( T )k x k,

Then the solutions of the IVP (1.5) is defined on I0 = R. No Proof. Example 1.7 (Global solutions and blow-up in finite time) 1. Consider the following IVP in R: (

x˙ = x, x (0) = x0 .

Its solution is given by φ(t, x0 ) = x0 et , which is globally defined for all t ∈ R. 2. Consider the following IVP in R: (

x˙ = x2 , x (0) = x0 > 0.

(1.13)

Its solution is given by φ(t, x0 ) =

x0 . 1 − x0 t

The solution only exists on the interval (−∞, 1/x0 ) and we have a blow up in finite time as t → 1/x0 . There is no way to extend this solution for t > 1/x0 . Therefore solutions might only exist locally in t, even for perfectly nice f .

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1.5 Prolongation of solutions 17

Fig. 1.4. Solutions of the IVP (1.13) ([Ol]).

I We conclude this Chapter with the following claim (see Theorem 1.1, Corollary 1.1, Theorems 1.5 and 1.8): Every IVP ( x˙ = f (t, x ), x ( t0 ) = x0 ,

where f ∈ C k ( I × M, Rn ), k > 1, has a unique local solution φ(t, t0 , x0 ) that is of class C k w.r.t. (t, t0 , x0 ). Moreover, the solution can be extended in time until it either reaches the boundary of M or blows up to infinity.

18 1.6

1 Initial Value Problems Exercises

Ch1.E1

(a) Let f , g : R → R be two globally Lipschitz continuous functions. Prove that their composition is globally Lipschitz continuous on R. (b) Prove that the real function f ( x ) :=

1 1 + x2

is globally Lipschitz continuous. (c) Use (a) and (b) to prove that the real function f ( x ) :=

1 1 + sin2 x

is globally Lipschitz continuous. (d) Prove that the scalar IVP (t > 0)    x˙ =  

e− t , 1 + sin2 x

x (0) = 1,

has a unique solution. :::::::::::::::::::::::: Ch1.E2 Consider the following IVP in R: (

x˙ = | x | p/q , x (0) = 0,

with p, q ∈ N \ {0}. (a) Prove that it has a unique solution if p > q. (b) Prove that it has an infinite number of solutions if p < q. (c) What can you say if p = q? :::::::::::::::::::::::: Ch1.E3 Consider the following IVP in R:  2 q   x˙ = x | x |, x2 + ε   x (0) = 0, with ε > 0. What can you say about existence and uniqueness of its solutions? Is the solution unique if ε = 0? :::::::::::::::::::::::: Ch1.E4 Consider the following IVP in R:  3   x˙ = x − x, 1   x (0) = . 2

1.6 Exercises

19

(a) What can you say about existence and uniqueness of its solutions? (b) Without solving the ODE, calculate limt→+∞ φ(t, 1/2), where φ(t, 1/2) is the solution. :::::::::::::::::::::::: Ch1.E5 Prove that the second-order scalar IVP ( x¨ = 2 x˙ − x + t − 2,

( x (0), x˙ (0)) = ( x0 , v0 ), has a unique solution for any choice of the initial values. :::::::::::::::::::::::: Ch1.E6 Consider the following IVP in R: (

x˙ = x α , x (0) = 0,

with α > 0. (a) Does it admit a unique solution? Find the solution(s). (b) Is there any essential modification of your conclusions if the initial condition is x (0) = 1? :::::::::::::::::::::::: Ch1.E7 Consider the following IVPs in R: ( x˙ = 2 t x, x (0) = 1,

(

x˙ = t + x, x (0) = 1.

For both of them construct the sequence of Picard iterations and obtain the explicit solution. :::::::::::::::::::::::: Ch1.E8 Consider the following ODE in R: x¨ − 2 t x˙ − 2 x = 0. (a) Find the solution by using a power series expansion around t = 0. (b) Solve the IVP (

x¨ = 2 t x˙ + 2 x,

( x (0), x˙ (0)) = (1, 0). :::::::::::::::::::::::: Ch1.E9 Prove that the solution of the scalar ODE x˙ = 1 + x14 diverges to infinity in a finite time, irrespective of the initial conditions x (0) ∈ R. ::::::::::::::::::::::::

20

1 Initial Value Problems

Ch1.E10 Consider the following IVP in R:     x˙ = 1 x2 − 1 , 2  x (0) ∈ R. (a) Find the solution. (b) Find the intervals (t− , t+ ) ⊆ R on which the solution is defined. (c) What happens if the solution approaches t− from the right and t+ from the left? :::::::::::::::::::::::: Ch1.E11 Consider the following ODE in R: x˙ = −2 t x2 . (a) Find the solution (this solution will depend on one arbitrary real constant, say α ∈ R). (b) Depending on α, find the maximal solutions. :::::::::::::::::::::::: Ch1.E12 Consider the following IVP in R: (

x˙ = 1 +

p

| x |,

x (0) = 0. Find the solution and the intervals of existence. :::::::::::::::::::::::: Ch1.E13 Consider the following IVP in R: (

x˙ = x − x2 , x ( t0 ) = x0 .

(a) Find the solution. (b) Find the intervals (t− , t+ ) ⊆ R on which the solution is defined. (c) What happens if the solution approaches t− from the right and t+ from the left? :::::::::::::::::::::::: Ch1.E14 Consider the following IVP in R:   x˙ = x − 4 , x  x (0) = −1. (a) Find the solution. (b) Find the maximal interval centered at x = 0 where the unique solution is defined and continuously differentiable. ::::::::::::::::::::::::

1.6 Exercises

21

Ch1.E15 Let A be a constant real n × n matrix. Define cosh( A) :=

e A + e− A , 2

sinh( A) :=

e A − e− A . 2

(a) Show that for all t ∈ R there holds: d cosh(tA) = A sinh(tA), dt

d sinh(tA) = A cosh(tA). dt

(b) Verify that the function a, b ∈ Rn , t ∈ R,

x (t) := cosh(tA) a + sinh(tA) b, satisfies the following second-order ODE in Rn : x¨ − A2 x = 0.

:::::::::::::::::::::::: Ch1.E16 Consider the following IVP in R: (

x˙ = x + f (t), x (0) ∈ R,

where f ∈

C1 (R, R).

(a) Find the solution. (b) Use the result in (a) to prove that the IVP of n ODEs ( x˙ k = xk−1 + xk , xk (0) = 1, where k = 1, . . . , n and x0 ≡ 0, admits the solution φk (t, 1) = et

k −1 j t

∑

j =0

j!

,

k = 1, . . . , n.

Compute the limit limn→∞ φn (t, 1). ::::::::::::::::::::::::

2 Continuous Dynamical Systems 2.1

Introduction

I A dynamical system is the mathematical formalization of the notion of deterministic process. • The future and past states of many physical, chemical, biological, ecological, economical, and even social systems can be predicted to a certain extent by knowing their present state and the laws governing their time evolution. • Provided these laws do not change in time, the behavior of such systems can be considered as completely defined by their initial state.

I The main ingredients to define a dynamical system are: 1. A complete metric set M, called phase space, containing all possible states of the system. According to the dimension of M, the dynamical system is either finiteor infinite-dimensional. In the finite-dimensional case, M is usually an open subset of Rn or a smooth manifold (which, roughly speaking, is an abstract surface that locally looks like a linear space, see Definition 4.1). 2. A time variable t, which can be continuous (t ∈ R, R+ ) or discrete (t ∈ Z, Z+ ). 3. A law of time evolution of states in M defined in terms of time evolution operator. The time evolution provides a change in the state of the system. The time evolution operator must satisfy certain conditions to be defined later. Example 2.1 (Mechanical systems and the planar pendulum) • In the canonical Hamiltonian formulation of classical mechanics, the state of an isolated system with n degrees of freedom is characterized by a 2 n-dimensional real vector:

(q, p) := (q1 , . . . , qn , p1 , . . . , pn ) ∈ M ⊆ R2n , where qi is the i-th coordinate and pi is the corresponding momentum. The time evolution of (q, p) is described by a system of 2 n first-order ODEs (Hamilton equations):  ∂H (q, p)   ,   q˙ i = ∂pi  ∂H (q, p)    p˙ i = − , ∂qi with i = 1, . . . , n. Here the differentiable function H : M → R is the Hamiltonian function of the system. • Assume n = 1 and p2 H (q, p) := + cos q, (q, p) ∈ M := [0, 2 π ) × R. 2

23

24

2 Continuous Dynamical Systems Then Hamilton equations are (

q˙ = p, p˙ = − sin q.

