Random Differential Equations in Scientific Computing 9788376560267, 9788376560250

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mathematics seminary-part i t

lecture

exercises

informatics seminary-part

workshop

teamworkk of mathe ematics & in nformatics

all stu udents have e to deal wiith all topicss (and d not just the eir own sem minary talk)

dynamical systems for deterministic & randomly perturbed (ordinary) differential equations

algorithms of scientific computing

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Space-Discretization (Finite Differences)

Path-Wise Solution Concept for R(O)DEs

Partial Differential Equation with Stochastic Effects (RPDE or SPDE)

Finite-Dimensional System of R(O)DE

Finite-Dimensional System of an Infinite Family of ODEs

Decrease Mesh-Size

Compatibility Conditions 1) all solutions of the ODE family are defined on a common time interval 2) all solutions are stochastic processes

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3NAH@R 1HFNHO  %KNQH@M 5XSS Problem Formulation/ Reduction & Motivation

§ 3 & § 4 RODEs

§ 5 Additional Examples

§ 2 RPDEs

Background Materials & Review § 1 Stochastic Processes Part II: Path-Wise ODEs § 6 ODE Theory § 7 ODE Numerics § 8 Dynamical Systems

Theory & Simulation of Random (Ordinary) Differential Equations

Part III: Fourier & Co. § 9 Fourier Transform

§ 12 Linear RODEs I

§ 10 Noise Spectra § 11 Space Filling Curves

§ 13 Linear RODEs II

§ 14 Simulation of RODEs

§ 15 Stability of RODEs

Holistic Theory: § 16 Random Dynamical Systems

The Workshop Projects § 17 The Workshop Idea § 18 The Workshop Project

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exp(0.8t) a = 0.8, b = 0.4 a = 0.8, b = 0.25 a = 0.8, b = 0.1 68,2% conf. int. for b = 0.25

0.0

1.0

0.5

1.5

1.0

exp(−0.8t) a = − 0.8, b = 0.4 a = − 0.8, b = 0.25 a = − 0.8, b = 0.1 68.2% conf. int. for b = 0.25

0.2

0.4

0.6

0.8

1.0

3.0

Time [t]

0.0

0.6

1.0

A

2.5

exp(0.1t) a = 0.1, b = 4.4 a = 0.1, b = 2.25 a = 0.1, b = 1.1 68.2% conf. int. for b = 2.25

1.5 1.0 0.5 0.0

0.0

0.5

1.0

1.5

0.8

Time [t]

2.0

2.5

0.4

@ exp(0.8t) a = 0.8, b = 4.4 a = 0.8, b = 2.25 a = 0.8, b = 1.1 68.2% conf. int. for b = 2.25

2.0

0.2

3.0

0.0

0

1

2 Time [t]

3

4

0

1

B

2 Time [t]

3

4

C

%HFTQD   /@SGR NE SGD &DNLDSQHB !QNVMH@M ,NSHNM ENQ a = 0.8 X0 = 1 @ a = −0.8 X0 = 2 A a = 0.8 X0 = 1 B @MC a = 0.1 X0 = 1 (M @ @MC A SGD U@KTDR NE b @QD SGD R@LD b = 0.4, 0.25, 0.1 @MC SGD 68.2 BNMƥCDMBD HMSDQU@K ENQ SGD b = 0.25 O@SGR HR RGNVM @R VDKK @R SGD DWODBS@SHNM U@KTD X0 DWO(at) (M B @MC C SGD U@KTDR NE b @QD SGD R@LD b = 4.4, 2.25, 1.1 @MC SGD 68.2 BNMƥCDMBD HMSDQU@K ENQ SGD b = 2.25 O@SGR HR RGNVM @R VDKK @R SGD DWODBS@SHNM U@KTD X0 DWO(at)

 4MCDQ VG@S BNMCHSHNMR @QD SVN RSNBG@RSHB OQNBDRRDR HMCHRSHMFTHRG@AKD  6G@S @QD SGD BG@Q@BSDQHRSHBR NE &@TRRH@M OQNBDRRDR  'NV B@M VD TSHKHYD , 3+ ! SN RHLTK@SD Q@MCNL U@QH@AKDR RSNBG@RSHB OQN BDRRDR @MC SGDHQ OQNODQSHDR @R VDKK @R SGD ENKKNVHMF JDX BNMBDOSR  σ @KFDAQ@R OQNA@AHKHSX LD@RTQDR @MC OQNA@AHKHSX RO@BDR  1@MCNL U@QH@AKDR @R VDKK @R SGDHQ CDMRHSX CHRSQHATSHNM @MC LNLDMS ETMB SHNMR

6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS  (MCDODMCDMBD @MC BNMCHSHNM@K DWODBS@SHNM  "NMUDQFDMBD HM CHRSQHATSHNM HM OQNA@AHKHSX HM SGD r SG LD@M @KLNRS RTQD BNMUDQFDMBD @MC RTQD BNMUDQFDMBD  2SNBG@RSHB OQNBDRRDR SGDHQ BNMSHMTHSX @MC HMCHRSHMFTHRG@AHKHSX  *NKLNFNQNVŗR ETMC@LDMS@K @MC BNMSHMTHSX SGDNQX  ,@QSHMF@KDR RTODQ L@QSHMF@KDR @MC ƥKSQ@SHNMR @MC  &@TRRH@M OQNBDRRDR 3GHR BG@OSDQ HR RSQTBSTQDC @R ENKKNVR (M 2DB   VD RS@QS VHSG SGD ETM C@LDMS@K BNMBDOSR NE Q@MCNL U@QH@AKDR FDMDQ@SDC σ @KFDAQ@R @MC CDMRHSX ETMBSHNMR -DWS 2DBSHNM   CHRBTRRDR LNLDMSR NE Q@MCNL U@QH@AKDR KHJD SGD DWODBS@SHNM U@KTD @MC U@QH@MBD @R VDKK @R HMSDFQ@SHNM VHSG QDRODBS SN OQNA@ AHKHSX LD@RTQDR (M 2DB   SGD DRRDMSH@K BNMBDOSR NE HMCDODMCDMBD NE Q@M CNL U@QH@AKDR @MC BNMCHSHNM@K OQNA@AHKHSHDR @MC BNMCHSHNM@K DWODBS@SHNM @QD RSTCHDC (M O@QSHBTK@Q HS HR GDQD SG@S VD FHUD SGD U@QHNTR CDƥMHSHNMR NE BNMUDQ FDMBD NE Q@MCNL U@QH@AKDR ,NQDNUDQ HM 2DB   VD FHUD SGD A@RHB CDƥMH SHNMR @MC BNMBDOSR NE BNMSHMTNTR RSNBG@RSHB OQNBDRRDR SNFDSGDQ VHSG @ AQHDE CHRBTRRHNM NE &@TRRH@M OQNBDRRDR %HM@KKX 2DBSHNM   VQ@OR TO SGD BNMSDMSR NE SGHR BG@OSDQ 3UHUHTXLVLWHV 2NLD OQD JMNVKDCFD NM OQNA@AHKHSX SGDNQX @MC RSNBG@RSHB OQNBDRRDR @QD GDKOETK 7HDFKLQJ 5HPDUNV 3GNTFG K@ADKDC BG@OSDQ  VD BDQS@HMKX CN MNS RTFFDRS SN RS@QS @ BNTQRD ENQ ADFHMMHMF FQ@CT@SD RSTCDMSR VHSG SGD A@RHBR OQDRDMSDC HM SGHR BG@OSDQ @R HSR BNMSDMSR @QD GD@UHKX KN@CDC VHSG SDBGMHB@K CDƥMHSHNMR SG@S @QD MNS UDQX LNSHU@SHMF ENQ SGD RSTCDMS HMSDQDRSDC HM @OOKHB@SHNMR (M UHDV NE NTQ SNO CNVM @OOQN@BG VD @RRTLD SGD BNMBDOSR NE SGHR BG@OSDQ @R OQD QDPTHRHSDR SN AD BNMRHCDQDC HM @ KDBSTQD @ESDQ BG@OSDQR  NQ  VGDM QDPTHQDC %NQ @ KDBSTQD BK@RR HS RDDLR SN AD LNRS @OOQNOQH@SD SN FHUD SGHR BG@OSDQ @R @ GNLDVNQJ @MC CHRBTRR RNLD QDKDU@MS DWDQBHRDR SNFDSGDQ HM SGD BK@RRQNNL

  1@MCNL 5@QH@AKDR &DMDQ@SDC σ KFDAQ@R @MC #DMRHSX %TMBSHNMR 6HSG QDRODBS SN SGD JDX DKDLDMSR @MC MNS@SHNMR NE OQNA@AHKHSX SGDNQX VD RS@QS VHSG SGD HMSQNCTBSHNM NE Q@MCNL U@QH@AKDR @MC DRODBH@KKX SGD σ @KFDAQ@R SGDX FDMDQ@SD -DWS CDMRHSX @MC CHRSQHATSHNM ETMBSHNMR VHKK AD CHRBTRRDC ENKKNVDC AX SGD CDƥMHSHNM NE BDMSQ@K LNLDMSR @MC LNLDMS FDMDQ@SHMF ETMBSHNMR %H M@KKX VD CDƥMD VG@S VD LD@M AX HMSDFQ@SHNM VHSG QDRODBS SN @ OQNA@AHKHSX LD@RTQD @MC FHUD RNLD TRDETK HMDPT@KHSHDR



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF

   "NMSHMTHSX ,D@RTQDR @MC /QNA@AHKHSX 2O@BDR +DS TR ƥQRS QDB@KK SGD CDƥMHSHNM NE '±KCDQ @MC +HORBGHSY BNMSHMTHSX @R VDKK @R SG@S NE C k,α ETMBSHNMR BE ENQ HMRS@MBD :< O  #DƥMHSHNM   '±KCDQ @MC +HORBGHSY "NMSHMTHSX C k,α ETMBSHNMR +DS (X,  · X ) (Y,  · Y ) AD MNQLDC RO@BDR @MC 0 < α ≤ 1 ETMBSHNM f : X → Y HR B@KKDC JOREDOO\ +·OGHU FRQWLQXRXV NE NQCDQ α HE SGDQD HR @ ONRHSHUD BNMRS@MS C RTBG SG@S f (x) − f (y)Y ≤ Cx − yαX ∀ x, y ∈ X .   f HR B@KKDC ORFDOO\ +·OGHU FRQWLQXRXV NE NQCDQ α HE HS R@SHRƥDR SGD BNMCHSHNM   NM DUDQX ANTMCDC RTARDS NE X f HR B@KKDC JOREDOO\ RU ORFDOO\ /LSVFKLW] FRQWLQXRXV HE HS HR FKNA@KKX NQ KNB@KKX '±KCDQ BNMSHMTNTR NE NQCDQ α = 1 f HR B@KKDC @ C k,α ETMBSHNM HE HS HR k SHLDR BNMSHMTNTRKX CHƤDQDMSH@AKD @MC SGD k SG CDQHU@SHUDR @QD KNB@KKX '±KCDQ BNMSHMTNTR NE NQCDQ α ENQ RNLD k ∈ N 3GD BDMSQ@K OQNAKDL HM LD@RTQD SGDNQX HR SN ƥMC @ LD@RTQD UNKTLD ENQ @R L@MX DKDLDMSR NE SGD ONVDQ RDS P(Rd ) @R ONRRHAKD RTBG SG@S SGHR LD@ RTQD UNKTLD HR @CCHSHUD SQ@MRK@SHNM HMU@QH@MS @MC MNQL@KHYDC R SGDQD HR MN RNKTSHNM SN CDƥMD @ LD@RTQD UNKTLD ENQ @KK DKDLDMSR NE P(Rd ) VD G@UD SN QDRSQHBS NTQRDKUDR SN RODBH@K RTA RDS RXRSDLR #DƥMHSHNM   σ KFDAQ@ +DS Ω AD @ MNMDLOSX RDS P(Ω) HR B@KKDC σDOJHEUD HE

BNKKDBSHNM NE RDSR A ⊂

Ş A HR @ DOJHEUD H D  Ŕ Ω ∈ A Ŕ A ∈ A ⇒ Ac ∈ A @MC A, B ∈ A ⇒ A ∪ B ∈ A Ş ∀n ∈ N : An ∈ A ⇒ ∪n∈N An ∈ A 3QHUH@K DW@LOKDR ENQ σ @KFDAQ@R @QD A = {∅, Ω} @MC A = P(Ω) LNQDNUDQ ENQ @MX A ⊂ Ω SGD σ @KFDAQ@ OQNODQSHDR NE A = {∅, A, Ac , Ω} @QD D@RHKX UDQHƥDC (M O@QSHBTK@Q HE E HR @ BNKKDBSHNM NE RTARDSR NE Ω SGDM SGD RL@KKDRS σ @KFDAQ@ FDMDQ@SDC AX E @MC CDMNSDC AX σ(E) HR CDƥMDC @R  σ(E) := {A : E ⊂ A @MC A HR @ σ @KFDAQ@ NM Ω} . %NQ HMRS@MBD SGD RL@KKDRS σ @KFDAQ@ BNMS@HMHMF @KK NODM RTARDSR NE Rd HR B@KKDC SGD %RUHO σDOJHEUD CDMNSDC AX B d NQ RHLOKX AX B HE SGD CHLDMRHNM d QDPTHQDR MN RODBHƥB LDMSHNMHMF +DS Ω AD @ MNMDLOSX RDS @MC E ⊂ P(Ω) 3GD RDS RXRSDL E HR B@KKDC LQWHUVHFWLRQVWDEOH HE ∀ E1 , E2 ∈ E ⇒ E1 ∩ E2 ∈ E . 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS .AUHNTRKX DUDQX σ @KFDAQ@ HR HMSDQRDBSHNM RS@AKD +DS Ω AD @ MNMDLOSX RDS @MC A AD @ σ @KFDAQ@ NM Ω 3GD O@HQ (Ω, A) HR B@KKDC PHDVXUDEOH VSDFH @MC SGD DKDLDMSR NE A @QD B@KKDC PHDVXUDEOH VHWV #DƥMHSHNM   ,D@RTQ@AKD %TMBSHNM +DS (A, A) @MC (B, B) AD LD@RTQ@AKD RO@BDR ETMBSHNM f : A → B HR B@KKDC ABPHDVXUDEOH HE f −1 (B) ⊂ A %NQ HMRS@MBD DUDQX BNMSHMTNTR ETMBSHNM f : X → Y ADSVDDM SVN LDSQHB NQ SNONKNFHB@K RO@BDR X @MC Y HR LD@RTQ@AKD #DƥMHSHNM   ,D@RTQD @MC /QNA@AHKHSX ,D@RTQD +DS Ω AD @ MNMDLOSX RDS @MC A AD @ σ @KFDAQ@ NM Ω 3GDM @ RDS ETMBSHNM μ NM A HR B@KKDC @ PHDVXUH HE Ş μ(A) ∈ [0, ∞] ENQ @KK A ∈ A Ş μ(∅) = 0 Ş μ HR σ @CCHSHUD H D  ENQ @MX CHRINHMS BNKKDBSHNM NE RDSR A1 , A2 , · · · ∈ A VHSG ∪n∈N An ∈ A HS GNKCR SG@S   ∞   An = μ(An ) . μ n∈N

n=1

,NQDNUDQ @ LD@RTQD μ HR B@KKDC @ SUREDELOLW\ PHDVXUH HE HS @CCHSHNM@KKX R@SHR ƥDR Ş μ(Ω) = 1 LD@RTQD μ NM @ LD@RTQ@AKD RO@BD (Ω, F) HR B@KKDC σƲQLWH HE SGDQD DWHRS E1 , E2 , · · · ∈ F  O@HQVHRD CHRINHMS R S Ω = ∪n∈N En @MC μ(En ) < ∞ ENQ @KK n ∈ N ,NQDNUDQ ENQ SVN LD@RTQDR μ ν NM @ LD@RTQ@AKD RO@BD (Ω, F) SGD LD@RTQD ν HR B@KKDC DEVROXWHO\ FRQWLQXRXV VHSG QDRODBS SN μ HE DUDQX μ MTKKRDS HR @ ν MTKKRDS 3GD MNS@SHNM ENQ SGHR OQNODQSX HR ν  μ (E μ HR @ LD@RTQD NM SGD σ @KFDAQ@ A NE @ LD@RTQ@AKD RO@BD (Ω, A) SGDM SGD SQHOKDS (Ω, A, μ) HR B@KKDC PHDVXUHVSDFH (M O@QSHBTK@Q #DƥMHSHNM   /QNA@AHKHSX 2O@BD +DS Ω AD @ MNMDLOSX RDS @MC A AD @ σ @KFDAQ@ NM Ω 3GD SQHOKDS (Ω, A, P) HR B@KKDC SUREDELOLW\ VSDFH HE P HR @ OQNA@ AHKHSX LD@RTQD NM SGD LD@RTQ@AKD RO@BD (Ω, A) +DS (Ω, A, P) AD @ OQNA@AHKHSX RO@BD ONHMSR ω ∈ Ω @QD TRT@KKX @CCQDRRDC @R VDPSOH SRLQWV @MC @ RDS A ∈ A HR B@KKDC HYHQW GDQDAX P(A) CDMNSDR SGD SUREDELOLW\ NE SGD DUDMS A 



(M @M DWSDMCDC BNTQRD NM LD@RTQD SGDNQX SGD OQNODQSX NE RDS RXRSDLR SN AD HMSDQRDBSHNM RS@AKD LNSHU@SDR SGD CHRBTRRHNM NE #XMJHM RXRSDLR RDD D F  :< OO 

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF B := {ω : X(ω) = y}

X(ω1)

Ω X(ω2)

ω1 A := {ω : X(ω) = x}

ω2

R y

x X(ω) = x for an element ω of A

P(A) = p = P({ω : X(ω) = x})

0

p

1

%HFTQD   1DK@SHNM ADSVDDM @ Q@MCNL U@QH@AKD X @MC HSR OQNA@AHKHSX ETMBSHNM ENK KNVHMF :< O 

OQNODQSX VGHBG HR SQTD DWBDOS ENQ @M DUDMS NE OQNA@AHKHSX YDQN HR R@HC SN GNKC DOPRVW VXUHO\ @AAQDUH@SDC Ś@ R Ś NQ DOPRVW HYHU\ZKHUH @AAQDUH@SDC Ś@ D Ś

   1@MCNL 5@QH@AKDR @MC SGD σ KFDAQ@R 3GDX &DMDQ@SD 3UREDELOLWLHV @QD @ RDS ETMBSHNMR SG@S @RRHFM @ MTLADQ ADSVDDM 0 @MC 1 SN @ RDS NE ONHMSR NE SGD R@LOKD RO@BD Ω BE :< OO  3GDHQ CNL@HM HR SGD VHW RI HYHQWV NE @ Q@MCNL DWODQHLDMS @MC SGDHQ Q@MFD HR BNMS@HMDC HM SGD HMSDQ U@K [0, 1] Q@MCNL U@QH@AKD HR @KRN @ ETMBSHNM VGNRD Q@MFD HR @ RDS NE QD@K MTLADQR ATS VGNRD CNL@HM HR SGD RDS NE R@LOKD ONHMSR ω ∈ Ω L@JHMF TO SGD VGNKD R@LOKD RO@BD Ω MNS RTARDSR NE Ω RDD %HF   #DƥMHSHNM   1@MCNL 5@QH@AKD +DS (Ω, A, P) AD @ OQNA@AHKHSX RO@BD 3GDM @ ETMBSHNM X : Ω → Rd HR B@KKDC UDQGRP YDULDEOH HE ENQ D@BG !NQDK RDS B ∈ B ⊂ Rd X −1 (B) = {ω ∈ Ω : X(ω) ∈ B} ∈ A . ( D  @ Q@MCNL U@QH@AKD HR @ Rd U@KTDC A LD@RTQ@AKD ETMBSHNM NM @ OQNA@AHKHSX RO@BD (Ω, A, P) 6D TRT@KKX VQHSD X @MC MNS X(ω) 3GHR ENKKNVR SGD BTRSNL VHSGHM OQNA@ AHKHSX SGDNQX NE LNRSKX MNS CHROK@XHMF SGD CDODMCDMBD NE Q@MCNL U@QH@AKDR NM SGD R@LOKD ONHMS ω ∈ Ω 6D @KRN CDMNSD P(X −1 (B)) @R P(X ∈ B) SGD OQNA@AHKHSX SG@S X HR HM B ∈ B $W@LOKD   (MCHB@SNQ @MC 2HLOKD %TMBSHNMR @QD 1@MCNL 5@QH@AKDR +DS A ∈ A 3GDM SGD LQGLFDWRU IXQFWLRQ NE A 1 HE ω ∈ A IA (ω) := 0 HE ω ∈ /A 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS HR @ Q@MCNL U@QH@AKD n ,NQD FDMDQ@KKX HE A1 , A2 , . . . , An ∈ A @QD CHRINHMS RDSR RTBG SG@S Ω = ∪˙ i=1 Ai @MC a1 , a2 , . . . , am ∈ R SGDM X =

n 

ai I A i

i=1

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n  XCP := ai P(Ai ) . Ω

i=1

 (E MDWS X HR @ MNM MDF@SHUD Q@MCNL U@QH@AKD VD CDƥMD



XCP := RTO Y CP . Ω

Y ≤X, Y RHLOKD Ω

 %HM@KKX HE X : Ω → R HR @ Q@MCNL U@QH@AKD VD VQHSD



+ XCP := X CP − X − CP , Ω

Ω

Ω

OQNUHCDC @S KD@RS NMD NE SGD HMSDFQ@KR NM SGD QHFGS G@MC RHCD HR ƥMHSD 'DQD VD TRDC SGD ONRHSHUD O@QS X + := L@W(X, 0) @MC SGD MDF@SHUD O@QS X − := LHM(X, 0) NE X RN SG@S VD G@UD X = X + − X −



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF -DWS RTOONRD X = (X (1) , X (2) , . . . , X (d) )T : Ω → Rd HR @ UDBSNQ U@KTDC Q@M CNL U@QH@AKD 3GDM 





(1) (2) (d) XCP := X CP, X CP, . . . , X CP . Ω

Ω

Ω

Ω

6D VHKK @RRTLD SGD TRT@K QTKDR ENQ SGDRD HMSDFQ@KR SN GNKC ,NQDNUDQ VD B@M VQHSD SGD DWODBSDC U@KTD E(X) @MC SGD U@QH@MBD Var(X) NE @ UDBSNQ U@KTDC Q@MCNL U@QH@AKD X @R



E(X) = XCP @MC Var(X) = |X − E(X)|2 CP , Ω

Ω

VGDQD | · | CDMNSDR SGD $TBKHCD@M MNQL .ARDQUD HM O@QSHBTK@Q SG@S Var(X) = E(|X − E(X)|2 ) = E(|X|2 ) − |E(X)|2 . 3GD FNNC SGHMF @ANTS CDƥMHMF LD@M @MC U@QH@MBD VHSG QDRODBS SN OQNA@ AHKHSX LD@RTQDR HR SG@S HS @KKNVR TR SN TRD SGD R@LD RXLANKR @MC ENQLTK@R ENQ ANSG BNMSHMTNTR @MC CHRBQDSD Q@MCNL U@QH@AKDR .MD L@X MNSD SG@S VHSG QD RODBS SN CHRBQDSD Q@MCNL U@QH@AKDR @MC SGDHQ CHRBQDSD OQNA@AHKHSX LD@RTQDR BNTMSHMF LD@RTQDR SGD @ANUD HMSDFQ@KR ADBNLD RTLR +DS X ∼ N (0, 1) 4SHKHYHMF , 3+ ! VD DRSHL@SD SGD LD@M @MC U@QH@MBD NE X 2 t 4 `M/MUR-RyyyyVc v 4 tXkc K 4 K2MUvVc p 4 p`UvVc K 4 yXNNdek p 4 kXyRky

  (MCDODMCDMBD @MC "NMCHSHNM@K $WODBS@SHNM +DS (Ω, A, P) AD @ OQNA@AHKHSX RO@BD @MC A, B ∈ A AD SVN DUDMSR VHSG P(B) > 0 6D V@MS SN ƥMC @ QD@RNM@AKD CDƥMHSHNM NE P(A|B) = ŚSGD OQNA@AHKHSX NE A FHUDM BŚ 2TOONRD RNLD ONHMS ω ∈ Ω HR RDKDBSDC Ś@S Q@MCNLŚ @MC VD @QD SNKC ω ∈ B 6G@S SGDM HR SGD OQNA@AHKHSX SG@S ω ∈ A @KRN 2HMBD VD JMNV ω ∈ B VD B@M QDF@QC B @R ADHMF @ MDV OQNA@AHKHSX RO@BD ˜ := B DMCNVDC VHSG SGD SQ@BD σ @KFDAQ@ A˜ := {C ∩ B : 3GDQDENQD VD CDƥMD Ω ˜ := P  RN SG@S P( ˜ Ω) ˜ = 1 3GDM SGD OQNA@AHKHSX C ∈ A} @MC SGD LD@RTQD P P(B) SG@S ω KHDR HM A HR P(A ∩ B) ˜ . P(A|B) = P(A) = P(B) 3GHR LNSHU@SDR SGD ENKKNVHMF CDƥMHSHNM 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   0 &HUD SGD CDƥMHSHNMR NE SGD DWODBSDC U@KTD E(X) @MC SGD U@QH@MBD Var NE @ QD@K U@KTDC Q@MCNL U@QH@AKD X 6G@S CN SGDRD SVN BNMBDOSR HKKTRSQ@SD 0 +DS SGD CDMRHSX ETMBSHNM f (x) NE @ Q@MCNL U@QH@AKD x AD FHUDM @R f (x) := RHM(x) x ∈ [0, 12 π] "NLOTSD SGD DWODBSDC U@KTD @MC U@QH@MBD NE x 0 +DS SGD CDMRHSX ETMBSHNM f (x) NE @ Q@MCNL U@QH@AKD x AD FHUDM @R f (x) := 6x − 6x2  x ∈ [0, 1] "NLOTSD SGD DWODBSDC U@KTD @MC U@QH@MBD NE x 0 6GX @QD LNLDMS FDMDQ@SHMF ETMBSHNMR TRDETK (KKTRSQ@SD XNTQ @MRVDQ VHSG SGD DW@LOKD NE MNQL@KKX CHRSQHATSDC Q@MCNL U@QH@AKDR 0 6G@S CNDR "GDAXRDUŗR HMDPT@KHSX RS@SD 0 +DS (Ω, A, P) AD @ OQNA@AHKHSXRO@BD @MC X AD @ QD@K U@KTDC Q@MCNL U@QH @AKD 6G@S CNDR SGD RXLANK Ω XCP LD@M 'NV CNDR SGHR QDK@SD SN E(X) @MC Var #DƥMHSHNM   "NMCHSHNM@K /QNA@AHKHSX +DS (Ω, A, P) AD @ OQNA@AHKHSX RO@BD @MC A, B ∈ A AD SVN DUDMSR VHSG P(B) > 0 3GDM SGD FRQGLWLRQDO SUREDELOLW\ P(A|B) NE A FHUDM B HR CDƥMDC @R P(A|B) :=

P(A ∩ B) . P(B)

-NV VG@S RGNTKC HS LD@M SN R@X ŚA @MC B @QD HMCDODMCDMSŚ 3GHR RGNTKC LD@M P(A|B) = P(A) RHMBD OQDRTL@AKX @MX HMENQL@SHNM SG@S SGD DUDMS B NBBTQQDC HR HQQDKDU@MS HM CDSDQLHMHMF SGD OQNA@AHKHSX SG@S A G@R NBBTQQDC 3GTR P(A) = P(A|B) =

P(A ∩ B) ⇒ P(A ∩ B) = P(A) · P(B) , P(B)

HE P(B) > 0 6D S@JD SGHR ENQ SGD CDƥMHSHNM  DUDM HE P(B) = 0 #DƥMHSHNM   3VN (MCDODMCDMS $UDMSR +DS (Ω, A, P) AD @ OQNA@AHKHSX RO@BD 3VN DUDMSR A @MC B @QD B@KKDC LQGHSHQGHQW HE P(A ∩ B) = P(A) · P(B) . 



5HRT@KKX HMCDODMCDMBD NE SVN DUDMSR A @MC B LD@MR SG@S SGD Q@SHN P(A) SN P(Ω) = 1 HR SGD R@LD @R SGD Q@SHN P(A ∩ B) SN P(B) NQ LNQD RKNOOX SGD A SN Ω HR SGD R@LD @R SGD O@QS NE A HM B SN B

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF

A

A B

B Ω = [0, 1]2

%HFTQD   M DW@LOKD VHSG ŚFDNLDSQHB OQNA@AHKHSHDRŚ λ(Ω) = 1 λ(A) = λ(B) = 0.25 R S  λ(A ∩ B) = 0.252 3GTR A @MC B @QD HMCDODMCDMS ,NQDNUDQ λ(Ac ) = λ(B c ) = 1 − 0.2.5 = 0.75 λ(Ac ∩ B) = λ(B \ (A ∩ B)) = 0.25 − 0.252 = 0.1875 @MC λ(Ac )λ(B) = 0.75 · 0.25 = 0.1875 H D  Ac @MC B @QD HMCDODMCDMS

3GHR BNMBDOS @MC HSR Q@LHƥB@SHNMR @QD SGD G@KKL@QJR NE OQNA@AHKHSX SGDNQX (S HR D@RX SN BGDBJ SG@S HE A @MC B @QD HMCDODMCDMS SGDM RN @QD Ac @MC B NQ KHJDVHRD Ac @MC B c  RDD %HF   D F 3VN DUDMSR A @MC B @QD B@KKDC PXWXDOO\ H[FOXVLYH HE A ∩ B = ∅ .MD B@M RGNV SG@S SVN DUDMSR B@MMNS AD ANSG HMCDODMCDMS @MC LTST@KKX DWBKTRHUD TMKDRR NMD NE SGD SVN HR @ MTKK RDS 3GD ENKKNVHMF CDƥMHSHNM DWSDMCR SGD BNMBDOS NE HMCDODMCDMBD ƥQRS SN @M @QAHSQ@QX MTLADQ NE DUDMSR TMCDQ BNMRHCDQ@SHNM SGDM SN σ @KFDAQ@R @MC K@RS SN Q@MCNL U@QH@AKDR #DƥMHSHNM   (MCDODMCDMBD NE $UDMSR σ KFDAQ@R @MC 1@MCNL 5@QH @AKDR +DS (Ω, A, P) AD @ OQNA@AHKHSX RO@BD Ş +DS A1 , A2 , . . . AD DUDMSR HM (Ω, A, P) 3GDRD DUDMSR @QD LQGHSHQGHQW HE ENQ @KK BGNHBDR NE 1 ≤ k1 < k2 < · · · < kn  HS GNKCR SG@S P(Ak1 ∩ Ak2 ∩ · · · ∩ Akn ) = P(Ak1 ) · P(Ak2 ) · · · · · P(Akn ) . Ş +DS Ai ⊂ A i = 1, 2, . . .  AD σ @KFDAQ@R 3GD {Ai }∞ i=1 @QD LQGHSHQGHQW HE ENQ @KK BGNHBDR NE 1 ≤ k1 < k2 < · · · < kn @MC NE DUDMSR Ai ∈ Ai  HS GNKCR SG@S P(Ak1 ∩ Ak2 ∩ · · · ∩ Akn ) = P(Ak1 ) · P(Ak2 ) · · · · · P(Akn ) . Ş +DS Xi : Ω → Rd  i = 1, 2, . . .  AD Q@MCNL U@QH@AKDR 3GD Q@MCNL U@QH@AKDR X1 , X2 , . . . @QD LQGHSHQGHQW HE ENQ @KK HMSDFDQR k ≥ 2 @MC @KK BGNHBDR NE !NQDK RDSR B1 , B2 , . . . , Bk ⊂ Rd  HS GNKCR SG@S P(X1 ∈ B1 , X2 ∈ B2 , . . . , Xk ∈ Bk ) = P(X1 ∈ B1 )·P(X2 ∈ B2 )·...·P(Xk ∈ Bk ) . 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS 3GHR HR DPTHU@KDMS SN R@XHMF SG@S SGD σ @KFDAQ@R {A(Xi )}∞ i=1 FDMDQ@SDC AX SGD Xi @QD HMCDODMCDMS -DWS VD FHUD SGD BNMMDBSHNM SN CHRSQHATSHNM ETMBSHNMR @MC CDMRHSHDR NE HM CDODMCDMS Q@MCNL U@QH@AKDR 3GDNQDL   (MCDODMCDMBD "QHSDQH@ ENQ #HRSQHATSHNM @MC #DMRHSX %TMBSHNM /HW (Ω, A, P) EH D SUREDELOLW\ VSDFH 7KH UDQGRP YDULDEOHV X1 , . . . , Xn : Ω → R ZLWK GLVWULEXWLRQ IXQFWLRQV FXi RI Xi  i = 1, . . . , n DUH LQGHSHQGHQW LI DQG RQO\ LI FX1 ,...,Xn (x1 , . . . , xn ) = FX1 (x1 ) · ... · FXn (xn ) ,   IRU DOO xi ∈ Rd  i = 1, . . . , n 0RUHRYHU LI WKH UDQGRP YDULDEOHV X1 , . . . , Xn KDYH GHQVLWLHV  LV HTXLYD OHQW WR fX1 ,...,Xn (x1 , . . . , xn ) = fX1 (x1 ) · ... · fXn (xn )

∀ xi ∈ Rd  i = 1, . . . , n ,

ZKHUH WKH IXQFWLRQV fXi DUH WKH GHQVLWLHV FRUUHVSRQGLQJ WR Xi  3URRI %NQ SGD @RRDQSHNM V Q S SGD CHRSQHATSHNMR KDS TR @RRTLD ƥQRS SG@S SGD Xi  i = 1, . . . , n @QD HMCDODMCDMS 3GDM FX1 ,...,Xn (x1 , . . . , xn ) = P(X1 ≤ x1 , . . . , Xn ≤ xn ) = P(X1 ≤ x1 ) · ... · P(Xn ≤ xn ) = FX1 (x1 ) · ... · FXn (xn ) -DWS VD OQNUD SGD @RRDQSHNM V Q S SGD CDMRHSHDR 'DQD VD OQNUD SGD BNM UDQRD RS@SDLDMS ENQ SGD B@RD SG@S @KK Q@MCNL U@QH@AKDR G@UD CDMRHSHDR 2DKDBS Ai ∈ A(Xi ) i = 1, . . . , n 3GDM Ai = Xi−1 (Bi ) ENQ RNLD Bi ∈ B 'DMBD P(A1 ∩ · · · ∩ An ) = P(X1 ∈ B1 , . . . , Xn ∈ Bn )

= fX1 ,...,Xn (x1 , . . . , xn )Cx1 . . . Cxn B1 ×···×Bn

=

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 i=1

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i=1

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P(Ai ) .

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 3GDNQDL   (MCDODMCDMBD (LOKHB@SHNMR ENQ ,D@M @MC 5@QH@MBD /HW (Ω, A, P) EH D SUREDELOLW\ VSDFH DQG X1 , . . . , Xn : Ω → Rd DUH LQGHSHQGHQW UDQGRP YDULDEOHV Ţ ,I E(|Xi |) < ∞ IRU i = 1, . . . , n WKHQ E(|X1 · ... · Xn |) < ∞

DQG

E(X1 · ... · Xn ) = E(X1 ) · ... · E(Xn ) .

Ţ ,I Var(Xi ) < ∞ IRU i = 1, . . . , n WKHQ Var(X1 + · · · + Xn ) = Var(X1 ) + · · · + Var(Xn ) . 3URRI %NQ SGD ƥQRS O@QS KDS TR RTOONRD SG@S D@BG Xi HR ANTMCDC @MC G@R @ CDMRHSX 3GDM  n 

 E Xi x1 · ... · xn fX1 ,...,Xn (x1 , . . . , xn )Cx1 . . . Cxn = Rd

i=1

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n 



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E(Xi ) ,

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  Var(X1 + X2 ) = X1 + X2 − (μ1 + μ2 )2 CP Ω

=

Ω

(X1 − μ1 )2 CP +

Ω

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=0

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ENQ @KK w ∈ V .

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||Y || :=

Ω

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1/2

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(X, Y ) := XY CP = E(XY ) . Ω



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ENQ @KK W ∈ V .