(2.1)

They are equivalent to the Newton equation q¨ = − sin q. The (Hamiltonian) flow of Hamilton equations is Φt : (0, (q(0), p(0))) 7→ (q(t), p(t)), where (0, (q(0), p(0))) is the initial state at t = 0 and (q(t), p(t)) is the solution of (2.1), defining the state at time t > 0. Physically, the flow defines the time evolution of the angular displacement q from the vertical position of a unit-mass planar pendulum and the time evolution of the angular velocity q˙ = p.

Fig. 2.1. Planar pendulum ([Ku]).

2.2

Definition of a dynamical system

I Let us recall the following facts from Group Theory: • A group is a set G together with an operation ∗ : G × G → G, called group law of G, that combines any two elements g and h to form another element g ∗ h ∈ G. The pair (G, ∗) must satisfy four axioms: 1. Closure: g ∗ h ∈ G for all g, h ∈ G. 2. Associativity: ( g ∗ h) · k = g ∗ (h ∗ k ) for all g, h, k ∈ G. 3. Existence of identity element: there exists a unique element e ∈ G, such that for every element g ∈ G, the equation g ∗ e = e ∗ g = g holds. 4. Existence of inverse element: for each g ∈ G, there exists an element h ∈ G such that g ∗ h = h ∗ g = e. • A semigroup is a set G together with an operation ∗ : G × G → G, such that only the group axioms 1. and 2. hold true. Thus G need not have an identity element and its elements need not have inverses within G.

2.2 Definition of a dynamical system

25

• Let M be a set and G be a group. The (left) group action of G on a set M is a map Φ g : G × M → M : ( g, x ) 7→ Φ g ( x ), such that the following axioms hold: 1. There holds:
Φ g∗h ( x ) = Φ g (Φh ( x )) ≡ Φ g ◦ Φh ( x )

∀ g, h ∈ G, x ∈ M.

(2.2)

2. Existence of identity element: there exists a unique element e ∈ G, such that Φe ( x ) = x

∀ x ∈ M.

(2.3)

From these two axioms, it follows that for every g ∈ G, the map Φ g is a bijective map from M to M (its inverse being the map which maps x to Φ g−1 ( x )).

I We now give a general definition of a (finite-dimensional) dynamical system. Definition 2.1 Let M be either (a subset of) the Euclidean space or a real finite-dimensional smooth manifold. We say that M is the phase space . Let G be a number set; an element g ∈ G is a time variable. A dynamical system is a triple {Φ g , G, M } defined in terms of a (semi)group action Φ g : G × M → M : ( g, x ) 7→ Φ g ( x ),

(2.4)

which is called time evolution operator. 1. If G is a group then {Φ g , G, M } is an invertible dynamical system. 2. If G is a subset of R (containing 0) then {Φ g , G, M } is a continuous dynamical system. 3. If G is a subset of Z (containing 0) then {Φ g , G, M } is a discrete dynamical system.

I Note that in Definition 2.1 G is a subset of R or Z containing 0. The group law of G is the addition, i.e., g ∗ h := g + h, and G is abelian, i.e., g + h = h + g for all g, h ∈ G. Therefore (2.2) reads Φ g+h ( x ) = Φ g (Φh ( x )) = Φh (Φ g ( x )) = Φh+ g ( x )

∀ g, h ∈ G, x ∈ M.

I So far time evolution operators are defined only in a formal way. The typical examples of invertible continuous and discrete dynamical systems are:

26

2 Continuous Dynamical Systems • {Φt , R, M }, where M ⊂ Rn (is compact). Here Φt is the global (i.e., G = R) flow of an autonomous differentiable IVP, namely the one-parameter family of diffeomorphisms (i.e., Φt is a bijection and both Φt and its inverse are differentiable) Φt : R × M → M : (t, x ) 7→ Φt ( x ) := φ(t, x ), where φ(t, x ) is the unique solution of the IVP with Φ0 ( x ) = φ(0, x ) = x ∈ M. Here x plays the role of the initial value of the IVP. (a) Property (2.2) reads Φt+s ( x ) = (Φt ◦ Φs )( x )

∀ t, s ∈ R, x ∈ M,

which means that the state at time t + s when starting at x is identical to the state at time t when starting at Φs ( x ).

Fig. 2.2. Property (2.2) ([Ku]).

(b) Property (2.3),

Φ0 ( x ) = x

∀ x ∈ M,

says that x is the prescribed state at time t = 0. (c) The invertibility property says that

(Φt ◦ Φ−t )( x ) = x

∀ t ∈ R, x ∈ M.

In such a situation one says that Φt is a one-parameter global Lie group of diffeomorphisms on Rn . One says that Φt is a one-parameter local Lie group of diffeomorphisms on Rn if instead of G = R one has only a subset of R (containing 0). • {Φk , Z, M }, M ⊂ Rn , where Φk ≡ Φk is the k-th iterate of a homeomorphism (i.e., Φ is a bijection and both Φ and its inverse are continuous) Φ : M → M,

2.2 Definition of a dynamical system

27

namely Φk ( x ) := (Φ · · ◦ Φ})( x ), | ◦ ·{z

k ∈ N, x ∈ M,

k times

k

Φ ( x ) := (Φ |

−1

◦ ·{z · · ◦ Φ−}1 )( x ),

−k ∈ N, x ∈ M.

|k| times

(a) Property (2.2) reads Φi+ j ( x ) = (Φi ◦ Φ j )( x )

∀ i, j ∈ Z, x ∈ M,

which means that the (i + j)-th iterate is identical to the composition of the i-th and j-th iterates. (b) Property (2.3), Φ0 ( x ) = x says that

Φ0

∀ x ∈ M,

is the identity map.

I One can consider also dynamical systems whose future states for t > 0 (resp. k > 0) are completely determined by their initial state at t = 0 (resp. k = 0), but the history for t < 0 (resp. k < 0) cannot be reconstructed. Such (noninvertible) dynamical systems are described by semigroup actions defined only for t > 0 (resp. k > 0). In the continuous time case one says that Φt is a semiflow. 2.2.1

Orbits, invariant sets, invariant functions and stability

I Let {Φ g , G, M} be a dynamical system. In particular g ≡ t ∈ R for a continuous (invertible and globally defined) dynamical system and g ≡ k ∈ Z for a discrete (invertible and globally defined) dynamical system. Definition 2.2 An orbit starting at x0 ∈ M is the ordered subset of M defined by

O( x0 ) := { x ∈ M : x = Φ g ( x0 ) ∀ g ∈ G}.

I Remarks: • Orbits of a continuous dynamical system are curves in M parametrized by the time t and oriented by the direction of time advance. • Orbits of a discrete dynamical system are sequences of points in M enumerated by increasing integers.

28

2 Continuous Dynamical Systems

Fig. 2.3. Continuous orbits ([Ku]).

• If y0 = Φ g ( x0 ) for some g ∈ G, then the sets O( x0 ) and O(y0 ) coincide. In particular different orbits are disjoint.

I The simplest orbits are fixed points and cycles. Definition 2.3 1. A point xe ∈ M is called a fixed point (or equilibrium point) if xe = Φ g ( xe) for all g ∈ G. 2. A cycle (or periodic orbit) is an orbit O( x0 ) such that each point x ∈ O( x0 ) satisfies Φ g+T ( x ) = Φ g ( x ) with some fixed T ∈ G, for all g ∈ G. The minimal T with this property is the period of the cycle.

I If a system starts its evolution at x0 on the cycle, it will return exactly to this point after every T units of time. The system exhibits periodic oscillations. • Cycles of a continuous dynamical system are closed curves in M. • Cycles of a discrete dynamical system are finite sets of points x0 , Φ ( x0 ), Φ2 ( x0 ), . . . , Φ T ( x0 ) = x0 ,

T ∈ N.