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XCP = ZCP ENQ @KK A ∈ S . A

A

2HMBD Z ∈ V HR S LD@RTQ@AKD VD RDD SG@S Z HM E@BS HR E(X|S) @R CDƥMDC HM NTQ D@QKHDQ CHRBTRRHNM 3G@S HR E(X|S) = projV (X) . 6D BNTKC SGDQDENQD @KSDQM@SHUDKX S@JD SGD K@RS HCDMSHSX @R @ CDƥMHSHNM NE BNM CHSHNM@K DWODBS@SHNM 3GHR ONHMS NE UHDV @KRN L@JDR HS BKD@Q SG@S Z = E(X|S) RNKUDR SGD KD@RS RPT@QDR OQNAKDL ||Z − X|| = LHM ||Y − X|| , Y ∈V

@MC RN E(X|S) B@M AD HMSDQOQDSDC @R SG@S S LD@RTQ@AKD Q@MCNL U@QH@AKD VGHBG HR SGD ADRS KD@RS RPT@QDR @OOQNWHL@SHNM NE SGD Q@MCNL U@QH@AKD X

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n=1

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P

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(M NQCDQ SN CDQHUD SGD TMCDQKXHMF DPT@SHNMR NE LNSHNM @ RDS NE /#$ ENQ DK@R SHB ANCHDR VD @QD @OOKXHMF SGD U@QH@SHNM@K OQHMBHOKD SN SGD ETMBSHNM@K NE DM DQFX NE SGD RXRSDL +DS Ω CDMNSD SGD CNL@HM NE HMSDQDRS H D SGD DK@RSHB ANCX BE %HF   @MC ∂Ω = Γ0 ∪ Γ1 ,   VGDQD Γ0 CDMNSDR SGD #HQHBGKDS ANTMC@QX O@QS VHSG OQDRBQHADC CHROK@BD LDMSR @MC Γ1 QDOQDRDMSR SGD -DTL@MM O@QS NE SGD ANTMC@QX ∂Ω VGDQD RTQ E@BD ENQBDR @QD OQDRBQHADC @R ANTMC@QX BNMCHSHNMR QDRODBSHUDKX 3GD HMMDQ DMDQFX NQ RSQ@HM DMDQFX NE @M DK@RSHB ANCX HR CDƥMDC @R

1 W := σ : εdx . 2 Ω 'DQD σ : ε = tr{σ T ε} QDOQDRDMSR SGD CNTAKD BNMSQ@BSHNM NE SGD SVN SDMRNQR BE :< ENQ @ MHBD QDOQDRDMS@SHNM NE SDMRNQ B@KBTKTR 3NFDSGDQ VHSG UNKTLD @MC RTQE@BD ENQBDR SGD RSQ@HM DMDQFX W CDƥMDR SGD ONSDMSH@K DMDQFX U NE SGD RXRSDL



1 U= σ : εdx − uT βdx − uT τ ds ,   2 Ω Ω Γ1    =W

VHSG SGD CDMRHSX NE UNKTLD ENQBDR β @MC SGD RTQE@BD RSQDRR UDBSNQ τ %HM@KKX VD TRD SGD JHMDSHB DMDQFX

1 T = ρu˙ T u˙ 2 Ω

 

@R SGD HMSDFQ@SDC RPT@QD NE SGD UDKNBHSX u˙ VDHFGSDC VHSG SGD BNMRS@MS L@RR CDMRHSX ρ 3GD U@QH@SHNM@K OQHMBHOKD BG@Q@BSDQHRDR SGD LNSHNM NE @M DK@RSHB ANCX UH@ SGD QDPTHQDLDMS SG@S SGD ETMBSHNM@K

t1

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t0

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∀x ∈ Ω ,

 

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t1 J(θ) := (T (u + θv) − U (u + θv)) dt ,   t0

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d J(θ)|θ=0 dθ

 

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( ' ( u2i−1,j − 2u2ij + u2i+1,j u2i,j−1 − 2u2ij + u2i,j+1 +μ + = ˙ ξ h2y h2x ' ( u1i+1,j+1 − u1i−1,j+1 − u1i+1,j−1 + u1i−1,j−1 +(λ + μ) .   4hx hy '

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⎛ '

(⎞ u2i+1,j − u2i−1,j u1ij − u1i,j−1 + ⎜μ ⎟ 2hx hy ⎜ ⎟ ⎜ ⎟ 2 2 1 1 ⎝ uij − ui,j−1 ui+1,j − ui−1,j ⎠ ξ +λ hy 2hx  

Ş 1HFGS ANTMC@QX n(x) = (1, 0)T ⎞ ⎛ ⎞ ⎛ u2i,j+1 − u2i,j−1 u1ij − u1i−1,j ∂u2 ∂u1 +λ ⎟ ⎜ ξ ⎜ ξ ∂x + λ ∂x ⎟ hx 2hy ⎟ ⎜ ' ( 1 2 ⎟ ⎜  = ˙ ⎜ u2 − u2 τ (x, t) = ⎝ ⎟ 1 1 − u u ∂u2 ∂u1 ⎠ ⎝ ij i−1,j i,j+1 i,j−1 ⎠ μ + μ + ∂x1 ∂x2 hx 2hy  



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/QNAKDLR "K@RRHƥB@SHNM ☼ D@RX  D@RX VHSG KNMFDQ B@KBTK@SHNMR  @ KHSSKD AHS CHƧBTKS  BG@KKDMFHMF $WDQBHRD   :☼< 2SQ@HM 3DMRNQ TMCDQ 1HFHC !NCX ,NSHNMR 2GNV SG@S SGD &QDDM 2S 5DM@MS RSQ@HM SDMRNQ E CDƥMDC HM   HR HMU@QH@MS TMCDQ QHFHC ANCX LNSHNMR $WDQBHRD   :☼< %HMHSD #HƤDQDMBDR HM #  #DQHUD SGD ENKKNVHMF  ƥMHSD CHƤDQDMBD %# @OOQNWHL@SHNMR ENQ SGD ƥQRS NQCDQ CDQHU@SHUD NE @ # ETMBSHNM u(x) ∈ C 3 (x − h, x + h) u(x + h) − u(x) ,   ∂ + u(x) := h u(x) − u(x − h) ,   ∂ − u(x) := h u(x + h) − u(x − h) .   ∂ 0 u(x) := 2h 6G@S HR SGD BNQQDRONMCHMF NQCDQ NE @OOQNWHL@SHNM NE D@BG U@QH@MS  Ŕ  6GX 'HMS 4RD SGD 3@XKNQ DWO@MRHNMR NE u(x ± h) 

%# @OOQNWHL@SHNM ENQ SGD RDBNMC NQCDQ CDQHU@SHUD NE SGD # ETMBSHNM u(x) ∈ C 4 (x − h, x + h) HR CDƥMDC HM SGD ENKKNVHMF L@MMDQ u(x + h) − 2u(x) + u(x − h) .   ∂ − ∂ + u(x) := h2 Ş #DQHUD   UH@ SGD 3@XKNQ DWO@MRHNM 6G@S HR SGD NQCDQ NE @OOQNW HL@SHNM Ş #DQHUD   HM @ CHQDBS V@X TRHMF  Ŕ 

$WDQBHRD   :< %HMHSD #HƤDQDMBDR HM # 6D BNMRHCDQ SGD SVN CHLDMRHNM@K /NHRRNM $PT@SHNM Δu(x, y) = − RHM(πx) · RHM(πy) · 2π 2 , u(x, y) = 0 ,

(x, y) ∈ Ω := [0, 1]2 ,   (x, y) ∈ Γ := ∂Ω ,

VGDQD Δu(x, y) QDOQDRDMSR SGD +@OK@BH@M Δu(x, y) =

∂2u (x, y) ∂x2

+

 

∂2u (x, y) ∂y 2



OOKX SGD # %# @OOQNWHL@SHNMR NE /QNAKDL   SN CDQHUD @ %# @OOQNWH L@SHNM ENQ SGD # +@OK@BH@M Δu(x, y)



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF  +DS Ωh := {(ih, jh) | i = 1, . . . , N, j = 1, . . . , N }

 

AD @ LDRG NE N HMMDQ FQHC ONHMSR HM D@BG CHQDBSHNM NM SGD TMHS RPT@QD [0, 1]2 ENQ @ FHUDM LDRG RHYD hx = hy = h := 1/(N + 1) Ş #Q@V @ RJDSBG GNV SGD LDRG Ωh KNNJR Ş 6GHBG MDHFGANTQHMF ONHMSR @QD MDBDRR@QX HM NQCDQ SN DU@KT@SD SGD %# @OOQNWHL@SHNM NE S@RJ  ENQ @ FHUDM ONHMS (i, j) Ş 2JDSBG SGD L@SQHW A NE SGD KHMD@Q RXRSDL NE DPT@SHNMR Auh = b SG@S HR BQD@SDC AX @OOKXHMF SGD %# NE S@RJ  SN SGD FHUDM OQNAKDL  Ŕ  TRHMF SGD LDRG    6QHSD @ RGNQS , 3+ ! RBQHOS VGHBG HLOKDLDMSR SGD MTLDQHB@K RNKTSHNM NE  Ŕ  'HMS 4RD SGD , 3+ ! A@BJRK@RG NODQ@SNQ ř;Ś SN RNKUD SGD KHMD@Q RXRSDL NE DPT@SHNMR 'NV B@M XNT UDQHEX SG@S XNTQ HLOKDLDMS@SHNM HR BNQQDBS $WDQBHRD   :☼< %HMHSD #HƤDQDMBD OOQNWHL@SHNM NE CUDBSHNM 3DQLR (M BGDLHRSQX DMFHMDDQHMF @MC D@QSG RBHDMBDR @CUDBSHNM HR @ SQ@MRONQS LDBG @MHRL NE @ RTARS@MBD NQ BNMRDQUDC OQNODQSX AX @ ƦTHC CTD SN SGD ƦTHCŗR ATKJ LNSHNM M DW@LOKD NE @CUDBSHNM HR SGD SQ@MRONQS NE ONKKTS@MSR NQ RHKS HM @ QHUDQ AX ATKJ V@SDQ ƦNV CNVMRSQD@L 3GD @CUDBSHNM NODQ@SNQ HR FHUDM @R u · ∇ = u, ∇  VGDQD u HR @ FHUDM UDBSNQ ƥDKC  (M # SGD @OOKHB@SHNM NE SGD @CUDBSHNM NODQ@SNQ SN SGD UDBSNQ ƥDKC u HSRDKE QD@CR @R u(x) · ux (x) &HUD HSR RO@BD CHRBQDSHR@SHNM +LQW OOKX 3@XKNQ @OOQNWHL@SHNM @M@KNFNTR SN SGD CDQHU@SHNM NE SGD ENQ V@QC SHLD CHRBQDSHR@SHNM  (M # SGD @OOKHB@SHNM NE SGD @CUDBSHNM NODQ@SNQ SN SGD UDBSNQ ƥDKC u = (u1 , u2 ) HSRDKE QD@CR @R ) *   u1 (x, y) u1 (x, y) ∂x , = u2 (x, y) u2 (x, y) ∂y  u1 (x, y)∂x u1 (x, y) + u2 (x, y)∂y u1 (x, y) = . u1 (x, y)∂x u2 (x, y) + u2 (x, y)∂y u2 (x, y) &HUD HSR RO@BD CHRBQDSHR@SHNM

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3NAH@R 1HFNHO  %KNQH@M 5XSS $WDQBHRD   :<  /NHMS 2SDMBHK -DFKDBSHMF ANTMC@QX BNMCHSHNMR GNV CNDR SGD KHMD@Q RXRSDL @MC DRODBH@KKX SGD L@SQHW NE CHRBQDSHR@SHNM ENQ SGD # +@OK@BD DPT@SHNM KNNJ KHJD HE HMRSD@C NE SGD  ONHMS RSDMBHK SGD  ONHMS RSDMBHK Δu(x, y) ≈

1 (−u(x + 2h, y) + 16u(x + h, y) − u(x, y + 2h) 12h2 +16u(x, y + h) − 60u(x, y) + 16u(x − h, y) − u(x − 2h, y) +16u(x, y − h) − u(x, y − 2h))

HR TRDC NM @ QDFTK@Q FQHC VHSG h = Δx = Δy  2DS TO SGD L@SQHW NE CHRBQDSHR@SHNM ENQ SGD # +@OK@BD DPT@SHNM VHSG SGHR RSDMBHK  6G@S @QD SGD @CU@MS@FDR @MC CHR@CU@MS@FDR NE TRHMF @  ONHMS RSDMBHK +LQW 3GD  ONHMS RSDMBHK KD@CR SN @M @BBTQ@BX NE O(h4 )  #DQHUD SGD  ONHMS RSDMBHK VHSG SGD GDKO NE RTHS@AKD 3@XKNQ DWO@MRHNMR



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF

"G@OSDQ 

3DWK:LVH 6ROXWLRQV RI 52'(V 3GHR BG@OSDQ LNSHU@SDR @MC QHFNQNTRKX CHRBTRRDR ANSG DWHRSDMBD @MC TMHPTD MDRR NE O@SG VHRD RNKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR 6D RS@QS AX LNCDKKHMF DWSDQM@K @MC FQNTMC LNSHNM DWBHS@SHNMR AX LD@MR NE RSNBG@RSHB OQNBDRRDR VGHBG LNSHU@SDR SGD RSTCX NE Q@MCNL NQCHM@QX CHƤDQ DMSH@K DPT@SHNMR 1.#$R 3GDHQ RNKTSHNM DWHRSDMBD @MC TMHPTDMDRR BNMBDOSR @QD SGDM CHRBTRRDC SNFDSGDQ VHSG SGD BNQQDRONMCDMBD ADSVDDM RSNBG@RSHB @MC Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR (M O@QSHBTK@Q VD RSTCX SGD BNMCH SHNMR SG@S KD@C SN SGD DWHRSDMBD NE O@SG VHRD TMHPTD RNKTSHNMR 2NKTSHNMR HM SGD DWSDMCDC RDMRD @QD @M@KXRDC @R VDKK @R SGD CDODMCDMBD NE RNKTSHNMR NM O@Q@LDSDQR @MC HMHSH@K BNMCHSHNMR R @M DWBTQRHNM VD FHUD SGD DPT@SHNMR NE LNSHNM ENQ RHMFKD @MC LTKSH RSNQDX VHQDEQ@LD ATHKCHMFR

  *DX "NMBDOSR %NKKNVHMF NTQ OQNFQ@L SN 3GHR BG@OSDQ AQHMFR TR NMD RSDO BKNRDQ SN RNKUHMF Q@MCNL O@QSH@K CHƤDQDM SH@K DPT@SHNMR AX DPTHOOHMF TR VHSG SGD SGDNQDSHB@K A@BJFQNTMC JMNVKDCFD QDPTHQDC SN BNMUDQS SGDL A@BJ SN CDSDQLHMHRSHB NQCHM@QX CHƤDQDMSH@K DPT@ SHNMR RDD %HF   .TQ L@HM RNTQBD ENQ SGD RDS TO @MC CHRBTRRHNM NE Q@MCNL CHƤDQDMSH@K DPT@SHNMR HR SGD E@ATKNTR ANNJ :< 1@MCNL CHƤDQDMSH@K DPT@SHNMR NBBTQ PTHSD M@STQ@KKX HM OKDMSX NE QD@K RHST @SHNMR @R SGD ENKKNVHMF DW@LOKD HKKTRSQ@SDR $W@LOKD   /NKKTSHNM (MCTBDC !HNBGDLHB@K .WXFDM #DL@MC @MC #HRRNKUDC .WXFDM HM 2SQD@LR %NKKNVHMF :< O  HS G@R ADBNLD HMBQD@RHMFKX DU HCDMS SG@S SGD VNQKCŗR LNRS U@KT@AKD M@STQ@K QDRNTQBDR Ŕ @HQ @MC V@SDQ Ŕ @QD ADHMF DMC@MFDQDC AX SGD @BSHUHSHDR NE BHUHKHYDC L@M 3GD V@SDQ RTOOKX HR AD HMF DMC@MFDQDC AX SGD CHRONR@K NE NQF@MHB @MC NSGDQ V@RSD L@SDQH@KR HMSN M@STQ@K ANCHDR NE V@SDQ AX LTMHBHO@KHSHDR @MC HMCTRSQHDR 3GHR ONKKTSHNM G@R ADBNLD @ L@INQ BNMBDQM NE SGD RBHDMSHƥB BNLLTMHSX @MC U@QHNTR QDFTK@SNQX @FDMBHDR G@UD RODBHƥDC LHMHLTL KDUDKR ENQ CHRRNKUDC NWXFDM #. HM K@JDR @MC RSQD@LR 3GDRD LHMHLTL KDUDKR NE #. @QD DWSQDLDKX HLONQS@MS RHMBD HE #. E@KKR ADKNV @ BDQS@HM SGQDRGNKC U@KTD SGD ƥRG @MC NSGDQ KHUHMF NQF@MHRLR HM SGD ANCX NE V@SDQ L@X CHD

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3NAH@R 1HFNHO  %KNQH@M 5XSS 3GD RHST@SHNM VD @QD LNCDKKHMF HR SG@S NE @ ONKKTS@MS QTMMHMF HMSN @ RSQD@L VGHBG HR SQ@MRONQSDC CNVMRSQD@L RDD %HF   6D @RRTLD @ RSD@CX @MC TMHENQL RSQD@L ƦNV @MC SG@S SGD BNMCHSHNMR @S DUDQX BQNRR RDBSHNM @QD TM BG@MFDC VHSG SHLD -DWS KDS x(t) CDMNSD SGD AHNBGDLHB@K NWXFDM CDL@MC !.# NE ONKKTSHNM CDFQ@CHMF A@BSDQH@ HM SGD ANCX NE V@SDQ @S SGD CHRS@MBD t CNVMRSQD@L EQNL SGD ONKKTSHNM RNTQBD @MC y(t) SGD BNMBDMSQ@SHNM NE #. @S CHRS@MBD t CNVMRSQD@L 3GHQC KDS TR @RRTLD SG@S SGD @BST@K ONKKTSHNM @S @ CHRS@MBD t CNVMRSQD@L HR OQNONQSHNM@K SN !.# Ω

Ω

Ω

t

t t x

x

x

Space-Time-Realization-Cube

Space-Discretization (Finite Differences)

Path-Wise Solution Concept for R(O)DEs

Partial Differential Equation with Stochastic Effects (RPDE or SPDE)

Finite-Dimensional System of R(O)DE

Finite-Dimensional System of an Infinite Family of ODEs

Decrease Mesh-Size

Compatibility Conditions 1) all solutions of the ODE family are defined on a common time interval 2) all solutions are stochastic processes

%HFTQD   1DCTBSHNM EQNL @ FHUDM BNMSHMTTL LDBG@MHB@K Q@MCNL O@QSH@K CHƤDQDMSH@K DPT@SHNM SN @ E@LHKX NE CDSDQLHMHRSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNMR (M SGHR BG@OSDQ VD ENBTR NM SGD RDBNMC RSDO H D  SGD BNQQDRONMCDMBD ADSVDDM Q@MCNL CHƤDQDMSH@K DPT@SHNMR @MC CDSDQLHMHRSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNMR

%HM@KKX VD @RRTLD SG@S SGDQD @QD ƥUD L@INQ @BSHUHSHDR HM SGD RSQD@L RDD :< O   3GD ONKKTSHNM !.# @MC #. @QD CDBQD@RDC AX SGD @BSHNM NE A@BSDQH@ @S @ Q@SD QDK@SDC SN SGD @LNTMS NE ONKKTSHNM OQDRDMS VHSG OQNONQSHNM@KHSX BNMRS@MS k1 > 0  3GD CHRRNKUDC NWXFDM HR HMBQD@RDC CTD SN QD@DQ@SHNM @S @ Q@SD OQNONQ SHNM@K SN SGD CHRRNKUDC NWXFDM CDƥBHS H D SGD #. R@STQ@SHNM BNMBDMSQ@ SHNM LHMTR SGD @BST@K #. BNMBDMSQ@SHNM VHSG OQNONQSHNM@KHSX BNMRS@MS k2 > 0  3GD ONKKTSHNM HR NMKX CDBQD@RDC AX RDCHLDMS@SHNM @MC @CRNQOSHNM @S @ Q@SD OQNONQSHNM@K SN SGD @LNTMS NE ONKKTSHNM OQDRDMS VHSG OQNONQSHNM@KHSX BNMRS@MS k3 > 0  3GD ONKKTSHNM HR HMBQD@RDC EQNL RL@KK RNTQBDR @KNMF SGD RSQDSBG NE RSQD@L VHSG Q@SD xa > 0 HMCDODMCDMS NE SGD @LNTMS NE ONKKTSHNM OQDRDMS



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF

distance t0 = 0 distance t

%HFTQD   2JDSBG NE SGD RHST@SHNM CDRBQHADC HM DW@LOKD   @ ONKKTS@MS HR QTMMHMF HMSN @ RSQD@L @MC HR SQ@MRONQSDC CNVMRSQD@L

 3GD CHRRNKUDC NWXFDM HR CDBQD@RDC @S @ Q@SD dO VGHBG L@X G@UD ONRHSHUD NQ MDF@SHUD U@KTDR @MC QDOQDRDMSR SGD MDS BG@MFD HM CHRRNKUDC NWXFDM CTD SN SGD !DMSG@K CDL@MC @MC QDROHQ@SHNM @MC OGNSNRXMSGDRHR NE OK@MSR (M :< SGD ENKKNVHMF LNCDK ENQ !.# @MC #. HR BNMRHCDQDC 0 = −ux(t) ˙ − (k1 + k3 ) x(t) + xa , 0 = −uy(t) ˙ + k2 (ys − y(t)) − k1 x(t) − dO , VGDQD u > 0 HR SGD @UDQ@FD UDKNBHSX @KNMF SGD RSQDSBG @MC ys > 0 HR SGD R@ST Q@SHNM BNMBDMSQ@SHNM ENQ CHRRNKUDC NWXFDM 3GD HMHSH@K BNMCHSHNMR @QD CDMNSDC AX x(0) = x0 @MC y(0) = y0 'DQDAX SGD PT@MSHSHDR k1 , k2 , k3 , u, xa , ys , dO , x0 , y0 @QD MNS LNCDKKDC @R BNMRS@MSR ATS @QD LNQD QD@KHRSHB@KKX BNMRHCDQDC @R Q@MCNL U@QH@AKDR NQ t CDODMCDMS BNMSHMTNTR RSNBG@RSHB OQNBDRRDR VGHBG G@UD BDQS@HM OQNA@AHKHSX CHRSQHATSHNMR 2TBG Q@MCNLKX ODQSTQADC NQCHM@QX CHƤDQDMSH@K DPT@SHNMR FN ADXNMC SGD RBNOD NE SGD SGDNQX NE CDSDQLHMHRSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNMR @MC @S @ ƥQRS FK@MBD HS HR MNS SQHUH@K SN CDSDQLHMD @ MNSHNM NE @ RNKTSHNM ENQ RTBG OQNAKDLR SG@S B@QDETKKX L@JDR RDMRD ENQ SGD Q@MCNL ODQSTQA@SHNMR HMUNKUDC (M SGHR BG@OSDQ VD VHKK RGNV GNV SN BNMUDQS Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR SN NQCHM@QX CHƤDQDMSH@K DPT@SHNMR (M O@QSHBTK@Q VD VHKK FHUD DWHRSDMBD @MC TMHPTDMDRR OQNNER ENQ SGD RNKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR 3GDRD OQNNER @QD NE BNMRSQTBSHUD @KFNQHSGLHB M@STQD @MC VHKK ENQDRG@CNV SGD MTLDQHBR SG@S B@M AD @OOKHDC SN RHLTK@SD SGD RNKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR 6GDM QD@CHMF SGHR BG@OSDQ MNSD SGD @MRVDQR SN SGD ENKKNVHMF PTDRSHNMR

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3NAH@R 1HFNHO  %KNQH@M 5XSS  6G@S @QD SGD BNLLNMKX @BBDOSDC CHƤDQDMSH@K DPT@SHNM LNCDKR ENQ D@QSG PT@JD DWBHS@SHNMR  6G@S HR SGD CHƤDQDMBD ADSVDDM RSNBG@RSHB NQCHM@QX CHƤDQDMSH@K DPT@ SHNMR @MC Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR  'NV B@M RSNBG@RSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNMR VHSG BNMRS@MS CHƤT RHNM SDQLR AD SQ@MRENQLDC SN Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR  'NV B@M VD OQNUD DWHRSDMBD @MC TMHPTDMDRR NE Q@MCNL NQCHM@QX CHƤDQ DMSH@K DPT@SHNMR @MC VG@S CN SGD MNSHNMR NE DWHRSDMBD @MC TMHPTDMDRR LD@M VGDM RSNBG@RSHB ODQSTQA@SHNMR @QD OQDRDMS  'NV CN SGD RNKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR CDODMC NM O@Q@LDSDQR @MC HMHSH@K BNMCHSHNMR @R VDKK @R SGD ENKKNVHMF JDX BNMBDOSR  &@TRRH@M VGHSD MNHRD @MC 6HDMDQ OQNBDRRDR HMBK SGDHQ BNMSHMTHSX @MC CHƤDQDMSH@AHKHSX OQNODQSHDR  3GD #NRR 2TRRL@MM  (LJDKKDQ 2BGL@KETRR BNQQDRONMCDMBD  /@SG VHRD DWHRSDMBD @MC TMHPTDMDRR NE RNKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR  2NKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR HM SGD DWSDMCDC RDMRD @MC  $PT@SHNMR NE LNSHNM ENQ RHMFKD @MC LTKSH RSNQDX VHQDEQ@LD ATHKCHMFR 3GHR BG@OSDQ HR RSQTBSTQDC @R ENKKNVR (M 2DB   VD LNCDK DWSDQM@K @MC FQNTMC LNSHNM DWBHS@SHNMR AX LD@MR NE RSNBG@RSHB OQNBDRRDR L@HMKX VGHSD MNHRD CQHUDM NMDR 3GHR LNSHU@SDR SGD RSTCX NE Q@MCNL NQCHM@QX CHƤDQDM SH@K DPT@SHNMR 3GDHQ RNKTSHNM DWHRSDMBD @MC TMHPTDMDRR BNMBDOSR @QD CHR BTRRDC HM 2DB   SNFDSGDQ VHSG SGD BNQQDRONMCDMBD ADSVDDM RSNBG@RSHB @MC Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR %NQ RHLOKHBHSX NE MNS@SHNM VD NE SDM RTOOQDRR SGD SDQL řNQCHM@QXŚ VGDM HS HR BKD@Q VGHBG JHMC NE CHƤDQDMSH@K DPT@SHNM VD LD@M -DWS HM 2DB   VD CHRBTRR SGD BNMCHSHNMR SG@S KD@C SN SGD DWHRSDMBD NE O@SG VHRD TMHPTD RNKTSHNMR 'DQD RNKTSHNMR HM SGD DWSDMCDC RDMRD @QD @M@KXRDC @R VDKK @R SGD CDODMCDMBD NE RNKTSHNMR NM O@Q@LDSDQR @MC HMHSH@K BNMCHSHNMR R @M DWBTQRHNM 2DBSHNM   LNCDKR CDSDQLHMHRSHB@KKX SGD DPT@SHNMR NE LNSHNM ENQ RHMFKD @MC LTKSH RSNQDX VHQDEQ@LD ATHKCHMFR %HM@KKX 2DBSHNM   VQ@OR TO SGD BNMSDMSR NE SGHR BG@OSDQ 3UHUHTXLVLWHV 3GD BNMSDMSR NE "G@O  @MC @ A@RHB JMNVKDCFD NE NQCHM@QX CHƤDQDMSH@K DPT@SHNMR @QD QDPTHQDC ENQ SGHR BG@OSDQ 2NLD OQD JMNVKDCFD NM RSNBG@RSHB OQNBDRRDR VNTKC AD FNNC 3GD QDPTHQDC , 3+ ! BNLL@MCR ENQ



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V0 (ω) = α(ω) + Lg0 Y10 (ω) ,

@MC W0 (ω, Y0 , Y ) = g0 (h0 Y10 − Y1 + Y20 − Y2 ) . !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   Ŕ /@QS  #DODMCDMBD NM /@Q@LDSDQR @MC (MHSH@K "NMCHSHNMR 0 &HUD &QNMV@KKŗR KDLL@ ENQ RSNBG@RSHB OQNBDRRDR @MC BNLO@QD HS SN HSR TRT@K CDSDQLHMHRSHB UDQRHNM 0 'NV CN SGD RNKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR CDODMC NM HMHSH@K U@KTDR @MC O@Q@LDSDQR 0 2JDSBG SGD OQNNE NE SGHR CDODMCDMBD @RRDQSHNM XNT ITRS TRDC 0 6GHBG BNMSHMTHSX HLOKHB@SHNMR CN SGDRD CDODMCDMBHDR NM HMHSH@K U@KTDR @MC O@Q@LDSDQR G@UD

  $WBTQRHNM #DSDQLHMHRSHB #DRBQHOSHNM NE SGD 5HAQ@SHNMR NE 2HMFKD  ,TKSH 2SNQDX !THKCHMFR %NKKNVHMF :< OO  SGD ENQBDR @BSHMF NM SGD L@RR NM SNO NE SGD VDHFGSKDRR EQ@LD NE SGD NMD RSNQDX ATHKCHMF RGNVM HM %HF   @ @QD SGD H[WHUQDO IRUFH p(t) @R VDKK @R SGD GDPSLQJ QDRHRSHMF ENQBD fD (t) @MC SGD HODVWLF NQ LQHODVWLF QDRSHMF ENQBD fS (t) NE SGD RSQTBSTQD 3GD DWSDQM@K ENQBD p HR ENQ SGD LNLDMS S@JDM SN AD ONRHSHUD HM SGD CHQDBSHNM NE SGD x @WHR 3GD CHROK@BDLDMS u(t) SGD UDKNBHSX u(t) ˙ @MC SGD @BBDKDQ@SHNM u ¨(t) @QD @KRN ONRHSHUD HM SGD CHQDBSHNM NE SGD x @WHR 3GD C@LOHMF fD  @MC



&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF DK@RSHB HMDK@RSHB fS  ENQBDR @QD @BSHMF M SGD NOONRHSD CHQDBSHNM ADB@TRD SGDX @QD HMSDQM@K ENQBDR @MC QDRHRS SGD UDKNBHSX @MC CDENQL@SHNM QDRODBSHUDKX 3GD QDRTKS@MS ENQBD @KNMF SGD x @WHR HR p − fD − fS @MC ƥM@KKX -DVSNMŗR RDBNMC K@V NE LNSHNM FHUDR p − fS − fD = m¨ u

NQ

m¨ u + f D + fS = p .

 

-NV VD VNTKC KHJD SN TMCDQRS@MC SGD QDRSHMF ENQBDR fD @MC fS 3GHR VHKK AD CNMD HM SGD ENKKNVHMF RDBSHNMR

   5HAQ@SHNMR NE @ 2HMFKD 2SNQDX !THKCHMF 3GD RS@MC@QC LNCDK ENQ LDBG@MHB NRBHKK@SHNMR HR SG@S NE @ L@RR m > 0 ADHMF BNMMDBSDC SN @ V@KK AX @ ROQHMF VHSG RSHƤMDRR k > 0 'NNJDŗR K@V F = −kx SNFDSGDQ VHSG -DVSNMŗR RDBNMC K@V NE LNSHNM F = m¨ x KD@C SN m¨ x + c˜x˙ + kx = F (t) ,

VHSG x(0) = x0 @MC x(0) ˙ = x1 ,

VGDQD Ş x CDMNSDR SGD CHROK@BDLDMS NE SGD L@RR ONHMS EQNL SGD DPTHKHAQHTL BN NQCHM@SD x0  Ş c˜x˙ CDMNSDR SGD UHRBNTR C@LOHMF NE SGD L@RR ONHMS @MC Ş F (t) HR @ SHLD CDODMCDMS DWSDQM@K ENQBD  2B@KHMF SHLD t UH@ t = m/kτ @MC SGD ONRHSHNM BNNQCHM@SD x UH@ x = x0 y VD NAS@HM SGD CHLDMRHNM EQDD ENQL NE SGD EQDD NRBHKK@SHNM VHSGNTS DWSDQM@K ENQBD @R  y¨(t) + cy(t) ˙ + y(t) = 0 , VHSG y(0) = 1 @MC y(0) ˙ = v = (x−1 0 x1 m/k , !−1  VGDQD c := c˜ m/k

-DWS SN SGD B@RD NE D@QSGPT@JD HMCTBDC NRBHKK@SHNMR RDDM HM LDBG@MHB DPTHKHAQHTL VD G@UD

R VD G@UD @KQD@CX

fS + fD + fI = 0 ENQ Ş SGD CDENQL@SHNM ENQBD fS = ku ˙ @MC Ş SGD C@LOHMF ENQBD fD = cu uT  Ş SGD L@RR HMDQSH@ fI = m¨

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3NAH@R 1HFNHO  %KNQH@M 5XSS x x x x x x x x xxxxxxxxxxxxxx x x x x x x x x x x x xxxxxxx x xxxxxxx x xxxxxxx x xxxxxxx xxxxxxx xxxxxxx xxxxxxx

u(t)

x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x 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u(t)

damper with damping constant c

mass m

spring with spring constant k

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@

A

%HFTQD   2JDSBG NE @ NMD RSNQDX ATHKCHMF RTAIDBS SN @M DWSDQM@K DWBHS@SHNM u(t) RTBG SG@S SGD SNO ƦNNQ NE SGD ATHKCHMF RVHMFR @ @MC NE SGD @M@KNFNTR L@RR C@LODQ RXRSDL A

VGDQD uT = u + ug CDMNSDR SGD SNS@K CHROK@BDLDMS NE SGD ONRHSHNM BNNQCHM@SD u RTAIDBS SN SGD HMƦTDMBD NE SGD D@QSGPT@JD HMCTBDC RNHK LNSHNM ug 'DQD NMKX SGD QDK@SHUD CHROK@BDLDMS NE SGD L@RR ONHMS EQNL SGD HMHSH@K ONRHSHNM G@R @M HMƦTDMBD NM SGD CDENQL@SHNM @MC SGD C@LOHMF ENQBDR 'DMBD m¨ u + cu˙ + ku = −m¨ ug . %NQ HMRS@MBD VHSG SGD ODQHNCHB@K FQNTMC DWBHS@SHNM A > 0 !  u ¨g = −A RHM ω k/mt @MC MDFKDBSHMF C@LOHMF VD FDS m¨ u + ku = A RHM ω F@HM RB@KHMF t =



!  k/mt .

m/kτ @MC u = (A/k)y KD@CR SN y¨(t) + y(t) = RHM(ωt) .

6D BNMRHCDQ @ NMD RSNQDX ATHKCHMF SG@S HR @S QDRS @S t = 0 @MC KDS X(t) t ≥ 0 CDMNSD SGD QDK@SHUD GNQHYNMS@K CHROK@BDLDMS NE HSR QNNE VHSG QDRODBS SN SGD FQNTMC 3GDM A@RDC TONM @M HCD@KHYDC KHMD@Q LNCDK SGD QDK@SHUD CHROK@BD LDMS X(t) RTAIDBS SN FQNTMC @BBDKDQ@SHNMR HR FNUDQMDC AX x ¨(t) + 2ζω0 x(t) ˙ + ω02 x(t) = −y(t) ,



ENQ t ≥ 0 .

  &KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 2

10

0.5 Implicit Euler Starting point Endpoint Heun

0.4

Explicit Euler Implicit Euler Heun

1

10

0

0.3

10

0.2

10

0.1

10

−1

Accuracy

−2

0

−0.1

−3

10

−4

10

−0.2

−5

10

−0.3 −6

10

−0.4 −7

10

−0.5 −0.025

0

10

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

1

10

0.025

@

2

3

10

10

4

10

5

10

Number of steps

A

%HFTQD   @ 2NKTSHNM ENQ SGD NMD RSNQDX DWBHS@SHNM LNCDK   HM SGD x x OG@RD ˙ OK@MD @MC A BNMUDQFDMBD NE SGD CHƤDQDMS LDSGNCR BNLO@QDC SN @ GHFG NQCDQ RNKT SHNM BNLOTSDC VHSG , 3+ !

%NQ SGHR DPT@SHNM KDS SGD O@Q@LDSDQR AD FHUDM @BBNQCHMF SN :< O  @R ω0 = 20 HM rad/s @MC ζ = 0.05 +DS TR ƥQRS CHROK@X SGD RNKTSHNM NE   HM SGD x x OG@RD ˙ OK@MD VHSG HMH SH@K BNMCHSHNMR x(0) = 0 @MC x˙ = 0 H D SGD ATHKCHMF HR @S QDRS @S t = 0 RDD %HF   @ 6D RDD SG@S SGD @BBDKDQ@SHNM x˙ HMBQD@RDR PTHBJKX @MC @KSDQM@SDR ADSVDDM ONRHSHUD @MC MDF@SHUD @BBDKDQ@SHNM 3GD RNKTSHNMR OQNCTBDC AX SGD CDSDQLHMHRSHB $TKDQ @MC 'DTM RBGDLD ADG@UD RHLHK@QKX HM %HFTQD   A SGD BNMUDQFDMBD NE SGD JMNVM LDSGNCR HR HKKTRSQ@SDC 3GD BNMUDQFDMBD Q@SD NE SGD HLOKHBHS @MC DWOKHBHS $TKDQ HR O(Δh) ATS SGD HLOKHBHS $TKDQ OQNCTBDR @M TRDETK RNKTSHNM DUDM ENQ RL@KK RSDO RHYDR 'DTMŗR LDSGNC G@R @ BNMUDQFDMBD Q@SD NE O(Δh2 ) VGHBG HR BKD@QKX HKKTRSQ@SDC HM SGD BNMUDQFDMBD OKNS +HJD SGD DWOKHBHS $TKDQ 'DTMŗR LDSGNC MDDCR @ LHMHL@K RSDO RHYD SN OQNCTBD @M @BBT Q@SD RNKTSHNM

   5HAQ@SHNMR NE @ ,TKSH 2SNQDX !THKCHMF (M SGD B@RD NE @ d RSNQDX ATHKCHMF d = 1, 2, . . .  VD G@UD SG@S SGD ENQBDR Fj SG@S @BS NM @ ƦNNQ j B@M AD ROKHS HMSN SGNRD QDRTKSHMF EQNL @ BNLONMDMS SG@S ADKNMFR SN SGD ƦNNQ @ANUD Fjj+1  @MC NMD SG@S ADKNMFR SN SGD ƦNNQ ADKNV HS

Fjj−1  H D 

Fj = Fjj−1 + Fjj+1 ,

VGDQD VD RDS Fdd+1 = 0 @R SGDQD HR MN DWSDQM@K ENQBD @BSHMF NM SGD QNNE @MC F10 DPT@K SN SGD ENQBDR HMCTBDC AX SGD D@QSGPT@JD 3GHR KD@CR SN SGD ENKKNV

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3NAH@R 1HFNHO  %KNQH@M 5XSS HMF ENQLR NE SGD CDENQL@SHNM @MC C@LOHMF ENQBDR VGDQD CHROK@BDLDMSR @QD LD@RTQDC QDK@SHUD SN SGD j SG ƦNNQ Ş #DENQL@SHNM ENQBD (j)

fS = kj (uj − uj−1 )+kj+1 (uj − uj+1 ) = −kj uj−1 +(kj +kj+1 )uj −kj+1 uj+1 . Ş #@LOHMF ENQBD (j)

fD = cj (u˙ j − u˙ j−1 )+cj+1 (u˙ j − u˙ j+1 ) = −cj u˙ j−1 +(cj +cj+1 )u˙ j −cj+1 u˙ j+1 . %NQ u := (u1 , u2 , . . . , ud )T SGD CHLDMRHNM EQDD DPT@SHNM HM L@SQHW UDBSNQ MNS@ SHNM QD@CR @R u ¨ + C u˙ + Ku = F (t) ,   VHSG @ SHLD CDODMCDMS DWSDQM@K ENQBD F BNQQDRONMCHMF SN SGD D@QSGPT@JD DW BHS@SHNM @MC VGDQD D F  ⎛ ⎞ k1 + k2 −k2 ⎜ −k2 ⎟ k2 + k3 −k3 ⎜ ⎟ ⎜ ⎟ −k3 k3 + k4 −k4 K = ⎜ ⎟ ⎜ ⎟

⎝ ⎠

−ki +ki @MC C @M@KNFNTR (E MDBDRR@QX L@RRDR @QD HMBKTCDC HMSN SGD LNCDK UH@ CH@F NM@K L@SQHBDR  %HFTQD   CHROK@XR @ RHLTK@SHNM QTM NE @ SGQDD RSNQDX ATHKCHMF VHSG HMHSH@K CHROK@BDLDMS u0 = (0.1, 0.02, −0.1)T @MC BNMRS@MSR k1 = k2 = k3 = 1 @MC c1 = c2 = c3 = 0.5 3GD DƤDBSR NE C@LOHMF @QD BKD@QKX UHRHAKD

  "G@OSDQŗR 2TLL@QX (M SGHR BG@OSDQ VD F@HMDC HMRHFGS HMSN SGD LNCDKKHMF NE RDHRLHB @BSHUHSHDR CTD SN SGD @OOKHB@SHNM NE KHMD@Q RSNBG@RSHB CHƤDQDMSH@K DPT@SHNMR (LONQS@MS LNC DKR VDQD SGD *@M@H 3@IHLH @MC SGD "KNTFG /DMYHDM ƥKSDQ VGHBG ANSG TRD @CCH SHUD VGHSD MNHRD @R SGD DRRDMSH@K CQHUHMF SDQL 3GHR @CCHSHUD CDODMCDMBD NM VGHSD MNHRD @KKNVDC TR TRHMF SGD #NRR 2TRRL@MM (LJDKKDQ 2BGL@KETRR BNQQDRONMCDMBD SN QDVQHSD SGDRD RSNBG@RSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNMR @R .QMRSDHM 4GKDMADBJ OQNBDRR CQHUDM Q@M CNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR 3GHR LNSHU@SDC SGD @M@KXSHB RSTCX NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR 6D CHRBTRRDC DWHRSDMBD @MC TMHPTDMDRR NE O@SG VHRD RNKTSHNMR NE Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNMR AX BNMUDQSHMF SGDL SN CDSDQLHMHRSHB NQCH M@QX CHƤDQDMSH@K DPT@SHNMR HM @ O@SG VHRD RDMRD 3GTR HM NQCDQ SN F@HM TMHPTD



&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 −1

−0.8

−0.6

−0.4

−0.2

@ 3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 −1

−0.8

−0.6

−0.4

−0.2

B 3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

−0.8

−0.6

−0.4

−0.2

0

D

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

C

3.5

0 −1

0

A

0.2

0.4

0.6

0.8

1

0 −1

−0.8

−0.6

−0.4

−0.2

0

E

%HFTQD   2HLTK@SHNM NE SGD NRBHKK@SNQX LNUDLDMS NE @ RSQTBSTQ@KKX C@LODC SGQDD RSNQDX ATHKCHMF RS@QHMF VHSG @ RKHFGS CHROK@BDLDMS EQNL SGD DPTHKHAQHTL ONRHSHNM

O@SG VHRD DWHRSDMBD D@BG DKDLDMS NE SGD ω CDODMCDMS E@LHKX NE NQCHM@QX CHE EDQDMSH@K DPT@SHNMR HR QDPTHQDC SN G@UD @ TMHPTD RNKTSHNM SG@S SGDRD RNKT

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3NAH@R 1HFNHO  %KNQH@M 5XSS SHNMR RG@QD @ BNLLNM SHLD HMSDQU@K NE DWHRSDMBD @MC SG@S SGD SGTR CDQHUDC ω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ƥB@SHNMR @R x(t) ˙ = −at (ω)x(t) + αt (ω) , y(t) ˙ = −bt (ω)y(t) − ct (ω)x(t) − βt (ω) , VGDQD at , bt , ct , αt , βt @QD RTHS@AKD ANTMCDC BNMSHMTNTR RSNBG@RSHB OQNBDRRDR CDƥMDC ENQ t ∈ [t0 , T ) NM SGD OQNA@AHKHSX RO@BD (Ω, A, P) VGDQD T = ∞ L@X GNKC OOKXHMF 3GDNQDL   SGD ENKKNVHMF BNMCHSHNMR G@UD SN AD ETKƥKKDC HM NQCDQ SN FT@Q@MSDD SGD DWHRSDMBD NE @ O@SG VHRD TMHPTD RNKTSHNM NE SGHR RXRSDL NE Q@MCNL CHƤDQDMSH@K DPT@SHNMR  3GD ETMBSHNMR f1 (x, y, t, ω) = −at (ω)x(t) + αt (ω) @MC f2 (x, y, t, ω) = −bt (ω)y(t) − ct (ω)x(t) − βt (ω) LTRS AD A LD@RTQ@AKD ENQ @KK (x, y, t) ∈ R × R × [t0 , T )  f1 (x, y, t, ω) @MC f2 (x, y, t, ω) LTRS AD BNMSHMTNTR NM R × R × [t0 , T ) ENQ @KLNRS @KK ω ∈ Ω %NQ SGHR LNCDK SGHR HR HLLDCH@SDKX UDQHƥDC  %NQ @KLNRS @KK ω ∈ Ω SGDQD LTRS AD @ QD@K BNMSHMTNTR ETMBSHNM L(t, ω) NM [t0 , T ) RTBG SG@S (f1 (x1 , y1 , t, ω) − f1 (x2 , y2 , t, ω))2 + (f2 (x1 , y1 , t, ω) − f2 (x2 , y2 , t, ω))2 = (at (x1 − x2 ))2 + (bt (y1 − y2 ) + ct (x1 − x2 ))2 ≤ (a2t + c2t ) (x1 − x2 )2 + b2t (y1 − y2 )2  ≤ L(t, ω) (x1 − x2 )2 + (y1 − y2 )2 , R NTQ L@MHOTK@SHNMR @K VGDQD i ∈ [t0 , T ) @MC x1 , x2 , y1 , y2 ∈ R QD@CX RGNV RTBG @ ETMBSHNM L(t, ω) HMCDDC DWHRSR AX RDSSHMF L(t, ω) ≥  L@W{a2t + c2t , b2t } 3GTR NTQ ONKKTSHNM RSQD@L LNCDK G@R @ TMHPTD O@SG VHRD RNKTSHNM NM [t0 , T ) ENQ @MX HMHSH@K BNMCHSHNM (x0 , y0 , t0 ) ∈ S1 × S1 × [t0 , T )



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0

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QFTD VGDSGDQ Xt HR @ 6HDMDQ OQNBDRR

+LQW !X DLOKNXHMF SGD OQNODQSHDR NE SGD CHRSQHATSHNM NE 6HDMDQ OQNBDRRDR SGHR DWDQBHRD B@M AD CNMD VHSGNTS DK@ANQ@SD B@KBTK@SHNM $WDQBHRD   :☼< "NLOKDSHNM SN @ B d × A ,D@RTQ@AKD %TMBSHNM 2GNV SGD @RRDQSHNM NE +DLL@   ( D  KDS B d SGD OQNCTBS !NQDK σ @KFDAQ@ NM Rd  Ω1 ∈ A @MC f (x, ω) : Rd × Ω → Rm AD BNMSHMTNTR NM Rd ENQ ω ∈ Ω1 @R VDKK @R A LD@RTQ@AKD HM ω ENQ @KK x ∈ Rd 3GDM SGDQD DWHRSR @ B d × A LD@RTQ@AKD ETMBSHNM g(x, ω) RTBG SG@S f (x, ω) = g(x, ω) ENQ (x, ω) ∈ Rd × Ω1 $WDQBHRD   :☼< 3GD !K@BJ #DQL@M 3NX ,NCDK @R @ 1@MCNL #HƤDQDMSH@K $PT@SHNM +DS 0 ≤ t ≤ T !K@BJ #DQL@M @MC 3NX CDUDKNODC @ CHRBQDSD SHLD LNCDK ENQ SGD RGNQS SDQL HMSDQDRS Q@SD r (SR DPTHU@KDMS HM BNMSHMTNTR SHLD HR SGD RSNBG@RSHB CHƤDQDMSH@K DPT@SHNM C KM (rt ) = a(t)Ct + bCWt . 1DVQHSD SGHR DPT@SHNM @R @ Q@MCNL CHƤDQDMSH@K DPT@SHNM AX @OOKXHMF SGD #NRR 2TRRL@MM  (LJDKKDQ 2BGL@KETRR BNQQDRONMCDMBD 4MCDQ VGHBG BNMCHSHNMR NM SGD O@Q@LDSDQR G@R SGHR Q@MCNL CHƤDQDMSH@K DPT@SHNM @ TMHPTD O@SG VHRD RNKTSHNM $WDQBHRD   :☼< 3GD 'TKK 6GHSD ,NCDK @R @ 1@MCNL #HƤDQDMSH@K $PT@SHNM +DS 0 ≤ t ≤ T 'TKK @MC 6GHSD CDUDKNODC @ LNCDK ENQ SGD RGNQS SDQL HMSDQ DRS Q@SD r HM VGHBG SGD KNMF QTM LD@M HR RODBHƥDC UH@ SGD MNM Q@MCNL SHLD CDODMCDMS ETMBSHNM a : [0, T ] → R 3GD BNQQDRONMCHMF RSNBG@RSHB CHƤDQDMSH@K DPT@SHNM ENQ SGHR LNCDK QD@CR @R Crt = (a(t) − αrt ) Ct + bCWt , VGDQD α, b ∈ R \ {0} @MC r0 ∈ R HR JMNVM 1DVQHSD SGHR DPT@SHNM @R @ Q@MCNL CHƤDQDMSH@K DPT@SHNM AX @OOKXHMF SGD #NRR 2TRRL@MM  (LJDKKDQ 2BGL@KETRR BNQQDRONMCDMBD 4MCDQ VGHBG BNMCHSHNMR NM SGD O@Q@LDSDQR G@R SGHR Q@MCNL CHƤDQDMSH@K DPT@SHNM @ TMHPTD O@SG VHRD RNKTSHNM



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$WDQBHRD   :☼< 3GD "KNTFG /DMYHDM ,NCDK @R @ 1@MCNL #HƤDQDMSH@K $PT@SHNM (M 2DB    VD CHRBTRRDC SGD "KNTFG /DMYHDM LNCDK ENQ FQNTMC LNSHNM DW BHS@SHNMR RDD DPT@SHNM   1DVQHSD SGHR RDBNMC NQCDQ DPT@SHNM @R @ ƥQRS NQCDQ RXRSDL NE RSNBG@RSHB CHƤDQDMSH@K DPT@SHNMR @MC SGDM BNMUDQS HS SN SGD BNQQDRONMCHMF RXRSDL NE Q@MCNL CHƤDQDMSH@K DPT@SHNM AX @OOKXHMF SGD #NRR 2TRRL@MM  (LJDKKDQ 2BGL@KETRR BNQQDRONMCDMBD 4MCDQ VGHBG BNMCHSHNMR NM SGD O@Q@LDSDQR G@R SGHR Q@MCNL CHƤDQDMSH@K DPT@SHNM RXRSDL @ TMHPTD O@SG VHRD RNKTSHNM $WDQBHRD   :☼< %TMBSHNMR NE /@SG 6HRD 2NKTSHNMR Ŕ /@QS  +DS Xt AD @ O@SG VHRD RNKTSHNM NE   NM I {t1 , . . . , tk } ⊂ I @M @QAHSQ@QX O@ Q@LDSDQ RDS @MC ϕ @M @QAHSQ@QX !NQDK LD@RTQ@AKD ETMBSHNM 2GNV SG@S SGD ETMBSHNM ϕ(Xt1 , . . . , Xtk ) HR A LD@RTQ@AKD $WDQBHRD   :☼< %TMBSHNMR NE /@SG 6HRD 2NKTSHNMR Ŕ /@QS  +DS Xt AD @ O@SG VHRD RNKTSHNM NE   NM I  2GNV SG@S @KLNRS @KK QD@KHR@SHNMR NE Xt NM I @QD BNMSHMTNTR ETMBSHNMR  +DS ψ AD @M @QAHSQ@QX BNMSHMTNTR ETMBSHNMR NM Rd 2GNV SG@S SGD ETMBSHNMR RTOt∈I ψ(Xt ) @MC HMEt∈I ψ(Xt ) VGHBG @QD CDƥMDC NM Ω @QD DPTHU@KDMS SN A LD@RTQ@AKD ETMBSHNMR 

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3NAH@R 1HFNHO  %KNQH@M 5XSS $WDQBHRD   :☼< 1#$ /QNODQSHDR R CHRBTRRDC HM 2DB    :< BNMRHCDQDC SGD ENKKNVHMF LNCDK HM NQCDQ SN CDRBQHAD HM RNLD PT@KHS@SHUD L@MMDQ SGD M@STQD NE @M D@QSGPT@JD CHRSTQA@MBD # n ENQ t ≥ 0 j=1 taj DWO(−αj t) BNR(ωj t + Θj ) , y(t) = , 0, ENQ t < 0 VGDQD aj  αj @MC ωj @QD FHUDM QD@K ONRHSHUD MTLADQR @MC SGD O@Q@LDSDQR Θj @QD HMCDODMCDMS Q@MCNL U@QH@AKDR TMHENQLKX CHRSQHATSDC NUDQ @M HMSDQU@K NE KDMFSG 2π +DS TR @RRTLD @ NMD RSNQDX ATHKCHMF SG@S HR @S QDRS @S t = 0 @MC KDS X(t) t ≥ 0 CDMNSD SGD QDK@SHUD GNQHYNMS@K CHROK@BDLDMS NE HSR QNNE VHSG QDRODBS SN SGD FQNTMC 3GDM A@RDC TONM @M HCD@KHYDC KHMD@Q LNCDK SGD QDK@SHUD CHR OK@BDLDMS X(t) RTAIDBS SN FQNTMC @BBDKDQ@SHNMR HR FNUDQMDC AX x ¨(t) + 2ζω0 x(t) ˙ + ω02 x(t) = −y(t) ,

ENQ t ≥ 0 .

 

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n 

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j=1

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3NAH@R 1HFNHO  %KNQH@M 5XSS  6G@S @ANTS SGD QDUDQRD QD SGDQD Q@MCNL CHƤDQDMSH@K DPT@SHNMR SG@S G@UD P NQ L R RNKTSHNMR ATS MN O@SG VHRD RNKTSHNMR  'NV CNDR SGHR @KK QDK@SD SN (S¯ RNKTSHNMR NE RSNBG@RSHB CHƤDQDMSH@K DPT@ SHNMR 3GHR BG@OSDQ HR RSQTBSTQDC @R ENKKNVR 2DBSHNM   CHRBTRRDR SGD MNSHNM NE P RNKTSHNMR @MC RTAIDBS SN VGHBG BNMCHSHNMR @ FHUDM Q@MCNL CHƤDQDMSH@K DPT@ SHNM G@R @ TMHPTD P RNKTSHNM -DWS 2DBSHNM   QDB@KKR SGD DRRDMSH@K BNMBDOSR NE LD@M RPT@QD B@KBTKTR VGHBG K@X SGD ENTMC@SHNM ENQ SGD ENKKNVHMF CHRBTR RHNMR (M 2DB   SGD BNMBDOS NE LD@M RPT@QD RNKTSHNMR HR RSTCHDC SNFDSGDQ VHSG SGD BNMCHSHNMR VGHBG @ FHUDM Q@MCNL CHƤDQDMSH@K DPT@SHNM LTRS ONRRDRR HM NQCDQ SN G@UD @ TMHPTD LD@M RPT@QD RNKTSHNM R @M DWBTQRHNM 2DBSHNM   OQNUHCDR @ AQHDE OQHLDQ NM (S¯ŗR RSNBG@RSHB HMSDFQ@SHNM VHSG QDRODBS SN VGHSD MNHRD VGHBG HR DRRDMSH@KKX @ LD@M RPT@QD B@KBTKTR %HM@KKX 2DBSHNM   VQ@OR TO SGD BNMSDMSR NE SGHR BG@OSDQ 3GD L@HM RNTQBD ENQ SGHR BG@OSDQ HR :< BG@O SDQ 

  P 2NKTSHNMR NE 1@MCNL #HƤDQDMSH@K $PT@SHNMR $MFHMDDQHMF @MC SGD M@STQ@K RBHDMBDR TRD SGD BNMBDOSR NE O@SG VHRD RNKTSHNMR SN CDRBQHAD OQNBDRRDR NM Rd @R O@SG VHRD RNKTSHNMR NE CXt = f (Xt (·), t, ω) , X˙ t := Ct

Xt (·) ∈ Rd ,

 

VGHBG LD@MR SG@S SGDX @QD RNKTSHNMR NE SGD MNM @TSNMNLNTR CDSDQLHMHRSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNM x˙ =

Cx = Fω (x, t) , Ct

x := Xt (ω) ∈ Rd

 

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Xt (·) ∈ Rd ,

t ∈ I.

.E BNTQRD O@SG VHRD RNKTSHNMR NE   @QD @KRN P RNKTSHNMR ATS P RNKTSHNMR CN MNS MDDC SN AD O@SG VHRD RNKTSHNMR @R SGD ENKKNVHMF DW@LOKD RGNVR $W@LOKD    P 2NKTSHNM 3G@S HR -NS @ /@SG 6HRD 2NKTSHNM BE :< OO  +DS Z AD @ TMHENQLKX CHRSQHATSDC Q@MCNL U@QH@AKD NM (0, 1) 3GD RSNBG@RSHB OQNBDRR Xt VHSG SGD QD@KHR@SHNMR # 0, t = Z(ω) · n−1 , ENQ n = 1, 2, . . . Xt (ω) = Z(ω) · t2 , DKRDVGDQD HR @ P RNKTSHNM NE SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM X˙ t = 2 · Z · t NM [0, ∞) 3GHR B@M AD UDQHƥDC @R ENKKNVR %HW t > 0 @MC KDS . / Ωt := ω ∈ Ω : Z(ω)n−1 = t ENQ @M n ∈ N . 3GDM P(Ωt ) = 1 @MC HE ω ∈ Ω \ Ωt SGDQD HR @M n∗ (t, ω) ∈ N RTBG SG@S t · n∗ < Z(ω) < t · (n∗ + 1) 'DMBD SGDQD HR @ QD@K U@KTD H > 0 RTBG SG@S (t + h) · n∗ < Z(ω) < (t + h) · (n∗ + 1) ENQ @KK h ∈ (−H, H) GNKCR "NMRDPTDMSKX ω ∈ / Ωt+h ENQ @KK h ∈ (−H, H) 6D ƥM@KKX FDS Xt+h (ω) − Xt (ω) Z(ω)(t + h)2 − Z(ω)t2 CXt (ω) = KHL = KHL = 2Z(ω)t h→0 h→0 Ct h h t (ω) = 0 3GTR NM I = [0, ∞) VD ENQ @KK ω ∈ / Ωt %NQ t = 0 VD SQHUH@KKX G@UD CXCt G@UD SG@S Xt HR @ P RNKTSHNM .M SGD NSGDQ G@MC HM @M @QAHSQ@QX HMSDQU@K [0, b) b > 0 DUDQX QD@KHR@SHNM Xt (ω) G@R BNTMS@AKX L@MX t U@KTDR tn = n−1 Z(ω) ENQ VGHBG HS HR CHRBNMSHMT NTR @MC CNDR MNS G@UD @ CDQHU@SHUD 3GTR Xt HR MNS @ O@SG VHRD RNKTSHNM NE SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM X˙ t = 2 · Z · t MNS DUDM HM SGD DWSDMCDC RDMRD

R SGHR DW@LOKD HKKTRSQ@SDR QD@KHR@SHNMR NE P RNKTSHNMR B@M AD PTHSD HQQDF TK@Q 3GNTFG P RNKTSHNMR VHSG BNMSHMTNTRKX CHƤDQDMSH@AKD QD@KHR@SHNMR @QD O@SG VHRD RNKTSHNMR SNN 3GDNQDL   P 2NKTSHNMR @QD /@SG 6HRD 2NKTSHNMR HE 3GDHQ 1D@KHR@SHNMR @QD /@SG 6HRD C 1  )RU DOPRVW DOO ω ∈ Ω OHW f (x, t, ω) EH FRQWLQXRXV RQ Rd × I DQG APHDVXUDEOH IRU HYHU\ Ʋ[HG (x, t) /HW Xt EH D PVROXWLRQ RI  RQ I VXFK WKDW LW KDV D SDWKZLVH FRQWLQXRXV SDWKZLWK GHULYDWLYH X˙ t RQ I 7KHQ Xt LV D SDWKZLVH VROXWLRQ RI   WRR 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS 3URRI %NKKNVHMF :< O  VD G@UD SG@S @BBNQCHMF SN +DLL@   SGDQD I

HR @ O@SG VHRD BNMSHMTNTR RSNBG@RSHB OQNBDRR Zt RTBG SG@S Zt = f (Xt , t, ω) I

,NQDNUDQ SGDQD HR @ O@SG VHRD BNMSHMTNTR RSNBG@RSHB OQNBDRR Yt RTBG SG@S Yt = X˙ t @MC CTD SN SGD CDƥMHSHNM NE @ P RNKTSHNM Xt  Yt = ˆ X˙ t =f ˆ (Xt , t, ω)=Z ˆ t  ENQ t ∈ I 3GD O@SG VHRD BNMSHMTHSX NE Yt @MC Zt MNV HLOKX I I I X˙ t = Yt = Zt = f (Xt , t, ω) .

3GHR RGNVR SGD @RRDQSHNM 3N OQNODQKX RDS TO SGD MNSHNM NE @ LD@M RPT@QD NQ PT@CQ@SHB LD@M q.m. RNKTSHNM NE @ Q@MCNL CHƤDQDMSH@K DPT@SHNM VD ƥQRS QDB@KK RNLD HLONQS@MS QD RTKSR NE LD@M RPT@QD @M@KXRHR ENQ RDBNMC NQCDQ RSNBG@RSHB OQNBDRRDR !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   0 &HUD SGD CDƥMHSHNM NE @ P RNKTSHNM 0 'NV @QD P RNKTSHNMR @MC O@SG VHRD RNKTSHNMR BNMMDBSDC 0 'NV HR SGHR BNMMDBSHNM OQNUDM 0 &HUD @M DW@LOKD ENQ @ Q@MCNL CHƤDQDMSH@K DPT@SHNM SG@S G@R @ P RNKTSHNM ATS MN O@SG VHRD RNKTSHNM

  1DUHDV ,D@M 2PT@QD M@KXRHR NE 2DBNMC .QCDQ /QNBDRRDR %NQ @ RDPTDMBD {Xn }n∈N ⊂ L2d BNMUDQFDMBD SNV@QCR X ∈ L2d HM PHDQVTXDUH NQ TXDGUDWLFPHDQ q.m. HR CDƥMDC UH@   q.m. KHL Xn = ˆ X :⇔ KHL E Xn − X 2 = 0 . n→∞

n→∞

RSNBG@RSHB OQNBDRR Xt  t ∈ I VHSG Xt ∈ L2d ENQ @KK t ∈ I HR B@KKDC @ VWRFKDVWLF SURFHVV RI VHFRQG RUGHU NQ RGNQS OQNBDRR NE RDBNMC NQCDQ %NQ SGD QDL@HMCDQ NE SGHR RDBSHNM KDS Xt @KV@XR CDMNSD @ OQNBDRR NE RDBNMC NQCDQ 3GD OQNBDRR Xt HR B@KKDC q.m.FRQWLQXRXV @S t HE ˆ Xt q.m. KHL Xt+h = h→0



&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF GNKCR (E SGDQD DWHRSR @ RSNBG@RSHB OQNBDRR Yt NE RDBNMC NQCDQ RTBG SG@S ˆ Yt , q.m. KHL h1 (Xt+h − Xt ) = h→0

b SGDM Yt HR B@KKDC q.m.GHULYDWLYH NE Xt 1HDL@MM q.m.LQWHJUDO −a Xt g(t)Ct VHSG @ QD@K ETMBSHNM g : I → R HR CDƥMDC @M@KNFNTRKX SN SGD TRT@K HMSDFQ@K @R SGD KHLHS m  q.m. KHL g(tj )Xtj (tj+1 − tj ) m→∞

j=1

SNFDSGDQ VHSG @ O@QSHSHNMHMF RDPTDMBD a = t1 < t2 < · · · < tm+1 = b @MC L@W(tj+1 − tj ) → 0 OQNUHCDC SGHR KHLHS DWHRSR @MC HR HMCDODMCDMS NE SGD O@QSHSHNMHMF RDPTDMBD TRDC (E ENQ DUDQX RDPTDMBD am → −∞ HS GNKCR SG@S

b KHL − Xt g(t)Ct = X ∈ L2d , m→∞ a m

b SGDM VD B@KK X = −−∞ Xt g(t)Ct @M LPSURSHU q.m.LQWHJUDO %HM@KKX HE q.m. KHLm→∞ Xm =X ˆ SGDM KHLm→∞ E(Xm ) = E(X) GNKCR (M O@QSHBTK@Q q.m. CHƤDQDMSH@SHNM @MC 1HDL@MM q.m. HMSDFQ@SHNM SGTR BNLLTSD VHSG S@JHMF SGD DWODBS@SHNM OQNUHCDC SGD CDQHU@SHUD NQ SGD HMSDFQ@K DWHRSR QD RODBSHUDKX (M SGD QDL@HMCDQ NE SGHR RDBSHNM VD @QD NMKX BNMRHCDQHMF RDBNMC NQCDQ (1) (d) OQNBDRRDR Xt = (Xt , . . . , Xt )T ∈ Rd SG@S G@UD U@MHRGHMF LD@M @MC ENQ VGHBG ΓX (t, s) CDMNSDR SGD UDBSNQ U@KTDC @TSN BNQQDK@SHNM ETMBSHNMR VHSG (l) (l) (l) SGDHQ BNLONMDMSR FHUDM @R ΓX (t, s) = E(Xs Xt ) l = 1, 2, . . . , d %NQ OQN BDRRDR Yt RTBG SG@S E(Yt ) = 0 CDƥMD Xt := Yt − E(Yt )  3GD ENKKNVHMF MNM BNLOQDGDMRHUD KHRS RTLL@QHYDR RNLD HLONQS@MS QDRTKSR NM RDBNMC NQCDQ RSNBG@RSHB OQNBDRRDR BE :< OO  @R VDKK @R :< :< @MC : 0 LW KROGV WKDW     KHL P Δ2n − E Δ2n ≥ ε = 0 .

 

n→∞

7KLV W\SH RI FRQYHUJHQFH LV FDOOHG BNMUDQFDMBD HM OQNA@AHKHSX 3URRI $PT@SHNM   HR @ BNMRDPTDMBD NE "GDAXRDUŗR BE /QNONRH    HMDPT@KHSX   SHNM   SG@S QD@CR HM NTQ RDSSHMF @R P Δ2n − E Δ2n ≥ ε ≤ ε−2 Var Δ2n 6D G@UD n    2 !  Var Δ2n = Var Wti − Wti−1 i=1

=

n  



Wti − Wti−1

E

i=1

≤

n  i=1



E



Wti − Wti−1

4 !

4 !

− E

(∗)

= √



Wti − Wti−1

n 2πΔt



2 !!2



 x2 Cx x4 DWO − 2Δt −∞ ∞

&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF = 3n (Δt)2 =

3T 2 → 0, n

@R n → ∞ ,

@R SGD 6HDMDQ HMBQDLDMSR Wti − Wti−1 @QD HMCDODMCDMS MNQL@KKX CHRSQHATSDC Q@MCNL U@QH@AKDR 3GD HCDMSHSX (∗) HR CTD SN SGD DPT@K RO@BHMF NE SGD O@QSHSHNM ONHMSR @MC SGD DWOQDRRHNM NE SGD ENTQSG LNLDMS NE @ MNQL@KKX CHRSQHATSDC Q@MCNL U@QH@AKD AX HSR CHRSQHATSHNM 'DMBD SGD @RRDQSHNM   ENKKNVR R @ BNMRDPTDMBD VD G@UD SG@S n  2 !    P = T, KHL Δ2n = KHL E Δ2n = KHL E Wti − Wti−1

n→∞

n→∞

n→∞

 

i=1

GNKCR @MC GDMBD KHL

n 

n→∞

  P Wti−1 Wti − Wti−1 =

1 2 WT

− 12 T .

i=1

6GDM VD BNMRHCDQ SGD RPT@QDC 6HDMDQ HMBQDLDMSR @MC SGTR @ JHMC NE LD@M RPT@QD KHLHS VD B@M řS@LDŚ SGD TMANTMCDC U@QH@SHNM NE SGD 6HDMDQ /QNBDRR @MC F@HM @ OQNODQ CDƥMHSHNM NE RSNBG@RSHB HMSDFQ@KR 3GHR B@M AD @OOKHDC SN OQNODQKX RDS TO SGD (S¯ RSNBG@RSHB HMSDFQ@K RDD D F  :< MNSGDQ BNMRD PTDMBD NE S@JHMF KHLHSR HM SGD LD@M HR SG@S SGD KHLHS MN KNMFDQ CDODMCR ONHMS VHRD NM SGD ω ∈ Ω ATS GNKCR ENQ SGD VGNKD Ω -NSD VD VHKK RDS TO SGD RSNBG@RSHB (S¯ HMSDFQ@K VHSG QDRODBS SN SGD KHLHS HM OQNA@AHKHSX .E BNTQRD @R ITRS HMCHB@SDC @ LD@M RPT@QD KHLHS VHKK VNQJ SNN (MCDDC (S¯ŗR B@KBTKTR HR HMSQHMRHB@KKX @ LD@M RPT@QD B@KBTKTR AX UHQSTD NE (S¯ŗR ENQLTK@ -DWS SN SGD RDBNMC SDQL HM   n 

  Wti Wti − Wti−1 =

1 2

i=1

n 

2 !  Wt2i − Wt2i−1 + Wti − Wti−1

i=1 2 1 2 WT

=

P

+ 12 Δ2n −→

2 1 2 WT

+ 12 T .

 

3GHR ƥM@KKX KD@CR SN KHL Snλ =

KHL

n→∞

n→∞

P

1 2 WT

=

n  

   (1 − λ)Wti−1 + λWti · Wti − Wti−1

i=1

  + λ − 12 T .

.AUHNTRKX CHƤDQDMS BGNHBDR NE SGD ONHMSR HM SGD O@QSHSHNM HMSDQU@KR ENQ SGD DU@KT@SHNM NE SGD HMSDFQ@MC KD@C SN CHƤDQDMS QDRTKSR (S HR @OO@QDMS EQNL SGDRD 

1DB@KK E(X(ω)) =

6HFWLRQ 

 Ω

X(ω)CP(ω)



3NAH@R 1HFNHO  %KNQH@M 5XSS B@KBTK@SHNMR SG@S SGD TRT@K QTKDR NE CHƤDQDMSH@K @MC HMSDFQ@K B@KBTKTR @QD MNS RTHS@AKD SN G@MCKD RSNBG@RSHB HMSDFQ@KR HM @ RSQ@HFGSENQV@QC L@MMDQ 3N L@JD RDMRD NE @ RSNBG@RSHB HMSDFQ@K HM SGD V@X NTSKHMDC @ANUD VD G@UD SN CDƥMD SGD BGNHBD NE SGD DU@KT@SHNM QTKD TOEQNMS H D  SGD U@KTD NE λ 4RT@KKX NMKX SVN SXODR NE DU@KT@SHNM QTKDR @QD VHCDKX TRDC  λ = 0 VGHBG KD@CR SN (S¯ŗR HMSDQ OQDS@SHNM NE SGD RSNBG@RSHB HMSDFQ@K ŕ @MC HR SGD NMKX ONRRHAKD BGNHBD RTBG SG@S SGD HMSDFQ@K HR @ L@QSHMF@KD ŕ @MC λ = 12 VGHBG KD@CR SN 2SQ@SNMNUHBGŗR HMSDQOQDS@SHNM NE SGD RSNBG@RSHB HMSDFQ@K ŕ @MC HR SGD NMKX ONRRHAKD BGNHBD RTBG SG@S SGD QTKDR NE BK@RRHB@K B@KBTKTR QDL@HM U@KHC !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM    Ŕ (MSDFQ@SHNM VHSG 1DRODBS SN 6GHSD -NHRD 1DB@KK SGD DRRDMSH@K RSDOR NE SGD @ANUD DWONRHSHNM @MC SGDHQ JDX QDPTHQDLDMSR  0 6GHBG NE SGD ENKKNVHMF OQNODQSHDR NE SGD 1HDL@MM RTLR ni=1 Wξ (Wti − Wti−1 ) VHSG ξ ∈ [ti−1 , ti ] @QD MNS U@KHC HM SGD KHLHS L@Wi (ti − ti−1 ) < ! (m) (m) (m) NE  (MCDODMCDMBD NE SGD RDPTDMBD 0 = t0 < t1 < · · · < tn = T m∈N

O@QSHSHNMR NE [0, T ]  (MCDODMCDMBD NE SGD BGNHBD NE SGD DU@KT@SHNM ONHMS ξ ∈ [ti−1 , ti ]

0 6G@S @CCHSHNM@K BNMUDMSHNMR @QD RDS HM RSNBG@RSHB HMSDFQ@SHNM SGDNQX SN L@JD TO ENQ SGD QN@CAKNBJ QN@CAKNBJR RS@SDC HM 0 0 6G@S @QD SGD U@KTDR NE (ti − ti−1 )(Wti − Wti−1 ) @MC (Wti − Wti−1 ) HM OQNA @AHKHSX @R VDKK @R HM DWODBS@SHNM

   (MSQNCTBHMF SGD # (S¯  2SQ@SNMNUHBG 2SNBG@RSHB (MSDFQ@K +DS Wt  t ≥ 0 AD @ 1 CHLDMRHNM@K 6HDMDQ OQNBDRR NM @ OQNA@AHKHSX RO@BD (Ω, F, P) +DS Ft ⊂ F  t ≥ 0 AD @M HMBQD@RHMF E@LHKX NE σ @KFDAQ@R HM F H D  Fs ⊂ Ft ⊂ F HE s < t RTBG SG@S ENQ @KK t ≥ 0 HS GNKCR Ş A(Ws : 0 ≤ s ≤ t) ⊆ Ft  @MC Ş A(Wt+s − Wt : s ≥ 0) HR HMCDODMCDMS NE Ft



(M :< :< @MC :< @ ATMBG NE ETQSGDQ QTKDR U@QXHMF EQNL (S¯ŗR @MC 2SQ@SNMNUHBGŗR HMSDQOQDS@SHNM @QD BNMRHCDQDC



&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF .MD B@M S@JD ENQ HMRS@MBD SGD 6HDMDQ ƥKSQ@SHNM Ft := A(Ws : 0 ≤ s ≤ t) @R RTBG @M HMBQD@RHMF E@LHKX @MC RGNTKC HMENQL@KKX SGHMJ NE Ft @R řBNMS@HMHMF @KK HMENQL@SHNM @U@HK@AKD @S SHLD tŚ Ft := A(Ws : 0 ≤ s ≤ t) HR SGD RL@KKDRS E@L HKX NE σ @KFDAQ@R VHSG SGDRD OQNODQSHDR 'DQD Ft HR FDMDQ@SDC AX SGD 6HDMDQ OQNBDRR (S HR NESDM MDBDRR@QX @MC CDRHQ@AKD SN @TFLDMS A(Ws : 0 ≤ s ≤ t) VHSG NSGDQ DUDMSR SG@S @QD HMCDODMCDMS NE A(Ws : 0 ≤ s < ∞) ENQ HMRS@MBD HMHSH@K BNMCHSHNMR (M SGD B@RD NE RSNBG@RSHB CHƤDQDMSH@K DPT@SHNMR VD VHKK TRT@KKX S@JD Ft := A(Ws , X0 : 0 ≤ s ≤ t) VGDQD X0 HR @ Q@MCNL U@QH@AKD HMCDODMCDMS NE A(Ws : 0 ≤ s < ∞) 3GD σ @KFDAQ@ A(Ws : 0 ≤ s ≤ t) HR B@KKDC SGD KLVWRU\ NE SGD 6HDMDQ OQNBDRR TO SN @MC HMBKTCHMF SHLD t 3GD σ @KFDAQ@ A(Wt+s − Wt : s ≥ 0) HR SGD IXWXUH NE SGD 6HDMDQ OQNBDRR ADXNMC SHLD t RSNBG@RSHB OQNBDRR f (t, ω) CDƥMDC ENQ 0 ≤ t ≤ T < ∞ HR B@KKDC DGDSWHG VHSG QDRODBS SN Ft HE ENQ D@BG t ∈ [0, T ] f (t, ω) HR Ft LD@RTQ@AKD 2SNBG@RSHB OQNBDRRDR f (t, ω) SG@S @QD @C@OSDC SN SGD 6HDMDQ ƥKSQ@SHNM Ft  @QD HMCDODMCDMS NE SGD HMBQDLDMSR NE SGD 6HDMDQ OQNBDRR Wt,ω řHM SGD ETSTQDŚ ( D  f (t, ω) HR HMCDODMCDMS NE Wt+s,ω − Wt,ω ENQ @KK s > 0 %NQ HMRS@MBD HE f (x) HR @M HMSDFQ@AKD CDSDQLHMHRSHB ETMBSHNM SGDM SGD ETMBSHNMR f (Wt,ω ) @MC t 0 f (Ws,ω )Cs @QD Ft @C@OSDC #DƥMHSHNM   3GD "K@RR NE C@OSDC %TMBSHNMR 6D CDMNSD AX Mω2 [0, T ] SGD BK@RR NE Ft @C@OSDC RSNBG@RSHB OQNBDRRDR f (t, ω) NM SGD HMSDQU@K [0, T ] RTBG SG@S

T   E f 2 (s, ω) Cs < ∞ . 0

R LNSHU@SDC HM 2DB    HM NQCDQ SN FHUD SGD KHLHS NE SGD 1HDL@MM RTLR ENQ SGD RSNBG@RSHB HMSDFQ@K LD@MHMF HMCDODMCDMS NE @M @QAHSQ@QX BGNHBD NE SGD DU@KT@SHNM ONHMSR VD @KV@XR G@UD SN BK@QHEX TOEQNMS VGHBG DU@KT@SHNM ONHMSR VD @QD BNMRHCDQHMF 3GHR HR @M @CCHSHNM@K QTKD ENQ SGD RSNBG@RSHB HMSDFQ@K (M OQHMBHOKD SVN RTBG QTKDR @QD BNLLNMKX TRDC Ş λ = 0 $U@KT@SHNM @S SGD RS@QS ONHMS NE SGD O@QSHSHNM HMSDQU@KR (S¯ŗR HMSDQ OQDS@SHNM Ş λ = 12  $U@KT@SHNM HM SGD LHCCKD NE SGD O@QSHSHNM HMSDQU@KR 2SQ@SNMNUHBGŗR HMSDQOQDS@SHNM 6D RS@QS VHSG RNLD ƥQRS CHRBTRRHNMR NM (S¯ŗR HMSDQOQDS@SHNM QTKD 3GD 2SNBG@RSHB (S¯ (MSDFQ@K (MSDFQ@SHNM VHSG QDRODBS SN VGHSD MNHRD HR CDƥMDC HM SGHR BK@RR Mω2 [0, T ] NE RSNBG@RSHB OQNBDRRDR @C@OSDC SN SGD 6HDMDQ ƥKSQ@SHNM (S¯ŗR BNMRSQTBSHNM NE SGD 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS HMSDFQ@K NE @ ETMBSHNM f (t, ω) ∈ Mω2 [0, T ] HR RHLHK@Q SN SGD λ HMSDFQ@K HM 2DB    VHSG λ = 0 @MC BGNNRDR SGD DU@KT@SHNM ONHMSR @S SGD RS@QS ONHMS NE SGD O@QSHSHNM HMSDQU@KR %NQ @MX O@QSHSHNM 0 = t0 < t1 < · · · < tn = T  VD ENQL SGD ,Wµ VXP Sn :=

n 

  f (ti−1 , ω) Wti ,ω − Wti−1 ,ω .

 

i=1

-NSD SG@S SGD HMBQDLDMS Wti ,ω −Wti−1 ,ω HR HMCDODMCDMS NE f (ti−1 , ω) ADB@TRD f (t, ω) HR Ft @C@OSDC .MD B@M RGNV RDD D F  :< SG@S ENQ @MX RDPTDMBD NE O@QSHSHNMR NE SGD HMSDQU@K RTBG SG@S L@Wi (ti − ti−1 ) → 0 SGD RDPTDMBD {Sn (t, ω)} BNMUDQFDR SN SGD R@LD KHLHS CDMNSDC

T 0

P

f (t, ω)CWt,ω :=

KHL

L@Wi (ti −ti−1 )→0

Sn

 

@MC B@KKDC ,Wµ LQWHJUDO NE f (t, ω) ,NQDNUDQ NMD B@M RGNV SG@S SGD BNMUDQ FDMBD HM   HR TMHENQL HM t VHSG OQNA@AHKHSX NMD H D NM @KLNRS DUDQX SQ@ IDBSNQX NE SGD 6HDMDQ OQNBDRR Wt,ω  RDD D F  :< 3GD (S¯ HMSDFQ@K HR @ Ft @C@OSDC RSNBG@RSHB OQNBDRR HM Ω (S S@JDR CHƤDQDMS U@KTDR NM CHƤDQDMS QD@KH R@SHNMR ω NE SGD SQ@IDBSNQHDR NE SGD 6HDMDQ OQNBDRR (M O@QSHBTK@Q HS HR D@RX SN RGNV SG@S ENQ @MX HMSDFQ@AKD CDSDQLHMHRSHB ETMBSHNM f  T

T 2 f (t)CWt ∼ N 0 , f (t) Ct 0

0

GNKCR 3GD NTSRS@MCHMF BG@Q@BSDQHRSHB NE SGD (S¯ HMSDFQ@K HR SG@S HS HR @ L@QSHMF@KD LNMF @KK λ HMSDFQ@KR HM 2DB    (S¯ŗR HMSDQOQDS@SHNM HR BG@Q@BSDQHRDC AX SGD E@BS SG@S @R @ ETMBSHNM NE SGD TOODQ KHLHS HS HR @ L@QSHMF@KD 3GHR B@M AD HKKTR SQ@SDC @R ENKKNVR RDD :< O  !X S@JHMF Xt := 12 Wt2 + (λ − 12 )t SGDM ENQ t ≥ s VD G@UD @KLNRS RTQDKX SG@S 1 1 1 1 E(Xt | A(Xu : u ≤ s)) = E(Wt2 | A( Wu2 + (λ − )u : u ≤ s)) + (λ − )t 2 2 2 2 1 1 1 2 1 2 = E(E(Wt |A(Wu : u ≤ s)) | A( Wu + (λ − )u : u ≤ s)) + (λ − )t 2 2 2 2 1 1 1 1 = E(E(Wt2 | A(Ws : s ≤ t)) | A( Wu2 + (λ − )u : u ≤ s)) + (λ − )t 2 2 2 2 1 1 1 2 1 2 = E(t − s + Ws | A( Wu + (λ − )u : u ≤ s)) + (λ − )t 2 2 2 2 1 2 1 1 = Ws + (t − s) + (λ − )t 2 2 2 = Xs + λ(t − s) ,



&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF VGDQD VD G@UD TRDC SGD U@QHNTR OQNODQSHDR NE SGD BNMCHSHNM@K DWODBS@SHNM @MC NE @ 6HDMDQ OQNBDRR 3GD OQNBDRR Xt HR SGDQDENQD @ L@QSHMF@KD  H D  E(Xt |A(Xu : 0 ≤ u ≤ s)) = Xs

@ R ,

HE @MC NMKX HE λ = 0 GDMBD ENQ (S¯ŗR BGNHBD NE SGD DU@KT@SHNM ONHMSR R VD G@UD @KQD@CX RDDM HM 2DB   

T 0

Wt Ct =

2 1 2 WT

− 12 T .

'DMBD (S¯ŗR HMSDFQ@K KD@CR SN @ QDRTKS CHƤDQDMS EQNL SG@S HLOKHDC AX BK@RRHB@K B@KBTKTR 3GD BNQQDBSHNM SDQL − 12 T HR DRRDMSH@K ENQ SGD L@QSHMF@KD OQNODQSX NE RNKTSHNMR NE (S¯ HMSDFQ@KR SGNTFG HS @KRN QDRTKSR HM řMDVŚ QTKDR NE B@KBTKTR (M O@QSHBTK@Q SGDRD MDV QTKDR ADBNLD @OO@QDMS VGDM CHRBTRRHMF RSNBG@RSHB (S¯ CHƤDQDMSH@KR 2SNBG@RSHB #HƤDQDMSH@KR  (S¯ŗR %NQLTK@ &HUDM @ RB@K@Q 6HDMDQ OQNBDRR Wt @MC @ RB@K@Q @S KD@RS SVHBD CHƤDQDMSH@AKD ETMBSHNM g(x) VG@S HR Cg(Wt ) %NQ SGHR OTQONRD KDS TR ENKKNV :< O  @MC BNMRHCDQ SGD 3@XKNQ DWO@MRHNM NE F g(Wt + CWt ) = g(Wt ) + gx (Wt )CWt + 12 gxx (Wt )(CWt )2 + . . .

Cg(Wt ) = g(Wt + CWt ) − g(Wt ) = gx (Wt )CWt + 12 gxx (Wt )(CWt )2 + . . .

(E Wt VNTKC AD @ CDSDQLHMHRSHB ETMBSHNM SGD SDQL (CWt )2 BNTKC AD MDFKDBSDC @R @ SDQL NE GHFGDQ NQCDQ (M SGD RSNBG@RSHB B@RD SGNTFG CWt = Wt+Ct − Wt

⇒

(CWt )2 = (Wt+Ct − Wt )2

GNKC @MC @BBNQCHMF SN SGD OQNODQSHDR NE SGD 6HDMDQ OQNBDRR VD FDS VHSG       P (CWt )2 = E (CWt )2 = E (Wt+Ct − Wt )2 = Ct . 'DMBD SGD SDQL (CWt )2 HR NE ƥQRS NQCDQ HM OQNA@AHKHSX @MC B@MMNS AD MD FKDBSDC (M O@QSHBTK@Q VD B@M ENQL@KKX RDS TO SGD ENKKNVHMF PXOWLSOLFDWLRQ WD EOH Ct · Ct = 0 , Ct · CWt = 0 , CWt · CWt = Ct . 