2.2 Definition of a dynamical system

29

Fig. 2.4. Continuous and discrete periodic orbits ([Ku]).

Definition 2.4 The phase portrait of {Φ g , G, M } is a partitioning of M into orbits.

I The phase portrait contains a lot of information on the behavior of a dynamical system. By looking at the phase portrait, we can determine number and types of asymptotic states to which the system tends as g → +∞ (and as g → −∞ if the system is invertible). Definition 2.5 1. An invariant set of {Φ g , G, M } is a subset S ⊂ M such that x0 ∈ S implies Φ g ( x0 ) ∈ S for all g ∈ G, or, equivalently, Φ g (S) ⊆ S for all g ∈ G. 2. An invariant function of {Φ g , G, M } is a function F : M → R such that F (Φ g ( x )) = F ( x )

∀ g ∈ G, x ∈ M.

I To represent an observable asymptotic state of a dynamical system, an invariant set must possess some “stability property”. Definition 2.6 Let S be a closed invariant set. 1. S is (Lyapunov) stable if for any sufficiently small neighborhood U ⊃ S there exists a neighborhood V ⊃ S such that Φ g ( x ) ∈ U for all x ∈ V and all g > 0. 2. S is attracting if there exists a neighborhood U ⊃ S such that Φ g ( x ) → S as g → +∞, for all x ∈ U.

30

2 Continuous Dynamical Systems 3. S is asymptotically stable if it is Lyapunov stable and attracting.

I Remarks: • If S is a fixed point or a cycle, then Definition 2.6 turns into the definition of stable fixed point or stable cycle. • There are invariant sets that are Lyapunov stable but not asymptotically stable and invariant sets that are attracting but not asymptotically stable.

I From now on we shall consider only continuous dynamical systems, meaning that our time will be a real variable. 2.3

Autonomous IVPs as continuous dynamical systems

I We now show how to realize more concretely a continuous dynamical system in terms of the unique solution of an autonomous IVP on M ⊆ Rn : ( x˙ = f ( x ), (2.5) x ( t0 ) = x0 , with f ∈ C k ( M, Rn ), k > 1. • Theorems 1.1 (see also Corollary 1.1) and 1.7 assure that (2.5) admits a unique maximal local solution φ(t, x0 ), x0 ∈ M, defined on a maximal interval Ix0 := (t− ( x0 ), t+ ( x0 )) ⊆ R. • Since (2.5) is autonomous it does not matter at what time t0 we specify the initial point x0 (time translational symmetry). Indeed, if φ(t, x0 ), t ∈ Ix0 , is a solution of (2.5), then so is ψ(t, x0 ) := φ(t + s, x0 ) with t + s ∈ Ix0 : ψ˙ (t, x0 ) = φ˙ (t + s, x0 ) = f (φ(t + s, x0 )) = f (ψ(t, x0 )). Therefore we can choose t0 = 0 and Ix0 always contains 0. • Define the set W :=

[

Ix0 × { x0 } ⊆ R × M.

(2.6)

x0 ∈ M

Then the flow (i.e., the continuous time evolution operator (2.4)) of (2.5) is the map Φt : W → M : (t, x0 ) 7→ Φt ( x0 ) := φ(t, x0 ), (2.7)

2.3 Autonomous IVPs as continuous dynamical systems

31

where φ(t, x0 ) is the maximal solution starting at x0 . We have: Φ0 ( x0 ) : = x0 , which is property (2.3). • The map Φt has the following property (cf. formula (2.2)): Φs+t ( x0 ) = Φs (Φt ( x0 )),

x0 ∈ M,

(2.8)

in the sense that if the r.h.s. is defined, so is the l.h.s., and they are equal. Indeed, suppose that s > t > 0 and that Φs (Φt ( x0 ) is defined, i.e., t ∈ Ix0 and s ∈ IΦt ( x0 ) . Define the map ψ : (t− ( x0 ), s + t] → M, by ( ψ (r ) : =

Φr ( x0 )

r ∈ ( t − ( x0 ), t ],

Φr−t (Φt ( x0 ))

r ∈ [t, t + s].

Then ψ is a solution and ψ(0) = Φ0 ( x0 ) = x0 . Hence s + t ∈ Ix0 . Moreover, Φt+s ( x0 ) = ψ(s + t) = Φs (Φt ( x0 )). Property (2.8) expresses the determinism of the system. • Setting s = −t in (2.8) we see that Φt is a local diffeomorphism with inverse Φ−t .

I Let us summarize some of the above results in the following Theorem, whose detailed proof is omitted (note that some of the claims have been justified above). Theorem 2.1 Consider the IVP (2.5). 1. For all x0 ∈ M there exists a maximal interval Ix0 ⊆ R containing 0 and a corresponding unique maximal solution of class C k w.r.t. t. 2. The set W defined by (2.6) is open. 3. The map Φt defined by (2.7) is a one-parameter local Lie group of diffeomorphisms (of class C k ) on M. No Proof.

32

2 Continuous Dynamical Systems

Example 2.2 (A one-dimensional continuous dynamical system) Consider the continuous dynamical system defined by the scalar IVP ( x˙ = x3 , x (0) = x0 . The flow is

Φt : W → R : (t, x0 ) 7→ Φt ( x0 ) := φ(t, x0 ) = q

x0 1 − 2 x02 t

where W := {(t, x0 ) : 2 t x02 < 1} ⊂ R2 ,

Ix0 : =

−∞,

1 2 x02

,

! .

I Let us now have a closer look at the geometric properties of the IVP (2.5). It turns out that the geometry of the vector field f is closely related to the geometry of the solutions of the IVP when the solutions are viewed as curves in M. One of the main goals of the geometric method is to derive qualitative properties of solutions directly from f without solving the IVP. • The solution t 7→ φ(t, x0 ) defines for each x0 ∈ M two different curves: 1. A solution curve (also called integral curve or trajectory curve): γ( x0 ) := {(t, x ) : x = Φt ( x0 ), t ∈ Ix0 } ⊂ Ix0 × M. 2. An orbit, which is the projection of γ( x0 ) onto M:

O( x0 ) := { x : x = Φt ( x0 ), t ∈ Ix0 } ⊂ M. Orbits can be splitted into forward (+) and backward orbits (-):

O+ ( x0 ) := { x : x = Φt ( x0 ), t ∈ (0, t+ ( x0 ))} , O− ( x0 ) := { x : x = Φt ( x0 ), t ∈ (t− ( x0 ), 0)} . One can see that O( x0 ) is a periodic orbit (cf. Definition 2.3) if and only if O+ ( x0 ) ∩ O− ( x0 ) 6= ∅.

Fig. 2.5. Orbits and solution curves ([Ku]).

2.3 Autonomous IVPs as continuous dynamical systems

33

Both curves γ( x0 ) and O( x0 ) are parametrized by time t and oriented by the direction of time advance. The phase portrait is the collective graph of orbits in M. • The map x 7→ f ( x ), x ∈ M, defines a vector field on M. (a) If we replace f by − f in (2.5) then Φt ( x0 ) 7→ Φ−t ( x0 ). (b) The zeros of f define the fixed points of Φt . Indeed, if f ( xe) = 0, then Φt ( xe) = xe (cf. Definition 2.3). (c) Each orbit O( x0 ) is tangent to f at each point x ∈ O( x0 ). Note that an orbit cannot cross itself because, in that case, we would have two different tangent vectors of the same vector field at the crossing point. On the other hand, a curve in M as in Fig. 2.6 can exist if the crossing point is a fixed point. Such a curve is the union of the four orbits.

Fig. 2.6. A curve in the phase portrait which consists of four orbits ([Ch]).

I We finally provide a characterization of invariant sets (cf. Definition 2.5) of continuous dynamical systems generated by an IVP (2.5). • A point x0 ∈ M is called (±)-complete if t± ( x0 ) = ±∞ and complete if Ix0 = R. • As a consequence of Theorem 1.8 one infers that if O+ ( x0 ) (resp. O− ( x0 )) lies in a compact subset D of M then x0 is (+)-complete (resp. (−)-complete). • The vector field f and its flow Φt are called complete if every point x ∈ M is complete, meaning that Φt is globally defined, i.e., W = R × M. Another way to formulate this is to say that compactly supported vector fields are complete: they generate one-parameter global Lie groups of diffeomorphisms on M. • A set S ⊂ M is called (±)-invariant if

O ± ( x0 ) ⊆ S

∀ x0 ∈ S,

and invariant if

O( x0 ) ⊆ S

∀ x0 ∈ S.

• We conclude saying that if S ⊂ M is a compact (±)-invariant set then all points in S are complete.

34

2 Continuous Dynamical Systems

Example 2.3 (Phase portrait of a scalar IVP) Consider a continuous dynamical system generated by a scalar IVP ( x˙ = f ( x ), x (0) = x0 , where f : R → R is a (smooth) function. Looking at the graph of the function y = f ( x ) in the ( x, y)plane, one concludes that any orbit is either a root of the equation f ( x ) = 0 or an open segment of the x-axis, bounded by such roots or extending to infinity. The orientation of a nontrivial (segment) orbit is determined by the sign of f ( x ) in the corresponding interval.

Fig. 2.6. Phase portrait of a scalar ODE ([Ku]).

Example 2.4 (Phase portrait of a planar IVP) Consider a continuous dynamical system generated by the planar IVP ( x˙ 1 = x22 , x˙ 2 = x1 ,

(2.9)

with given initial conditions ( x1 (0), x2 (0)) ∈ R2 . The main qualitative properties of the
phase portrait can be derived just by looking at the flow lines of the vector field f ( x1 , x2 ) = x22 , x1 in the ( x1 , x2 )plane

Fig. 2.7. Phase portrait of the IVP (2.9).

2.3 Autonomous IVPs as continuous dynamical systems 2.3.1

35

Flows, vector fields and invariant functions

I In this Subsection we are not interested in the uniqueness problem of solutions of IVPs. Therefore we relax our regularity conditions and we assume that all maps and functions are smooth, i.e., C ∞ . Furthermore, we will be mainly interested in properties of flows instead of IVPs. For this reason we slightly change our notation denoting the initial condition x0 of an IVP by x. Next Theorem 2.2 should clarify any further notational ambiguity. I Consider a one-parameter local Lie group of smooth diffeomorphisms: Φt : I × M → M : (t, x ) 7→ x (t, x ) := Φt ( x ),

(2.10)

where I ⊂ R contains 0. In particular we have: x (0, x ) = Φ0 ( x ) = x,

Φt+s ( x ) = Φt (Φs ( x )) = Φs (Φt ( x )),

(2.11)

for all x ∈ M and t, s ∈ I, t + s ∈ I.

I The following Theorem establishes a fundamental one-to-one correspondence between one-parameter local Lie groups of smooth diffeomorphisms (i.e., continuous dynamical systems) and solutions of autonomous smooth IVPs. Theorem 2.2 (Lie) Any one-parameter local Lie group of smooth diffeomorphisms (2.10) is equivalent to the solution of an autonomous IVP ( x˙ = f ( x ), (2.12) x (0, x ) = Φ0 ( x ) = x, where

d f ( x ) := Φ t ( x ). dt t=0

Proof. We proceed by steps. • Expanding the flow Φt ( x ) in powers of t around t = 0 we get d Φ t ( x ) + O ( t2 ) Φt ( x ) = x + t dt t=0

= x + t f ( x ) + O ( t2 ). • Consider formula Φt+s ( x ) = Φs (Φt ( x )). Expanding the l.h.s. in powers of s around s = 0 we get Φt+s ( x ) = Φt ( x ) + s

d Φ t ( x ) + O ( s2 ). dt

(2.13)

36

2 Continuous Dynamical Systems Expanding the r.h.s. in powers of s around s = 0 we get Φs (Φt ( x )) = Φt ( x ) + s f (Φt ( x )) + O(s2 ).

(2.14)

• Equating (2.13) and (2.14) we see that Φt ( x ) satisfies the IVP   d Φ ( x ) = f (Φ ( x )) , t t dt  Φ0 ( x ) = x, that is

(

x˙ = f ( x ) ,

(2.15)

x (0, x ) = x.

• The IVP (2.15) satisfies the conditions of Theorem 1.1 and therefore it admits a (unique) local solution x (t, x ) = Φt ( x ).



The Theorem is proved.

I Theorem 2.2 shows that the infinitesimal transformation Φ t ( x ) = x + t f ( x ) + O ( t2 ), contains the essential information to characterize a one-parameter local Lie group of smooth diffeomorphisms. This justifies the next definition. Definition 2.7 1. The infinitesimal generator of Φt is the linear differential operator n

v :=

∂

∑ fi (x) ∂xi ,

(2.16)

i =1

where f i ( x ) is the i-th component of f ( x ) :=

d Φ t ( x ). dt t=0

2. The action of v on a function F ∈ F ( M, R) defines the Lie derivative of F along v, denoted by Lv F: n

(Lv F )( x ) := v[ F ( x )] =

∂F

∑ fi (x) ∂xi

= h f ( x ), gradx F ( x ) i .

i =1

Here h ·, · i is the scalar product in Rn and   ∂F ∂F ,..., gradx F ( x ) := ∂x1 ∂xn

(2.17)

2.3 Autonomous IVPs as continuous dynamical systems

37

is the gradient of F w.r.t. x.

I Remarks: • v is the tangent vector to the orbit of Φt ( x ) at each point x ∈ M. In Differential Geometry v takes the name of (smooth) tangent vector field, or simply (smooth) vector field. Technically, it would be more precise to use the notation v| x (or v x ) instead of v and ∂/∂xi | x (or (∂/∂xi ) x ) instead of ∂/∂xi . It may happen that we call both v and f vector field. • Smooth vector fields v over M form a real vector space, denoted by X( M ), w.r.t. the operations

(v + w)| x = v| x + w| x

(λ v)| x = λ v| x ,

for all v, w ∈ X( M ) and λ ∈ R. We can multiply v by smooth functions as well, by the rule ( F v)| x := F ( x )v| x . • From (2.16) we see that v is regarded as a differential operator which naturally acts on smooth functions on M. Indeed, each smooth vector field v becomes a derivation on the algebra of smooth functions F ( M, R) when we define v[ F ] to be the element of F ( M, R) whose value at a point x ∈ M is the directional derivative (2.17) of F at x in the direction v| x . • Saying that v is a derivation on F ( M, R) means that v[ F G ] = v[ F ] G + F v[ G ]

∀ F, G ∈ F ( M, R),

which is called Leibniz rule. It can be proved any derivation on F ( M, R) arises in this fashion from a uniquely determined smooth vector field v.

I The next Theorem (claim 2.) shows how to use the infinitesimal generator v to find the explicit solution of the associated IVP (2.12). Theorem 2.3 Let Φt be the one-parameter local Lie group of smooth diffeomorphisms (2.10). Let v ∈ X( M ) be its infinitesimal generator. 1. There holds d (Lv F )( x ) = v[ F ( x )] = ( F ◦ Φt ) ( x ), dt t=0

38

2 Continuous Dynamical Systems for any F ∈ F ( M, R). In particular, d (Lv xi )( x ) = v[ xi ] = (Φt ( x ))i , dt t=0

i = 1, . . . , n.

2. There holds ( Lie series):

(Φt ( x ))i = exp (t v) xi :=

∞

tk k v [ x i ], k! k =0

∑

i = 1, . . . , n.

where vk := v vk−1 , v0 being the identity. Proof. We prove both claims. 1. From Theorem 2.2 we know that x (t, x ) = Φt ( x ) is the solution of the IVP (2.12). Hence by the chain rule we have:   d d Φ ( x ) F ◦ Φ ( x ) = grad F ( x ) , ( ) t t x dt t=0 dt t=0 = h gradx F ( x ), f ( x ) i = (Lv F )( x ). In particular, setting F ( x ) = xi we get

(Lv xi )( x ) = =

d v [ xi ] = ( x ◦ Φt ) ( x ) dt t=0 i d (Φt ( x ))i = f i ( x ). dt t =0

2. Expanding component-wise the flow Φt ( x ) in powers of t around t = 0 we get ∞ k t dk (Φt ( x ))i (Φt ( x ))i = ∑ k! dtk k =0 t =0 d t2 d2 = xi + t (Φt ( x ))i + (Φt ( x ))i + . . . dt t=0 2 dt2 t=0 Now we have:

d (Φt ( x ))i = v[ xi ], dt t=0

and

  d2 d 2 ( Φ ( x )) ( Φ ( x )) = v t t i = v [ v [ xi ]] = v [ xi ], i dt t=0 dt2 t=0 and, in general, dk (Φt ( x ))i = vk [ xi ]. dtk t =0

2.3 Autonomous IVPs as continuous dynamical systems

39

Both claims are proved.