3N OQNUD SGHR L@QSHMF@KD OQNODQSX ENQ ř@QAHSQ@QXŚ HMSDFQ@MCR RNLD DƤNQS G@R SN AD OTS HMSN CDƥMHMF LNQD OQDBHRDKX VG@S BNMCHSHNMR G@UD SN AD HLONRDC NM SGD HMSDFQ@MCR (K KTRSQ@SHUDKX SGD HMSDFQ@MCR G@UD SN AD RTBG SG@S ENQ D@BG SHLD SGDQD HR MN HMENQL@SHNM MDDCDC SG@S VHKK AD @BBDRRHAKD NMKX HM @ σ @KFDAQ@ FDMDQ@SDC AX RNLD 6HDMDQ OQNBDRR HM SGD ETSTQD

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3NAH@R 1HFNHO  %KNQH@M 5XSS 3GTR SGD ENQLTK@ Cg(Wt ) = gx (Wt )CWt + 12 gxx (Wt )Ct

 

GNKCR HM SGD RSNBG@RSHB B@RD 6D VHKK OQNUD SGHR QHFNQNTRKX UH@ (S¯ŗR ENQLTK@ !DKNV VD FHUD SGD CDƥMHSHNM NE @ (S¯ RSNBG@RSHB CHƤDQDMSH@K ENKKNVDC AX BHSHMF @ 1 CHLDMRHNM@K UDQRHNM NE (S¯ŗR ENQLTK@ HM NQCDQ SN GHFGKHFGS SGD CHƤDQ DMBDR SN BK@RRHB@K CHƤDQDMSH@K @MC HMSDFQ@K B@KBTKTR #DƥMHSHNM   2SNBG@RSHB #HƤDQDMSH@K +DS Xt  0 ≤ t ≤ T  AD @ RSNBG@RSHB OQNBDRR RTBG SG@S ENQ @MX 0 ≤ t1 < t2 ≤ T

X t2 − Xt1 =

t2

a(t)Ct +

t1

t2

b(t)CWt , t1

 VGDQD |a|, b ∈ Mω2 [0, T ] 3GDM VD R@X SG@S Xt G@R @ VWRFKDVWLF GLƱHUHQWLDO CXt NM [0, T ] FHUDM AX CXt = a(t)Ct + b(t)CWt . .ARDQUD SG@S Xt HR @M Ft @C@OSDC RSNBG@RSHB ETMBSHNM SG@S ADKNMFR SN Mω2 [0, T ] $PT@SHNM   @MC SGD BG@HM QTKD NE CDSDQLHMHRSHB HMSDFQ@K @MC CHƤDQDMSH@K B@KBTKTR HLLDCH@SDKX HLOKX (S¯ŗR BDKDAQ@SDC ENQLTK@ SGD @M@KNF SN SGD BG@HM QTKD HM BK@RRHB@K B@KBTKTR (M HSR RHLOKDRS ENQL HS QD@CR @R +DLL@   (S¯R %NQLTK@ # /HW GXt = a(t)Gt + b(t)GWt IRU t ∈ [0, T ] DQG u : R × [0, T ] → R EH RQFH FRQWLQXRXVO\ GLƱHUHQWLDEOH LQ t ≥ 0 DQG WZLFH FRQWLQXRXVO\ GLƱHUHQWLDEOH LQ x ∈ R 7KHQ u(Xt , t) KDV D VWRFKDVWLF GLƱHUHQWLDO JLYHQ E\   Gu(Xt , t) = ut (Xt , t) + a(t)ux (Xt , t) + 12 b2 (t)uxx (Xt , t) Gt   + b(t)ux (Xt , t)GWt , 3URRI 2DD :< OO  3GHR B@M AD DWSDMCDC SN ETMBSHNMR u : Rn → R @MC KD@CR TR SN SGD FDM DQ@K ENQLTK@SHNM NE ,WµśV IRUPXOD HMUNKUHMF  CHLDMRHNM@K RSNBG@RSHB OQNBDRRDR (1) (m) Xt , . . . , X t  (i)

3GDNQDL   (S¯R %NQLTK@ # /HW GXt = ai (t)Gt + bi (t)GWt  i = 1, 2, . . . , m DQG OHW u(x1 , . . . , xm , t) EH RQFH FRQWLQXRXVO\ GLƱHUHQWLDEOH LQ t ≥



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 0 DQG WZLFH FRQWLQXRXVO\ GLƱHUHQWLDEOH LQ x = (x1 , . . . , xm ) ∈ Rm  7KHQ (1) (m) u(Xt , . . . , Xt ) KDV D VWRFKDVWLF GLƱHUHQWLDO JLYHQ E\ ⎛ ⎞ m m   1 Gu(Xt , t) = ⎝ut (Xt , t) + ai (t)uxi (Xt , t) + bi (t)bj (t)uxi xj (Xt , t)⎠ Gt 2 i=1

+

m 

i,j=1

bi (t)uxi (Xt , t)GWt ,

 

i=1 (1)

(m) T ) 

ZKHUH Xt = (Xt , . . . , Xt 3URRI RDD :< OO 

R @ ƥQRS DW@LOKD VD BNMRHCDQ @ UDQX RHLOKHƥDC LNCDK ENQ ƥM@MBH@K L@Q JDSR SG@S B@M AD RDDM @R @ BQTCD @OOQNWHL@SHNM NE SGD LNCDK FHUDM AX %HRBGDQ !K@BJ @MC ,XQNM 2BGNKDR HM : 0 RQ ZKLFK WKH LQLWLDO YDOXH SUREOHP x˙ = F (t, x) ,

x(t0 ) = x0 ,

 

KDV D XQLTXH VROXWLRQ 0RUH SUHFLVHO\ /HW Q := Ia (t0 ) × Kb (x0 ) EH DQ DUELWUDU\ FRPSDFW FXERLG LQ U RQ ZKLFK F LV /LSVFKLW]FRQWLQXRXV LQ x ZLWK /LSVFKLW]FRQVWDQW L /HW δ ≤ a EH D SRVLWLYH QXPEHU VXFK WKDW δ F Q ≤ b DQG δL < 1 7KHQ WKH LQLWLDO YDOXH SUREOHP  KDV D XQLTXH VROXWLRQ ϕ RQ Iδ (t0 ) 7KLV VROXWLRQ OLHV LQ Kb (x0 ) VLQFH ZH KDYH ϕ(t) − x0 ≤ b , IRU DOO t ∈ Iδ (t0 ) ,   DQG LW LV WKH OLPLW RI WKH VHTXHQFH (ϕk )k∈N RI /HB@QC +HMCDK±E HSDQ@SHNM ETMBSHNMR ZKLFK DUH GHƲQHG DV

t ϕ0 (t) := x0 , ϕk+1 (t) := x0 + F (s, ϕ(s)) Gs .   t0

7KH VHTXHQFH (ϕk )k∈N FRQYHUJHV XQLIRUPO\ RQ Iδ (t0 ) 3URRI %NKKNVHMF :< OO  HS HR RTƧBHDMS SN BNMRSQTBS @ BNMSHMTNTR ETMB SHNM ϕ : Iδ (t0 ) → Kd RTBG SG@S ϕ R@SHRƥDR SGD HMDPT@KHSX   ENQ @KK t ∈ Iδ (t0 ) @R VDKK @R SGD HMSDFQ@K DPT@KHSX

t ϕ(t) = x0 + F (s, ϕ(s)) Cs .   t0

6D HMSDQOQDS   @R @ ƥWDC ONHMS HCDMSHSX 3GDQDENQD KDS M AD SGD RO@BD NE @KK BNMSHMTNTR ETMBSHNMR ψ : Iδ (t0 ) → Kd RTBG SG@S ψ(t) − x0 ≤ b ,

ENQ @KK t ∈ Iδ (t0 ) ,

@MC P AD SG@S L@OOHMF SG@S L@OR @ ETMBSHNM ψ ∈ M SN P ψ : Iδ (t0 ) → Kd AX

t (P ψ)(t) := x0 + F (s, ϕ(s)) Cs . t0

P ψ HR BNMSHMTNTR @MC R@SHRƥDR SGD HMDPT@KHSX & t & t

& &

&

(P ψ)(t) − x0 = & & F (s, ψ(s)) Cs& ≤ F (s, ψ(s)) Cs ≤ δ F Q ≤ b . t0

t0

3GTR ENQ ψ ∈ M VD @KRN G@UD P ψ ∈ M



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 6HSG SGD @HC NE P  SGD HMSDFQ@K DPT@KHSX   QD@CR @R SGD ƥWDC ONHMS HCDM SHSX P ϕ = ϕ (M NQCDQ SN @OOKX SGD !@M@BG ƥWDC ONHMS SGDNQDL VD HMSQNCTBD @ LDSQHB NM M @R ENKKNVR %NQ ψ1 , ψ2 ∈ M VD CDƥMD d (ψ1 , ψ2 ) := RTO { ψ1 (t) − ψ2 (t) : t ∈ Iδ (t0 )} . (M SGHR LDSQHB SGD RDPTDMBD (ψk )k∈N BNMUDQFDR HM M HE @MC NMKX HE HS BNMUDQFDR TMHENQLKX NM Iδ (t0 ) R Kb (x0 ) HR ANTMCDC HS ENKKNVR SG@S (M, d) HR @ BNLOKDSD LDSQHB RO@BD H D @ !@M@BG RO@BD ,NQDNUDQ P : M → M HR @ BNMSQ@BSHNM ADB@TRD & t & & & & d (P ψ1 , P ψ2 ) = RTO & (F (s, ψ1 (s)) − F (s, ψ2 (s))) Cs& & I

t0

t



≤ RTO F (s, ψ1 (s)) − F (s, ψ2 (s)) Cs

I

t0

I

t0

t

≤ RTO

L ψ1 (s) − ψ2 (s) Cs

≤ δ · L · d (ψ1 , ψ2 ) ,

VHSG δL < 1 .

3GTR !@M@BGŗR ƥWDC ONHMS SGDNQDL B@M AD @OOKHDC @MC SGDQD DWHRSR @ TMHPTD ETMBSHNM ϕ ∈ M RTBG SG@S P ϕ = ϕ 'DQD ϕ KHDR HM Kb (x0 ) @MC RNKUDR SGD HMHSH@K U@KTD OQNAKDL   $W@LOKD   "NMRSQTBSHNM NE @ 2NKTSHNM BE :< OO  +DS TR @OOKX SGD /HB@QC +HMCDK±E HSDQ@SHNM ENQ SGD QNS@SHNM ƥDKC v(x, y) = (−y, x) 6D @OOKX HS SN SGD HMHSH@K U@KTD OQNAKDL x˙ = −y , x˙ = x , 6D NAS@HM

 ϕ1 (t) = 

1 0

+

VHSG x(0) = 1 , VHSG y(0) = 0 .

t 0

t

0 1



 Cs =

1 t 



1 − 12 t2 ϕ2 (t) = + Cs = t 0  t  −s 1 1 − 12 t2 + Cs = ϕ3 (t) = 0 1 − 12 s2 t − 3!1 t3 0 1 0

−s 1



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3NAH@R 1HFNHO  %KNQH@M 5XSS   1 2k t 1 − 2!1 t2 + − · · · + (−1)k (2k)! . ϕ2k+1 (t) = 1 t − 3!1 t3 + − · · · + (−1)k (2k+1)! t2k+1 3GD BNLONMDMSR NE ϕ2k+1 @QD SGD 3@XKNQ ONKXMNLH@KR NE SGD RHMD ETMBSHNM @MC SGD BNRHMD ETMBSHNM QDRODBSHUDKX NE NQCDQ 2k + 1 (M SGD KHLHS k → ∞ SGD RNKTSHNM NE SGD HMHSH@K U@KTD OQNAKDL QDRTKSR @R  BNR(t) ϕ(t) = . RHM(t)

  

(MSDQKTCD 2NKUHMF .#$R 2XLANKHB@KKX VHSG , 3+ !

%NQ SGD RXLANKHB RNKTSHNM NE CHƤDQDMSH@K DPT@SHNMR VHSG , 3+ ! VD B@M TRD SGD /bQHp2 BNLL@MC BE :< (M NQCDQ SN B@KBTK@SD SGD RNKTSHNM NE SGD RB@K@Q NQCHM@QX CHƤDQDMSH@K DPT@SHNM x˙ = t · x SXOD == t 4 /bQHp2UǶ.t 4 i tǶ- ǶiǶV t 4 *R 2tTURfk ikV

, 3+ ! TRDR B@OHS@K # SN HMCHB@SD SGD CDQHU@SHUD @MC QDPTHQDR SG@S SGD DMSHQD DPT@SHNM @OOD@QR HM RHMFKD PTNSDR !X CDE@TKS , 3+ ! @RRTLDR t SN AD SGD !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   Ŕ /@QS ( 0 'NV @QD HMSDFQ@K BTQUDR @MC RNKTSHNMR NE NQCHM@QX CHƤDQDMSH@K DPT@SHNMR QDK@SDC 0 (M VG@S QDRODBS @QD NQCHM@QX CHƤDQDMSH@K DPT@SHNMR @MC HMSDFQ@K DPT@SHNMR DPTHU@KDMS 0 4MCDQ VGHBG BNMCHSHNMR @QD HMSDFQ@K BTQUDR TMHPTD 0 2JDSBG SGD OQNNE NE XNTQ @RRDQSHNM EQNL 0 0 2S@SD SGD SGDNQDL NE /HB@QC +HMCDK±E 0 2JDSBG SGD OQNNE NE SGD /HB@QC +HMCDK±E SGDNQDL 6G@S HR SGD DRRDMSH@K RSDO HM SGHR OQNNE 0



OOKX SGD /HB@QC +HMCDK±E HSDQ@SHNM SN RNKUD SGD HMHSH@K U@KTD OQNAKDL x˙ = tx x(0) = 1

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF HMCDODMCDMS U@QH@AKD RN VD BNTKC G@UD TRDC SGD BNLL@MC t 4 /bQHp2UǶ.t 4 i tǶV VHSGNTS @LAHFTHSX (E VD V@MS SN TRD SGD R@LD DPT@SHNM @ MTLADQ NE SHLDR VD L@X CDƥMD HS @R @ U@QH@AKD == 2[MR 4 Ƕ.v 4 t vǶ == v 4 /bQHp2U2[MR- ǶtǶV v 4 *R 2tTURfk tkV

3N RNKUD SGD BNQQDRONMCHMF HMHSH@K U@KTD OQNAKDL y  (x) = x · y VHSG QDRODBS SN y(1) = 1 VD TRD == v 4 /bQHp2U2[MR- ǶvURV4RǶ- ǶtǶV v 4 Rf2tTURfkV 2tTURfk tkV

NQ == BMBibR 4 ǶvURV4RǶc == v 4 /bQHp2U2[MR- BMBibR- ǶtǶV v 4 Rf2tTURfkV 2tTURfk tkV

-NV SG@S VD G@UD RNKUDC SGD CHƤDQDMSH@K DPT@SHNM VD L@X V@MS SN OKNS SGD RNKTSHNM SN FDS @ QNTFG HCD@ NE HSR ADG@UHNQ 'DQD VD HLLDCH@SDKX QTM HMSN SVN OQNAKDLR H SGD DWOQDRRHNM y VD FDS EQNL , 3+ ! HR MNS RTHSDC ENQ @QQ@X NODQ@SHNMR @MC HH y HR @ RXLANK NQ RXLANKHB NAIDBS %NKKNVHMF :< SGD ƥQRS NE SGDRD NARS@BKDR HR RSQ@HFGSENQV@QC SN ƥW AX @OOKXHMF SGD p2+i`Q`Bx2 BNLL@MC %NQ SGD RDBNMC VD DLOKNX SGD 2pH BNLL@MC SG@S DU@KT@SDR NQ DWDBTSDR SDWS RSQHMFR SG@S BNMRSHSTSD U@KHC , 3+ ! BNLL@MCR 'DMBD VD B@M TRD == t 4 HBMbT+2Uy-R-kyVc == x 4 2pHU p2+iQ`Bx2UvV Vc == THQiUt-xV

SN F@HM SGD QDRTKS RGNVM HM %HF   @ "NMRHCDQ MNV SGD RNKTSHNM NE SGD RDBNMC NQCDQ DPT@SHNM y  (x) + 8y  (x) + 2y(x) = BNR(x) ,

y(0) = 0 ,

y  (0) = 1 .

3GDM SGD ENKKNVHMF RDKE DWOK@M@SNQX , 3+ ! BNCD RNKUDR SGD DPT@SHNM @MC CQ@VR %HF   A == 2[Mk 4 Ƕ.kv Y 3 .v Y k v 4 +QbUtVǶc == BMBibk 4 ǶvUyV 4 y- .vUyV 4 RǶc == v 4 /bQHp2U2[Mk- BMBibk- ǶtǶV v 4 Rfe8 +QbUtV Y 3fe8 bBMUtV Y U@RfRjyY8jfR3ky R9URfkVV 2tTUU@9YR9URfkVV tV @ RfR3ky U8jYR9URfkVV R9URfkV 2tTU@U9YR9URfkVV tV == t 4 HBMbT+2Uy-R-kyVc == x 4 2pHU p2+iQ`Bx2UvV Vc == THQiUt-xV

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0.2

0.18

0.95

0.16 0.9 0.14 0.85

0.12

0.8

0.1

0.08

0.75

0.06 0.7 0.04 0.65

0.6

0.02

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0

0.1

0.2

@

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

%HFTQD   @ &Q@OG NE SGD RNKTSHNM y NE y  (x) = x · y VHSG QDRODBS SN y(1) = 1 @MC A FQ@OG NE SGD RNKTSHNM y NE y  (x) + 8y  (x) + 2y(x) = BNR(x) VHSG QDRODBS SN y(0) = 0 y  (0) = 1

%HM@KKX VD SN RNKUD @MC OKNS RNKTSHNMR SN SGD KHMD@Q RXRSDL x˙ = x + 2y − z ,

 

y˙ = x + z ,

 

z˙ = 4x − 4y + 5z ,

 

NE SGQDD BNTOKDC DPT@SHNM BE :< 6D ƥQRS ƥMC SGD FDMDQ@K RNKTSHNM @M@K NFNTRKX SN SGD RB@K@Q B@RD 'DQD VD ITRS G@UD SN AQ@BD D@BG DPT@SHNM HM HSR NVM O@HQ NE RHMFKD PTNS@SHNM L@QJR == (t- v- x) 4 /bQHp2UǶ.t 4 t Y k v @ xǶ- Ƕ.v 4 t Y xǶǶ.x 4 9 t @ 9 v Y 8 xǶV t 4 @*R 2tTUj iV Y k *R 2tTUk iV Y k *k 2tTUk iV @ k *k 2tTUiV @ Rfk *j 2tTUj iV Y Rfk *j 2tTUiV v 4 *R 2tTUj iV @ *R 2tTUk iV Y k *k 2tTUiV @ *k 2tTUk iV Y Rfk *j 2tTUj iV @ Rfk *j 2tTUiV x 4 @9 *R 2tTUk iV Y 9 *R 2tTUj iV @ 9 *k 2tTUk iV Y 9 *k 2tTUiV @ *j 2tTUiV Y k *j 2tTUj iV

.E BNTQRD HE VD TRD , 3+ ! SN CNTAKD BGDBJ @M@KXSHB RNKTSHNMR NE SGHR RXRSDL VD G@UD SN JDDO HM LHMC SG@S , 3+ !ŗR BGNHBD NE SGD BNMRS@MSR C1  C2 @MC C3 OQNA@AKX VNTKC MNS BNQQDRONMC SN NTQ NVM BGNHBD $ F  VD LHFGS G@UD C := −2C1 + 12 C3 RTBG SG@S SGD BNDƧBHDMSR NE DWO(t) HM SGD DWOQDRRHNM ENQ x @QD BNLAHMDC %NQSTM@SDKX SGDQD HR MN RTBG @LAHFTHSX VGDM SGD HMHSH@K U@KTDR @QD @RRHFMDC 3N RNKUD @ BNQQDRONMCHMF HMHSH@K U@KTD OQNAKDL D F VHSG x(0) = 1 y(0) = 2 @MC z(0) = 3 VD B@M TRD



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 25

20

25

20

15 z−axis

15

10

10

5

0 8

5

7

1.5

6

1.4 5

1.3 1.2

4

1.1

3

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

y−axis

@

1 2

0.9

x−axis

A

%HFTQD   @ &Q@OG NE SGD RNKTSHNMR x(t) y(t) @MC z(t) NE SGD RXRSDL     NUDQ SHLD VHSG QDRODBS SN SGD HMHSH@K BNMCHSHNMR x(0) = 1 y(0) = 2 @MC z(0) = 3 @MC T T A RNKTSHNM SQ@IDBSNQX HM SGD x y z RO@BD RS@QSHMF @S (x(0), y(0), z(0)) = (1, 2, 3)

== BMBibj 4 ǶtUyV 4 R- vUyV 4 k- xUyV 4 jǶc == (t- v- x) 4 /bQHp2UǶ.t 4 t Y k v @ xǶ- Ƕ.v 4 t Y xǶǶ.x 4 9 t @ 9 v Y 8 xǶ- BMBibjV t 4 @8fk 2tTUj iV Y e 2tTUk iV @ 8fk 2tTUiV v 4 8fk 2tTUj iV @ j 2tTUk iV Y 8fk 2tTUiV x 4 @Rk 2tTUk iV Y Ry 2tTUj iV Y 8 2tTUiV

%HM@KKX VD OKNS SGHR RNKTSHNM SN NAS@HM SGD QDRTKSR CHROK@XDC HM %HF   == == == == ==

i 4 HBMbT+2Uy- X8- k8Vc tt 4 2pHU p2+iQ`Bx2UtV Vc vv 4 2pHU p2+iQ`Bx2UvV Vc xx 4 2pHU p2+iQ`Bx2UxV Vc THQiUi- tt- Ƕ@FǶ- i- vv- Ƕ,FǶ- i- xx- ǶXFǶV

@MC THQijUtt- vv- xx-Ƕ@FǶV

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 SGD RSQHBSKX LNMNSNMNTRKX CDBQD@RHMF RNKTSHNM r : R+ → (1, ∞) VHSG 3 1 r(t) = 1 − DWO(−2t) VGDQD KHLt↓0 r(t) → ∞ @MC KHLt→∞ r(t) = 1 !X SHLD RGHESR VD NAS@HM @KK L@WHL@K RNKTSHNMR NE r˙ = r(1 − r2 ) r > 0 EQNL SGDRD SGQDD SXODR %HM@KKX VD BNLAHMD SGDRD RNKTSHNMR VHSG SGD RNKTSHNM ϕ(t) = t NE ϕ˙ = 1 SN NAS@HM SGD RNKTSHNMR NE   (M SGD B@RD  SGHR KD@CR SN ODQHNCHB RNKTSHNMR @MC HM SGD B@RDR  @MC  SN ROHQ@KR SG@S SDMC SNV@QCR SGD ODQHNCHB RNKTSHNM ENQ t → ∞ RDD %HF   KK ETQSGDQ RNKTSHNMR NE   @QD NAS@HMDC AX SHLD RGHESR

  

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!DRHCDR NQCHM@QX CHƤDQDMSH@K DPT@SHNMR NE SGD ENQL x˙ = F (t, x) SG@S HM UNKUD NMKX ƥQRS NQCDQ CDQHU@SHUDR DPT@SHNMR LTRS AD BNMRHCDQDC SG@S BNMS@HM GHFGDQ NQCDQ CDQHU@SHUDR M DW@LOKD ENQ RTBG @ GHFGDQ NQCDQ NQCHM@QX CHƤDQ DMSH@K DPT@SHNM HR SGD NRBHKK@SHNM DPT@SHNM x ¨ = −x (S B@M AD SQ@MRENQLDC SN @ RXRSDL NE ƥQRS NQCDQ AX RDSSHMF x1 := x @MC x2 := x˙ 1 3GHR KD@CR SN SGD RXRSDL x˙ 1 = x2 x ¨ = −x , ⇒ x˙ 2 = −x1 VGHBG HR QDK@SDC SN SGD QNS@SHNM ƥDKC v(x1 , x2 ) = (x2 , −x1 )



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF +DS f : I × Ω → K AD @ RB@K@Q U@KTDC ETMBSHNM NM @M HMSDQU@K I @MC @ NODM RDS Ω ⊂ Kd 3GD FDMDQ@K DUDQX d SG NQCDQ NQCHM@QX CHƤDQDMSH@K DPT@SHNM ! x(d) = f t, x, x,   ˙ . . . , x(d−1) , B@M AD SQ@MRENQLDC HMSN @ d CHLDMRHNM@K RXRSDL NE ƥQRS NQCDQ DPT@SHNMR ⎫ ⎞ ⎛ x˙ 1 = x2 ⎪ ⎪ x2 ⎪ ⎪ x˙ 2 = x3 ⎪ ⎟ ⎬ ⎜

⎟ ⎜

NQ HM UDBSNQ MNS@SHNM x ˙ = F (t, x) ⎟. ⎜

⎪ ⎝ xd ⎠ ⎪ ⎪ x˙ d−1 = xd ⎪ ⎪ f (t, x) ⎭ x˙ d = f (t, x1 , x2 , . . . , xd )   3GD ETMBSHNM ϕ : I → Kd  ϕ = (ϕ1 , ϕ2 , . . . , ϕd ) HR @M HMSDFQ@K BTQUD NE SGD (d−1) RXRSDL   HE @MC NMKX HE HS G@R SGD ENQL ϕ = (ϕ1 , ϕ˙ 1 , . . . , ϕ1 ) VGDQD ϕ1 : I → K HR @ RNKTSHNM NE   .MD L@X QDBNFMHYD SG@S F HR KNB@KKX +HORBGHSY BNMSHMTNTR HM x TMHENQLKX HM t HE @MC NMKX f G@R SGHR OQNODQSX 3GTR SGD DWHRSDMBD @MC TMHPTDMDRR SGDNQDLR ENQ RXRSDLR NE ƥQRS NQCDQ DPT@SHNMR @QD @KRN DWHRSDMBD @MC TMHPTDMDRR SGDN QDLR ENQ GHFGDQ NQCDQ NQCHM@QX CHƤDQDMSH@K DPT@SHNMR %NQ GHFGDQ NQCDQ DPT@ SHNMR NMD TMTRT@KKX S@JDR x(t0 ) SNFDSGDQ VHSG SGD U@KTDR x(t ˙ 0 ), . . . , xd−1 (t0 ) NE SGD ƥQRS d − 1 CDQHU@SHUDR @R HMHSH@K BNMCHSHNMR @S SGD HMHSH@K SHLD t0 !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   Ŕ /@QS ((( 0 &HUD @ BK@RRHƥB@SHNM NE SGD HMSDFQ@K BTQUDR SG@S B@M NBBTQ HM KNB@KKX +HORBGHSY BNMSHMTNTR @TSNMNLNTR CXM@LHB@K RXRSDLR 0 2JDSBG @MC CHRBTRR SGD OG@RD ONQSQ@HS NE r˙ = r(1 + r2 ) ,

@MC

ϕ˙ = 1 .

0 1DVQHSD SGD RDBNMC NQCDQ C@LODC #TƧMF NRBHKK@SNQ DPT@SHNM x ¨ + x˙ + x + ax3 = 0 a > 0 @R @ RXRSDL NE ƥQRS NQCDQ DPT@SHNMR x = 0 a, b, c > 0 @R @ 0 1DVQHSD SGD ENQSG NQCDQ DPT@SHNM x(4) + ax3 + bx˙ + c¨ RXRSDL NE ƥQRS NQCDQ DPT@SHNMR

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3NAH@R 1HFNHO  %KNQH@M 5XSS

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x(t) = x0 +

t

f (s, x(s))Cs .

 

t0

(M NSGDQ VNQCR HE ϕ HR @ RNKTSHNM NE x˙ = f (t, x) NM RNLD HMSDQU@K I ENQ VGHBG ϕ(t0 ) = x0  SGDM x(t) = ϕ(t) VHKK R@SHREX   NM I @MC BNMUDQRDKX "KD@QKX SGD HMSDFQ@K HM   L@JDR RDMRD ENQ L@MX ETMBSHNMR f SG@S @QD MNS BNMSHMTNTR 1DB@KK SG@S SGD BNMSHMTHSX NE f FT@Q@MSDDR SG@S @ RNKTSHNM NE x˙ = f (t, x) HR BNMSHMTNTRKX CHƤDQDMSH@AKD 3GTR HE @ BNMSHMTNTRKX CHƤDQDMSH@AKD ETMBSHNM HR MNS CDL@MCDC KHJD HM SGD B@RD NE ITLO OQNBDRRDR SGD BNMSHMTHSX QDRSQHBSHNM NM f B@M AD QDK@WDC 2TOONRD f HR @ QD@K U@KTDC MNS MDBDRR@QHKX BNMSHMTNTR ETMBSHNM CDƥMDC NM RNLD RTA RDS S NE SGD (t, x) RO@BD 3GDM VD B@M DWSDMC SGD MNSHNM NE RNKTSHNMR SN SGD CHƤDQDMSH@K DPT@SHNM x˙ = f (t, x) AX @CCQDRRHMF SGD ENKKNVHMF OQNAKDL %HMC @M @ARNKTSDKX BNMSHMTNTR ETMBSHNM ϕ CDƥMDC NM @ QD@K SHLD HMSDQU@K I RTBG SG@S  (t, ϕ(t)) ∈ S ENQ @KK t ∈ I  ϕ(t) ˙ = f (t, ϕ(t)) ENQ @KK t ∈ I DWBDOS NM @ RDS NE +DADRFTD LD@RTQD YDQN (E RTBG @M HMSDQU@K I @MC @ ETMBSHNM ϕ DWHRS SGDM ϕ HR R@HC SN AD SGD VROXWLRQ RI x˙ = f (t, x) LQ WKH H[WHQGHG VHQVH RQ I -NSHBD SG@S SGD @ARNKTSD BNMSHMTHSX NE @ RNKTSHNM FT@Q@MSDDR SGD DWHRSDMBD NE ϕ˙ @KLNRS DUDQXVGDQD H D  DWBDOS NM @ RDS NE +DADRFTD LD@RTQD YDQN RN SG@S QDPTHQDLDMS  L@JDR RDMRD (E f ∈ C(S) @MC ϕ HR @ RNKTSHNM NE x˙ = f (t, x) HM SGD DWSDMCDC RDMRD NM I SGDM VD B@M HLLDCH@SDKX BNMBKTCD ϕ˙ ∈ C(I) EQNL QDPTHQDLDMS  @MC SGDQDENQD SGD LNQD FDMDQ@K MNSHNM NE SGD CHƤDQDMSH@K DPT@SHNM @MC HSR RNKTSHNM HM SGD DWSDMCDC RDMRD QDCTBDR SN SGD BNLLNM CDƥMHSHNM NE @M NQCHM@QX CHƤDQDMSH@K DPT@SHNM VGDM f ∈ C(S) 4RT@KKX HS VHKK AD BKD@Q EQNL SGD BNMSDWS VGHBG LD@MHMF HR @SS@BGDC SN x˙ = f (t, x) @MC HSR RNKTSHNM ϕ @MC GDMBD HS VHKK Q@QDKX AD MDBDRR@QX SN @CC SGD OGQ@RD řHM SGD DWSDMCDC RDMRDŚ



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF

   3GD 3GDNQDL NE "@Q@SGDNCNQX R QDF@QCR SGD DWHRSDMBD NE RNKTSHNM NE x˙ = f (t, x) HM SGD DWSDMCDC RDMRD NM I "NMRS@MSHM "@Q@SGDNCNQX   G@R OQNUDC SGD ENKKNVHMF PTHSD FDMDQ@K SGDNQDL TMCDQ SGD @RRTLOSHNM SG@S f AD ANTMCDC AX @ +DADRFTD HMSDFQ@AKD ETMBSHNM NE t %NKKNVHMF :< OO  SGD OQNNE VHKK AD B@QQHDC NTS HM NMD RO@BD CHLDMRHNM NMKX HS VHKK AD BKD@Q VG@S LNCHƥB@SHNMR @QD QDPTHQDC HM SGD B@RD NE @ RXRSDL NE NQCHM@QX CHƤDQDMSH@K DPT@SHNMR RDD OQNAKDL   R VHKK CDMNSD SGD QDBS@MFKD . / R := (t, x) ∈ R2 : |t − τ | ≤ a , |x − ξ| ≤ b , VGDQD (τ, ξ) HR @ ƥWDC FHUDM ONHMS HM SGD (t, x) OK@MD @MC a, b @QD ONRHSHUD QD@K MTLADQR 3GDNQDL   3GDNQDL NE "@Q@SGDNCNQX /HW f EH GHƲQHG RQ R DQG VXSSRVH LW LV PHDVXUDEOH LQ t IRU HDFK Ʋ[HG x DV ZHOO DV FRQWLQXRXV LQ x IRU HDFK Ʋ[HG t ,I WKHUH H[LVWV D /HEHVJXHLQWHJUDEOH IXQFWLRQ m RQ WKH LQWHUYDO |t − τ | ≤ a VXFK WKDW |f (t, x)| ≤ m(t) , IRU DOO (t, x) ∈ R ,   WKHQ WKHUH H[LVWV D VROXWLRQ ϕ RI x˙ = f (t, x) LQ WKH H[WHQGHG VHQVH RQ VRPH LQWHUYDO |t − τ | ≤ β β > 0  VDWLVI\LQJ ϕ(τ ) = ξ 3URRI %NKKNVHMF :< OO  KDS TR BNMRHCDQ SGD B@RD t ≥ τ @R SGD RHST@SHNM HR RHLHK@Q VGDM t ≤ τ (E M HR CDƥMDC AX ⎧ ENQ @KK t < τ ⎪ ⎨ M (t) = 0

t ,   ⎪ m(s)Cs ENQ @KK τ ≤ t ≤ τ + a ⎩ M (t) = τ

SGDM HS HR BKD@Q SG@S M HR BNMSHMTNTR MNM CDBQD@RHMF @R m ≥ 0 AX   @MC M (τ ) = 0 'DMBD (t, ξ ± M (t)) ∈ R ENQ RNLD HMSDQU@K τ ≤ t ≤ τ + β ≤ τ + a VGDQD β HR RNLD ONRHSHUD BNMRS@MS "GNNRD @MX β > 0 ENQ VGHBG SGHR HR SQTD @MC CDƥMD SGD /HB@QC @OOQNWHL@ SHNMR ϕj  j = 1, 2, . . .  AX ⎧ ENQ @KK τ ≤ t ≤ τ + β/j ⎪ ⎨ ϕj (t) = ξ

t−β/j . ⎪ ⎩ ϕj (t) = ξ + f (s, ϕj (s))Cs ENQ @KK τ + β/j < t ≤ τ + β τ

  "KD@QKX ϕ1 HR CDƥMDC NM τ ≤ t ≤ τ + β ENQ HS HR SGD BNMRS@MS ξ %NQ @MX ƥWDC j ≥ 1 SGD ƥQRS ENQLTK@ HM   CDƥMDR ϕj NM τ ≤ t ≤ τ + β/j @MC RHMBD (t, ξ) ∈ R ENQ τ ≤ t ≤ β/j SGD RDBNMC ENQLTK@ HM   CDƥMDR ϕj @R @ 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS BNMSHMTNTR ETMBSHNM NM SGD HMSDQU@K τ + β/j < t ≤ τ + 2β/j %TQSGDQ NM SGHR K@SSDQ HMSDQU@K ! |ϕj (t) − ξ| ≤ M t − βj ,   AX UHQSTD NE   @MC   RRTLD SG@S ϕj HR CDƥMDC NM τ ≤ t ≤ τ + kβ/j ENQ 1 < k < j 3GDM SGD RDBNMC ENQLTK@ NE   CDƥMDR ϕj ENQ τ + kβ/j < t ≤ τ + (k + 1)β/j RHMBD JMNVKDCFD NE SGD LD@RTQ@AKD HMSDFQ@MC HR NMKX QDPTHQDC NM τ ≤ t ≤ τ + kβ/j KRN NM τ +kβ/j < t ≤ τ +(k+1)β/j SGD ETMBSHNM ϕj R@SHRƥDR   ADB@TRD NE   @MC   3GDQDENQD AX HMCTBSHNM   CDƥMDR @KK ϕj @R BNMSHMTNTR ETMBSHNMR NM τ ≤ t ≤ τ + β VGHBG R@SHREX ⎧ ϕj (t) = ξ ENQ @KK τ ≤ t ≤ τ + β/j ⎨ ! .   ⎩ |ϕj (t) − ξ| ≤ M t − β ENQ @KK τ + β/j < t ≤ τ + β j (E t1 @MC t2 @QD @MX SVN ONHMSR HM SGD HMSDQU@K [τ, τ + β] SGDM NM @BBNTMS NE     @MC  

! !

  |ϕj (t1 ) − ϕj (t2 )| ≤ M t1 − βj − M t2 − βj . 2HMBD M HR BNMSHMTNTR NM [τ, τ + β] HS HR TMHENQLKX BNMSHMTNTR SGDQD 3GHR HLOKHDR AX   SG@S SGD RDS {ϕj } HR TMHENQLKX ANTMCDC NM [τ, τ + β] "NM RDPTDMSKX HS ENKKNVR AX SGD KDLL@ NE QYDK@ RBNKH +DLL@   SG@S SGDQD DWHRSR @ RTA RDPTDMBD {ϕjk } VGHBG BNMUDQFDR TMHENQLKX NM [τ, τ + β] SN @ BNM SHMTNTR KHLHS ETMBSHNM ϕ @R k → ∞ %QNL   |f (t, ϕjk (t))| ≤ m(t) ,

ENQ @KK τ ≤ t ≤ τ + β ,

@MC RHMBD f HR BNMSHMTNTR HM x ENQ ƥWDC t f (t, ϕjk (t)) → f (t, ϕ(t)) ,

ENQ k → ∞ ,

ENQ DUDQX ƥWDC t ∈ [τ, τ + β] 3GDQDENQD +DADRFTDŗR CNLHM@SDC BNMUDQFDMBD SGDNQDL L@X AD @OOKHDC SN FHUD

t

t KHL f (s, ϕjk (s))Cs = f (s, ϕ(s))Cs   k→∞ τ

τ

ENQ @MX t ∈ [τ, τ + β] !TS

t

f (s, ϕjk (s))Cs − ϕjk (t) = ξ + τ



t t−β/j

f (s, ϕjk (s))Cs ,

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF VGDQD HS HR BKD@Q SG@S SGD K@SSDQ HMSDFQ@K U@MHRGDR @R k → ∞ 'DMBD KDSSHMF k → ∞ @MC @OOKXHMF   HS ENKKNVR SG@S

t

ϕ(t) = ξ +

f (s, ϕ(s))Cs ,

τ

EQNL VGHBG SGD @RRDQSHNM NE SGD SGDNQDL ENKKNVR @S NMBD (S HR HMSDQDRSHMF SN QDL@QJ SG@S SGD NQHFHM@K @OOQNWHL@SHNMR   LTRS BNM UDQFD SN @ RNKTSHNM HM SGD B@RD VGDQD @ TMHPTD RNKTSHNM HR JMNVM 3GHR RHST@ SHNM CNDR MNS @OOKX ENQ NQCHM@QX RTBBDRRHUD @OOQNWHL@SHNMR @R SGD ENKKNVHMF DW@LOKD RGNVR $W@LOKD   "NMSHMTHSX NE f HR -NS 2TƧBHDMS ENQ SGD "NMUDQFDMBD NE SGD 2TBBDRRHUD OOQNWHL@SHNMR BE :< O  +DS SGD QHFGS G@MC RHCD f NE @M NQCHM@QX CHƤDQDMSH@K DPT@SHNM x˙ = f (t, x) AD CDƥMDC AX ⎧ ⎪ 0 ENQ t = 0 @MC −∞ < x < +∞ ⎪ ⎪ ⎪ ⎪ ⎨ 2t ENQ 0 < t ≤ 1 @MC −∞ < x < 0 f (t, x) := ⎪ ENQ 0 < t ≤ 1 @MC 0 ≤ x ≤ t2 ⎪ ⎪ 2t − 4x/t ⎪ ⎪ ⎩ −2t ENQ 0 < t ≤ 1 @MC t2 < x < +∞ . .M SGD QDFHNM 0 ≤ t ≤ 1 −∞ < x < +∞ SGHR ETMBSHNM f HR BNMSHMTNTR @MC ANTMCDC AX SGD BNMRS@MS 2 %NQ SGD HMHSH@K ONHMS (τ, ξ) = (0, 0) SGD RTBBDRRHUD @OOQNWHL@SHNMR ϕ0 (t) = ξ ϕm+1 (t) = ξ +



t τ

f (s, ϕm (s))Cs ,

ENQ @KK m = 0, 1, 2, . . .

ADBNLD ENQ 0 ≤ t ≤ 1 ϕ0 (t) = 0 ,

ϕ2m−1 (t) = t2 ,

@MC

ϕ2m (t) = −t2 ,

ENQ @KK m = 1, 2, . . . .