I In summary, there are, in principle, two ways to construct explicitly Φt : 1. Integrate the IVP (2.12). 2. Express Φt in terms of its Lie series. Example 2.5 (Linear IVPs) 1. On Rn consider the infinitesimal generator n

v :=

∂

∑ ai ∂xi ,

ai ∈ R.

i =1

Then, for x ∈ Rn , the corresponding flow, x (t, x ) = Φt ( x ) = exp(t v) x =

n

∂ + O ( t2 ) 1 + t ∑ ai ∂x i i =1

! x = x + t a,

with a := ( a1 , . . . , an ), is the flow of the IVP ( x˙ = a, x (0, x ) = Φ0 ( x ) = x. 2. On Rn consider the infinitesimal generator v :=

n

n

i =1

j =1

∑ ∑ Aij x j

!

∂ , ∂xi

Aij ∈ R.

Then, introducing the n × n real matrix A := ( Aij )16i,j6n , the corresponding flow, x (t, x ) = Φt ( x ) = exp(t v) x = et A x, is the flow of the IVP

(

x˙ = A x, x (0, x ) = Φ0 ( x ) = x.

I A consequence of Theorem 2.3 is the following claim. Corollary 2.1 Let F ∈ F ( M, R). Then there holds F (Φt ( x )) = exp(t v) F ( x )

∀ t ∈ I, x ∈ M.

No Proof.

I According to Definition 2.5, an invariant function of Φt is a function F ∈ F ( M, R) such that ∀ t ∈ I, x ∈ M. (2.18) F (Φt ( x )) = F ( x )

40

2 Continuous Dynamical Systems • Condition (2.18) means that F is constant on every orbit of Φt . The hypersurface Sh : = { x ∈ M : F ( x ) = h }, where h ∈ R is a fixed constant (depending on the initial condition), is called level set of F. Therefore, if F is an invariant function under the action of Φt , then clearly every level set of F is an invariant set of Φt . However, it is not true that if the set of zeros of a function, { x ∈ M : F ( x ) = 0}, is an invariant set then the function itself is an invariant function. Nevertheless it can be proved that if every level set of F is an invariant set, then F is an invariant function. • It may happen that a given flow Φt admits more than one invariant function, say m functions F1 , . . . , Fm ∈ F ( M, R). These functions define m distinct invariant functions if they satisfy condition (2.18) and they are functionally independent on M, that is rank ( gradx F1 ( x ), . . . , gradx Fm ( x )) = m

∀ x ∈ M.

• The knowledge of invariant functions provides useful information about solutions of the corresponding IVP. In particular, if the number m of invariant functions is m = n − 1 then the IVP is solvable. Indeed, for any initial point the orbit through that point lies in the intersection of the n − 1 level sets Shi := { x ∈ M : Fi ( x ) = hi },

i = 1, . . . , n − 1,

with hi ∈ R being fixed by the initial condition. The IVP is then reduced to a one-dimensional problem which is solvable by separation of variables. The mechanism behind such solvability is called reduction to quadratures. • In the context of dynamical systems a function satisfying condition (2.18) is called integral of motion or conserved quantity. • The existence of one or more integrals of motion is not guaranteed a priori. Given a dynamical system, both continuous and discrete, it is not easy to find them (if exist). Often, one falls back on either physical intuition and guesswork. A deeper fact, due to E. Noether, is that integrals of motion are the result of underlying symmetry properties of the dynamical system (see Theorem 3.8). Example 2.6 (Invariant sets, invariant functions) 1. On R2 consider the infinitesimal generator ∂ ∂ v := a + , a ∈ R. ∂x1 ∂x2 This generates the flow Φt : (t, ( x1 , x2 )) 7→ ( x1 + a t, x2 + t). • The set

Sh := {( x1 , x2 ) ∈ R2 : F ( x1 , x2 ) := x1 − a x2 = h},

2.3 Autonomous IVPs as continuous dynamical systems

41

where h ∈ R is a constant, is evidently an invariant set of Φt being precisely the orbits of Φt . • The function F ( x1 , x2 ) := x1 − a x2 is an invariant function since F (Φt ( x1 , x2 )) = x1 + t a − a ( x2 + t) = x1 − a x2 = F ( x1 , x2 ), for all t ∈ R and ( x1 , x2 ) ∈ R2 . Indeed, every function F ( x1 − a x2 ) is invariant under Φt . 2. On R2 consider the infinitesimal generator v : = x1 This generates the flow

∂ ∂ + x2 , ∂x1 ∂x2

a ∈ R.


Φt : (t, ( x1 , x2 )) 7→ et x1 , et x2 .

• The set

S := {( x1 , x2 ) ∈ R2 : F ( x1 , x2 ) := x1 x2 = 0}

is evidently an invariant set of Φt , but the function F ( x1 , x2 ) := x1 x2 is not an invariant function. • The function F1 ( x1 , x2 ) :=

x1 x2

is an invariant function defined on R2 \ { x2 = 0}. Another invariant function is F2 ( x1 , x2 ) :=

x1 x2 x12 + x22

defined on R2 \ {(0, 0)}. Note however that F1 and F2 are functionally dependent, rank ( gradx F1 ( x1 , x2 ), gradx F2 ( x1 , x2 )) = 1. Indeed, F2 = F1 /(1 + F12 ).

I Here is an important characterization of integral of motions. Theorem 2.4 A function F ∈ F ( M, R) is an integral of motion of Φt if and only if

(Lv F )( x ) = 0

∀ x ∈ M.

Proof. From Corollary 2.1 we have: F (Φt ( x ))

∞

= exp (t v) F ( x ) = =

tk k ∑ k! v [ F(x)] k =0

F ( x ) + t v[ F ( x )] +

t2 2 v [ F ( x )] + . . . 2

If condition (2.18) holds true then v[ F ( x )] = (Lv F )( x ) = 0

(2.19)

42

2 Continuous Dynamical Systems

for all x ∈ M. Conversely, if (Lv F )( x ) = 0 for all x ∈ M then vk [ F ( x )] = 0 for all k = 2, . . . , ∞, which implies condition (2.18). 

I Condition (2.19) can be explicitly written as n

∂F

∑ fi (x) ∂xi

∀ x ∈ M.

=0

(2.20)

i =1

Therefore F ∈ F ( M, R) is an invariant function of Φt if and only if F is a solution to the homogeneous linear first-order partial differential equation (2.20).

I We give an illustration of the above notions with a couple of examples. Example 2.7 (Rotations in R2 ) Consider the one-parameter Lie group of smooth diffeomorphisms on R2 : Φt :

[0, 2 π ) × R2 (t, ( x1 , x2 ))

→ 7→

R2 ,

( x1 , x2 ) := ( x1 cos t + x2 sin t, − x1 sin t + x2 cos t).

Note that Φt satisfies properties (2.11) and it defines planar rotations of angle t. • The planar IVP which has Φt as flow is easily constructed. We have:  x˙ 1 = f 1 ( x1 , x2 ),    x˙ 2 = f 2 ( x1 , x2 ),    ( x1 (0, ( x1 , x2 )), x2 (0, ( x1 , x2 )) = ( x1 , x2 ), with

( f 1 ( x1 , x2 ), f 2 ( x1 , x2 ))

:=

d ( x cos t + x2 sin t, − x1 sin t + x2 cos t) dt t=0 1

=

( x 2 , − x 1 ).