3GD RDPTDMBD {ϕm (t)} G@R SVN BKTRSDQ U@KTDR ENQ D@BG t = 0 @MC GDMBD SGD RTBBDRRHUD @OOQNWHL@SHNMR CN MNS BNMUDQFD -NSD MDHSGDQ NE SGD SVN BNMUDQ FDMS RTA RDPTDMBDR {ϕ2m−1 } @MC {ϕ2m } BNMUDQFD SN @ RNKTSHNM ENQ ϕ˙ 2m−1 (t) = 2t = f (t, t2 ) ,

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ϕ˙ 2m (t) = −2t = f (t, −t2 ) .



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ENQ @KK t ∈ I ,

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ENQ @KK t ∈ I ,

GNKCR ENQ DUDQX NSGDQ RNKTSHNM ϕ NE x˙ = f (t, x) NM I ENQ VGHBG ϕ(t) = ξ SGDM ϕm HR B@KKDC PLQLPXP VROXWLRQ NE x˙ = f (t, x) NM I O@RRHMF SGQNTFG (τ, ξ) "KD@QKX SGD ETMBSHNMR ϕM @MC ϕm  HE SGDX DWHRS LTRS AD TMHPTD 3GD DWHRSDMBD NE ϕM @MC ϕm VHKK MNV AD CDLNMRSQ@SDC TMCDQ SGD "@Q@SGDNCNQX @RRTLOSHNMR 3GDNQDL   $WHRSDMBD NE @ ,@WHLTL  ,HMHLTL 2NKTSHNM /HW WKH K\ SRWKHVLV RI &DUDWKHRGRU\śV WKHRUHP 7KHRUHP  EH VDWLVƲHG 7KHQ WKHUH H[ LVWV D PD[LPXP VROXWLRQ ϕM DQG D PLQLPXP VROXWLRQ ϕm RI x˙ = f (t, x) RQ |t − τ | ≤ β SDVVLQJ WKURXJK (τ, ξ) 3URRI %NKKNVHMF :< OO  VD VHKK OQNUD SGD DWHRSDMBD NE ϕM NM [τ, τ + β] -NV @MX RNKTSHNM ϕ NE x˙ = f (t, x) O@RRHMF SGQNTFG SGD ONHMS (τ, ξ) LTRS R@SHREX

t ϕ(t) = ξ + f (s, ϕ(s))Cs   τ

@R E@Q @R HS DWHRSR @MC EQNL   HS ENKKNVR SG@S |ϕ(t1 ) − ϕ(t2 )| ≤ |M (t1 ) − M (t2 )|

 

ENQ @MX SVN ONHMSR t1 , t2 VGDQD ϕ DWHRSR VGDQD M HR CDƥMDC AX   2HMBD M HR BNMSHMTNTR   HLOKHDR AX SGD "@TBGX BQHSDQHNM ENQ BNMUDQ FDMBD SG@S SGD RNKTSHNM ϕ B@M AD BNMSHMTDC HE MDBDRR@QX SN SGD DMSHQD HMSDQU@K [τ, τ + β] L@JHMF TRD NE SGD "@Q@SGDNCNQX DWHRSDMBD SGDNQDL 3GTR @KK RNKT SHNMR NE x˙ = f (t, x) O@RRHMF SGQNTFG (τ, ξ) DWHRS NM [τ, τ + β] @MC LTRS R@SHREX   SGDQD %QNL SGD TMHENQL BNMSHMTHSX NE M NM [τ, τ + β] HS ENKKNVR EQNL



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF   SG@S SGD RDS NE @KK RNKTSHNMR {ϕ} NE x˙ = f (t, x) NM [τ, τ + β] HR @M DPTH BNMSHMTNTR RDS SG@S HR FHUDM @MX ε > 0 SGDQD DWHRSR @ δε > 0 HMCDODMCDMS NE t @MC SGD RNKTSHNM ϕ RTBG SG@S

ϕ(tˆ) − ϕ(t˜) < ε , VGDMDUDQ |tˆ − t˜| < δε ,   @MC tˆ, t˜ @QD HM [τ, τ + β] +DS Φ AD SGD ETMBSHNM CDƥMDC AX Φ(t) =

RTO {ϕ(t)}

t∈[τ,τ +β]

S@JDM NUDQ @KK RNKTSHNMR ϕ NE x˙ = f (t, x) NM [τ, τ + β] O@RRHMF SGQNTFG (τ, ξ) "KD@QKX Φ DWHRSR NM [τ, τ + β] @MC HR BNMSHMTNTR @MC GDMBD TMHENQLKX BNMSHMT NTR SGDQD 3GTR ENQ @MX FHUDM ε > 0 SGDQD HR @ δε > 0 RTBG SG@S MNS NMKX   HR SQTD ENQ SGHR δε  ATS @KRN ENQ tˆ, t˜ HM [τ, τ + β]

Φ(tˆ) − Φ(t˜) < ε , VGDMDUDQ |tˆ − t˜| < δε .   (S VHKK AD RGNVM SG@S Ψ HR @ RNKTSHNM NE x˙ = f (t, x) R@SHREXHMF Ψ(τ ) = ξ @MC HE ϕM HR CDƥMDC SN AD Φ HS HR BKD@Q SG@S SGHR ϕM VHKK R@SHREX SGD QDPTHQDLDMSR NE SGD SGDNQDL NM [τ, τ + β] %NQ @ FHUDM ε > 0 BGNNRD δε RTBG SG@S   @MC   GNKC 2TACHUHCD SGD HMSDQU@K [τ, τ + β] HMSN n HMSDQU@KR AX SGD ONHMSR τ = t0 < t1 < t2 < · · · < tn = τ + β HM RTBG @ V@X SG@S L@W(ti+1 − ti ) < δε . %NQ DUDQX ti  i = 0, 1, . . . , n − 1 BGNNRD @ RNKTSHNM ϕi NE x˙ = f (t, x) O@RRHMF SGQNTFG (τ, ξ) RN SG@S 0 ≤ Φ(ti ) − ϕi (ti ) < ε @MC ENQ i ≥ 1

ϕi (ti ) − ϕi−1 (ti ) ≥ 0 .

3GHR HR ONRRHAKD EQNL SGD CDƥMHSHNM NE Φ -NV ENQ SGD FHUDM ε CDƥMD SGD ETMBSHNM ϕε @R ENKKNVR +DS ϕε (t) = ϕn−1 (t) ,

ENQ tn−1 ≤ t ≤ tn = τ + β .

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3NAH@R 1HFNHO  %KNQH@M 5XSS ϕε (t) := ϕn−2 (t) NM [tn−2 , tn−1 ) "NMSHMTHMF HM SGHR V@X NMD B@M CDƥMD @ RN KTSHNM ϕε NE x˙ = f (t, x) NM [τ, τ + β] O@RRHMF SGQNTFG SGD ONHMS (τ, ξ) NAS@HMDC AX O@SBGHMF SNFDSGDQ RNKTSHNMR NE x˙ = f (t, x) @MC G@UHMF SGD OQNODQSX 0 ≤ Φ(ti ) − ϕε (ti ) < ε ,

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2HMBD SGD U@QH@SHNM NE Φ @MC ϕε HM D@BG HMSDQU@K [ti , ti+1 ] HR KDRR SG@M ε AX   @MC   SGDQD QDRTKSR EQNL   0 ≤ Φ(t) − ϕε (t) < 3ε ,

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3NAH@R 1HFNHO  %KNQH@M 5XSS !X BGNNRHMF γ RL@KK DMNTFG H ⊂ D MX RNKTSHNM ϕ NE x˙ = f (t, x) VGHBG RS@QSR NM SGD KDES UDQSHB@K RHCD t = t0 − γ NE H H D  |ϕ(t0 − γ) − ϕM,ξ (t0 − γ)| ≤ γ VHKK AX   QDL@HMR HM H @R t HMBQD@RDR 3GTR @MX RTBG RNKTSHNM B@M AD BNMSHMTDC SN t0 + γ !X BGNNRHMF ε HM   RN SG@S ε = γ HS ENKKNVR SG@S ENQ 0 < η − ξ < δε SGD RNKTSHNMR ϕM,η B@M AD BNMSHMTDC SN t0 + γ 3GHR HLOKHDR SGD DWHRSDMBD NE Φ NUDQ [τ, t0 + γ] VGHBG BNMSQ@CHBSR SGD @RRTLOSHNM NM t0 3GTR t0 > τ + α @MC SGDQDENQD Φ DWHRSR NUDQ [τ, τ + α] !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   0 6G@S HR LD@MS AX @M NQCHM@QX CHƤDQDMSH@K DPT@SHNM HM SGD DWSDMCDC RDMRD @MC HSR RNKTSHNM 0 2S@SD SGD SGDNQDL NE "@Q@SGDNCNQX 0 2JDSBG SGD OQNNE NE SGD SGDNQDL NE "@Q@SGDNCNQX @MC DWOK@HM VGX NMD G@R SN AD B@TSHNTR @ANTS SGD BNMUDQFDMBD NE RTBBDRRHUD @OOQNWHL@SHNMR 0 &HUD SGD CDƥMHSHNM NE @ L@WHLTL @MC @ LHMHLTL RNKTSHNM 0 6G@S B@M XNT R@X @ANTS SGD DWHRSDMBD NE @ L@WHLTL @MC @ LHMHLTL RNKTSHNM 0 2JDSBG SGD OQNNE NE SGD @RRDQSHNM XNT TRDC HM 0 0 6G@S B@M XNT R@X @ANTS SGD BNMSHMT@SHNM NE RNKTSHNMR ADXNMC @ SHLD HM SDQU@K (a, b) 0 2S@SD SGD BNMCHSHNMR SG@S @KKNV XNT SN QD BNMRSQTBS @ RNKTSHNM HE SGD L@W HLTL @MC LHMHLTL RNKTSHNMR @QD JMNVM RTBG SG@S @KK SGQDD NE SGDRD RN KTSHNMR G@UD SGD R@LD HMHSH@K ONHMS 0 2JDSBG SGD OQNNE NE SGD @RRDQSHNM XNT TRDC HM 0

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x˙ d = ad1 (t)x1 + · · · + add (t)xn + bd (t) NQ HM L@SQHW UDBSNQ MNS@SHNM x˙ = A(t)x + b(t) , VGDQD A : I → Kd×d @MC b : I → Kd @QD FHUDM L@OOHMFR NM @ HMSDQU@K I ⊂ R 3GD KHMD@Q NQCHM@QX CHƤDQDMSH@K DPT@SHNM HR B@KKDC KRPRJHQHRXV HE b ≡ 0 @MC QRQKRPRJHQHRXV VHSG LQKRPRJHQHLW\ b NSGDQVHRD

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t → (ϕ1 (t), ϕ2 (t), . . . , ϕd (t)) .

3GHR L@OOHMF Φ HR B@KKDC IXQGDPHQWDO PDWUL[ NE x˙ = Ax @MC AX BNMRSQTBSHNM HS NADXR SGD DPT@SHNM ˙ Φ(t) = A(t)Φ(t) . BBNQCHMF SN O@QS SVN NE "NQNKK@QX   SGD L@OOHMF Φ HR HMUDQSHAKD +DS ϕ ∈ L AD @MX NSGDQ RNKTSHNM SGDM HS B@M AD VQHSSDM @R @ KHMD@Q BNLAHM@SHNM ϕ = c1 ϕ1 + c2 ϕ2 + · · · + cd ϕd NE SGD DKDLDMSR NE SGD ETMC@LDMS@K RXRSDL @MC RB@K@QR ci ∈ K i = 1, 2, . . . , d 3GTR VHSG c := (c1 , c2 , . . . , cd )T VD G@UD ϕ(t) = Φ(t)c ,

c ∈ Kd .

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF %NQ k → ∞ VD G@UD

ϕ(t) = DWO(At)x0 ,

 

VGDQD DWO(At) HR SGD L@SQHW U@KTDC DWONMDMSH@K ETMBSHNM SG@S NADXR SGD NQ C CHM@QX CHƤDQDMSH@K DPT@SHNM Ct DWO(At) = A DWO(At) 3GHR @KKNVR TR SN UDQHEX HLLDCH@SDKX SG@S ϕ(t) = DWO(At)x0 RNKUDR SGD FHUDM HMHSH@K U@KTD OQNAKDL 3GD TRD NE , 3+ !ŗR L@SQHW DWONMDMSH@K HR RSQ@HFGSENQV@QC BE :< BG@O SDQ  $ F  ENQ SGD BNMRS@MS L@SQHW  1 1 A = −2 4 VD B@M B@KBTK@SD DWO(A) @R ENKKNVR ==  4 bvKU(R Rc @k 9)Vc == 2tTKUV Mb 4 ( k 2tTUkV@2tTUjV2tTUjV@2tTUkV) ( @k 2tTUjVYk 2tTUkV- @2tTUkVYk 2tTUjV)

(M O@QSHBTK@Q VD B@M HLLDCH@SDKX NAS@HM SGD RNKTSHNM NE x˙ = Ax VHSG HMHSH@K BNMCHSHNM x0 = (−4, 2) ==  4 bvKU(R Rc @k 9)Vc == bvKb i == 2tTKU iV (@9c k) Mb 4 ( e 2tTUj iV @ Ry 2tTUk iV) ( @Ry 2tTUk iV Y Rk 2tTUj iV)

3GD QDOQDRDMS@SHNM   NE SGD RNKTSHNM @R ϕ(t) = DWO(At)x0 HR VDKK RTHSDC ENQ SGDNQDSHB@K BNMRHCDQ@SHNMR KHJD CHRBTRRHNMR NM SGD FDNLDSQHB RSQTBSTQD NE SGD RNKTSHNM $W@LOKD   2NKTSHNMR NM @ 2OGDQD BE :< OO  (E A HR QD@K @MC @MSH RXLLDSQHB SGDM DUDQX RNKTSHNM ϕ NE x˙ = Ax KHUDR NM @ ROGDQD @QNTMC SGD NQHFHM H D  ϕ(t) 2 = ϕ(0) 2 ENQ @KK t ∈ R . 3GHR B@M AD RDDM @R ENKKNVR HE A HR QD@K @MC @MSH RXLLDSQHB SGDM DWO(At) HR NQSGNFNM@K DWO(At) (DWO(At))T = DWO((A + AT )t) = I 2TARSHSTSHMF SGD DKDLDMSR v1 , v2 , . . . , vn NE @M @QAHSQ@QX A@RHR NE SGD Kn HMSN   HM OK@BD NE x0  VD NAS@HM @ A@RHR NE SGD RNKTSHNM RO@BD L NE SGD GNLNFD MDNTR DPT@SHNM x˙ = Ax %NQ HMRS@MBD SGD B@MNMHB@K A@RHR UDBSNQR e1 , e2 , . . . , ed KD@C SN SGD BNKTLMR NE DWO(At) @R @ A@RHR NE L

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3NAH@R 1HFNHO  %KNQH@M 5XSS 4RT@KKX SGD BNLOTS@SHNM NE DWO(At) HR SHQDRNLD 3GDQDENQD HM NQCDQ SN NA S@HM @ A@RHR NE L VD BGNNRD @MNSGDQ O@SG A@RDC NM SGD )NQC@M MNQL@K ENQL NE A 3GD D@RHDRS @MC LNRS HLONQS@MS RHST@SHNM NBBTQR VGDM A G@R d KHMD@QKX HM CDODMCDMS DHFDMUDBSNQR @R HM SGD B@RD NE A ADHMF @ QD@K RXLLDSQHB L@SQHW NQ A G@UHMF d CHRSHMBS DHFDMU@KTDR +DLL@    !@RHR NE L "NMRHRSHMF NE $HFDMUDBSNQR NE A /HW v EH DQ HLJHQ YHFWRU RI A DQG λ EH LWV FRUUHVSRQGLQJ HLJHQYDOXH WKHQ ϕ v : R → Kd ,

ϕv (t) := DWO(λt)v ,

VROYHV WKH LQLWLDO YDOXH SUREOHP x˙ = Ax x(0) = v /HW v1 , v2 , . . . , vd EH OLQHDUO\ LQGHSHQGHQW HLJHQYHFWRUV RI A DQG λ1 , λ2 , . . . , λd EH WKHLU FRUUHVSRQGLQJ HLJHQ YDOXHV WKHQ ϕv1 , ϕv2 , . . . , ϕvd IRUP D IXQGDPHQWDO V\VWHP 3URRI %NKKNVHMF :< O  VD G@UD SG@S ϕv HR @ RNKTSHNM NE SGD GNLNFD MDNTR DPT@SHNM ADB@TRD ϕ˙ v = λ DWO(λt)v = DWO(λt)Av = Aϕv . ,NQDNUDQ SGD RNKTSHNMR ϕv1 , ϕv2 , . . . , ϕvd ENQL @ A@RHR NE L ADB@TRD SGDHQ U@K TDR ϕv1 (0), ϕv2 (0), . . . , ϕvd (0) ENQL @ A@RHR NE Kd (E A CNDR MNS G@UD @ ETKK RDS NE d KHMD@QKX HMCDODMCDMS DHFDMUDBSNQR @R HM SGD B@RD NE LTKSHOKD DHFDMU@KTDR SGDM @ ETMC@LDMS@K RXRSDL B@M AD BNMRSQTBSDC VHSG SGD @HC NE FDMDQ@KHYDC DHFDMUDBSNQR #DƥMHSHNM   &DMDQ@KHYDC $HFDMUDBSNQ UDBSNQ v ∈ C d  v = 0 HR B@KKDC JHQHUDOL]HG HLJHQYHFWRU RI WKH PDWUL[ A FRUUHVSRQGLQJ WR WKH HLJHQYDOXH λ HE SGDQD HR @ M@STQ@K MTLADQ s RTBG SG@S (A − λI)s v = 0 . 3GD RL@KKDRS MTLADQ s HR B@KKDC GHJUHH NE v 3GD FDMDQ@KHYDC DHFDMUDBSNQR NE CDFQDD NMD @QD SGD DHFDMUDBSNQR SGDL RDKE (E v HR @ FDMDQ@KHYDC DHFDMUDBSNQ NE CDFQDD s SGDM SGD UDBSNQR vs := v ,

vs−1 := (A − λI) v ,

...

v1 := (A − λI)s−1 v

@QD FDMDQ@KHYDC DHFDMUDBSNQR NE CDFQDD s s − 1 ş 1 QDRODBSHUDKX v1 HR @M DHFDMUDBSNQ @MC vi  i = 2, 3, . . . , s HR @ RNKTSHNM NE SGD DPT@SHNM (A − λI) vi = vi−1



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF $W@LOKD   &DMDQ@KHYDC $HFDMUDBSNQR NE @ 3 × 3 ,@SQHW BE :< O  %NQ SGD ENKKNVHMF L@SQHW A SGD MTLADQ 1 HR @M DHFDMU@KTD NE @KFDAQ@HB LTKSH OKHBHSX 3 ATS G@UHMF NMKX NMD DHFDMUDBSNQ M@LDKX e1  ⎞ ⎛ 1 2 3 A = ⎝ 0 1 2 ⎠. 0 0 1 ,NQDNUDQ (A − λI) e2 = 2e1 ,

(A − λI)2 e2 = 0

(A − λI) e3 = 3e1 + 2e2 ,

(A − λI)2 e3 = 4e1 ,

(A − λI)3 e3 = 0 .

H D  es HR @ FDMDQ@KHYDC DHFDMUDBSNQ NE CDFQDD s s = 1, 2, 3 3GD SGDNQDL NM SGD )NQC@M MNQL@K ENQL NE L@SQHBDR EQNL +HMD@Q KFDAQ@ HLOKHDR 3GDNQDL    !@RHR "NMRHRSHMF NE &DMDQ@KHYDC $HFDMUDBSNQR )RU DQ\ PD WUL[ A ∈ C d×d WKHUH LV D EDVLV RI C d FRQVLVWLQJ RI JHQHUDOL]HG HLJHQYHFWRUV WKDW FRQWDLQV IRU DQ\ kIROG HLJHQYDOXH λ H[DFWO\ k JHQHUDOL]HG HLJHQYHFWRUV v1 , v2 , . . . , vk  ZKHUH WKH GHJUHH RI vs LV DW PRVW s A@RHR h1 , h2 , . . . , hd NE FDMDQ@KHYDC DHFDMUDBSNQR FHUDR QHRD SN SGD ETMC@ LDMS@K RXRSDL DWO(At)h1  DWO(At)h2  . . .  DWO(At)hd NE SGD GNLNFDMDNTR DPT@SHNM x˙ = Ax -DWS VD @M@KXRD SGD BNMRSQTBSHNM NE RTBG @ RNKTSHNM ϕv : R → Cd ,

ϕv (t) := DWO(At)v ,

VGDQD v HR @ FDMDQ@KHYDC DHFDMUDBSNQ NE CDFQDD s BNQQDRONMCHMF SN SGD DHFDM U@KTD λ DWO (At) v = DWO (λIt) DWO ((A − λI) t) v = DWO (λt) ·

∞ 

1 k!

(A − λI)k tk v .

k=0

2HMBD (A − λI)k v = 0 ENQ k ≥ s SGD @ANUD HMƥMHSD RTL QDCTBDR SN @ ƥMHSD NMD @MC VD NAS@HM ϕv (t) = DWO (λt) pv (t) ,

VGDQD

pv (t) =

s−1 

1 k!

(A − λI)k tk v .

 

k=0

'DQD pv HR @ ONKXMNLH@K NE CDFQDD KDRR NQ DPT@K SN s − 1 SGD BNDƧBHDMSR k 1 d k! (A − λI) v NE VGHBG @QD UDBSNQR HM C %NQ s = 1 VD G@UD pv (t) = v +DS TR RTLL@QHYD SGD BNMRSQTBSHNM LDSGNCNKNFX HM SGD ENKKNVHMF SGDNQDL

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3NAH@R 1HFNHO  %KNQH@M 5XSS 3GDNQDL   !KTDOQHMS ENQ SGD "NMRSQTBSHNM NE %TMC@LDMS@K 2XRSDL ENQ x˙ = Ax 7KH IROORZLQJ SURFHGXUH OHDGV WR D IXQGDPHQWDO V\VWHP RI WKH OLQHDU KRPRJHQHRXV HTXDWLRQ x˙ = Ax  'HWHUPLQH DOO GLVWLQFW HLJHQYDOXHV λ1 , λ2 , . . . , λr RI A DQG WKHLU DOJHEUDLF PXOWLSOLFLWLHV k1 , k2 , . . . , kr  ZKHUH k1 + k2 + · · · + kr = d KROGV  )RU HDFK HLJHQYDOXH λρ  ρ = 1, 2, . . . , r ZLWK DOJHEUDLF PXOWLSOLFLW\ kρ FRQ VWUXFW kρ VROXWLRQV RI x˙ = Ax E\ Ţ ƲUVW GHWHUPLQLQJ WKH FRUUHVSRQGLQJ JHQHUDOL]HG HLJHQYHFWRUV v1 , v2 , . . . , vk  ZKHUH vs KDV GHJUHH OHVV RU HTXDO WR s DQG Ţ WKHQ FDOFXODWLQJ DFFRUGLQJ WR   WKHVH VROXWLRQV ϕvs (t) = DWO (λρ t) pvs (t) ,

s = 1, 2, . . . , kρ .

7KH REWDLQHG d VROXWLRQV IRUP D IXQGDPHQWDO V\VWHP RI x˙ = Ax $W@LOKD   %TMC@LDMS@K 2XRSDL ENQ @ 3 × 3 ,@SQHW BE :< O  R RDDM HM DW@LOKD   A G@R SGD 3 ENKC DHFDMU@KTD 1 VHSG BNQQDRONMCHMF DHFDMUDBSNQ e1 e2 HR @ FDMDQ@KHYDC DHFDMUDBSNQ NE CDFQDD 2 @MC e3 HR @ FDM DQ@KHYDC DHFDMUDBSNQ NE CDFQDD 3 BBNQCHMF SN NTQ BNMRSQTBSHNM AKTDOQHMS VD SGTR FDS SGD RNKTSHNMR ⎛ ⎞ 1 ϕ1 (t) = DWO (t) e1 = DWO (t) ⎝ 0 ⎠ 0 ⎞ ⎛ 2t ϕ2 (t) = DWO (I + (A − I) t) e2 = DWO (t) ⎝ 1 ⎠ , 0 ⎞ ⎛ 3t + 2t2 ! ⎠. 2t ϕ3 (t) = DWO I + (A − I) t + 12 (A − I)2 t2 e3 = DWO (t) ⎝ 1 'DQD ϕ1 (t), ϕ2 (t), ϕ3 (t) ENQL @ ETMC@LDMS@K RXRSDL

  %HQRS (MSDFQ@KR  .RBHKK@SHNMR %NKKNVHMF :< OO  ƥQRS HMENQL@SHNM @ANTS SGD SQ@IDBSNQHDR NE HMSDFQ@K BTQUDR B@M AD F@HMDC EQNL ƥQRS HMSDFQ@KR ƲUVW LQWHJUDO HR @ NE @ UDBSNQ ƥDKC



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   0 6G@S B@M XNT R@X @ANTS SGD DWHRSDMBD @MC TMHPTDMDRR NE SGD RNKTSHNMR NE x˙ = Ax + b VGDQD x, b ∈ Rd @MC A ∈ Rd×d  0 &HUD SGD CDƥMHSHNM NE @ ETMC@LDMS@K RXRSDL @MC NE @ ETMC@LDMS@K L@SQHW 0 2S@SD SGD SGDNQDL NE +HNTUHKKD @MC RJDSBG HSR OQNNE 0 #DRBQHAD SGD FDMDQ@K BNMRSQTBSHNM LDSGNC ENQ @ RNKTSHNM NE x˙ = Ax + b VGDQD x, b ∈ Rd @MC A ∈ Rd×d 0 2NKUD 

x˙ 1 x˙ 2



 =

−8 3 −18 7



x1 x2



 + DWO(−x)

5 12

.

v : Ω → Rd  Ω ⊂ Rd HR @ C 1 ETMBSHNM E : Ω → R SGD CDQHU@SHUD NE VGHBG U@MHRGDR @KNMF v ∂v E(x) =

d 

vi (x)∂i E(x) = 0 .

 

i=1

%NQ HMRS@MBD SGD ETMBSHNM E : R2 → R E(x, y) = x2 + y 2 HR @ ƥQRS HMSDFQ@K NE SGD QNS@SHNM ƥDKC v : R2 → R2  v(x, y) = (−y, x) 3GD HLONQS@MBD NE @ ƥQRS HMSDFQ@K E HR SG@S HS S@JDR @ BNMRS@MS U@KTD NM D@BG HMSDFQ@K BTQUD ϕ NE SGD UDBSNQ ƥDKC CTD SN SGD ENKKNVHMF HCDMSHSX C (E ◦ ϕ) (t) = E  (ϕ(t)) ϕ(t) ˙ = E  (ϕ(t)) v (ϕ(t)) = ∂v E (ϕ(t)) = 0 . Ct 3GDNQDL   (MSDFQ@K "TQUDR @MC +DUDK 2DSR NE %HQRS (MSDFQ@KR (YHU\ LQWH JUDO FXUYH RI v OLHV RQ D OHYHO VHW RI E (M SGD @ANUD DW@LOKD NE @ QNS@SHNM ƥDKC VD G@UD SG@S DUDQX HMSDFQ@K BTQUD = 0 KHDR NM @ BHQBKD E(x, y) = x2 + y 2 = r2 3GDQD HR MN FDMDQ@K LDSGNC SN CDSDQLHMD SGD ƥQRS HMSDFQ@K NE @ FHUDM UDBSNQ ƥDKC 3GNTFG @ RDS NE LDSGNCNKNFHB@K @OOQN@BGDR DM@AKD SGD BNLOTS@SHNM NE ƥQRS HMSDFQ@KR HM RODBHƥB BNMBQDSD B@RDR RDD 2DB    (M OGXRHB@K @OOKH B@SHNMR BNMRDQU@SHNM K@VR KHJD SGD BNMRDQU@SHNM NE DMDQFX GDKO SN DRS@AKHRG ƥQRS HMSDFQ@KR RDD 2DB    @MC   

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3NAH@R 1HFNHO  %KNQH@M 5XSS !DENQD VD BNMSHMTD SN CHRBTRR RODBHƥB DW@LOKDR VD FHUD SVN ETQSGDQ QD L@QJR HLONQS@MS ENQ SGD RSTCX NE KDUDK RDSR NE ƥQRS HMSDFQ@KR BE :< O  

MNM DLOSX KDUDK RDS E −1 (c) ENQ @ QDFTK@Q U@KTD c ∈ R HR @ (d − 1) CHLDMRHNM@K RTA L@MHENKC NM Ω ⊂ Rd %NQ @M DKDLDMS x ∈ Ω @S VGHBG SGD U@KTD E(x) HR QDFTK@Q SGD BNMCHSHNM   B@M AD HMSDQOQDSDC HM SGD RDMRD SG@S v(x) HR @ S@MFDMSH@K UDBSNQ SN SGD KDUDK RDS SGQNTFG x H D  v(x) KHDR HM SGD JDQMDK NE CE(x)

 +DS E AD @ C 2 ETMBSHNM @MC x0 @ MNM CDFDMDQ@SD BQHSHB@K ONHMS H D  @ ONHMS RTBG SG@S E  (x0 ) = 0 @MC MNM CDFDMDQ@SD 'DRRH@M E  (x0 ) 3GD RG@OD NE SGD KDUDK RDSR HM SGD UHBHMHSX NE x0 HR CDSDQLHMDC AX SGD DHFDMU@KTDR NE E  (x0 ) TO SN CHƤDNLNQOGHRLR @BBNQCHMF SN SGD +DLL@ NE ,NQRD RDD OQNAKDL   6D BHSD @ RODBH@K B@RD NE SGHR KDLL@ VGHBG G@R O@QSHBTK@Q HLONQS@MBD GDQD +DS @KK DHFDMU@KTDR NE E  (x0 ) AD ONRHSHUD SGDM SGDQD HR @ CHƤDN LNQOGHRL h : K → Ω0 NE @ A@KK K ⊂ Rd @QNTMC 0 SN @M DMUHQNMLDMS Ω0 ⊂ Ω NE x0 RTBG SG@S E ◦ h(ξ) = E(x0 ) + ξ12 + · · · + ξd2 . 3GTR +DS E  (x0 ) = 0 @MC E  (x0 ) > 0 SGDM SGDQD HR @M DMUHQNMLDMS Ω0 ⊂ Ω NE SGD ONHMS x0 RTBG SG@S DUDQX HMSDFQ@K BTQUD ϕ NE v SGQNTFG @ ONHMS x ∈ Ω0 KHDR HM @ L@MHENKC ⊂ Ω0 VGHBG HR CHƤDNLNQOGHB SN @ (d − 1) ROGDQD @MC G@R HMƥMHSD KHED SHLD 1DL@QJ   5DBSNQ %HDKCR NM ,@MHENKCR %HQRS HMSDFQ@KR M@STQ@KKX FHUD QHRD SN SGD RSTCX NE UDBSNQ ƥDKCR NM L@MHENKCR YHFWRU ƲHOG NQ WDQJHQWLDO ƲHOG RQ D PDQLIROG M ⊂ Rd HR @ L@OOHMF v SG@S @RRHFMR @ S@MFDMSH@K UDBSNQ v(x) SN D@BG DKDLDMS NE x ∈ M M HMSDFQ@K BTQUD SN RTBG @ ƥDKC HR @ BTQUD ϕ : I → M RTBG SG@S ϕ(t) ˙ = v (ϕ(t)) ENQ @KK t ∈ I (E ENQ HMRS@MBD M HR @ KDUDK RDS NE @ ƥQRS HMSDFQ@K NE SGD UDBSNQ ƥDKC V : Ω → Rd NM SGD NODM RDS Ω ⊂ Rd  SGDM v := V|M HR @ UDBSNQ ƥDKC NM M %HM@KKX VD FHUD SVN DW@LOKDR NE ƥQRS HMSDFQ@KR HM RXRSDLR EQNL OGXRHBR @MC NMD EQNL AHNKNFX DBNKNFX

  

OOKHB@SHNM  3GD &DMDQ@K .RBHKK@SHNM $PT@SHNM

%NKKNVHMF :< O  KDS U : Ω → R AD @ C 1 ETMBSHNM @ ONSDMSH@K NM @M NODM RDS Ω ⊂ Rd VGHBG G@R @M HRNK@SDC LHMHLTL @S x0 3GD DPT@SHNM x ¨ = −∇U T (x) ,



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF H D  SGD ƥQRS NQCDQ RXRSDL   C y x =: v(x, y) , = −∇U T (x) Ct y

 

G@R SGD DMDQFX ETMBSHNM E : Ω × Rd → R FHUDM AX E(x, y) :=

1 2

d 

yi2 + U (x)

i=1

@R @ ƥQRS HMSDFQ@K ADB@TRD ∂v E(x, y) = y T ∇U T (x) − ∇U (x)y = 0 . MX RNKTSHNM ϕ = (x, y) = (x, x) ˙ NE SGD RXRSDL   VGDQD x HR @ RNKTSHNM NE ˙ HR BNMRS@MS VHSG SGD x ¨ = −∇U T (x) KHDR HM @ KDUDK RDS NE E RTBG SG@S E(x, x) BNMRS@MS ADHMF CDSDQLHMDC AX SGD HMHSH@K BNMCHSHNM (x(0), x(0)) ˙ BNMRDQU@SHNM NE DMDQFX

  

OOKHB@SHNM  3GD #DSDQLHMHRSHB /DMCTKTL

2HMBD SGD BNMFDMH@K HMRHFGS NE )NG@MMDR *DOKDQ   2@MSNQHN 2@M SNQHN 2@MBSNQHTR   @MC &@KHKDN &@KHKDH   SN SHLD SGD OTKRD NE @ GTL@M UH@ @ ODMCTKTL RDD :< OO  SGHR D@RX SN OGXRHB@KKX QD@KHRD ATS G@QC SN @M@KXRD CDUHBD G@R F@HMDC DMNQLNTR HMSDQDRS ANSG EQNL SGD OTQDKX L@SGDL@SHB@K @R VDKK @R EQNL SGD @OOKHDC ONHMS NE UHDV %NKKNVHMF : η @MC SGTR a (y(t)) ≤ a (y(ε)) =: α < 0 BBNQCHMF SN SGD ƥQRS CHƤDQDMSH@K DPT@SHNM x(t) ≤ DWO (α(t − t0 )) x(ε) GNKCR ENQ SGDRD U@KTDR NE t !DB@TRD NE SGHR @MC SGD E@BS x(t) > ξ > 0 HS G@R SN GNKC SG@S t1 < ∞ 3GD CDƥMHSHNM NE t1 HS HLLDCH@SDKX HLOKHDR MNV SG@S x(t1 ) = ξ @MC ϕ(t1 ) = A1 

R @ANUD HM BNMBKTRHNM  NMD B@M RGNV SGD DWHRSDMBD NE O@Q@LDSDQ ONHMSR t1 < t2 < t3 < t4 RTBG SG@S ϕ(tk ) = Ak  k = 2, 3, 4 (M O@QSHBTK@Q NMD G@R VHSG T = t4 SG@S ϕ(T ) = A4 = A0 = ϕ(t0 ) GNKCR



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 4.5

4.5

4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0

0.5

1

@

1.5

2

2.5

3

3.5

4

4.5

A

%HFTQD   5DKNBHSX OKNS @ @MC RJDSBG NE SGD OG@RD ONQSQ@HS A NE SGD 5NKSDQQ@ +NSJ@ RXRSDL x˙ = (3 − 2y)x @MC y˙ = (x − 2)y RGNVHMF SGD DWHRSDMBD NE @M DPTHKHAQHTL @MC ODQHNCHB RNKTSHNMR

KSNFDSGDQ SGHR RGNVR SGD ENKKNVHMF SGDNQDL 3GDNQDL   2NKTSHNM NE SGD 5NKSDQQ@ +NSJ@ 2XRSDL (YHU\ PD[LPDO LQWH JUDO FXUYH ϕ ZLWK ϕ(0) ∈ R2+ OLHV IRU DOO WLPHV LQ WKLV TXDGUDQW DQG LV SHULRGLF !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   0 &HUD SGD CDƥMHSHNM NE @ ƥQRS HMSDFQ@K 0 'NV @QD ƥQRS HMSDFQ@KR @MC HMSDFQ@K BTQUDR BNMMDBSDC 0 &HUD SGD NQCHM@QX CHƤDQDMSH@K DPT@SHNM SG@S CDRBQHADR SGD CDSDQLHMHRSHB ODMCTKTL 2JDSBG @MC CHRBTRR HSR OG@RD ONQSQ@HS 0 &HUD SGD NQCHM@QX CHƤDQDMSH@K DPT@SHNM SG@S CDRBQHADR SGD 5NKSDQQ@ +NSJ@ RXRSDL 2JDSBG @MC CHRBTRR HSR OG@RD ONQSQ@HS

  .QCHM@QX #HƤDQDMSH@K $PT@SHNMR NM !@M@BG 2O@BDR %NKKNVHMF :< "G@O  VD DMC NTQ CHRBTRRHNM NM CDSDQLHMHRSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNMR VHSG +HORBGHSY SXOD BNMCHSHNMR ENQ SGD DWHRSDMBD @MC

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3NAH@R 1HFNHO  %KNQH@M 5XSS TMHPTDMDRR NE CDSDQLHMHRSHB NQCHM@QX CHƤDQDMSH@K DPT@SHNMR NM @QAHSQ@QX !@ M@BG RO@BDR 3GDRD +HORBGHSY SXOD BNMCHSHNMR CN MNS KD@C SN SGD LNRS QDƥMDC QDRTKSR ENQ RTBG DPT@SHNMR SGNTFG SGDX @QD RHLHK@Q SN VG@S V@R CHRBTRRDC HM SGD Kd RDSSHMF NE SGD OQDUHNTR RDBSHNMR 3GD @CU@MS@FD HR SG@S SGDX @QD @KQD@CX RTƧBHDMS SN TMCDQRS@MC @ K@QFD BK@RR NE OQNAKDLR QDK@SDC SN LD@M RPT@QD RNKTSHNMR NE Q@MCNL CHƤDQDMSH@K DPT@SHNMR @R HMSQNCTBDC HM "G@O  %QNL MNV NM VD KDS X AD @ !@M@BG RO@BD NUDQ K H D  @ BNLOKDSD MNQLDC UDBSNQ RO@BD NUDQ K "NLOKDSDMDRR HM SGHR QDF@QC LD@MR SG@S ENQ DUDQX "@TBGX RDPTDMBD {xn }∞ n=1 ⊂ X SGDQD DWHRSR @M DKDLDMS x ∈ X RTBG SG@S KHLn→∞ xn = x VHSG QDRODBS SN SGD BGNRDM MNQL ,NQDNUDQ KDS D ⊂ X RTBG SG@S x0 ∈ D J = [0, a] ⊂ R @M HMSDQU@K @MC f : I × D → X @ ETMBSHNM (M SGHR RDBSHNM VD @QD HMSDQDRSDC HM BNMSHMTNTRKX CHƤDQDMSH@AKD ETMBSHNMR x : [0, δ] → D ,

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∞ φ(x)xn Cx , un (0) = −∞

VGDQD u−1 (t) ≡ 0 @MC u0 (t) ≡ 1 +DS A AD SGD BNQQDRONMCHMF HMƥMHSD L@SQHW @MC BNMRHCDQ SGD RB@KD NE DLADCCDC !@M@BG RO@BDR ⎧ ⎫ ⎨ ⎬  ENQ 0 < α ≤ s ≤ β . Xs = u ∈ RN : |u|Xs = |uj | DWO(js)(j!)−1/2 < ∞ ⎩ ⎭ j≥1

.AUHNTRKX |u|Xs ≤ |u|Xs ENQ s < s @MC u ∈ Xs |A|L(Xs ,Xs ) ≤ M · (s − s )−1 ,

RHLOKD B@KBTK@SHNM XHDKCR

VHSG M := DWO(2β) · (1 + |a| + |b|) .

'DMBD SGD LNLDMSR NE φ @QD HM Xβ  @MC SGD LNLDMSR NE u @QD HM Xs ENQ 0 ≤ t ≤ (M · e)−1 (β − s) 6D L@X QDOK@BD SGD KHMD@Q QHFGS G@MC RHCD A(t)x + b(t) NE 3GDNQDL   AX @ MNM KHMD@Q ETMBSHNM f RTBG SG@S f : I × Xs → Xs HR BNMSHMTNTR @MC |f (t, x) − f (t, y)|Xs =

M · |x − y|Xs s − s

ENQ x, y ∈ Xs @MC t ∈ I .

%TQSGDQ FDMDQ@KHR@SHNMR @MC @OOKHB@SHNMR @QD FHUDM D F  HM :
12 δ 3/2 .MBD ϕn (t) ≥ t3/2 /6 HS RS@XR SGDQD @R KNMF @R t < 1/2000 (MCDDC ENQ t ≥ 4δ @MC @R KNMF @R ϕn (t) ≥ t3/2 /6 ϕ˙ n (t) > 1/2 ϕn (t − δ) − t > 12 (t − δ)3/2 − t > t3/5 /10 2HMBD t3/5 /10 > (C/Ct)(t3/2 /6) SGD QDRTKS ENKKNVR (E n HR NCC ϕn (t) < −t3/2 /6 ENQ 3δ < t < 1/2000



&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 6

2

1.5

4 1

2 0.5

0

0

−0.5

−2 −1

−4 −1.5

−6 −6

−4

−2

0

2

4

6

−2 −2

−1.5

−1

−0.5

@ 6

6

4

4

2

2

0

0

−2

−2

−4

−4

−6 −2

−1.5

−1

−0.5

0

0

0.5

1

1.5

2

A

0.5

1

1.5

2

−6 −6

−4

−2

B

0

2

4

6

2

4

6

C 6

2

1.5

4 1

2 0.5

0

0

−0.5

−2 −1

−4 −1.5

−2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−6 −6

−4

D

−2

0

E

%HFTQD   (MSDFQ@K BTQUD OKNSR ENQ OQNAKDL  

$WDQBHRD   :☼< OOKHB@SHNM NE SGD /HB@QC +HMCDK±E ,DSGNC Ŕ /S  %HMC SGD DW@BS RNKTSHNM NE SGD HMHSH@K U@KTD OQNAKDL x˙ = x2 ,

6HFWLRQ 

x(0) = 1 .



3NAH@R 1HFNHO  %KNQH@M 5XSS 2S@QSHMF VHSG x0 (t) = 1 @OOKX SGD /HB@QC +HMCDK±E HSDQ@SHNM SN B@KBTK@SD x1 (t) x2 (t) @MC x3 (t) @MC BNLO@QD SGDRD QDRTKSR VHSG SGD DW@BS RNKTSHNM $WDQBHRD   :☼< OOKHB@SHNM NE SGD /HB@QC +HMCDK±E ,DSGNC Ŕ /S  %HMC SGD DW@BS RNKTSHNM NE SGD HMHSH@K U@KTD OQNAKDL x˙ = 2t (1 + x) ,

x(0) = 0 .

2S@QSHMF VHSG x0 (t) = 0 B@KBTK@SD x1 (t) x2 (t) x3 (t) @MC x4 (t) @MC BNLO@QD SGDRD QDRTKSR VHSG SGD DW@BS RNKTSHNM $WDQBHRD   :☼< OOKHB@SHNM NE SGD /HB@QC +HMCDK±E ,DSGNC Ŕ /S  (S HR HMRSQTBSHUD SN RDD GNV SGD /HB@QC +HMCDK±E HSDQ@SHNM VNQJR VHSG @ BGNHBD NE SGD HMHSH@K @OOQNWHL@SHNM NSGDQ SG@M SGD BNMRS@MS ETMBSHNM x0 (t) = x0 OOKX SGD LDSGNC SN SGD HMHSH@K U@KTD OQNAKDL x˙ = t + x ,

x(0) = 1 .

VHSG  x0 (t) = DWO(t)  x0 (t) = 1 + t @MC  x0 (t) = BNR(t) $WDQBHRD   :☼< $WHRSDMBD NE 2NKTSHNMR 2GNV SG@S DUDQX HMHSH@K U@KTD OQNAKDL VHSG SGD CHƤDQDMSH@K DPT@SHNM x˙ = t · | RHM(tx)| G@R DW@BSKX NMD RNKTSHNM SG@S HR CDƥMDC NM SGD VGNKD NE R 3GD RNKTSHNMR VHSG x(0) = 0 CN MNS G@UD YDQNR $WDQBHRD   :☼< 2NKTSHNM NE @ 'NLNFDMDNTR +HMD@Q .#$ 2XRSDL SQ@MRONQS LDBG@MHRL @KKNVR SGD DWBG@MFD ADSVDDM SVN QDRDQUNHQR NE R@KSR HM GNLNFDMDNTR RNKTSHNM 3GHR RXRSDL HR CDRBQHADC AX x˙ 1 = −k1 x1 + k2 x2 , x˙ 2 = k1 x1 − k2 x2 , VGDQD xi CDMNSDR SGD @LNTMS NE R@KS HM QDRDQUNHQ i i = 1, 2 @MC SGD ki ŗR @QD BNMRS@MS DWBG@MFD Q@SDR "NLOTSD SGD RNKTSHNM VHSG QDRODBS SN SGD HMHSH@K BNMCHSHNMR x1 (0) = s1 @MC x2 (0) = s2  @MC CHRBTRR HSR ADG@UHNQ ENQ t → ∞ $WDQBHRD   :☼< 2NKTSHNM NE @M (MGNLNFDMDNTR +HMD@Q .#$ 2XRSDL Ŕ /S  &HUD SGD FDMDQ@K QD@K RNKTSHNM NE SGD ENKKNVHMF SVN CHLDMRHNM@K HMGNLNFD MDNTR RXRSDL NE KHMD@Q CHƤDQDMSH@K DPT@SHNMR     C x(t) 1 −1 x(t) RHM(t) = + , 0 2 y(t) BNR(t) Ct y(t)



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF VGDQD x, y : R → R @MC t ∈ R $WDQBHRD   :☼< 2NKTSHNM NE @M (MGNLNFDMDNTR +HMD@Q .#$ 2XRSDL Ŕ /S  +DS v ∈ C d AD @M DHFDMUDBSNQ NE SGD L@SQHW A ∈ Rd×d BNQQDRONMCHMF SN SGD DHFDMU@KTD λ ∈ C 2GNV SG@S ENQ SGD HMGNLNFDMDNTR CHƤDQDMSH@K DPT@SHNM x˙ = Ax + DWO (ωt) v @ O@QSHBTK@Q RNKTSHNM HR NAS@HMDC AX SGD @MR@SY x = β DWO (ωt) v ,

HE ω = λ

@MC x = βt DWO (ωt) v ,

HE ω = λ .

(M D@BG B@RD CDSDQLHMD SGD BNMRS@MS β %HM@KKX BNLOTSD SGD FDMDQ@K RNKTSHNM NE SGD CHƤDQDMSH@K DPT@SHNM ⎞ ⎛ ⎞ ⎛ 2 2 0 1 ⎠ ⎝ ⎝ 0 2 0 x + DWO (2t) 0 ⎠ . x˙ = 1 0 1 3 $WDQBHRD   :☼< (MSDFQ@K "TQUDR HM SGD /K@MD #DSDQLHMD @KK BNMRS@MS @MC @KK ODQHNCHB HMSDFQ@K BTQUDR NE SGD UDBSNQ ƥDKC v : R2 → R2 VHSG v(0, 0) = (0, 0)T @MC     2  1 y x 2 v(x, y) = + x + y RHM  , 2 2 −x y x +y HE (x, y) = (0, 0) ,NQDNUDQ RJDSBG SGD QDL@HMHMF HMSDFQ@K BTQUDR PT@KHS@SHUDKX $WDQBHRD   :☼< $PTHU@KDMBD NE (MSDFQ@K "TQUDR +DS v : Ω → Rd AD @ BNMSHMTNTR UDBSNQ ƥDKC NM @M NODM RDS Ω ⊂ Rd  @MC KDS α : Ω → R \ {0} 2GNV SG@S SGD SQ@BDR SQ@IDBSNQHDR NE SGD HMSDFQ@K BTQUDR NE SGD SVN UDBSNQ ƥDKCR v @MC αv BNHMBHCD ,NQD OQDBHRDKX (E ψ HR @ HMSDFQ@K BTQUD NE SGD UDBSNQ ƥDKC αv VHSG ψ(τ0 ) = x0 @MC HE τ = τ (t) HR @ SHLD SQ@MRENQL@SHNM CDƥMDC AX

τ α (φ (s)) Cs = t − t0 , τ0

SGDM ϕ := ψ ◦ τ HR @M HMSDFQ@K BTQUD NE SGD UDBSNQ ƥDKC v VHSG ϕ(τ0 ) = x0 $WDQBHRD   :☼< 3Q@MRONQSHMF SGD 3GDNQDL NE "@Q@SGDNCNQX HMSN ,TKSHOKD #HLDMRHNMR %NQLTK@SD @MC OQNNE SGD SGDNQDL NE "@Q@SGDNCNQX 3GDNQDL   ENQ RXRSDLR NE NQCHM@QX CHƤDQDMSH@K DPT@SHNMR HM SGD DWSDMCDC RDMRD

6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS $WDQBHRD   :☼< /QNUHMF SGD "NMSHMTHSX 3GDNQDL ENQ 2NKTSHNMR HM SGD $W SDMCDC 2DMRD %NKKNVHMF SGD OQNNE NE SGD BNMSHMT@SHNM 3GDNQDL   RGNV SG@S 3GDNQDL   GNKCR ENQ RNKTSHNMR HM SGD DWSDMCDC RDMRD ( D  ENQ RNKTSHNMR HM SGD DWSDMCDC RDMRD RNV SGD ENKKNVHMF (M @ CNL@HM D NE SGD (t, x) OK@MD KDS SGD ETMBSHNM f AD CDƥMDC LD@RTQ@AKD HM t ENQ ƥWDC x @MC BNMSHMTNTR HM x ENQ ƥWDC t +DS SGDQD DWHRS @M HMSDFQ@AKD ETMBSHNM m RTBG SG@S |f (t, x)| ≤ m(t) ENQ (t, x) ∈ D 3GDM FHUDM @ RNKTSHNM ϕ NE x˙ = f (t, x) ENQ t ∈ (a, b) HS HR SGD B@RD SG@S ϕ(b − 0) DWHRSR @MC HE (b, ϕ(b − 0)) ∈ D SGDM ϕ B@M AD BNMSHMTDC NUDQ (a, b + δ] ENQ RNLD δ > 0 RHLHK@Q QDRTKS GNKCR @S a 3GTR SGD RNKTSHNM ϕ B@M AD BNMSHMTDC TO SN SGD ANTMC@QX NE D ,NQDNUDQ SGD R@LD BNMSHMT@SHNM HR U@KHC ENQ @ L@WHLTL RNKTSHNM ϕM NQ @ LHMHLTL RNKTSHNM ϕm $WDQBHRD   :☼< .M SGD %NQL NE SGD %TMC@LDMS@K ,@SQHW #DSDQLHMD SGD ENQL NE SGD ETMC@LDMS@K L@SQHW NESDM B@KKDC :URQVNLPDWUL[ NE SGD ƥQRS NQCDQ RXRSDL @RRNBH@SDC SN @ d SG NQCDQ KHMD@Q NQCHM@QX CHƤDQDMSH@K DPT@SHNM $WDQBHRD   :< +DLL@ NE ,NQRD (M SGD UHBHMHSX NE @ RS@SHNM@QX ONHMS a @ C 2 ETMBSHNM f : Rd → R VHSG MNM CDFDMDQ@SD 'DRRH@M f  (a) B@M AD @OOQNWHL@SDC AX SGD PT@CQ@SHB ENQL (x − a)T f  (a)(x − a) 3GD KDLL@ NE ,NQRD SDKKR TR SG@S f HSRDKE HR KNB@KKX @MC HM RTHS@AKD BNNQCHM@SDR SGHR PT@CQ@SHB ENQL 3GHR KDLL@ HR DRRDMSH@K ENQ SGD L@SGDL@SHB@K ƥDKC NE CHƤDQDMSH@K SNONKNFX +DLL@   +DLL@ NE ,NQRD /HW f : U → R EH D C 2 IXQFWLRQ LQ D YLFLQLW\ U RI 0 ∈ Rd VXFK WKDW f (0) = 0 f  (0) = 0 DQG QRQGHJHQHUDWH +HVVLDQ f  (0) 7KHQ WKHUH LV D GLƱHRPRUSKLVP ϕ : U0 → V RI DQ HQYLURQPHQW U0 RI 0 WR DQ HQYLURQPHQW V RI 0 VXFK WKDW   2 f ◦ ϕ−1 (y) = y12 + · · · + yk2 − yk+1 + · · · + yd2 . /QNNE SGD KDLL@ NE ,NQRD +LQW 2S@QS AX BNMRSQTBSHMF @ RXLLDSQHB L@SQHW A(x) ∈ Rd×d SGD BNDƧBHDMSR NE VGHBG @QD C ∞ ETMBSHNMR RTBG SG@S f (x) =

1 T 2 x A(x)x ,

A(0) = f  (0) .

%NQ x MD@Q 0 SGD L@SQHW A(x) HR HMUDQSHAKD -NV RDS B(x) := A(0)A(x)−1 3GDM B(0) = E 6HSG SGD GDKO NE @ ONVDQ RDQHDR BNMRSQHBS @ CHƤDQDMSH@AKD L@OOHMF



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF Q : U0 → Rn×n RTBG SG@S Q2 = B !DB@TRD BA = AB T VD G@UD QA = AQT s @MC VHSG S := Q−1 HS ENKKNVR A = SA(0)S T ,NQDNUDQ CDƥMD ψ(x) := S(x)T  @MC SGTR VD NAS@HM f (x) = 12 ψ(x)T A(x)ψ(x) . #TD SN ψ  (0) = E SGD ETMBSHNM ψ G@R @M HMUDQRHNM ϕ @S 0 6HSG SGHR VD FDS (f ◦ ϕ) (y) =

1 T 2 y A(0)y .

%HM@KKX SGHR K@RS PT@CQ@SHB ENQL B@M AD SQ@MRENQLDC SN OQHMBHO@K @WHR ENQL AX @ KHMD@Q SQ@MRENQL@SHNM $WDQBHRD   :☼< .M 2DBNMC .QCDQ 2XRSDLR %NQ SGD RNKTSHNMR NE @ GNLNFDMDNTR RDBNMC NQCDQ KHMD@Q CHƤDQDMSH@K DPT@SHNM VHSG BNMSHMTNTR BNDƧBHDMSR NM SGD HMSDQU@K I RGNV SGD ENKKNVHMF  $UDQX RNKTSHNM CHƤDQDMS EQNL SGD MTKK RNKTSHNMR G@R RHLOKD YDQNR NMKX @MC SGD RDS NE HSR YDQNR CNDR MNS G@UD @ BTLTK@SHNM ONHMS HM I  (E (ϕ, ψ) HR @ ETMC@LDMS@K RXRSDL SGDM SGDQD HR @ YDQN NE ψ ADSVDDM @MX SVN YDQNR NE ϕ 3GHR QDRTKS HR RNLDSHLDR B@KKDC VHSDUDWLRQ WKHRUHP +LQW ϕψ˙ − ψ ϕ˙ G@R MN YDQNR $WDQBHRD   :☼< .M SGD 5NKSDQQ@ +NSJ@ 2XRSDL 2GNV SG@S SGD 5NKSDQQ@ +NSJ@ RXRSDL   G@R @KRN ƥQRS HMSDFQ@KR HM SGD SGQDD PT@CQ@MSR R+ × R−  R− × R+  @MC R− × R− '@R SGD 5NKSDQQ@ +NSJ@ RXRSDL ODQHNCHB RNKTSHNMR HM NMD NE SGDRD SGQDD PT@CQ@MSR +LQW #HRBTRR VGDSGDQ SGD KDUDK RDSR NE SGD ƥQRS HMSDFQ@KR @QD BNLO@BS $WDQBHRD   :☼< .RBHKK@SHNMR NE @ 2OGDQHB@K !TNX ROGDQHB@K ATNX NE Q@CHTR r ƦN@SR G@KE RTALDQFDC HM V@SDQ (E HS HR RTOOQDRRDC RKHFGSKX @ QDRSNQHMF ENQBD DPT@K SN SGD VDHFGS SN SGD CHROK@BDC V@SDQ OQDRRDR HS TOV@QC @MC HE HS HR SGDM QDKD@RDC HS VHKK ANA TO @MC CNVM %HMC SGD ODQHNC NE NRBHKK@SHNM HE SGD EQHBSHNM NE SGD V@SDQ HR MDFKDBSDC $WDQBHRD   :☼< .RBHKK@SHNMR 4MCDQ &Q@UHSX 2TOONRD SG@S @ RSQ@HFGS STMMDK HR CQHKKDC SGQNTFG SGD D@QSG SGQNTFG @MX SVN ONHMSR NM SGD RTQE@BD (E SQ@BJR @QD K@HC SGDM Ŕ MDFKDBSHMF EQHBSHNM Ŕ @ SQ@HM OK@BDC HM SGD STMMDK @S NMD DMC VHKK QNKK TMCDQ SGD D@QSG TMCDQ HSR NVM VDHFGS RSNO @S SGD NSGDQ DMC @MC QDSTQM 2GNV SG@S SGD SHLD QDPTHQDC ENQ @ BNLOKDSD QNTMC SQHO HR SGD R@LD ENQ @KK RTBG STMMDKR @MC DRSHL@SD HSR U@KTD $WDQBHRD   :< "NMSHMTNTR $WSDMRHNMR NE "NMSHMTNTR ,@OOHMFR /QNUD +DLL@   +DS X, Y AD !@M@BG RO@BDR Ω ⊂ X BKNRDC @MC f : Ω → Y BNMSHMTNTR 3GDM SGDQD HR @ BNMSHMTNTR DWSDMRHNM f˜ : X → Y RTBG SG@S f˜(X) ⊂ BNMU(f (Ω)) VGDQD BNMU(f (Ω)) CDMNSDR SGD BNMUDW GTKK NE f (Ω) 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS $WDQBHRD   :☼< %HKKHMF SGD &@OR HM $W@LOKD   $W@LOKD   HR Q@SGDQ E@RS O@BDC %HKK SGD F@OR NE HSR NTSKHMD



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF

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5HFDS 6LPXODWLRQ RI 2UGLQDU\ 'LƨHUHQWLDO (TXDWLRQV 3GD QDKDU@MS @RODBSR NE SGD MTLDQHB@K RHLTK@SHNM NE NQCHM@QX CHƤDQDMSH@K DPT@SHNMR @QD CHRBTRRDC "K@RRHB@K DWOKHBHS NMD RSDO LDSGNCR RTBG @R SGD DW OKHBHS $TKDQ NQ 1TMFD *TSS@ RBGDLDR @QD OQDRDMSDC ADENQD LNSHU@SHMF HLOKHBHS @OOQN@BGDR ENQ RSHƤ .#$R U@QHDSX NE DW@LOKD HLOKDLDMS@SHNMR RGNV SGD ADG@UHNTQ NE SGD CHƤDQDMS RBGDLDR @OOKHDC SN CHƤDQDMS HMHSH@K U@KTD OQNA KDLR 3GD AQHDE CHRBTRRHNM NE SGD -DVL@QJ E@LHKX NE RBGDLDR @MC NE RXL OKDBSHB LDSGNCR VHCDMR SGD RBNOD NE SGHR BG@OSDQ SN @OOQN@BGDR SG@S @QD SXO HB@KKX MNS SQD@SDC HM BNTQRD BTQQHBTK@ ATS SG@S OQNUHCD TRDETK ED@STQDR VNQSG ADHMF NM SGD Q@C@Q HM SGD BNMSDWS NE 1.#$ RHLTK@SHNMR

  *DX "NMBDOSR ,@MX LDSGNCR DWHRS ENQ NAS@HMHMF @M@KXSHB@K RNKTSHNMR NE NQCHM@QX CHƤDQDMSH@K DPT@SHNMR R VD G@UD RDDM HM "G@O  SGDRD @OOQN@BGDR @QD OQHL@QHKX KHLHSDC SN RODBH@Kŕ@MC SXOHB@KKX RHLOKDŕSXODR NE .#$R 6GDM @OOKHB@AKD SGDX OQN CTBD @ RNKTSHNM HM SGD ENQL NE @ MHBD @M@KXSHB ENQLTK@ (M OQ@BSHB@K OQNAKDLR VD EQDPTDMSKX DMBNTMSDQ .#$R SG@S @QD MNS @LDM@AKD SN RNKTSHNM AX RODBH@K LDSGNCR @MC @ MTLDQHB@K @OOQNWHL@SHNM ENQ RNKTSHNMR LTRS AD ENTMC ,NQD NUDQ DUDM HE @M @M@KXSHB@K RNKTSHNM B@M AD NAS@HMDC @ MTLDQHB@K RNKTSHNM NE SGD NQCHM@QX CHƤDQDMSH@K DPT@SHNM L@X AD OQDEDQ@AKD HM BDQS@HM B@RDR DROD BH@KKX HE SGD @M@KXSHB@K RNKTSHNM HR UDQX BNLOKHB@SDC @MC SGTR BNLOTS@SHNM@KKX DWODMRHUD SN DU@KT@SD -TLDQHB@K LDSGNCR ENQ NQCHM@QX CHƤDQDMSH@K DPT@SHNMR L@X AD CDQHUDC HM U@QHNTR V@XR .MD @OOQN@BG HR SN TRD SGD KHMJ ADSVDDM NQCHM@QX CHƤDQDMSH@K DPT@SHNMR @MC HMSDFQ@SHNM RDD : OO < D F  (MSDFQ@SHMF ANSG RHCDR NE SGD HMHSH@K U@KTD OQNAKDL RDD $P   ADKNV EQNL s SN s + h VD NAS@HM

s+h

s+h y(t)Ct ˙ = f (t, y(t))Ct , s

s

@MC FDS SGD HMSDFQ@K DPT@SHNM NE SGD BNQQDRONMCHMF .#$ NM [s, s + h] @R

s+h f (t, y(t))Ct .   y(s + h) = y(s) + s

OOQNWHL@SHMF SGD HMSDFQ@K VHSG RNLD MTLDQHB@K HMSDFQ@SHNM RBGDLD QDRTKSR HM ENQLTK@D ENQ SGD @OOQNWHL@SD RNKTSHNM NE .#$R 3GTR @ RSQNMF QDK@SHNM &KDSWHU 



3NAH@R 1HFNHO  %KNQH@M 5XSS ADSVDDM MTLDQHB@K PT@CQ@STQD @MC MTLDQHB@K HMSDFQ@SHNM ENQ RNKUHMF NQCH M@QX CHƤDQDMSH@K DPT@SHNMR DWHRSR (M SGD B@RD VGDM SGD @OOQNWHL@SHNM NE  s+h f (t, y(t))Ct CNDR MNS HMUNKUD SGD U@KTD y(s + h) SGD U@KTD NMD HR SQXHMF s SN @OOQNWHL@SD ENQ SGD MDWS SHLD RSDO SGD BNQQDRONMCHMF LDSGNCR @QD B@KKDC H[SOLFLW PHWKRGV 5HBD UDQR@ HE y(s + h) HR HM SGD ENQLTK@ VD B@KK SGDRD LPSOLFLW PHWKRGV 2TBG RBGDLDR SXOHB@KKX HLOKX SGD RNKTSHNM NE @ MNM KHMD@Q RXRSDL NE DPT@SHNMR HM D@BG SHLD RSDO VGHBG L@JDR SGDRD LDSGNCR BNLOTS@SHNM@KKX LNQD DWODMRHUD "DQS@HM BHQBTLRS@MBDR NQ OQNAKDLR L@X QDPTHQD SGD TRD NE HLOKHBHS RBGDLDR 6GHKD QD@CHMF SGHR BG@OSDQ MNSD SGD @MRVDQR SN SGD ENKKNVHMF PTDRSHNMR  'NV B@M .#$ RXRSDLR AD SQD@SDC MTLDQHB@KKX  6G@S B@SDFNQHDR NE OQNAKDLR DWHRS @MC VGHBG RODBHƥB LDSGNCR CN SGDX QDPTHQD  6GX CN RN L@MX CHƤDQDMS HMSDFQ@SHNM RBGDLDR DWHRS 'NV CN SGDX CHE EDQ @R VDKK @R SGD ENKKNVHMF JDX BNMBDOSR  "NMRHRSDMBX NE .#$ HMSDFQ@SHNM RBGDLDR  "NMUDQFDMBD NE .#$ HMSDFQ@SHNM RBGDLDR  $WOKHBHS UR HLOKHBHS LDSGNCR 3GD QDL@HMCDQ NE SGHR BG@OSDQ G@R SGD ENKKNVHMF RSQTBSTQD 6D OQDRDMS @ QD RTKS BNMBDQMHMF SGD BNMCHSHNM NE HMHSH@K U@KTD OQNAKDLR HM 2DB   (M 2DB   @ RTQUDX NM BK@RRHB@K NMD RSDO LDSGNCR HR FHUDM 6D AQHDƦX CHRBTRR RSHƤ .#$R @R @ LNSHU@SHNM ENQ HLOKHBHS RBGDLDR @MC OQDRDMS SGD HLOKHBHS $TKDQ LDSGNC @MC SGD SQ@ODYNHC@K QTKD HM 2DB   3GD E@LHKX NE -DVL@QJ LDSGNCR HM 2DB   BNMRSQTBSDC ENQ RDBNMC NQCDQ RXRSDLR HM RSQTBSTQ@K CXM@LHBR HR @M @KSDQM@SHUD SN ENQLTK@SHMF SGD OQNAKDL @R @ RXRSDL NE .#$R (M 2DB   @ AQHDE DWBTQRHNM KD@CR TR SN RXLOKDBSHB LDSGNCR @ B@SDFNQX NE MNM RS@MC@QC RBGDLDR VGHBG ONRRDRR BDQS@HM TRDETK BNMRDQU@SHNM OQNODQSHDR %HM@KKX 2DB SHNM   RTLL@QHRDR SGD BNMSDMSR NE SGHR BG@OSDQ

  &DMDQ@K RODBSR NE -TLDQHB@K 2NKTSHNM NE .#$R %NQ SGD R@JD NE BNLO@SHAHKHSX VHSG BK@RRHB@K KHSDQ@STQD HM SGD ƥDKC NE MTLDQ HBR ENQ .#$R VD MNV @KSDQ SGD MNS@SHNM HM SGD CDƥMHSHNM NE SGD HMHSH@K U@KTD OQNAKDL (5/   y˙ = f (t, y(t)),



t ∈ [t0 , te ]

 

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF y(t0 ) = y0 ,

 

ENQ f : [t0 , te ] × Rm → Rm ADHMF RTƧBHDMSKX RLNNSG -NSD SG@S VD NLHS SGD UDBSNQ NE O@Q@LDSDQR λ NE $P   HM SGHR CDƥMHSHNM RHMBD HS CNDR MNS G@UD @MX HLO@BS NM SGD MTLDQHB@K RBGDLDR CDQHUDC ADKNV 3GD BNMRHCDQ@SHNM NE ƥQRS NQCDQ RXRSDLR HR RTƧBHDMS RHMBD DUDQX RXRSDL NE GHFGDQ NQCDQ B@M AD QDENQLTK@SDC SN ENQL@KKX AD NE ƥQRS NQCDQ RDD 2DB    'DMBD MTLDQHB@K RBGDLDR SG@S @QD @AKD SN CD@K VHSG RXRSDLR NE DPT@SHNMR L@X @KRN AD @OOKHDC SN .#$R BNMS@HMHMF GHFGDQ NQCDQ CDQHU@SHUDR %NKKNVHMF :< SGD ENKKNVHMF SGDNQDL RGNVR GNV LTBG SVN RNKTSHNMR NE SGD .#$ RXRSDL B@M CHƤDQ CDODMCHMF NM SGD CHƤDQDMBD HM SGDHQ HMHSH@K U@KTDR 3GHR BNQQDRONMCR SN @M@KXRHMF SGD BNMCHSHNM NE SGD HMHSH@K U@KTD OQNAKDL 3GDNQDL   )RU WZR VROXWLRQV y, z RI WKH 2'(   ZLWK GLƱHUHQW LQLWLDO YDO XHV DQG IRU DOO t, t0  LW KROGV ||y(t) − z(t)|| ≤ ||y(t0 ) − z(t0 )|| · eL·|t−t0 |

 

ZLWK 0 < L ∈ R 3URRI +DS t0 < t D(t) := ||y(t) − z(t)|| HR @ BNMSHMTNTR RB@K@Q ETMBSHNM 'DMBD t E(t) := e−Lt t0 D(t)dt HR BNMSHMTNTRKX CHƤDQDMSH@AKD 3GDQDENQD HS GNKCR   D(t) = eLt E(t) = LeLt E(t) + eLt E  (t) .

 

%QNL   VHSG s = t0 ENQ y(t) @MC z(t) EQNL SGD SQH@MFKD HMDPT@KHSX @MC SGD +HORBGHSY BNMSHMTHSX   VD G@UD

t

t D(t) ≤ ||y0 − z0 || + ||f (τ, y(τ )) − f (τ, z(τ ))||dτ ≤ ||y0 − z0 || + LD(τ )dτ . t0

t0

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E(t) =

E  (t) ≤ ||y0 − z0 ||Le−Lt , t   E  (τ )dτ ≤ ||y0 − z0 || e−Lt0 − e−Lt .

t0

-NSD SG@S E(t0 ) = 0 AX CDƥMHSHNM 4RHMF SGDRD DRSHL@SDR NE SGD QHFGS G@MC RHCD NE   QDRTKSR HM ! D(t) ≤ ||y0 − z0 || · eL(t−t0 ) − 1 + 1 .   6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS (M B@RD NE x < x0  NMD TRDR −x @R HMCDODMCDMS U@QH@AKD @MC −f @R QHFGS G@MC RHCD NE SGD BNQQDRONMCHMF .#$ $P   QDOQDRDMSR @M TOODQ ANTMC ENQ SGD DQQNQ NQ BNMCHSHNM %NQ MTLDQHB@K RNKTSHNMR VD DMBNTMSDQ HMDW@BS HMHSH@K U@KTDR CTD SN QNTMC NƤ DQQNQR @MCNQ CHRBQDSHR@SHNM DQQNQR 'DMBD SGD QDKH@AHKHSX NE RNKTSHNMR LD@ RTQDC UH@   CHLHMHRGDR VHSG te >> t0 CTD SN SGD DWONMDMSH@K SDQL ,NQD @BBTQ@SD DQQNQ QDOQDRDMS@SHNMR DWHRS ATS @QD LNQD BNLOKDW SN CDQHUD @MC DU@K T@SD (M SGD ENKKNVHMF RDBSHNMR CHƤDQDMS CHRBQDSHR@SHNM LDSGNCR ENQ .#$R VHKK AD CHRBTRRDC KK NE SGDL QDKX NM @ CHRBQDSHR@SHNM NE SGD SHLD HMSDQU@K TMCDQ BNM RHCDQ@SHNM CDMNSDC AX t0 < t1 < . . . < tn = te  hn = tn+1 −tn HR B@KKDC SGD SHLD RSDO RHYD VGHBG VD @RRTLD SN AD DPTHCHRS@MS HM SGD ENKKNVHMF H D hi = h ∀i %NQ SGD MTLDQHB@K HMSDFQ@SHNM NE .#$R VD MDDC SN BNLOTSD @OOQNWHL@SD RN KTSHNMR y n ≈ y(tn ) KK LDSGNCRŕHM SGD ENKKNVHMF @MC HM FDMDQ@KŕB@M AD ENQLTK@SDC TRHMF @M TOC@SD ETMBSHNM Φf  y n+1 = y n + h · Φf (h, yh ),

n = 0, 1, . . . .

 

-NSD SG@S Φf HM   MDDCR SN RNLDGNV DU@KT@SD SGD QHFGS G@MC RHCD f NE SGD .#$   @MC CDODMCR NM SGD SHLD RSDO RHYD h @MC @ UDBSNQ yh QDOQDRDMSHMF k OQDUHNTRKX BNLOTSDC @OOQNWHL@SHNMR H D yh = (y n , y n−1 , . . . , y n−k+1 ) 3GD ƥDKC HR MNV OQDO@QDC SN CHRBTRR RNLD LDSGNCR HM CDS@HK 6D ƥQRS OQDRDMS DWOKHBHS NMD RSDO LDSGNCR ADENQD S@BJKHMF HLOKHBHS RBGDLDR !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   0 6G@S HR SGD A@RHB HCD@ SN CDQHUD MTLDQHB@K RBGDLDR ENQ .#$R TRHMF SGD QDK@SHNM   0 6GX @QD HLOKHBHS RBGDLDR BNLOTS@SHNM@KKX LNQD DWODMRHUD 0 6G@S HR SGD OQNAKDL NE SGD TOODQ ANTMC ENQ SGD BNMCHSHNM HM $P   ENQ K@QFD U@KTDR NE t 0 6G@S CNDR SGD TOC@SD ETMBSHNM Φ NE @ MTLDQHB@K RBGDLD FDMDQ@KKX CD ODMC NM



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6D MNV OQDRDMS SGQDD VDKK JMNVM QDOQDRDMS@SHUDR NE DWOKHBHS NMD RSDO LDSGNCR ENQ .#$R SGD DWOKHBHS $TKDQ LDSGNC 'DTMŗR LDSGNC @MC SGD BK@R RHB@K 1TMFD *TSS@ RBGDLD 6D RGNV , 3+ ! ETMBSHNMR TRDC SN BNLO@QD SGD SGQDD LDSGNCR HM 2DB   

   $WOKHBHS $TKDQ ,DSGNC 3GD DWOKHBHS $TKDQ LDSGNC @KRN JMNVM @R SGD ENQV@QC $TKDQ %$ LDSGNC HR @ RSQ@HFGSENQV@QC @OOQN@BG SN BNLOTSD MTLDQHB@K @OOQNWHL@SHNMR NE .#$R .MD V@X NE LNSHU@SHMF SGHR ƥQRS NQCDQ RBGDLD HR SN @OOQNWHL@SD SGD SHLD CDQHU@SHUD NE SGD HMHSH@K U@KTD OQNAKDL   TRHMF ENQV@QC ƥMHSD CHƤDQDMBDR y˙ = ˙

y n+1 − y n hn

 

ENQ SHLD RSDOR n @MC n + 1 "NLAHMHMF SGHR VHSG @ KHMD@Q @OOQNWHL@SHNM NE SGD TMJMNVM ETMBSHNM y @S SHLD RSDO tn+1 TRHMF SGD CDQHU@SHUD DU@KT@SDC @S SGD OQDUHNTR SHLD RSDO tn QDRTKSR HM SGD DWOKHBHS $TKDQ RBGDLD y n+1 = y n + hn · f (tn , y n ) .

 

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s+h f (t, y(t))Ct ≈ f (s, y(s)) · h , s

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hn · [f (tn , y n ) + f (tn+1 , y n + hn · f (tn , y n ))] .   2

-NSD SG@S VD HMUDRS @M @CCHSHNM@K DU@KT@SHNM NE f HM D@BG SHLD RSDO %NQ RHL OKD OQNAKDLR SGHR CNDR MNS QDOQDRDMS @ L@INQ NUDQGD@C ATS VGDM HS BNLDR SN LNQD BNLOKDW RBDM@QHNR SGHR L@X HMSQNCTBD RTARS@MSH@K @CCHSHNM@K BNRSR HM SGD B@RD NE .#$ HMSDFQ@SHNM HMRHCD /#$R @ ETMBSHNM DU@KT@SHNM HMUNKUDR SGD BNLOTS@SHNM NE @KK CHRBQDSD RO@SH@K NODQ@SNQR NM SGD VGNKD FQHC 3GD @C U@MS@FD NE 'DTMŗR LDSGNC NUDQ SGD DWOKHBHS $TKDQ RBGDLD HR NMD NQCDQ NE BNM



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s 



 

bj Ynj .

j=1 

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s 



aij Ynj ,



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∀i = 1, . . . ,  s.

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= yn  = y n + h2n Y1   = y n + h2n Y2   = y n + hn Y 3 



Y1  Y2  Y3  Y4

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.

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x ˜˙ = A˜ x,

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x(0) = x0

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ş H D  SGDHQ @KFDAQ@HB @MC FDNLDSQHB LTKSHOKHBHSX @QD DPT@K



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF SN SGD DHFDMU@KTDR λ NE A RTBG SG@S SGDHQ QD@K O@QSR RG@QD SGD R@LD RHFM SGD XQVWDEOH VXEVSDFH HR RO@MMDC AX SGNRD VHSG 1D(λ) > 0 SGD VWDEOH VXEVSDFH AX SGNRD VHSG 1D(λ) < 0 @MC SGD FHQWHU VXEVSDFH AX SGNRD VHSG 1D(λ) = 0 %HFTQD   RJDSBGDR SGD U@QHNTR SXODR NE CXM@LHBR NE OK@M@Q RXRSDLR VHSG QDRODBS SN SGD RS@AHKHSX OQNODQSHDR NE SGD DPTHKHAQH@ SGDQDHM RDD :< O  %HM@KKX VD B@M DWSDMC SGDRD BNMBDOSR SN MNM KHMD@Q RXRSDLR @S KD@RS KN B@KKX @MC NMKX ENQ SGNRD DPTHKHAQH@ ENQ VGHBG SGD L@SQHW NE SGD KHMD@QHYDC RXR SDL G@R MN DHFDMU@KTDR VHSG U@MHRGHMF QD@K O@QS H D  MN BDMSDQ RTARO@BD HR OQDRDMS #DƥMHSHNM   'XODQANKHB $PTHKHAQHTL +DS x∗ AD @M DPTHKHAQHTL NE SGD NQ CHM@QX CHƤDQDMSH@K DPT@SHNM x˙ = f (x) f ∈ C 1 (Rd , Rd ) 3GDM x∗ HR B@KKDC K\ SHUEROLF HE 1D(λ) = 0 ENQ @KK DHFDMU@KTDR λ NE #f (x∗ ) (E MNS x∗ HR B@KKDC QRQ K\SHUEROLF 3GD MDWS SGDNQDL SDKKR TR SG@S KNB@KKX @QNTMC GXODQANKHB DPTHKHAQH@ SGD CX M@LHBR NE SGD KHMD@QHYDC @MC MNM KHMD@Q RXRSDL @QD DPTHU@KDMS 3GDNQDL   '@QSL@M &QNAL@MR 3GDNQDL NE +HMD@QHYDC 2S@AHKHSX &RQ VLGHU WKH G\QDPLFDO V\VWHP JLYHQ E\ WKH DXWRQRPRXV GLƱHUHQWLDO HTXDWLRQ x˙ = f (x) RQ Rd ZLWK f ∈ C 2 (Rd , Rd ) DQG x∗ ∈ Rd EHLQJ DQ HTXLOLEULXP SRLQW RI WKLV V\VWHP /HW Φ(t; x) GHQRWH WKH ƳRZ RI WKLV V\VWHP DQG Ψ(t; x) WKH ƳRZ RI WKH OLQHDUL]HG V\VWHP x˙ = 'f (x∗ )(x − x∗ ) ,I x∗ LV K\SHUEROLF WKHQ WKHUH LV DQ RSHQ VXEVHW U ⊂ Rd VXFK WKDW x∗ ∈ U  DQG D KRPHRPRUSKLVP h : U → h(U) ⊂ Rn VXFK WKDW Ψ(t; h(x)) = h(Φ(t; x)) ZKHQHYHU x ∈ U  3URRI 2DD :< 2DB   NQ : OO Ŕ< 3GHR KNB@K RS@AHKHSX BNMRDQUHMF QDK@SHNMRGHO ADSVDDM KHMD@Q @MC MNM KHMD@Q RXRSDLR @KKNVR TR SN DWSDMC SGD BNMBDOSR NE SGD RS@AKD TMRS@AKD @MC BDMSDQ RTARO@BD SN MNM KHMD@Q CXM@LHB@K RXRSDLR 3GD BNQQDRONMCHMF ORFDO VWDEOH XQVWDEOH @MC FHQWHU PDQLIROG NE SGD MNM KHMD@Q RXRSDL @QD HMU@QH@MS @MC S@M FDMS SN SGDHQ BNTMSDQO@QS RTARO@BDR 3GD RS@AKD @MC TMRS@AKD L@MHENKCR @QD TMHPTDKX CDSDQLHMDC @MC G@UD SGD R@LD CHLDMRHNM @R SGDHQ BNTMSDQO@QSR 3GNTFG '@QSL@M &QNAL@MŗR SGDNQDL NE KHMD@QHYDC RS@AHKHSX CNDR MNS QDB NLLDMC @MXSGHMF ENQ SGD MNM GXODQANKHB B@RD H D  SGD B@RD NE DHFDMU@K TDR VHSG U@MHRGHMF QD@K O@QS .MD B@M TRD BDMSDQ L@MHENKC LDSGNCR NQ +X@OTMNU ETMBSHNMR +X@OTMNUŗR RDBNMC LDSGNC SN F@HM LNQD HMENQL@SHNM 

  

LNQD FDMDQ@K @OOQN@BG HR FHUDM AX RN B@KKDC MNQL@K ENQLR VGDQD MNS NMKX SGD KHMD@Q @O OQNWHL@SHNM NE SGD MNMKHMD@Q RXRSDL HR BNMRHCDQDC ATS @KRN GHFGDQ NQCDQ @OOQNWHL@SHNMR RDD D F  :< 1DB@KK KRPHRPRUSKLVP HR @ AHIDBSHUD BNMSHMTNTR L@OOHMF VHSG BNMSHMTNTR HMUDQRD ş RDD D F :< 2DB   NQ :< O  3G    @MC O  3G    şRDD :< 2DB   NQ : OO < D F 

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Stable Continuum of Equilibria

No Flow Situation

Unstable Continuum of Equilibria

3NAH@R 1HFNHO  %KNQH@M 5XSS

A= a 0 0 -b

a0 0b

Stable Node (2. Kind) A= 0 1 0 0

Center

Unstable Focus

A=

a1 0a

A=

A = ib 0 0 ib

Stable Node (3. Kind)

Unstable Node (3. Kind)

A = a1+ib1 0 0 a2+ib2

A=

A = -a 0 0 -a

-a 0 0 -b

Stable Focus

Unstable Node (2. Kind)

Unstable Continuum of Equilibria

A= a 0 0 a

A = -a 0 0 0

Stable Node (1. Kind)

A= 0 0 0 0

Saddle Point

Unstable Node (1. Kind)

A= a 0 0 0

A = -a1+ib1 0 0 -a2+ib2

asymptotically (Lyapunov) stable (Lyapunov) stable unstable direction of the flow

A=

-a 1 0 -a

Real constants: a, a1, a2, b, b1, b2 > 0

%HFTQD   &Q@OGHB@K BK@RRHƥB@SHNM NE DPTHKHAQH@ HM OK@M@Q RXRSDLR VHSG QDRODBS SN SGDHQ RS@AHKHSX OQNODQSHDR RDD :< O 



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF (M SGD QDL@HMCDQ NE SGHR RDBSHNM VD CHRBTRR SVN LDSGNCR SG@S @KKNV TR SN CDBHCD HE SGD RODBSQTL σ(A) NE @ d × d L@SQHW A KHDR HM SGD KDES G@KE OK@MD H D  HE SGD +X@OTMNU DWONMDMSR NE A KD@C SN KHMD@Q RS@AHKHSX 3GDRD LDSGNCR @QD JMNVM @R SGD 1NTSG 'TQVHSY RS@AHKHSX BQHSDQHNM @MC SGD @OOKHB@SHNM NE SGD +NYHMRJHH LD@RTQD

   3GD 1NTSG 'TQVHSY 2S@AHKHSX "QHSDQHNM 6D ENKKNV :< OO  ENQ SGD NTSKHMD NE SGD 1NTSG 'TQVHSY RS@AHKHSX BQHSD QHNM RDD :< OO  @MC HSR HLOQNUDLDMS SGD +HDM@QC "GHO@QS RS@AHKHSX BQHSDQHNM RDD :< OO  %NQ SGD CDQHU@SHNM NE SGD 1NTSG 'TQVHSY BQHSDQHNM VD BNMRHCDQ SGD ONKXMN LH@K f (z) = a0 z n + b0 z n−1 + a1 z n−2 + b1 z n−3 + . . . ,

(a0 = 0) ,

!DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   Ŕ /@QS ( 0 6G@S B@M XNT R@X @ANTS SGD RS@AHKHSX NE SGD NQHFHM @S KHMD@Q RXRSDLR VHSG QDRODBS SN SGD RHFM NE SGD QD@K O@QSR NE SGD DHFDMU@KTDR NE SGD RXRSDL L@SQHW 0 &HUD RNLD QDOQDRDMS@SHUD OG@RD RO@BD OKNSR ENQ SGD CHƤDQDMS SXODR NE RS@AHKHSX OQNODQSHDR BNLAHM@SHNMR SG@S B@M NBBTQ HM OK@M@Q RXRSDLR 0 6G@S HR @ GXODQANKHB DPTHKHAQHTL 0 2S@SD SGD SGDNQDL NE '@QSL@M @MC &QNAL@M 0 #DSDQLHMD SGD RS@AHKHSX NE SGD MTKK RNKTSHNM NE SGD C@LODC MNM KHMD@Q ODMCTKTL x˙ = y ,

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@MC

y˙ = − RHM(x) − 0.1y .