• The infinitesimal generator of the flow Φt is v : = x2

∂ ∂ − x1 . ∂x1 ∂x2

• The Lie series of Φt reproduces the solution of the original IVP. Note that v0 [ x1 ]

v[v[v[ x1 ]]]

= = = =

v[v[v[v[ x1 ]]]]

=

v [ x1 ] v[v[ x1 ]]

x1 , x2 , v2 [ x1 ] = − x1 , v3 [ x1 ] = − x2 , v4 [ x1 ] = x1 ,

and, in general, v2k+1 [ x1 ] = (−1)k x2 ,

v2k [ x1 ] = (−1)k x1 ,

k ∈ N0 .

(2.21)

2.3 Autonomous IVPs as continuous dynamical systems

43

This leads to

(Φt ( x ))1

∞

=

exp (t v) x1 =

∑

k =0 ∞

=

x1

∑

k =0

=

tk k v [ x1 ] k!

∞ (−1)k t2k (−1)k t2k+1 + x2 ∑ (2k)! (2k + 1)! k =0

x1 cos t + x2 sin t.

Similarly for (Φt ( x ))2 = − x1 sin t + x2 cos t. • An integral of motion F of Φt can be found by solving the linear homogeneous partial differential equation ∂F ∂F (Lv F )( x1 , x2 ) = x2 − x1 = 0, ∂x1 ∂x2 whose general solution (as one may expect from some geometric intuition) is   F ( x1 , x2 ) = F x12 + x22 . • It is evident that F1 ( x1 , x2 ) := x12 + x22 ,

  F2 ( x1 , x2 ) := log x12 + x22 ,

are two functionally dependent integrals of motion.

Example 2.8 (Construction of integrals of motion) Let Φt be a flow acting on R3 with infinitesimal generator  ∂  ∂ ∂ . + x1 + 1 + x32 v : = − x2 ∂x1 ∂x1 ∂x3 Note that this corresponds to the IVP  x˙ 1 = − x2 ,       x˙ 2 = x1 ,   x˙ 3 = 1 + x23 ,     xi (0, ( x1 , x2 , x3 )) = xi , i = 1, 2, 3. • From condition (2.19) we have that F = F ( x1 , x2 , x3 ) is an integral of motion of Φt if and only if   ∂F ∂F ∂F − x2 + x1 + 1 + x32 = 0, ∀ x ∈ R3 . ∂x1 ∂x2 ∂x3 This linear homogeneous partial differential equation has a corresponding characteristic system of ODEs given by dx dx dx3 − 1 = 2 = . x2 x1 1 + x32 • The first of these two ODEs,

−

dx1 dx = 2, x2 x1

is easily solved to give x12 + x22 − c21 = 0,

c1 ∈ R.

Therefore the first invariant function is F1 ( x1 , x2 ) := x12 + x22 .

44

2 Continuous Dynamical Systems • Now set x12 = c21 − x22 and solve the second of the two ODEs, dx2 q

=

c21 − x22

dx3 , 1 + x32

to find

x2 = arctan x3 + c2 , c2 ∈ R. c1 A straightforward manipulation gives a second invariant function: arcsin

F2 ( x1 , x2 , x3 ) :=

x1 x3 − x2 , x2 x3 + x1

which is functionally independent on F1 ( x1 , x2 ).

I Let Φt and Ψs , t, s ∈ I, be two distinct one-parameter local Lie groups of smooth diffeomorphisms on M whose infinitesimal generators are respectively given by n

v :=

∂

n

w :=

f ( x ) :=

d Φ t ( x ), dt t=0

g( x ) :=

(2.22)

i =1

i =1

with

∂

∑ gi (x) ∂xi ,

∑ fi (x) ∂xi ,

d Ψ s ( x ). ds s=0

In general the flows Φt and Ψs do not commute, namely

(Φt ◦ Ψs )( x ) 6= (Ψs ◦ Φt )( x ). Example 2.9 (Rotations and translations in R2 ) Consider the following one-parameter Lie groups of smooth diffeomorphisms on R2 : Φt : (t, ( x1 , x2 )) 7→ ( x1 cos t + x2 sin t, − x1 sin t + x2 cos t), and Ψs : (s, ( x1 , x2 )) 7→ ( x1 + s, x2 ). Note that Φt corresponds to planar rotations of angle t, while Ψs corresponds to translations in the x1 -direction. • The corresponding infinitesimal generators are:

v : = x2

∂ ∂ − x1 , ∂x1 ∂x2

w :=

∂ . ∂x1

• As it is clear from Fig. 2.8 the flows do not commute:

(Φt ◦ Ψs )( x1 , x2 ) 6= (Ψs ◦ Φt )( x1 , x2 ).

2.3 Autonomous IVPs as continuous dynamical systems

45

Fig. 2.8. Non-commutativity of translations and rotations in R2 ([FaMa]).

I We introduce a new object which will give a measure of the lack of commutativity of two flows (see Theorem 2.5). Definition 2.8 The Lie bracket of two vector fields v, w ∈ X( M ), defined in (2.22), is the vector field denoted by [ v, w ] and defined by ! n ∂ fi ∂ ∂gi − gj (x) [ v, w ] := ∑ f j ( x ) ∂x ∂x ∂x j j i i,j=1 n

=

∂

∑ (Lv gi − Lw fi ) (x) ∂xi .

(2.23)

i =1

I Remarks: • It is not difficult to check that the Lie bracket (2.23) satisfies the following properties: 1. (Bi)linearity: [ λ1 v + λ2 w, r ] = λ1 [ v, r ] + λ2 [ w, r ], 2. Skew-symmetry: [ v, w ] = −[ w, v ], 3. Jacobi identity: [ v, [ w, r ] ]+ (v, w, r) = 0. for all v, w, r ∈ X( M ) and λ1 , λ2 ∈ R. Here (v, w, r) means cyclic permutation of (v, w, r).

46

2 Continuous Dynamical Systems • In general, a vector space V equipped with a bilinear mapping [·, ·] : V × V → V satisfying properties 1., 2. and 3. defines a Lie algebra (V, [·, ·]) (see Chapter 4, Subsection 4.2.7, for further details on Lie algebras). Therefore we call (X( M), [·, ·]), with [·, ·] defined by (2.23), the Lie algebra of smooth vector fields. Indeed, (X( M ), [·, ·]) is the Lie algebra of the one-parameter local Lie group Φt .

I Let us give the following useful result. Lemma 2.1 Let v, w ∈ X( M ). Then   L[ v,w ] F ( x ) = ((Lv Lw − Lw Lv ) F ) ( x )

∀ F ∈ F ( M, R).

Proof. Let F ∈ F ( M, R). We have:

((Lv Lw − Lw Lv ) F ) ( x ) = (Lv (Lw F )) ( x ) − (Lw (Lv F )) ( x ) ! !! n ∂F ∂ ∂F ∂ gj (x) − gi ( x ) f j (x) = ∑ fi (x) ∂xi ∂x j ∂xi ∂x j i,j=1 n

= =

∂g j ∂F ∂ f j ∂F ∂2 F ∂2 F ∑ fi (x) gj (x) ∂xi ∂x j + fi (x) ∂xi ∂x j − gi (x) f j (x) ∂xi ∂x j − gi ∂xi ∂x j i,j=1  n  ∂g j ∂ f j ∂F f ( x ) − g ( x ) ∑ i ∂xi i ∂xi ∂x j i,j=1 n

=

∂F

∑ [ v, w ] j ∂x j

!

  = L[ v,w ] F ( x ).

j =1



The Theorem is proved.

I We are now in the position to show that the Lie bracket of two vector fields v, w ∈ X( M ) gives a measure of the degree of non-commutativity of the corresponding flows Φt , Ψs . Theorem 2.5 Let Φt and Ψs , t, s ∈ I, be two distinct one-parameter groups of smooth diffeomorphisms on M whose infinitesimal generators are v, w ∈ X( M ). Then

(Φt ◦ Ψs )( x ) = (Ψs ◦ Φt )( x )

∀ t, s ∈ I, x ∈ M,

2.3 Autonomous IVPs as continuous dynamical systems

47

if and only if

[ v, w ] = 0

∀ x ∈ M.