3NAH@R 1HFNHO  %KNQH@M 5XSS VHSG QD@K BNDƧBHDMSR B@KKDC +XUZLW] PDWUL[ ⎛ b0 b1 b2 ⎜ a0 a1 a2 ⎜ ⎜ 0 b0 b 1 ⎜ H := ⎜ 0 a a 0 1 ⎜ ⎜ 0 0 b0 ⎝



a0 , b0 , . . . 3GD ENKKNVHMF RPT@QD L@SQHW NE NQCDQ n HR ... ... ... ... ...



bn−1 an−1 bn−2 an−2 bn−3



⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

# VGDQD

ak = 0 HE k > bk = 0

HE k >

0n1 2

0 n−1 1 . 2

6D VHKK SQ@MRENQL SGD 'TQVHSY L@SQHW RTBBDRRHUDKX SN TOODQ SQH@MFTK@Q ENQL AX &@TRRH@M DKHLHM@SHNM 6D BNMƥMD NTQ @SSDMSHNM SN SGD QDFTK@Q B@RD VGDQD −1 b0 = 0 c0 := a1 − a0 b1 b−1 0 = 0 d0 := b1 − b0 c1 c0 = 0 DSB %HQRS @ESDQ LTK −1 SHOKXHMF SGD NCC MTLADQDC QNVR AX a0 b0 VD RTASQ@BS SGDRD EQNL SGD BNQQD RONMCHMF QNVR NE DUDM MTLADQ 03GTR 1 SGD QNVR NE DUDM MTLADQ G@UD DMSQHDR c0 , c1 , . . .  VGDQD ck = 0 HE k > n2 − 1 -DWS @ESDQ LTKSHOKXHMF SGD QNVR NE DUDM MTLADQ AX b0 c−1 0 VD RTASQ@BS SGDRD EQNL SGD BNQQDRONMCHMF QNVR NE NCC MTLADQ 3GHR KD@CR SN ⎛ ⎞ ⎞ ⎛ b0 b1 b2 . . . bn−1 b0 b1 b2 . . . bn−1 ⎜ 0 c0 c1 . . . cn−2 ⎟ ⎜ 0 c0 c1 . . . cn−2 ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ 0 b0 b1 . . . bn−2 ⎟ ⎜ 0 0 d0 . . . dn−3 ⎟ ⎜ ⎟ ⎟ ⎜ H  ⎜ 0 0 c ... c ⎟  ⎜ 0 0 c0 . . . cn−3 ⎟ . 0 n−3 ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 b0 . . . bn−3 ⎟ ⎜ 0 0 d0 . . . dn−4 ⎟ ⎝ ⎠ ⎠ ⎝











OOKXHMF SGHR OQNBDCTQD ETQSGDQ VD nSG NQCDQ L@SQHW ⎛ b0 ⎜ 0 ⎜ R := ⎜ 0 ⎝



TKSHL@SDKX @QQHUD @S @M TOODQ SQH@MFTK@Q ⎞ b1 b2 . . . c0 c 1 . . . ⎟ ⎟ , 0 d0 . . . ⎟ ⎠



VGHBG VD RG@KK B@KK SGD 5RXWK PDWUL[ #DƥMHSHNM   $PTHU@KDMBD NE ,@SQHBDR AX ,HMNQR 3VN n × n L@SQHBDR A @MC B @QD R@HC SN AD HTXLYDOHQW HE @MC NMKX HE ENQ @MX 1 ≤ k ≤ n SGD BNQQD RONMCHMF kSG NQCDQ LHMNQR NM SGD ƥQRS k QNVR NE SGDRD L@SQHBDR @QD DPT@K H D HM SGD MNS@SHNM NE ODQLTS@SHNMR $ % $ % 1 2 ... k 1 2 ... k A = B , i 1 i2 . . . i k i1 i2 . . . i k ENQ 1 ≤ ik ≤ n @MC k = 1, 2, . . . , n



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 2HMBD SGD kSG NQCDQ LHMNQR NM SGD ƥQRS k QNVR k = 1, 2, . . . , n CN MNS BG@MFD SGDHQ U@KTDR HE NMD RTASQ@BSR EQNL @MX QNVR @MNSGDQ QNV LTKSHOKHDC AX @M @Q AHSQ@QX BNMRS@MS SGD 'TQVHSY @MC 1NTSG L@SQHBDR @QD DPTHU@KDMS $ % $ % 1 2 ... k 1 2 ... k H = R , i1 i 2 . . . i k i1 i2 . . . ik ENQ 1 ≤ ik ≤ n @MC k = 1, 2, . . . , n 3GD DPTHU@KDMBD NE SGD 'TQVHSY @MC 1NTSG L@SQHBDR ODQLHSR TR SN DWOQDRR @KK NE SGD DKDLDMSR NE R HM SDQLR NE SGD LHMNQR NE SGD 'TQVHSY L@SQHW H @MC SGDQDENQD HM SDQLR NE BNDƧBHDMSR NE SGD FHUDM ONKXMNLH@K f (z) 6D NAS@HM $ % $ % $ % 1 1 2 1 2 3 H = b0 , H = b 0 c0 , H = b 0 c0 d0 , 1 1 2 1 2 3 $ % $ % $ % 1 2 1 2 3 1 = b 0 c1 , H = b 0 c0 d1 , H = b1 , H 1 3 1 2 4 2 $ % $ % $ % 1 2 1 2 3 1 = b 0 c2 , H = b 0 c0 d2 , H = b2 , H 1 4 1 2 5 3





. 3GD RTBBDRRHUD OQHMBHO@K LHMNQR NE SGD 'TQVHSY L@SQHW H @QD TRT@KKX B@KKDC SGD +XUZLW] GHWHUPLQDQWV @MC CDMNSDC AX $ % 1 = b0 , Δ1 = H 1 $ %  1 2 b0 b1 Δ2 = H = CDS = a 1 b0 − a 0 b1 = b 0 c 0 , 1 2 a0 a1 @MC RN NM SN

⎛ $

Δn = H

1 2 ... n 1 2 ... n

%

⎜ ⎜ ⎜ = CDS ⎜ ⎜ ⎝

b 0 b1 b2 a0 a1 a 2 0 b 0 b1 0 a0 a1



... ... ... ...



bn−1 an−1 bn−2 an−2

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

(M SGD QDFTK@Q B@RD @KK NE SGD PT@MSHSHDR b0 , c0 , d0 , . . . @QD CDƥMDC @MC CHƤDQ DMS EQNL YDQN 3GHR HR BG@Q@BSDQHRDC AX SGD HMDPT@KHSHDR Δ1 = b0 = 0 ,

Δ2 = b0 c0 = 0 ,

Δ3 = b0 c0 d0 = 0 ,

...

Δn = 0 .

(M O@QSHBTK@Q b0 = Δ1 ,

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c0 =

Δ2 , Δ1

d0 =

Δ3 , Δ2

... .



3NAH@R 1HFNHO  %KNQH@M 5XSS 3GDNQDL   1NTSG 'TQVHSY 3GDNQDL ,Q WKH UHJXODU FDVH WKH QXPEHU k RI URRWV RI WKH UHDO SRO\QRPLDO f (z) = a0 z n + . . . ZKLFK OLH LQ WKH ULJKW KDOISODQH LV JLYHQ E\ WKH IRUPXOD  Δ2 Δ3 Δn k = V a0 , Δ1 , , ,..., Δ1 Δ2 Δn−1 RU HTXLYDOHQWO\ k = V (a0 , Δ1 , Δ3 , . . . ) + V (1, Δ2 , Δ4 , . . . ) , ZKHUH V (x1 , . . . , xn ) GHQRWHV WKH QXPEHU RI VLJQ FKDQJHV LQ WKH VHTXHQFH x1 , . . . , x n  3URRI 3GD OQNNE HR A@RDC HM 1NTSGŗR BQHSDQHNM BE :< OO  3GD RODBH@K B@RD HM VGHBG @KK QNNSR NE f (z) KHD HM SGD KDES G@UD OK@MD 1D(z) < 0 ENKKNVR EQNL SGD 1NTSG 'TQVHSY 3GDNQDL ENQ k = 0 "NQNKK@QX   1NTSG 'TQVHSY 2S@AHKHSX "QHSDQHNM ,Q WKH UHJXODU FDVH LQ RU GHU IRU DOO URRWV RI WKH UHDO SRO\QRPLDO f (z) = a0 z n + . . . a0 = 0 WR KDYH QHJDWLYH UHDO SDUWV LW LV QHFHVVDU\ DQG VXƴFLHQW WKDW WKH LQHTXDOLWLHV # IRU n RGG a 0 Δn > 0 a0 Δ1 > 0, Δ2 > 0, a0 Δ3 > 0, Δ4 > 0, . . . Δn > 0 IRU n HYHQ   KROG ,I a0 > 0 WKHVH FRQGLWLRQV UHGXFH WR Δi > 0 IRU i = 1, 2, . . . , n QD@K ONKXMNLH@K f (z) SGD BNDƧBHDMSR NE VGHBG R@SHREX SGD BNMCHSHNMR HM   HR @ QD@K ONKXMNLH@K VGNRD QNNSR @KK G@UD MDF@SHUD QD@K O@QSR @MC HR B@KKDC +XUZLW] SRO\QRPLDO (E VD HMSQNCTBD SGD TRT@K MNS@SHNM ENQ SGD BNDƧBHDMSR NE SGD ONKXMNLH@K f (z) = αn z n + αn−1 z n−1 + αn−2 z n−2 + . . . + α0 ,

VHSG αn > 0 ,

SGDM SGD 1NTSG 'TQVHSY BNMCHSHNMR LX AD QD VQHSSDM HM SGD QDFTK@Q B@RD @R SGD ENKKNVHMF CDSDQLHM@MS@K HMDPT@KHSHDR  αn−1 αn−3 Δ1 = αn−1 > 0 , Δ2 = > 0, αn αn−2 ⎞ ⎛ αn−1 αn−3 αn−5 an−2 αn−4 ⎠ > 0 , Δ3 = ⎝ αn 0 αn−1 αn−3



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF @MC RN NM SN

⎛ ⎜ ⎜ Δn = ⎜ ⎝

⎞ αn−1 αn−3 αn−5 . . . αn an−2 αn−4 . . . ⎟ ⎟ > 0. 0 αn−1 αn−3 . . . ⎟ ⎠



(E SGD CDSDQLHM@MS BNMCHSHNMR Δi > 0 i = 1, 2, . . . , n @QD R@SHRƥDC SGDM SGD ONKXMNLH@K f (z) L@X AD VQHSSDM @R @ OQNCTBS NE αn AX E@BSNQR NE SGD ENQL z + γ2 @MC z 2 + γ1 z + γ0 VHSG β0 , β1 , β2 > 0 3GTR @KK BNDƧBHDMSR NE f (z) LTRS AD ONRHSHUD H D αi > 0 i = 1, 2, . . . , n (M BNMSQ@RS SN SGD CDSDQLHM@MS BNMCHSHNMR Δi > 0 i = 1, 2, . . . , n SGD BNDE ƥBHDMS BNMCHSHNMR αi > 0 i = 1, 2, . . . , n @QD MDBDRR@QX ATS MNS RTƧBHDMS ENQ @KK SGD QNNSR NE f (z) SN KHD HM SGD KDES G@KE OK@MD 'NVDUDQ NMBD SGD BNDƧBHDMS BNMCHSHNMR ai > 0 i = 1, 2, . . . , n @QD R@SHRƥDC SGD CDSDQLHM@MS BNMCHSHNMR Δi > 0 i = 1, 2, . . . , n @QD MN KNMFDQ HMCDODMCDMS %NQ HMRS@MBD VGDM n = 4 SGD 1NTSG 'TQVHSY BNMCHSHNMR QDCTBD SN SGD RHMFKD HMDPT@KHSX Δ3 > 0 VGDM n = 5 SN SGD O@HQ NE HMDPT@KHSHDR Δ2 > 0 @MC Δ4 > 0 @MC VGDM n = 6 SN SGD O@HQ NE HMDPT@KHSHDR Δ3 > 0 SNFDSGDQ VHSG Δ5 > 0 %NKKNVHMF :< OO  SGHR BHQBTLRS@MBD V@R RSTCHDC AX SGD %QDMBG L@SGDL@SHBH@MR +HDM@QC @MC "GHO@QS @MC KDC SGDL HM  SN SGD CHRBNUDQX NE RS@AHKHSX BQHSDQH@ CHƤDQDMS EQNL SGD 1NTSG 'TQVHSY BQHSDQHNM 3GDNQDL   3GD +HDM@QC "GHO@QS 2S@AHKHSX "QHSDQH@ 1HFHVVDU\ DQG VXƴ FLHQW FRQGLWLRQV IRU WKH UHDO SRO\QRPLDO f (z) = αn z n + αn−1 z n−1 + αn−2 z n−2 + · · ·+α0 ZLWK αn > 0 WR KDYH RQO\ URRWV ZLWK QHJDWLYH UHDO SDUWV PD\ EH H[SUHVVHG LQ DQ\ RI WKH IRXU IROORZLQJ IRUPV  α0 , α2 , α4 , · · · > 0 DQG Δ1 , Δ3 , · · · > 0  α0 , α2 , α4 , · · · > 0 DQG Δ2 , Δ4 , · · · > 0  α0 , α1 , α3 , · · · > 0 DQG Δ1 , Δ3 , · · · > 0 RU  α0 , α1 , α3 , · · · > 0 DQG Δ2 , Δ4 , · · · > 0 3URRI M DKDF@MS OQNNE HR A@RDC NM SGD RN B@KKDC "@TBGX HMCDW @MC '@MJDK RDPTDMBDR RDD :< OO  "KD@QKX SGDRD ENTQ +HDM@QC "GHO@QS RS@AHKHSX BQHSDQH@ G@UD @M @CU@MS@FD NUDQ SGD 1NTSG 'TQVHSY BNMCHSHNMR HM SG@S SGDX HMUNKUD @ANTS G@KE @R L@MX CDSDQ LHM@MS HMDPT@KHSHDR %QNL 3GDNQDL   HS ENKKNVR SG@S ENQ @ QD@K ONKXMNLH@K HM VGHBG @KK BNDƧ BHDMSR NQ DUDM NMKX RNLD NE SGDL M@LDKX α0 , α2 , α4 , . . . NQ α0 , α1 , α3 , . . .  @QD ONRHSHUD SGD 1NTSG 'TQVHSY CDSDQLHM@MS HMDPT@KHSHDR Δi > 0 i = 1, 2, . . . , n 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS @QD MNS HMCDODMCDMS (M O@QSHBTK@Q SGD ONRHSHUHSX NE SGD 'TQVHSY CDSDQLHM@MSR NE NCC NQCDQ HLOKHDR SG@S NE SGD 'TQVHSY CDSDQLHM@MSR NE DUDM NQCDQ @MC BNM UDQRDKX

  

3GD +NYHMRJHH ,D@RTQD @MC 2S@AHKHSX

(M SGHR DWONRHSHNM NM SGD +NYHMRJHH ,D@RTQD VD ENKKNV : 0 7KHQ A LV VWDEOH LI DQG RQO\ LI μ(A[2] ) < 0 IRU VRPH /R]LQVNLLPHDVXUH μ RQ RN ×N ZLWK N = d2  $W@LOKD   2S@AHKHSX NE @ /@Q@LDSDQ #DODMCDMS ,@SQHW RDD :< O  6D RGNV SG@S SGD 3 × 3 L@SQHW ⎛ ⎞ −1 −t2 −1 ⎠ t A(t) = ⎝ t −t − 1 2 2 t 1 −t − 1 HR RS@AKD ENQ @KK t > 0 %QNL 3@AKD   VD QD@C NƤ SGD RDBNMC @CCHSHUD BNL ONTMC L@SQHW ⎞ ⎛ −2 − t −t 1 ⎠. 1 −2 − t2 −t2 A[2] (t) = ⎝ 2 2 −t t −2 − t − t A[2] (t) HR CH@FNM@KKX CNLHM@MS HM HSR QNVR 'DMBD KDS μ AD SGD +NYHMRJHH LD@RTQD VHSG QDRODBS SN SGD MNQL x ∞ = RTO{|x1 |, |x2 |, |x3 |} 3GDM μ(A[2] (t)) = −1 < 0 ,NQDNUDQ CDS(A(t)) = −2t5 − 3t3 − 2t2 − t − 1 < 0 ENQ t > 0 3GD RS@AHKHSX NE A(t) ENKKNVR EQNL 3GDNQDL  

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3NAH@R 1HFNHO  %KNQH@M 5XSS '@QSL@M @MC &QNAL@M @ BK@RRHƥB@SHNM NE DPTHKHAQH@ HM OK@M@Q RXRSDLR VHSG QDRODBS SN SGDHQ RS@AHKHSX OQNODQSHDR @R VDKK @R SGD SDBGMHPTDR ENQ SGD CDSDQ LHM@SHNM NE SGD ONRHSHNM NE +X@OTMNU DWONMDMSR NE @ KHMD@Q RXRSDL KHJD SGD 1NTSG 'TQVHSY BQHSDQHNM NE SGD +NYHMRJHH LD@RTQD LDSGNC

/QNAKDLR "K@RRHƥB@SHNM ☼ D@RX  D@RX VHSG KNMFDQ B@KBTK@SHNMR  @ KHSSKD AHS CHƧBTKS  BG@KKDMFHMF $WDQBHRD   :☼< .M (MU@QH@MBD 2GNV SGD @RRDQSHNM RS@SDC HM #DƥMHSHNM   H D  SG@S ENQ ONRHSHUDKX HMU@QH@MS RDSR B SGDHQ BKNRTQD B @MC HMSDQHNQ int(B) @QD ONRHSHUDKX HMU@QH@MS SNN $WDQBHRD   :☼< 2S@AHKHSX UH@ +X@OTMNU %TMBSHNMR  2GNV SG@S SGD YDQN RNKTSHNM NE SGD RXRSDL x˙ = −x − xy 2 ,

y˙ = −y − x2 y ,

HR FKNA@KKX @RXLOSNSHB@KKX RS@AKD AX FTDRRHMF @ RTHS@AKD +X@OTMNU ETMBSHNM  (MUDRSHF@SD SGD RS@AHKHSX NE SGD YDQN RNKTSHNM NE x˙ = −xy − x ,

y˙ = y 3 − xy 3 + xy − y ,

AX TRHMF SGD ETMBSHNM V (x, y) = −x − KM(1 − x) − y − KM(1 − y) KNB@KKX @QNTMC (x, y) = (0, 0) !DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   Ŕ /@QS (( 0 2S@SD SGD 1NTSG 'TQVHSY RS@AHKHSX BQHSDQHNM 0 &HUD SGD CDƥMHSHNM NE SGD +NYHMRJHH LD@RTQD 0 6G@S B@M XNT R@X @ANTS RS@AHKHSX NE SGD MTKK RNKTSHNM TRHMF SGD +NYHMRJHH LD@RTQD 0 2JDSBG SGD OQNNE NE SGD @RRDQSHNM XNT TRDC HM 0



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF $WDQBHRD   :☼< :< 6G@S 1D@KKX '@OODMDC @S SGD /@QHR /D@BD 3@KJR 3GD NQHFHM@K OK@M CDUDKNODC AX 'DMQX *HRRHMFDQ @MC +D #TB 3GN SN RDSSKD SGD 5HDSM@LDRD V@Q HR CDRBQHADC ADKNV (S V@R @FQDDC SG@S  LHKKHNM 2NTSG 5HDS M@LDRD @MSR @MC  LHKKHNM -NQSG 5HDSM@LDRD @MSR VNTKC AD OK@BDC HM SGD A@BJX@QC NE SGD /QDRHCDMSH@K O@K@BD HM /@QHR @MC AD @KKNVDC SN ƥFGS HS NTS ENQ @ KNMF ODQHNC NE SHLD (E SGD 2NTSG 5HDSM@LDRD @MSR CDRSQNXDC MD@QKX @KK SGD -NQSG 5HDSM@LDRD @MSR SGDM 2NTSG 5HDSM@L VNTKC QDS@HM BNMSQNK NE @KK NE HSR K@MC (E SGD -NQSG 5HDSM@LDRD @MSR VDQD UHBSNQHNTR SGDM -NQSG 5HDSM@L VNTKC S@JD NUDQ @KK NE 2NTSG 5HDSM@L (E SGDX @OOD@QDC SN AD ƥFGSHMF SN @ RS@MCNƤ SGDM 2NTSG 5HDSM@L VNTKC AD O@QSHSHNMDC @BBNQCHMF SN SGD OQNONQSHNM NE @MSR QDL@HMHMF -NV SGD 2NTSG 5HDSM@LDRD @MSR CDMNSDC AX S @MC SGD -NQSG 5HDSM@LDRD MSR CDMNSDC AX N  BNLODSD @F@HMRS D@BG NSGDQ @BBNQCHMF SN SGD ENKKNVHMF CHƤDQDMSH@K DPT@SHNMR dS dt dN dt

= =

1 1 S− S×N 10 20 1 1 1 N− N2 − S×N 100 100 100

   

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF 

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x ∈ C 2 (R, R) ,

VHSG BNMRS@MSR c = 0.64 @MC d = 15.56  %HM@KKX BNMRHCDQ SGD O@Q@LDSDQR c @MC d EQNL O@QS A SN AD @QAHSQ@QX H D  (c, d) ∈ R2 #DƥMD @ ETMBSHNM CDODMCHMF NM c @MC d SG@S HR ONRHSHUD HE SGD NQHFHM HR @RXLOSNSHB@KKX RS@AKD HM DWDQBHRD A YDQN HE SGD NQHFHM HR RS@AKD @MC MDF@SHUD HE SGD NQHFHM HR TMRS@AKD /KNS SGD FQ@OG NE SGHR ETMBSHNM @F@HMRS SGD c d OK@MD $WDQBHRD   :☼< OOKHB@SHNM NE SGD +NYHMRJHH ,D@RTQD (M @M @QSHBKD AX 1TOO @MC 2BGDTQKD @ QDCTBDC LNCDK ENQ ƥRG IDKKXƥRG HMSDQ @BSHNMR HR OQNONRDC VGDQD ƥRG ADHMF @RRTLDC SN QDOQDRDMS SGD CNLHM@MS OQDC@SNQX RODBHDR EDDCHMF NM IDKKXƥRG 3GHR LNCDK HR FHUDM AX SGD ENKKNVHMF RDS NE BNTOKDC MNM KHMD@Q NQCHM@QX CHƤDQDMSH@K DPT@SHNMR   y x x˙ = c + x − x2 , @MC y˙ = β 1 − d y, 1+y 1+y VGDQD c ∈ R @MC β, d > 0 @QD O@Q@LDSDQR @MC x CDMNSDR SGD ƥRG ONOTK@SHNM VGDQD@R y RS@MCR ENQ SGD IDKKXƥRG ONOTK@SHNM QDRODBSHUDKX  2GNV SG@S SGD NQHFHM HR @M DPTHKHAQHTL @MC FHUD SGD KHMD@QHR@SHNM NE SGD DPT@SHNMR @S SGHR ONHMS 

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y˙ = − RHM(x) .

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.

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t Yt = Y 0 + Xt Ct . 0

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X(0) = X0 ,

 

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RHNM RSDO (E VD QDVQHSD SGD ANTMC@QHDR NE SGDRD RTAHMSDQU@KR HM @ PT@SDQM@QX QDOQDRDMS@SHNM HMRSD@C NE @ CDBHL@K NMD D@BG ANQCDQ SQ@MRK@SDR SN NMD CHFHS {0, . . . , 3} NE SGD PT@SDQM@QX RXRSDL 01 21 03 1 4 , 4 = [04 .1, 04 .2] , 4 , 1 = [04 .3, 14 .0] %TQSGDQLNQD DUDQX RTARPT@QD NE SGD S@QFDS CNL@HM Q BNMS@HMR @ RB@KDC SQ@MRK@SDC @MC QNS@SDC NQ QDƦDBSDC 'HKADQS BTQUD RDKE RHLHK@QHSX NE SGD BTQUD #DƥMHMF ENTQ A@RHB L@OOHMFR Hi , i ∈ {0, . . . , 3}, SG@S ODQENQL DW@BSKX SGDRD SQ@MRENQL@SHNMR QDRTKSR HM   1    1 x x 0 0 0 2 2 H0 := 1 , H1 := + 1 , 1 0 0 y y 2 2 2  H2 :=

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∞ Yt0 = mY 0 (t) + χ−1 DWO (−aτ ) RHM (χτ ) Ut−τ Cτ   0

HR @ ODQHNCHB @MC ENQ c = 0 RS@SHNM@QX O@SG VHRD @MC LD@M RPT@QD RNKTSHNM NE   VHSG SGD R@LD ODQHNC @R Zt 'DQD

∞ mY 0 (t) = χ−1 c DWO (−aτ ) RHM (χτ ) RHM (α(t − τ )) Cτ = cβ RHM (α(t − γ)) , 0

VGDQD c ∈ R HR @ BNMRS@MS β −2 = (b − α)2 + (2a)2 α2  @MC  2aα 1 . γ = α @QBS@M b − α2

 

6HSG   EQNL "G@O   @MC   SGD BNU@QH@MBD ETMBSHNM NE Yt0 QD@CR @R

∞ CY 0 (t, t + τ ) = CY 0 (τ ) = DWO (iλτ ) |A(iλ)|−2 Φ(λ)Cλ −∞

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!DB@TRD CY 0 (τ ) |A(iλ)| @MC Φ(λ) @QD QD@K ETMBSHNMR HS GNKCR SG@S

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF (E SGD RODBSQ@K CDMRHSX Φ(λ) HR FHUDM MTLDQHB@KKX SGDM SGD HMSDFQ@K HM   B@M AD BNLOTSDC MTLDQHB@KKX "N@QRD @OOQNWHL@SHNMR ENQ SGD BNU@QH@MBD @MC U@QH@MBD ETMBSHNMR @QD NA S@HMDC ENQ RL@KK C@LOHMF BNDƧBHDMSR a HE Φ(λ) HR RTƧBHDMSKX RLNNSG @MC CNDR MNS ONRRDRR RG@QO OD@JR (M SGHR B@RD SGD ETMBSHNM |A(iλ)|−2 DWGHAHSR @ OD@J @S λ = b @MC LNQDNUDQ SGD DRRDMSH@K BNMSQHATSHNM SN SGD HMSDFQ@K   BNLDR EQNL @ RL@KK HMSDQU@K @QNTMC λ = b 3GTR HS @OOQNWHL@SDKX GNKCR SG@S

∞  −1 CY 0 (τ ) ≈ Φ(b) BNR (λτ ) (λ2 − b)2 + (2aλ)2 Cλ , −∞



@MC σY 0 (t) ≈ Φ(b)

∞ −∞



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t→∞

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t→∞ t→∞

−mX (t)mX (t + s) + mY 0 (t)mY 0 (t + s)| ≤     2 1/2   2 1/2   1/2 ! = 0. + CY 0 (0)E Vt+s + CY 0 (0)E Vt2 KHL E Vt2 E Vt+s

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∞ τ DWO (−aτ ) RHM (α(t − τ )) Cτ . mY 0 (t) = 0

 

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X0 = X˙ 0 = 0 ,

 

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X0 = X˙ 0 = 0 ,

 

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X0 = X˙ 0 = 0 ,

 

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X0 = X˙ 0 = 0 ,

 

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xg (0) = x˙ g (0) = 0 .

 &HUD u ¨g HM SDQLR NE @ Q@MCNL CHƤDQDMSH@K DPT@SHNM AX @OOKXHMF SGD #NRR 2TRRL@MM  (LJDKKDQ 2BGL@KETRR BNQQDRONMCDMBD  (R u ¨g @ RSQHBS θ ODQHNCHB OQNBDRR $WDQBHRD   :☼< "NMUDQFDMBD 3NV@QCR /DQHNCHB 2NKTSHNMR Ŕ /@QS  +DS Zt AD @ RTHS@AKD RSQHBSKX 2 ODQHNCHB O@SG VHRD BNMSHMTNTR RSNBG@RSHB OQN BDRR +DS TR BNMRHCDQ SGD Q@MCNLKX ENQBDC ,@SGHDT DPT@SHNM x ¨ + (a − 2b BNR(2t)) x + Zt = 0 , VGDQD a, b > 0 6G@S B@M XNT R@X @ANTS SGD DWHRSDMBD NE θ ODQHNCHB RNKTSHNMR HM SGHR RXRSDL @MC SGD BNMUDQFDMBD SNV@QCR SGDL



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF $WDQBHRD   :< "NMUDQFDMBD 3NV@QCR /DQHNCHB 2NKTSHNMR Ŕ /@QS  +DS Zt AD @ RTHS@AKD RSQHBSKX θ ODQHNCHB LD@M RPT@QD BNMSHMTNTR RSNBG@RSHB OQNBDRR +DS TR BNMRHCDQ SGD Q@MCNLKX ENQBDC #TƧMF DPT@SHNM x ¨ + ax˙ + bx + cx3 − d BNR(θt) + Zt = 0 , VGDQD a, b, c, d > 0 6G@S B@M XNT R@X @ANTS SGD DWHRSDMBD NE θ ODQHNCHB RNKT SHNMR HM SGHR RXRSDL @MC SGD BNMUDQFDMBD SNV@QCR SGDL $WDQBHRD   :☼< 2S@SHNM@QX /QNBDRRDR  &HUD @M DW@LOKD NE @ RSQHBSKX RS@SHNM@QX O@SG VHRD LD@M RPT@QD BNMSHM TNTR RSNBG@RSHB OQNBDRR  +DS {Xn }n∈N AD @ RDS NE TMBNQQDK@SDC Q@MCNL U@QH@AKDR VHSG U@MHRGHMF LD@M @MC U@QH@MBD 1 2GNV SG@S {Xn } HR @ RSQHBSKX RS@SHNM@QX OQNBDRR  +DS Xt := A1 + A2 t VGDQD A1 , A2 @QD HMCDODMCDMS Q@MCNL U@QH@AKDR VHSG E(Ai ) = ai @MC Var(Ai ) = σi2  ENQ i = 1, 2 2GNV SG@S {Xn } HR MNS RS@SHNM@QX $WDQBHRD   :☼< 1NLDN @MC )TKHDSŗR +NUD Ƥ@HQ  +DS TR CHRBTRR SGD Q@MCNL ODQSTQA@SHNM NE @ RHLOKD LNCDK ENQ KNUD @Ƥ@HQR BE :< @MC :< OO  1NLDN HR HM KNUD VHSG )TKHDS ATS HM NTQ UDQRHNM NE SGHR RSNQX )TKHDS HR @ ƥBJKD KNUDQ 3GD LNQD 1NLDN KNUDR GDQ SGD LNQD )TKHDS V@MSR SN QTM @V@X @MC GHCD !TS VGDM 1NLDN FDSR CHRBNTQ@FDC @MC A@BJR NƤ )TKHDS ADFHMR SN ƥMC GHL RSQ@MFDKX @SSQ@BSHUD .M SGD NSGDQ G@MC 1NLDN SDMCR SN DBGN GDQ GD V@QLR TO VGDM RGD KNUDR GHL @MC FQNVR BNKC VGDM RGD G@SDR GHL +DS R(t) = 1NLDNŗR KNUD G@SD ENQ )TKHDS @S SHLD t J(t) = )TKHDSŗR KNUD G@SD ENQ 1NLDN @S SHLD t /NRHSHUD U@KTDR NE R @MC J RHFMHEX KNUD MDF@SHUD U@KTDR RHFMHEX G@SD 3GDM @ Q@MCNLKX ODQSTQADC LNCDK ENQ SGDHQ RS@Q BQNRRDC QNL@MBD HR R˙ = aJ + Ot ,

@MC

J˙ = −bR ,

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3NAH@R 1HFNHO  %KNQH@M 5XSS $WDQBHRD   :☼< 1NLDN @MC )TKHDSŗR +NUD Ƥ@HQ  R @ BNMSHMT@SHNM NE OQNAKDL   KDS TR BNMRHCDQ SGD ENQDB@RS ENQ KNUDQR FNUDQMDC AX SGD Q@MCNLKX ODQSTQADC FDMDQ@K KHMD@Q RXRSDL R˙ = aR + bJ + Ot ,

@MC J˙ = cR + dJ ,

VGDQD SGD O@Q@LDSDQR a, b, c, d L@X G@UD DHSGDQ RHFM $RODBH@KKX HM SGD CD SDQLHMHRSHB RDSSHMF @ BGNHBD NE RHFMR RODBHƥDR QNL@MSHB RSXKDR R BNHMDC AX 2SQNF@SY BE :< O  SGD BGNHBD NE a, b > 0 LD@MR SG@S 1NLDN HR @M řD@ FDQ AD@UDQŚ Ŕ GD FDSR DWBHSDC AX )TKHDSŗR KNUD ENQ GHL @MC HR ETQSGDQ ROTQQDC NM AX GHR NVM @ƤDBSHNM@SD EDDKHMFR ENQ GDQ (SŗR DMSDQS@HMHMF SN M@LD SGD NSGDQ SGQDD QNL@MSHB RSXKDR @MC SN OQDCHBS SGD NTSBNLDR ENQ SGD U@QHNTR O@HQHMFR %NQ HMRS@MBD B@M @ řB@TSHNTR KNUDQŚ a < 0 b > 0 ƥMC SQTD KNUD VHSG @M D@FDQ AD@UDQ 6G@S B@M XNT R@X @ANTS SGD @Ƥ@HQR HM SGD Q@MCNLKX ODQSTQADC RDSSHMF $WDQBHRD   :☼< 1NLDN @MC )TKHDSŗR +NUD Ƥ@HQ  (M @ RSTCX RNBHNKNFHRS ENTMC SG@S VNLDM VDQD AKHMC SN SGD LHMCRDS NE SGDHQ NOONRHSD RDW EQHDMCR ADB@TRD EDL@KDR FDMDQ@KKX VDQD MNS @SSQ@BSDC SN SGDHQ L@KD EQHDMCR SGDX @RRTLDC SG@S SGHR K@BJ NE @SSQ@BSHNM V@R LTST@K R @ QDRTKS LDM BNMRHRSDMSKX řNUDQDRSHL@SDCŚ SGD KDUDK NE @SSQ@BSHNM EDKS AX SGDHQ EDL@KD EQHDMCR Ŕ @MC VNLDM BNMRHRSDMSKX řTMCDQDRSHL@SDCŚ SGD KDUDK NE @SSQ@BSHNM EDKS AX SGDHQ L@KD EQHDMCR  !@RDC NM OQNAKDL   RDS TO @ CDSDQLHMHRSHB LNCDK ENQ SGHR ADG@UHNQ AX RODBHEXHMF SGD QHFGS RHFM BNLAHM@SHNMR NE SGD BNDƧBHDMSR HMUNKUDC  (MBNQONQ@SD Q@MCNL DƤDBSR SG@S S@JD HMSN @BBNTMS SG@S LDM @MC VNLDM @QD MNS FNNC HM QD@CHMF SGDHQ NOONRHSD RDW EQHDMCR  6G@S B@M XNT R@X @ANTS SGD DWHRSDMBD NE RS@SHNM@QX RNKTSHNMR HM SGHR B@RD @MC SGD @RXLOSNSHB BNMUDQFDMBD SNV@QCR RTBG RNKTSHNMR HE SGDX DWHRS $WDQBHRD   :☼< 1NLDN @MC )TKHDSŗR +NUD Ƥ@HQ  "NMSHMTHMF OQNAKDLR   @MC   VD ENKKNV :< O  @MC FHUD RNLD ETQSGDQ PTDRSHNMR QDK@SDC SN SGD KNUD @Ƥ@HQ CXM@LHBR OQDRDMSDC HM OQNAKDL   (M D@BG NE SGD ENKKNVHMF RBDM@QHNR OQDCHBS SGD BNTQRD NE SGD KNUD @E E@HQ CDODMCHMF NM SGD RHFMR @MC SGD QDK@SHUD RHYDR NE a @MC b .E BNTQRD XNT @QD DMBNTQ@FDC SN TRD @ 6HDMDQ OQNBDRR NQ @ EQ@BSHNM@K !QNVMH@M LNSHNM @R Q@MCNL ODQSTQA@MBD SNN @MC @KSDQ SGD RXRSDL @BBNQCHMFKX  .TS NE SNTBG VHSG SGDHQ NVM EDDKHMFR 2TOONRD 1NLDN @MC )TKHDS QD@BS SN D@BG NSGDQ ATS MNS SN SGDLRDKUDR H D  R˙ = aJ @MC J˙ = bR 6G@S G@OODMR



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF  %HQD @MC V@SDQ #N NOONRHSDR @SSQ@BS −bR − aJ

M@KXYD R˙ = aR + bJ @MC J˙ =

 /D@R HM @ ONC 2TOONRD 1NLDN @MC )TKH@ @QD QNL@MSHB BKNMDR H D  R˙ = aR + bJ @MC J˙ = bR + aJ #N SGDX DWODBS ANQDCNL NQ AKHRR  1NLDN SGD QNANS -NSGHMF BNTKC DUDQ BG@MFD SGD V@X 1NLDN EDDKR @ANTS )TKHDS H D  R˙ = 0 @MC J˙ = aR+bJ #NDR )TKHDS DMC TO KNUHMF GHL NQ G@SHMF GHL $WDQBHRD   :☼< "NMUDQFDMBD 3NV@QCR 2S@SHNM@QX 2NKTSHNMR HM @ +1" "HQBTHS 2XRSDL "NMRHCDQ SGD Q@MCNLKX ODQSTQADC BHQBTHS DPT@SHNM LI¨+ RI˙ + I/C + Ot  VGDQD L, R, C > 0 @MC Ot HR @M .QMRSDHM 4GKDMADBJ OQNBDRR 2GNV SG@S SGDQD HR @ RS@SHNM@QX RNKTSHNM @MC SG@S @KK NSGDQ RNKTSHNMR SDMC SNV@QCR HS

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  *DX "NMBDOSR (M SGHR BG@OSDQ VD CHRBTRR SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM CXt = At Xt + Zt , Ct

 

VGDQD Zt HR @ d CHLDMRHNM@K RSNBG@RSHB UDBSNQ OQNBDRR @MC At HR @ RSNBG@RSHB d × d L@SQHW VHSG DKDLDMSR SG@S @QD RSNBG@RSHB OQNBDRRDR 2STCXHMF DPT@SHNM   HR BNMRHCDQ@AKX G@QCDQ SG@M SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM   EQNL "G@O  VHSG @ MNM Q@MCNL L@SQHW A(t) (M FDMDQ@K HS HR MNS ONRRHAKD SN FHUD RHLOKD BKNRDC ENQL ENQLTK@R ENQ SGD RS@SHRSHB@K BG@Q@BSDQHRSHBR NE SGD RN KTSHNM CDODMCHMF NM SGD RS@SHRSHB@K BG@Q@BSDQHRSHBR NE SGD BNDƧBHDMSR NE   1@SGDQ VD @HL ENQ PT@KHS@SHUD QDRTKSR KHJD SGD @RXLOSNSHB ADG@UHNQ NE SGD QD @KHR@SHNMR NQ SGD LNLDMSR NE SGD RNKTSHNM RODBH@K B@RD NE   HR FHUDM VGDM SGD BNDƧBHDMSR @QD @OOQNWHL@SDKX VGHSD MNHRD OQNBDRRDR ,@MX VNQJR SQD@S SGHR SXOD NE CHƤDQDMSH@K DPT@SHNMR DH SGDQ AX TSHKHYHMF @OOQNWHL@SD %NJJDQ /K@MBJ DPT@SHNMR NQ (S¯  2SQ@SNMNUHBG RSNBG@RSHB CHƤDQDMSH@K DPT@SHNMR RDD :< :< :< $W@LOKD   1@MCNLHYDC "NDƧBHDMSR @S @ ,NMNC *HMDSHBR ,NCDK BE :t1

ˆ Xt2 − Xt1  ≤

@BBNQCHMF SN

∞ E (A + Fτ  Xτ ) Cτ t0

t2

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t1 ,t2 →∞ , t2 >t1

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1 t



t0 t t0

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VGDQD A HR @ BNMRS@MS d×d L@SQHW Ft @ RSNBG@RSHB d×d L@SQHW OQNBDRR @MC C(t) @ QD@K d × d L@SQHW ETMBSHNM 6D CDMNSD SGD K@QFDRS @MC RL@KKDRS DHFDMU@KTD NE @ L@SQHW B AX λL@W (B) @MC λLHM (B) QDRODBSHUDKX 3GDNQDL   $WONMDMSH@K #DB@X NE 2NKTSHNMR 5DQRHNM  /HW WKH IROORZLQJ FRQGLWLRQV EH VDWLVƲHG  7KH PDWUL[ SURFHVV Ft LV VWULFWO\ VWDWLRQDU\ SDWKZLVH FRQWLQXRXV DQG HU JRGLF  7KH PDWUL[ IXQFWLRQ C(t) LV FRQWLQXRXV RQ I  7KHUH LV D V\PPHWULF SRVLWLYH GHƲQLWH PDWUL[ B VXFK WKDW 1 E (ρ1 (t0 )) + KHL t→∞ t − t0 ZKHUH DQG



t t0

ρ2 (τ )Gτ ≤ −ε ,

ε > 0,

 

  ρ1 (t) = λL@W AT + FtT + B (A + Ft ) B −1   ρ2 (t) = λLHM C T (t) + BC(t)B −1 .