Proof. We prove only that [ v, w ] = 0 is a necessary condition for the commutativity of the flows Φt and Ψs . • Let F ∈ F ( M, R). Define the smooth function ∆ ≡ ∆(t, s, x ) := F ((Ψs ◦ Φt ) ( x )) − F ((Φt ◦ Ψs ) ( x )) . Note that ∆(0, 0, x ) = 0 and if Φt ◦ Ψs = Ψs ◦ Φt then ∆ ≡ 0 for all t, s ∈ I. • Consider the Taylor expansion of ∆ around (t, s) = (0, 0): ∂∆ ∂∆ +s ∆ = t ∂t (t,s)=(0,0) ∂s (t,s)=(0,0) ! 2 1 2 ∂2 ∆ ∂2 ∆ 2 ∂ ∆ + t +s +2st 2 ∂s∂t (t,s)=(0,0) ∂t2 (t,s)=(0,0) ∂s2 (0,0)

+ O(s2 t, s t2 ) ∂2 ∆ = st + O(s2 t, s t2 ), ∂s∂t (t,s)=(0,0) because ∆(t, 0, x ) = ∆(0, s, x ) = 0. • By Theorem 2.3 (claim 1.) we know that ∂ F (Φt ◦ Ψs ( x )) = (Lv F )(Ψs ( x )) ∂t t =0

so that

∂ ∂ F (Φt ◦ Ψs ( x )) = (Lw (Lv F ))( x ), ∂t s=0 ∂t t=0

and similarly ∂ ∂ F (Ψs ◦ Φt ( x )) = (Lv (Lw F ))( x ). ∂t t=0 ∂s s=0 • Subtracting the last two equations we have:   ∂2 ∆ = L L − L L F ( x ) = L F ( x ), (( ) ) v w w v [ v,w ] ∂s∂t (0,0) where we used Lemma 2.1.

48

2 Continuous Dynamical Systems • Therefore,

  ∆ = s t L[ v,w ] F ( x ) + O(s2 t, s t2 ) = O(s2 t, s t2 )

if [ v, w ] = 0.



The claim is proved. Example 2.10 (Rotations and translations in R2 ) Consider the non-commuting flows on R2 of Example 2.9: Φt : (t, ( x1 , x2 )) 7→ ( x1 cos t + x2 sin t, − x1 sin t + x2 cos t), and

Ψs : (s, ( x1 , x2 )) 7→ ( x1 + s, x2 ).

Their infinitesimal generators are given respectively by v : = x2

∂ ∂ , − x1 ∂x1 ∂x2

w :=

∂ . ∂x1

It is easy to verify that [ v, w ] 6= 0. Take F ∈ F (R2 , R) and compute     ∂F ∂F ∂ ∂ ∂ ∂F − x1 − x2 − x1 [ v, w ][ F ( x )] = x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x1 ∂x2 ∂2 F ∂2 F ∂F ∂2 F ∂2 F − x2 2 + + x1 − x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x12 ∂x1

=

x2

=

∂F , ∂x2

that is

[ v, w ] =

2.3.2

∂ . ∂x2

Evolution of phase space volume

I Let D ⊂ M be a compact set. Then the volume of D is defined by the Riemann integral Z Vol( D ) := dx. D

Here dx := dx1 ∧ · · · ∧ dxn is the volume n-form on M.

I Consider a one-parameter global Lie group of smooth diffeomorphisms: Φt : R × D → D : (t, x ) 7→ x (t, x ) := Φt ( x ), whose infinitesimal generator is n

v :=

∂

∑ fi (x) ∂xi ,

i =1

with

d Φ t ( x ). f ( x ) := dt t=0

2.3 Autonomous IVPs as continuous dynamical systems

49

Then D (t) := Φt ( D ) and the volume of D at time t is Vol( D )(t) :=

Z

dx := dx1 ∧ · · · ∧ dx n .

dx, D (t)

Fig. 2.9. Time evolution of the volume Vol( D )(t) ([MaRaAb]).

Theorem 2.6 (Liouville) If

n

∂f

∑ ∂xi

div x f ( x ) :=

∀ x ∈ M,

=0

i =1

then Φt preserves Vol( D ), that is

∀ t ∈ R.

Vol( D )(t) = Vol( D )

Proof. We proceed by steps. • We have: Z

Vol( D )(t) :=

Z

dx = D (t)

δ(t, x ) dx,

D

where  δ(t, x ) := det

∂Φt ( x ) ∂x



is the determinant of the n × n Jacobian matrix of Φt . Note that δ(0, x ) = 1. • Therefore,

Y

R

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E

M

O

E

)

G

t

(

)

L

A

D

C

I

(

l

R

o

E

V

H

P

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M

O

R

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F

0

(

S

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P

A

D

M

(

l

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E

V

L

B

A

R

G

E

T

N

I

2

Z d d Vol( D )(t) = δ(t, x ) dx. dt t=s D dt t=s

50

2 Continuous Dynamical Systems • Note that d δ(t, x ) dt t=s

= =

    ∂Φs ( x ) d ∂Φt ( x ) det det dt t=s ∂Φs ( x ) ∂x   d ∂Φt ( x ) δ(s, x ), det dt t=s ∂Φs ( x )

where, using Φt+s ( x ) = Φt (Φs ( x )),   ∂Φt ( x ) d = det dt t=s ∂Φs ( x )

= =

  d ∂Φt−s (Φs ( x )) det dt t=s ∂Φs ( x ) d δ(t − s, Φs ( x )) dt t=s d δ(t, Φs ( x )). dt t=0

• Therefore the claim follows if we prove that d δ(t, x ) = 0 dt t=0 for all x ∈ M. • Recalling that Φ t ( x ) = x + t f ( x ) + O ( t2 ), we have ∂Φt ( x ) = 1n + t A + O ( t2 ), ∂x

A := ( Aij )16i,j6n ,

Aij :=

∂ fi . ∂x j

• For any n × n matrix A we have det (1n + t A) = 1 + t Trace A + O(t2 ). • Therefore, δ(t, x ) = 1 + t Trace A + O(t2 ) = 1 + t div x f ( x ) + O(t2 ). • Hence we get d δ(t, x ) = div x f ( x ). dt t=0 The Theorem is proved.

I We now give the following fundamental claim.



2.3 Autonomous IVPs as continuous dynamical systems

51

Theorem 2.7 (Poincar´e) Let D ⊂ M be a compact set. Consider a one-parameter global Lie group of smooth diffeomorphisms Φt : R × D → D which is volume preserving, i.e.,

∀ t ∈ R.

Vol( D )(t) = Vol( D )

Then for each open set U ⊂ D and for each fixed time s > 0 there exists x ∈ U and t > s such that Φt ( x ) ∈ U. Proof. We proceed by steps. • Fix a time s > 0 and consider the set Φs (U ). If Φs (U ) ∩ U 6= ∅ then the claim follows since, by continuity, Φs (U ) intersects U for t > s sufficiently close to s. • Suppose Φs (U ) ∩ U = ∅. Define Un := Φns (U ), n ∈ N0 , U0 = U. Suppose that Un ∩ U = ∅ ∀ n ∈ N. (2.24) This implies Un ∩ Um = ∅

∀ n > m ∈ N.

(2.25)

• If (2.25) is not satisfied for some n > m > 0 then this would imply Un ∩ Um 6= ∅. But this implies Un−1 ∩ Um−1 6= ∅. Indeed, if there exists x ∈ Un ∩ Um and Un−1 ∩ Um−1 = ∅ then one would have x = Φs ( xn ) and x = Φt ( xm ) with xn ∈ Un−1 and xm ∈ Um−1 . But this violates the unicity of solutions. • By iteration of this argument we would end up with Un−m ∩ U 6= ∅, which contradicts our hypothesis (2.24). • Condition (2.24) implies that 



∑

n∈N0

Vol(Un ) = Vol 

[

Un  .

n∈N0

Our map Φt is, by assumption, volume preserving, meaning that Vol(Un ) = Vol(U )

∀n ∈ N.

Therefore we find:  ∞=

∑

n∈N0

Vol(Un ) = Vol 

 [

Un  6 Vol( D ) < ∞,

n∈N0

where the last inequality follows from the compactness of D. We obtained a contradiction.

52

2 Continuous Dynamical Systems



The Theorem is proved.

I Remarks: • Theorem 2.7 shows that any arbitrary neighborhood U (even a single point x) in a compact space, will evolve, under the action of a volume preserving flow, in such a way that U will return arbitrarily close to any point it starts from. The time it takes the map to return depends on the map, and on the size of the neighborhood U. In general, this time may be extremely large.

Fig. 2.10. A Poincar´e recurrence ([Ge]).

• Reformulation of Theorem 2.7: Let D ⊂ M be a compact set. Consider a oneparameter global Lie group of smooth diffeomorphisms Φt : R × D → D which is volume preserving. Then, for any x ∈ D there exists ε > 0 and xe ∈ D such that if k x − xek < ε then k x − Φt ( xe)k < ε for some t > 0. Example 2.11 (Maxwell Gedankenexperiment) Suppose to have a box partitioned into two chambers, say A and B. Suppose that A contains a gas, while B is empty. We make a hole in the wall separating A and B. Then it is reasonable to expect that after some time the gas will be uniformly distributed in A and B.

Fig. 2.11. Maxwell Gedankenexperiment ([Ge]).

2.3 Autonomous IVPs as continuous dynamical systems

53

• We are under the conditions of Theorem 2.7. Indeed, the volume of the box is finite and, assuming elastic and regular interactions between molecules, the conservation of the global energy assures that molecular velocities are limited. Then the phase space is a compact space and Theorem 2.7 says that there exists a time s for which all molecules constituting the gas will come back to a configuration which is close to the initial configuration, that is all molecules in the chamber A! • The resolution of this paradoxical situation lies in the fact that s is longer than the duration of the solar system’s existence (...billions of years). Furthermore, one of the assumption we used in this ideal experiment is that the system is isolated (i.e., no external perturbations are admitted). This assumption is not realistic, especially on long times.

2.3.3

Stability of fixed points and Lyapunov functions

I One of the major tasks of dynamical systems theory is to analyze the long-time behavior of a dynamical system. Of course, one might try to solve this problem by brute force, merely computing many orbits numerically (by simulations). However, the most useful aspect of the theory is that one can predict some features of the phase portrait without actually solving the IVP. The simplest example of such information is the number and the positions of fixed points. I Let Φt be a one-parameter global Lie group of smooth diffeomorphisms, Φt : R × M → M. Let v ∈ X( M ) be the infinitesimal generator of Φt . We know from Definition 2.3 that a point xe ∈ M is a fixed point of the dynamical system {Φt , R, M } if xe = Φt ( xe) for all t ∈ R. Such a point is uniquely defined by the vanishing of the function f in the corresponding IVP, i.e., f ( xe) = 0. I We now adapt Definition 2.6 to the case of a dynamical system {Φt , R, M}. Definition 2.9 Let xe ∈ M be a fixed point of Φt . 1. xe is (Lyapunov) stable if for any ε > 0 there exists δ(ε) > 0 such that whenever k x − xek < δ(ε) then kΦt ( x ) − xek < ε for all t > 0. e such that 2. xe is attracting if there exists a neighborhood of xe, say M, lim kΦt ( x ) − xek = 0

t→+∞

e ∀ x ∈ M.

(2.26)

e where (2.26) holds true is called basin of attraction The maximal open set M of xe. 3. xe is asymptotically stable if it is stable and attracting. Equivalently, if it is stable and there exists δ > 0 such that whenever k x − xek < δ then limt→+∞ Φt ( x ) = xe.

54

2 Continuous Dynamical Systems 4. xe is unstable if it is not stable.

Fig. 2.12. Stable and asymptotically stable fixed point ([Ol1]).

I Remarks: • In Definition 2.9, item 1., we assumed that Φt ( x ) with initial point x ∈ Rn such that k x − xek < δ(ε), is globally defined for t > 0. In fact, Φt ( x ) admits a limited maximal interval of existence Ix only if the solution of the corresponding IVP leaves every compact D ⊂ M in a finite time. Therefore, to require kΦt ( x ) − xek < ε for all t ∈ Ix ∩ R+ automatically implies Ix ∩ R+ = R+ .

x

δ

�x

R

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l

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ε

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(

t

Φ

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ε

(

δ

�x

ε

)

x

(

t

Φ

2

2

• A fixed point xe can be attractive but not asymptotically stable. It is sufficient that xe is unstable.

Fig. 2.13. An attracting fixed point which is unstable ([BuNe]).

• An important consequence of Theorem 2.7 is that if M is compact and Φt is a volume-preserving (global) flow, then Φt cannot possess asymptotically stable fixed points.

2.3 Autonomous IVPs as continuous dynamical systems

55

Example 2.12 (Stability analysis of a scalar IVP) Consider the dynamical system defined by the flow Φt of the scalar IVP
( x˙ = x 1 − x2 , x (0) = x0 .

(2.27)


• The fixed points are given by the solutions of the algebraic equation f ( x ) := x 1 − x2 = 0: xe1 = −1,

xe2 = 0,

xe3 = 1.

• Let x < −1. As x increases, the graph of the function f ( x ) switches from positive to negative at xe1 = −1 which proves its stability. Any solution with x0 < 0 will end up, asymptotically, at xe1 = −1. • Similarly, the graph goes from negative to positive at xe2 = 0 establishing its instability. • The last fixed point xe3 = 1 is stable because f ( x ) again changes from positive to negative there. Any solution with x0 > 0 will end up, asymptotically, at xe3 = 1.

Fig. 2.14. Plot of the function f ( x ) := x (1 − x2 ) and solution curves of (2.27) ([Ol1]). • The only solution that does not end up at ±1 is the unstable stationary solution φ(t) ≡ 0 for all t. Any perturbation of it will force the solutions to choose one of the stable fixed points.

I Definition 2.9 does not provide a practical tool to detect the stability of a fixed point. The most practical way to investigate the stability of fixed points is based on the use of the so called Lyapunov functions. The direct method of Lyapunov functions finds its motivation in a physical observation. Systems which have damping, viscosity and/or frictional effects do not typically possess integrals of motion. From a physical standpoint, these effects imply that the total energy of the system is a decreasing function of time. In other words, asymptotically, the system returns to a (stable) configuration, where its energy has a minimum and the extra energy has been dissipated away. I Here is the mathematical definition of a Lyapunov function. Definition 2.10 e be a neighborhood of xe. A function Let xe ∈ M be a fixed point of Φt and M

56

2 Continuous Dynamical Systems e R) is called Lyapunov function of Φt if the following conditions hold: F ∈ C1 ( M, e \ { xe}. 1. F ( xe) = 0 and F ( x ) > 0 for all x ∈ M e 2. (Lv F )( x ) 6 0 for all x ∈ M. e \ { xe} then F is called strict Lyapunov function. If (Lv F )( x ) < 0 for all x ∈ M

I The following Theorem gives sufficient conditions for stability and asymptotic stability of fixed points of Φt . It is customary to call this way of detecting stability Lyapunov direct method. Theorem 2.8 (Lyapunov) e be a neighborhood of xe. Let xe ∈ M be a fixed point of Φt and M e R) then xe is stable. 1. If there exists a Lyapunov function F ∈ C1 ( M, e R) then xe is asymptot2. If there exists a strict Lyapunov function F ∈ C1 ( M, ically stable.

Proof. We prove only the first claim. eε ⊂ M e a neighborhood of radius ε > 0 and center xe. Define • Let M α(ε) := min F ( x ), eε x ∈∂ M

e \ { xe}. so that α(ε) > 0 because F ( x ) > 0 for all x ∈ M • Define the set  U :=

 e ε : F ( x ) < 1 α(ε) , x∈M 2

e R). Therefore there exists a which is open due to the fact that F ∈ C1 ( M, e neighborhood Mδ ⊂ U with δ < ε. Note that the open set U is not necessarily connected, but there exists always a connected component which contains xe e δ. and M

p

S

e

m

e

r

r

e

a

e

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