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t→∞

KROGV IRU HYHU\ SDWKZLVH VROXWLRQ  DQG HYHU\ α < ε 3URRI %NKKNVHMF :< OO  VD @RRTLD SG@S D @MC B @QD SVN QD@K RXLLDSQHB d × d L@SQHBDR @MC SG@S B HR ONRHSHUD CDƥMHSD 3GDM VD G@UD ENQ SGD LHMHLTL @MC L@WHLTL NE SGD PT@CQ@SHB ENQLR   xT Dx = λLHM DB −1 , T x∈Rd \{0} x Bx

 

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LHM

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF (E Xt HR SGD MTKK RNKTSHNM SGDM SGD @RRDQSHNM SQHUH@KKX ENKKNVR (M NQCDQ SN NAS@HM @ BNMSQ@CHBSHNM KDS Xt AD RNLD NSGDQ RNKTSHNM CHƤDQDMS EQNL SGD MTKK RNKTSHNM @MC BNMRHCDQ SGD ETMBSHNM v(x) = xT Bx .

 

6HSG v˙ t (x) = xT



  AT + FtT + C T (t) B + B (A + Ft + C(t)) x

 

DPT@SHNMR   @MC   HLOKX 1 Cv(Xt ) I v˙ t (Xt ) I = ≤ v(Xt ) Ct v(Xt )

v˙ t (x) = ρ(t) , x∈Rd \{0} v(x)

 

L@W

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(E D1 , D2 @MC B @QD QD@K RXLLDSQHB L@SQHBDR @MC HE B HR ONRHSHUD CDƥMHSD SGDM         λL@W (D1 + D2 ) B −1 ≤ λL@W D1 B −1 + λmax D2 B −1 GNKCR CTD SN   !DB@TRD NE   DPT@SHNM   HLOKHDR ρ(t) ≤ ρ1 (t) + ρ2 (t) .   @MC   KD@C SN   I v(Xt ) ≤ v(X0 ) DWO (t − t0 )

1 t − t0



t t0

1 ρ1 (τ )Cτ + t − t0

 

t t0

ρ2 (τ )Cτ

.

 

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t

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t0

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3NAH@R 1HFNHO  %KNQH@M 5XSS 1DL@QJ   4OODQ !NTMCR ENQ E(ρ2 ) +DS C(t) G@UD SGD ENQL C(t) = C1 (t)+ C2 (t) VGDQD C1 (t) HR @ BNMSHMTNTR ODQHNCHB ETMBSHNM VHSG ODQHNC θ 3GDM     ρ2 (t) ≤ λL@W CtT (t) + BC1 (t)B −1 + λL@W C2T (t) + BC2 (t)B −1 GNKCR @MC ENQ HMRS@MBD

t   1 1 θ KHL ρ2 (τ )Cτ ≤ λL@W C1T (τ ) + BC1 (τ )B −1 Cτ t→∞ t − t0 t θ 0 0   + RTO λL@W C2T (τ ) + BC2 (τ )B −1 τ ∈[t0 ,∞)

 

HR U@KHC !DB@TRD NE λL@W (D) ≤ D VD F@HM EQNL   SG@S

t & 1 1 θ& &C1T (τ ) + BC1 (τ )B −1 & Cτ KHL ρ2 (τ )Cτ ≤ t→∞ t − t0 t θ 0 0 & & + RTO &C2T (τ ) + BC2 (τ )B −1 & .  τ ∈[t0 ,∞)

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N 

ci (t)Ci ,

i=1

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N  

≤

 T   T  −1 −1 c+ + c− i (t)λL@W Ci + BCi B i (t)λLHM Ci + BCi B

i=1

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#

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.

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t 1 ρ2 (τ )Cτ t − t0 t0

θi N       1 ≤ |ci (τ )| Cτ λL@W CiT + BCi B −1 − λLHM CiT + BCi B −1 . 2θi 0 i=1  

t 1 (E HM   VD QDOK@BD SGD KHLHS KHLt→∞ ρ2 (τ )Cτ VHSG HSR ANTMCR t − t0 t0 EQNL     @MC   QDRODBSHUDKX VD F@HM HLOKHB@SHNMR ENQL 3GD NQDL   VGHBG L@X AD UDQHƥDC LNQD D@RX KHLt→∞

%TQSGDQ BNMRDPTDMBDR EQNL 3GDNQDL   @QD NAS@HMDC AX TRHMF TOODQ ANTMCR NM ρ1 1DL@QJ   4OODQ !NTMCR ENQ ρ1  (MDPT@KHSX   HLOKHDR     ρ1 (t) ≤ λL@W AT + BAB −1 + λLHM FtT + BFt B −1 .

 

(E A HR @ RS@AKD L@SQHW H D @KK DHFDMU@KTDR NE SGD L@SQHW A G@UD MDF@SHUD QD@K O@QSR SGDM SGD L@SQHW B B@M ENQ HMRS@MBD AD BGNRDM @R SGD RNKTSHNM NE SGD L@SQHW DPT@SHNM AT B + BA = I , RDD :< !DB@TRD NE   λL@W AT + BAB −1 =

1 λL@W (B)

 

HMDPT@KHSX   KD@CR SN ρ1 (t) ≤ −

  1 + λL@W FtT + BFt B −1 λL@W (B)

 

& & 1 + &FtT + BFt B −1 & . λL@W (B)

 

NQ ρ1 (t) ≤ −

OOKXHMF SGD SQ@MRENQL@SHNM x = B −1/2 y VD FDS SGD DPT@SHNM     xT FtT B + BFt x y T B −1/2 FtT B 1/2 + B 1/2 Ft B −1/2 y = L@W , L@W xT Bx yT y x∈Rd \{0} y∈Rd \{0} @MC SGTR SG@S !   λL@W FtT + BFt B −1 = λL@W B −1/2 FtT B 1/2 + B 1/2 Ft B −1/2 . 6HFWLRQ 

 



3NAH@R 1HFNHO  %KNQH@M 5XSS %QNL   @MC   HS ENKKNVR SG@S & & 1 & & + &B −1/2 FtT B 1/2 + B 1/2 Ft B −1/2 & . ρ1 (t) ≤ − λL@W (B)

 

3GD BNMCHSHNM &! & & & E &B −1/2 FtT B 1/2 + B 1/2 Ft B −1/2 &
0 ,

 2 4 (b − α1 )2 + α2−1 α2 + α12 − 1 − Zt + 2α1 (b − α1 )  

@MC   λL@W C T + BCB −1 =

1 √ a |RHM (ωt)| , α2

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t

2π/ω 1 aω 2a ρ2 (τ )Cτ ≤ |RHM (ωt)| Ct = √ . KHL √ t∈∞ t − t0 t 2π α π α2 2 0 0 ρ1 (t) ADBNLDR LHMHL@K ENQ α1 = b ,

α2 = 1 − b2 ,

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α1 = b ,

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√

1 2 2, √ 1 2 2.

 

/KTFFHMF   HMSN   @MC   VD FDS VHSG   SGD ENKKNVHMF BNMCHSHNM √ √ 2a E (|Zt |) ≤ (2b − ε) 1 − b2 − , HE b ≤ 12 2 , π  

 √  2a 2 , HE b ≥ 12 2 . E Zt + 1 − 2b ≤ (2b − ε)b − π 6HSG α1 = b @MC α2 = b2 + 1 VD NAS@HM @S E (Zt ) = 0 VHSG   @MC   EQNL   AX @OOKXHMF SGD "@TBGX 2BGV@QY HMDPT@KHSX @KRN SGD BNMCHSHNM    2 2a 2 2 E Zt ≤ (2b − ε) 1 + b − − 4b4   π



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%HFTQD   2HLTK@SHNMR NE SGD RNKTSHNM OQNBDRR NE SGD C@LODC NRBHKK@SHNM DPT@SHNM   VHSG Zt = Ot  b = 0.1 @MC c(t) = t HM @ @R VDKK @R c(t) = RHM(t) HM A

@MC HM SGD B@RD a = 0

  E Zt2 ≤ 4b2 − ε∗ .

QDRODBSHUDKX $W@LOKD    2DBNMC .QCDQ 1@MCNL #HƤDQDMSH@K $PT@SHNM VHSG 2SNBG@R SHBHSX @S SGD 5DKNBHSX "NLONMDMS BE :< OO  -DWS KDS TR RSTCX SGD RDB NMC NQCDQ Q@MCNL CHƤDQDMSH@K DPT@SHNM Y¨t + (2b + Zt + c(t)) Y˙ t + Yt = 0 ,

 

VGDQD Zt @F@HM HR @ RS@SHNM@QX OQNBDRR HM SGD M@QQNV RDMRD SG@S HR DQFNCHB @MC O@SG VHRD BNMSHMTNTR NM I RTBG SG@S E (Zt ) = 0 3GD ETMBSHNM c(t) HR BNMSHMTNTR @MC ODQHNCHB VHSG ODQHNC θ @MC RTBG SG@S



1 θ 1 θ c(τ )Cτ = 0 , @MC |c(τ )| Cτ = c . θ 0 θ 0 %HF   RGNVR RNLD RHLTK@SHNMR NE SGHR C@LODC NRBHKK@SHNM DPT@SHNM   VHSG Zt = Ot RTBG SG@S COt = −Ot Ct + CWt  b = 0.1 @MC SVN CHƤDQDMS U@KTDR NE SGD ETMBSHNM c M@LDKX c(t) = t HM %HF   @ @MC c(t) = RHM(t) HM %HF   A 3GD RHLTK@SHNMR VDQD BNLOTSDC VHSG SGD @UDQ@FDC $TKDQ LDSGNC SG@S VD VHKK CHRBTRR HM "G@O  VHSG @ RSDO RHYD h = 3 · 10−3 6D QD VQHSD DPT@SHNM   HM SGD ENQL   VHSG XtT = (Yt , Y˙ t ) @MC   0 0 0 1 , A = , F t = Zt 0 −1 −1 −2b 

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3NAH@R 1HFNHO  %KNQH@M 5XSS OOKXHMF SGD UDQX R@LD L@SQHW B @R HM DW@LOKD   VD FDS   ρ1 (t) = −2b−Zt + (Zt + 2b − 2α1 )2 +α2−1 α2 + α12 − 1 + α1 Zt + 2α1 (b − α1 )   @MC 3  T    α2 λL@W C + BCB −1 − λLHM C T + BCB −1 = 1+ 1 .   α2   @MC   HLOKX 1 t→∞ t − t0



t

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t0

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c 2

1+

α12 . α2

 

ρ1 (t) ADBNLDR LHMHL@K ENQ α1 = b , α1 = √

1 , 2 b +1

α2 = 1 − b2 ,

√ HE b ≤ 12 ( 5 − 1) ,

b2 , b2 + 1

√ HE b ≥ 12 ( 5 − 1) .

α2 =

3GTR VHSG   @MC   VD NAS@HM EQNL   SGD BNMCHSHNM √ E (|Zt |) ≤ (2b − ε) 1 − b2 − 12 c ,

√ HE b ≤ 12 ( 5 − 1)

 −1/2

! −1/2 1 

≤ (2b − ε)b 1 + b2 − 2c , E Zt + 2b − 2 1 + b2

√ HE b ≥ 12 ( 5 − 1) 3GD "@TBGX 2BGV@QY HMDPT@KHSX @MC SGD U@KTDR α1 =

b , 1 + b2

@MC

α2 = 1 −

b2 (1 + b2 )2

@OOKHDC SN   KD@C VHSG   @MC   @KRN SN SGD BNMCHSHNM   E Zt2 ≤ NQ SN

 −1  1/2 1 !2 −2 6  (2b − ε) 1 + b2 1 + b 2 + b4 − 2 c − 4 1 + b2 b ,   −2   − ε∗ E Zt2 ≤ 4b4 1 + b4 1 + b2

HM SGD B@RD c = 0 QDRODBSHUDKX



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3NAH@R 1HFNHO  %KNQH@M 5XSS GNKCR 3GTR ENQ SGD ETMC@LDMS@K L@SQHW Φt NE SGD GNLNFDMDNTR DPT@SHNM VHSG Φt0 = I DRSHL@SDR NE SGD ENQL & & I &Φt Φ−1 & ≤ h DWO (−ε(t − τ )) , τ

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R HM

Z˙ t = f (Zt + Ot ) + Ot AD FHUDM VGDQD Ot HR @ RS@SHNM@QX .QMRSDHM 4GKDMADBJ OQNBDRR @MC @RRTLD SG@S f R@SHRƥDR SGD NMD RHCDC CHRRHO@SHUD +HORBGHSY BNMCHSHNM x − y , f (x) − f (y) ≤ −Lx − y2 ,

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3NAH@R 1HFNHO  %KNQH@M 5XSS ENQ @KK x, y ∈ Rd @MC RNLD L > 0 3GDM ENQ @MX SVN RNKTSHNMR Z1 (t) @MC Z2 (t) NE SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM VD G@UD C Ct Z1 (t)

− Z2 (t)2 = 2Z1 (t) − Z2 (t) ,

C Ct Z1 (t)

−

C Ct Z2 (t)

= 2Z1 (t) − Z2 (t) , f (Z1 (t) + Ot ) − f (Z2 (t) + Ot ) = 2(Z1 (t) + Ot ) − (Z2 (t) + Ot ) , f (Z1 (t) + Ot ) − f (Z2 (t) + Ot ) ≤ −2LZ1 (t) − Z2 (t)2 , EQNL VGHBG HS ENKKNVR SG@S Z1 (t) − Z2 (t)2 ≤ DWO(−2Lt)Z1 (0) − Z2 (0)2 → 0 ,

@R t → ∞ .

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OOKHB@SHNM NE RSNBG@RSHB RS@AHKHSX SN SGD CDSDQLHMHRSHB RS@AHKHSX NE RXR SDLR RTAIDBS SN BNMSHMTNTRKX @BSHMF Q@MCNL ODQSTQA@SHNMR

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ONHMS VHRD OQNA@AHKHRSHB RHYD QDK@SHNM Ut < Vt @R (p)

Ut < Vt



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF ENQ t ∈ I @MC Ut

(I,p)


p (p)

(I,p)

VGHBG GNKCR NM SGD VGNKD NE I "KD@QKX Ut < Vt ENQ @KK t ∈ I HLOKHDR Ut < Vt ,NQDNUDQ CTQHMF NTQ CHRBTRRHNM NE RS@AHKHSX VD VHKK EQDPTDMSKX @OOKX SGD ENKKNVHMF DRSHL@SD +DLL@    &QNVSG (MDPT@KHSX ENQ "NMSHMTNTR &@TRRH@M /QNBDRRDR /HW α, β, γ > 0 DQG Zt  t ∈ I := [t0 , ∞) EH D SDWKZLVH DV ZHOO DV q.m.FRQWLQXRXV *DXVVLDQ YHFWRUYDOXHG SURFHVV VXFK WKDW

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3URRI 2DD :< 3GHR BG@OSDQ HR RSQTBSTQDC @R ENKKNVR 2DB   RSTCHDR SGD U@QHNTR MNSHNMR NE RS@AHKHSX NE SGD MTKK RNKTSHNM NE @ Q@MCNL NQCHM@QX CHƤDQDMSH@K DPT@SHNM VHSG @ ENBTR NM O@SG VHRD DPTH RS@AHKHSX h  P  @MC W RS@AHKHSX 3GD QDK@SHNM RGHO ADSVDDM SGDRD BNMBDOSR @QD CHRBTRRDC @MC SGD QDRTKSR NE "G@O  NM SGD O@SG VHRD RS@AHKHSX NE KHMD@Q Q@MCNL CHƤDQDMSH@K DPT@SHNMR VHSG RSNBG@RSHB BN DƧBHDMSR @QD QD EQ@LDC HM SGD BNMSDWS NE SGDRD BNMBDOSR -DWS 2DB   DW SDMCR SGD CDSDQLHMHRSHB +X@OTMNU LDSGNC SN Q@MCNL CHƤDQDMSH@K DPT@SHNMR (M O@QSHBTK@Q A@RDC NM RTHS@AKD +X@OTMNU ETMBSHNMR MDBDRR@QX BNMCHSHNMR ENQ h RS@AHKHSX @MC O@SG VHRD DPTH RS@AHKHSX @QD FHUDM GDQD R @M DWBTQRHNM 2DB   GNKCR QDRTKSR BNMBDQMHMF SGD RS@AHKHSX NE CDSDQLHMHRSHB RXRSDLR RTA IDBS SN CHƤDQDMS BK@RRDR NE BNMSHMTNTRKX @BSHMF Q@MCNL ODQSTQA@SHNMR %HM@KKX 2DB   VQ@OR TO SGD BNMSDMSR NE SGHR BG@OSDQ

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3NAH@R 1HFNHO  %KNQH@M 5XSS  4MHPTD DWHRSDMBD NE RNKTSHNMR 3GDQD HR @ TMHPTD RNKTSHNM x(t, x0 , t0 , ω) NM I ENQ @MX HMHSH@K BNMCHSHNM (x0 , t0 ) ∈ Rd × I  "NMSHMTHSX NE RNKTSHNMR %NQ ƥWDC (t, ω) SGD RNKTSHNM x(t, x0 , t0 , ω) HR @ BNM SHMTNTR UDBSNQ U@KTDC ETMBSHNM NM Rd × I  (RNK@SDC DPTHKHAQH@ @S YDQN 3GDQD HR @ MDHFGANQGNNC Uω  {0} @QNTMC x = 0 RTBG SG@S f (x, t, ω) = 0 t ∈ I GNKCR HM Uω HE @MC NMKX HE x = 0 6D @RRTLD SG@S ENQ @MX HMHSH@K BNMCHSHNM (X0 , t0 ) ∈ Sd × I SGDQD HR @ TMHPTD O@SG VHRD RNKTSHNM Xt (X0 , t0 ) NE SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM CXt = f (Xt , t, ω) . Ct I

  I

3GTR Xt (X0 , t0 ) = x(t, X0 (ω), t0 , ω) 3GD O@SG VHRD RNKTSHNM Xt (0, t0 ) = 0 VHSG QDRODBS SN SGD HMHSH@K BNMCHSHNM (0, t0 ) HR B@KKDC QXOOVROXWLRQ NQ WULYLDO VROXWLRQ NE  

  

3GD 9NN NE 2SNBG@RSHB 2S@AHKHSX "NMBDOSR

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF  @RXLOSNSHB@KKX RS@AKD  @RXLOSNSHB@KKX RS@AKD HM SGD VGNKD  TMHENQLKX RS@AKD  DWONMDMSH@KKX RS@AKD NQ  RS@AKD TMCDQ BNMSHMTNTRKX @BSHMF CHRSTQA@MBDR QDRODBSHUDKX 6D FDS @ RSQNMF MNSHNM NE RS@AHKHSX HE VD B@M BGNNRD SGD BNMRS@MSR δ1 , δ2 , . . . SG@S NBBTQ HM SGDRD RS@AHKHSX CDƥMHSHNMR HMCDODMCDMSKX NE ω #DƥMHSHNM   /@SG 6HRD $PTH 2S@AHKHSX 3GD MTKK RNKTSHNM NE SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM   NM I HR B@KKDC SDWKZLVH HTXLVWDEOH HE ENQ DUDQX ε > 0 SGDQD HR @ δ > 0 RTBG SG@S I

Xt (X0 , t0 ) < ε , ˆ ENQ @KK X0 ∈ Sd VHSG X0  0 RTBG SG@S KHL Xt (X0 , t0 ) = ˆ 0

t→0

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3NAH@R 1HFNHO  %KNQH@M 5XSS ENQ @KK t ∈ I @MC X0 ∈ Sd VHSG E(h(X0 )) < δ (E @CCHSHNM@KKX SGDQD HR @ η > 0 RTBG SG@S KHL E (h(Xt (X0 , t0 ))) = 0

t→∞

ENQ @KK X0 ∈ Sd VHSG E(h(X0 )) < η SGDM SGD MTKK RNKTSHNM NE   HR B@KKDC DV\PSWRWLFDOO\ hVWDEOH 3GD MTKK RNKTSHNM NE   HR B@KKDC H[SRQHQWLDOO\ hVWDEOH HE SGDQD @QD ONRH SHUD BNMRS@MSR a b @MC η RTBG SG@S E (h(Xt (X0 , t0 ))) ≤ aE(h(X0 )) DWO(−b(t − t0 )) GNKCR ENQ @KK 0 ≤ t0 ≤ t @MC @KK X0 ∈ Sd VHSG E(h(X0 )) < η (E h(x) = x SGDM h RS@AHKHSX HR NESDM B@KKDC m RS@AHKHSX ENQ VWDELOLW\ LQ WKH PHDQ M@KNFNTRKX HE h(x) = x2  SGDM h RS@AHKHSX HR NESDM B@KKDC q.m. RS@AHKHSX ENQ VWDELOLW\ LQ WKH TXDGUDWLF PHDQ NQ PHDQ VTXDUH VWDELOLW\ #DƥMHSHNM   P 2S@AHKHSX 3GD MTKK RNKTSHNM NE SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM   NM I HR B@KKDC PVWDEOH RS@AKD HM OQNA@AHKHSX HE ENQ DUDQX ε > 0 @MC DUDQX p ∈ (0, 1) SGDQD @QD MTLADQR δ > 0 @MC γ ∈ (0, 1) RTBG SG@S (p)

Xt (X0 , t0 ) < ε (γ)

ENQ @KK t ∈ I @MC X0 ∈ Sd VHSG X0  < δ #DƥMHSHNM   W 2S@AHKHSX 3GD MTKK RNKTSHNM NE SGD Q@MCNL CHƤDQDMSH@K DPT@SHNM   NM I HR B@KKDC W VWDEOH RSNBG@RSHB@KKX RS@AKD HE ENQ DUDQX ε > 0 @MC DUDQX p ∈ (0, 1) SGDQD @QD MTLADQR δ > 0 @MC γ ∈ (0, 1) RTBG SG@S Xt (X0 , t0 )

(I,p)


π t→∞ (χ)

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Xt (X0 , t0 ) < ε (γ)

ENQ @KK t ∈ I @MC X0 ∈ M VHSG X0  < δ 3GD MNSHNMR NE RSNBG@RSHB RS@AHKHSX CDRBQHADC @ANUD DM@AKD TR SN RSTCX SGD RS@AHKHSX OQNODQSHDR VGHBG SGD MTKK RNKTSHNM NE @ Q@MCNL CHƤDQDMSH@K DPT@SHNM DWGHAHSR (M SGD CDSDQLHMHRSHB KHLHS H D  HE f (x, t, ω) @MC X0 (ω) @QD HMCDODMCDMS NE ω D@BG NE SGD RS@AHKHSX BNMBDOSR OQNUHCDC HM CDƥMHSHNMR   SN   AD BNLDR @ AXVNQC ENQ SGD RS@AHKHSX BNMBDOSR CHRBTRRDC HM 2DB  3GDRDR RSNBG@R SHB RS@AHKHSX BNMBDOSR @QD SGTR QD@KKX FDMDQ@KHR@SHNMR NE SGD CDSDQLHMHRSHB NMDR

  

1DK@SHNMR !DSVDDM SGD #HƤDQDMS 2S@AHKHSX -NSHNMR

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X0 2 t DWO (−Z(t − t0 )) . t20

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X0 t2 (1 − DWO (−(t − t0 ))) . · t20 (t − t0 )

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3NAH@R 1HFNHO  %KNQH@M 5XSS

path-wise equi-stability

path-wise stability

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W-stability

P-stability

%HFTQD   "NMMDBSHNMR ADSVDDM SGD CHƤDQDMS MNSHNMR NE SGD RSNBG@RSHB RS@AHKHSX BNMBDOSR ENQ Q@MCNL CHƤDQDMSH@K DPT@SHNMR

/QNONRHSHNM   /@SG 6HRD $PTH 2S@AHKHSX (LOKHDR W  h 2S@AHKHSX /HW WKH QXOOVROXWLRQ RI HTXDWLRQ  EH SDWKZLVH HTXLVWDEOH 7KHQ LW LV W  VWDEOH DQG WKXV PVWDEOH DV ZHOO DV hVWDEOH ZLWK UHVSHFW WR GHWHUPLQLVWLF LQLWLDO FRQGLWLRQV M ⊂ Rd  WRR 3URRI %NKKNVHMF :< OO  SGD ƥQRS O@QS HR D@RX 6D BNMSHMTD VHSG SGD RDBNMC O@QS +DS SGD MTKK RNKTSHNM NE   AD O@SG VHRD DPTH RS@AKD R HM SGD OQNNE NE SGD OQDUHNTR KDLL@ KDS h(ε) := RTOx∈Rd , x ≤ε h(x) %NQ FHUDM ε > 0 BGNNRD ε > 0 @MC δ > 0 RTBG SG@S h(ε ) < ε @MC I

ˆ 3GDM SGD DRSHL@SD Xt (X0 , t0 ) < ε ENQ X0  0 DQG VXƴFLHQWO\ VPDOO ρ > 0 LW KROGV IRU D b > 0 WKDW   t E DWO a Lτ Gτ ≤ b DWO(ρ(t − t0 )) , t0 ∈ I , t ≥ t0 . t0

0RUHRYHU OHW M EH WKH VHW RI DOO UDQGRP YDULDEOHV IURP Sd WKDW DUH LQGHSHQGHQW RI WKH VWRFKDVWLF SURFHVV Lt  t ∈ I 7KHQ WKH QXOOVROXWLRQ RI WKH UDQGRP HTXD WLRQ  LV H[SRQHQWLDOO\ hVWDEOH ZLWK UHVSHFW WR M  ZKHUH h(x) = x2r  3URRI %NKKNVHMF :< O  CTD SN BNMCHSHNM  @MC SNFDSGDQ VHSG *Q@RNURJHHŗR SGDNQDL 3GDNQDL   SGDQD HR @ QD@K ETMBSHNM w(x, t) CDƥMDC NM Rd × I VHSG BNMSHMTNTR O@QSH@K CDQHU@SHUDR ∂x w @MC ∂t w NM Rd × I RTBG SG@S w G@R SGD OQNODQSHDR   RS@SDC HM 3GDNQDL   !DB@TRD NE OQNODQSHDR   EQNL 3GDNQDL   @MC BNMCHSHNM  VD NAS@HM Rd ×I

w(x, ˙ t, ω) = ∂t w + (∂x w)T (f (x, t) + g(x, t, ω))

≤

−c3 x2 + Lt c4 x2 ,  

VGHBG HLOKHDR w(x, ˙ t, ω) −1 VHSG ηt := c4 c−1 1 L t − c3 c 2 KD@CR SN SGD HMDPT@KHSX

Rd ×I

≤

 

ηt w(x, t) ,

M@KNFNTRKX SN SGD OQNNE NE +DLL@   SGHR 

I0

w(Xt , t) ≤ w(X0 , t) DWO



t

ητ Cτ

.

 

t0

%QNL SGD OQNODQSHDR   NE 3GDNQDL   @MC   SNFDSGDQ VHSG h(x) = x2r HS ENKKNVR SG@S  t −r r h(Xt ) ≤ c1 c2 h(X0 ) DWO r ητ Cτ , t0 ∈ I , t ≥ t0 .   t0

−1 (E BNMCHSHNM  HR R@SHRƥDC VHSG ρ < c3 c−1 2 · r @MC a > bc1 · r @MC HE α ∈ (ρ, c3 c−1 2 · r) GNKCR SGDM   KD@CR SN   t −r r Lτ Cτ DWO (−α(t − t0 )) h(t) ≤ c1 c2 h(t0 )E DWO a t0

r ≤ c−r 1 c2 h(t0 ) DWO (−(α − ρ)(t − t0 )) , t0 ∈ I , lt ≥ t0 .  

3GHR RGNVR SGD @RRDQSHNM 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS %HM@KKX VD RSTCX SGD KHMD@Q Q@MCNL CHƤDQDMSH@K DPT@SHNM VHSG RSNBG@RSHB BNDƧBHDMSR X˙ t = At Xt ,   VGDQD At HR @ O@SG VHRD BNMSHMTNTR OQNBDRR NM I 6D RDS v(x, t) = xT Bx VHSG @ RXLLDSQHB ONRHSHUD CDƥMHSD L@SQHW B %NKKNVHMF SGD R@LD BNMRHCDQ@SHNMR @R HM SGD OQNNE NE 3GDNQDL   VD NAS@HM HMDPT@KHSX   VHSG SGD OQNBDRR ηt = λL@W [ATt + BAt B −1 ] . +DS M AD SGD RDS NE @KK HMHSH@K BNMCHSHNMR X0 ∈ L2d SG@S @QD HMCDODMCDMS NE SGD L@SQHW OQNBDRR At  t ∈ I #TD SN λLHM [B]x2 ≤ v(x, t) λL@W [B]x2 , 3GDNQDL   +DLL@   @MC +DLL@   HLOKX 3GDNQDL   ,D@M 2PT@QD 2S@AHKHSX V Q S M  /HW WKHUH EH D V\PPHWULF SRV LWLYH GHƲQLWH PDWUL[ B VXFK WKDW   t T −1 λL@W [Aτ + BAτ B ]Gτ ≤ 1 , t ∈ I0 .   y(t0 , t) = E DWO t0

7KHQ WKH QXOOVROXWLRQ RI  LV m.s.VWDEOH ZLWK UHVSHFW WR M  ,I DGGLWLRQDOO\ KHLt→∞ y(t0 , t) = 0 KROGV WKHQ WKH QXOOVROXWLRQ RI  LV DV\PSWRWLFDOO\ m.s.VWDEOH ZLWK UHVSHFW WR M  0RUHRYHU LI y(t0 , t) ≤ c · DWO (−d(t − t0 )) ,

t0 ∈ I ,

t > t0

DOVR KROGV WKHQ WKH QXOOVROXWLRQ RI  LV H[SRQHQWLDOO\ m.s.VWDEOH ZLWK UHVSHFW WR M  M@KNFNTRKX SN SGD BNQNKK@QHDR SN 3GDNQDL   NMD L@X FDS LNQD OQ@BSHB@K BNMCHSHNMR AX QDOK@BHMF ηt VHSG TOODQ ANTMCR %NQ HMRS@MBD HE At = A + Ft GNKCR VHSG @ L@SQHW A SGD DHFDMU@KTDR NE VGHBG @KK G@UD MDF@SHUD QD@K O@QS @MC HE VD BGNNRD B @R SGD RNKTSHNM NE SGD L@SQHW DPT@KHSX AT B + BA = −I SGDM VD FDS & T & I T −1 & & . ηt ≤ −λ−1 L@W [B] + Ft + BFt B %NQ DPT@SHNM   VD SGTR NAS@HM SGD ENKKNVHMF RTƧBHDMS BNMCHSHNM   t & T & T −1 & & Fτ + BFτ B Cτ ≤ DWO (λL@W [B](t − t0 )) , t ∈ I0 . E DWO t0



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF $W@LOKD    2DBNMC .QCDQ 1@MCNL #HƤDQDMSH@K $PT@SHNM VHSG 2SNBG@R SHB /DQSTQA@SHNM @S SGD /NRHSHNM 3DQL BE :< O  +DS TR CHRBTRR SGD RDB NMC NQCDQ Q@MCNL CHƤDQDMSH@K DPT@SHNM ¨ t + 2bX˙ t + (1 + Zt ) Xt = 0 , X

 

VHSG b > 0 @MC @ O@SG VHRD BNMSHMTNTR RSNBG@RSHB OQNBDRR Zt OOKXHMF SGD RNKTSHNM LDSGNCR EQNL "G@O  VD FDS ηt = λL@W [ATt + BAt B −1 ] # −2b + |Zt |(1 − b2 )−1/2 = −2b + |2b2 − 1 − Zt |b−1

HE 2b2 ≤ 1 HE 2b2 ≥ 1

ENQ @ BDQS@HM BGNHBD NE B %NQ SGD LD@M RPT@QD RS@AHKHSX NE SGD MTKK RNKTSHNM NE   ! ! !     E |Xt |2 + E |X˙ t |2 < ε @S E |X0 |2 + E |X˙ 0 |2 < δ VD GDMBD G@UD SGD ENKKNVHMF RTƧBHDMS BNMCHSHNMR  

t 2 −1/2 u1 (t) = E DWO (1 − b ) |Zτ |Cτ ≤ DWO (2b(t − t0 )) , t0

@S 2b2 ≤ 1 NQ  

t u2 (t) = E DWO b−1 |2b2 − 1 − Zτ |Cτ ≤ DWO (2b(t − t0 )) ,

t ∈ I0  

t ∈ I0

t0

  @S 2b2 ≥ 1 %NQ SGD DWONMDMSH@K LD@M RPT@QD RS@AHKHSX NE SGD MTKK RNKTSHNM NE   @ RTƧBHDMS BNMCHSHNM HR SGD DWHRSDMBD NE ONRHSHUD MTLADQR c @MC d RTBG SG@S ui (t) ≤ c · DWO ((2b − d)(t − t0 )) ,

t 0 ∈ I0 ,

t ≥ t0

VHSG i = 1 @S 2b2 ≤ 1 @MC i = 2 @S 2b2 ≥ 1 QDRODBSHUDKX -DWS KDS TR @RRTLD SG@S Zt HR @ &@TRRH@M OQNBDRR VHSG LD@M mZ (t) BNU@QH @MBD CZ (τ, t) @MC U@QH@MBD σZ2 (t) = CZ (t, t) RRTLD ENQ @KK t ≥ 0

∞ 2 2 CZ (τ, t)Cτ ≤ γ . |mZ (t)| ≤ α , σZ (t) ≥ β , @MC 0

(M SGD B@RD 2b2 ≤ 1 VD NAS@HM EQNL   SNFDSGDQ VHSG +DLL@   SGD RTƧBHDMS BNMCHSHNM !−1   α + β + 2 1 − b2 γ ≤ 2b 1 − b2   6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS ENQ LD@M RPT@QD RS@AHKHSX (M O@QSHBTK@Q HE !−1   α + β + 2 1 − b2 γ < 2b 1 − b2

 

GNKCR SGDM SGD MTKK RNKTSHNM NE   HR DUDM DWONMDMSH@KKX LD@M RPT@QD RS@AKD (M SGD B@RD 2b2 ≥ 1 VD @OOKX +DLL@   SN SGD &@TRRH@M OQNBDRR ZtT = Zt + 1 − 2b2 3GDM @M@KNFNTRKX SN   VD NAS@HM SGD RTƧBHDMS BNMCHSHNM

E (Zt ) + 1 − 2b2 < 2b2 − β − (2b)−1 γ − ε , t ∈ I   VHSG @M @QAHSQ@QX ε > 0 +DS E (Zt ) ≥ m 2PT@QHMF ANSG RHCDR NE   VD FDS @ RTƧBHDMS BNMCHSHNM ENQ SGD DWON MDMSH@K LD@M RPT@QD RS@AHKHSX NE SGD MTKK RNKTSHNM NE   M@LDKX 2b2 ≥ 1 ,

2b2 > Q ,

α2 + 2m + 1 − Q2 < 4b2 (1 + m − Q) ,  

VGDQD Q := β + (2b)−1 γ (E Zt HR @ RS@SHNM@QX &@TRRH@M OQNBDRR RTBG SG@S mZ (t) = 0 ,

σ 2 (t) = σ 2 ,

@MC

CZ (τ, t) = σ 2 DWO (−ρ|t − τ |) ,

SGDM EQNL   @MC   VD FDS SGD ENKKNVHMF RTƧBHDMS BNMCHSHNMR ENQ DWONMDMSH@K LD@M RPT@QD RS@AHKHSX !−1   σ + σ 2 2ρ 1 − b2 < 2b 1 − b2 , HE 2b2 ≤ 1 @MC q < 1,

4b2 >

1 − q2 1−q

HE 2b2 ≥ 1 VHSG q := σ + σ 2 (2bρ)−1

  

+X@OTMNU %TMBSHNMR @MC /@SG 6HRD $PTH 2S@AHKHSX

+DS K AD @M @QAHSQ@QX ƥWDC ONRHSHUD MTLADQ 6D R@X SG@S @ QD@K U@KTDC ETMB SHNM v(x, t, ω) NM Rd × I × Ω ADKNMFR SN SGD BK@RR F (K) HE SGD ENKKNVHMF GNKCR  v(0, t, ω)=0 ˆ t ∈ I0  3GDQD HR @ BNMSHMTNTR QD@K U@KTDC @MC LNMNSNMHB@KKX HMBQD@RHMF ETMBSHNM ϕ NM [0, K] RTBG SG@S ϕ(0) = 0 @MC |v(x, t, ω)|

d ×I RK 0

≥

ϕ (x) ,

d := {x ∈ Rd : x < K} VGDQD RK



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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF d RTBG SG@S ψ(0) = 0  3GDQD HR @ BNMSHMTNTR QD@K U@KTDC ETMBSHNM ψ NM RK @MC d RK

v(x, t, ω) ≤ ψ(x) . 3GDNQDL   /@SG 6HRD $PTH 2S@AHKHSX /HW WKHUH EH D IXQFWLRQ v WKDW EH ORQJV WR WKH FODVV F (K) VXFK WKDW IRU HYHU\ X0 ∈ γ(K) WKH IXQFWLRQ Vt = v(Xt (X0 , t0 ), t, ω) KDV WKH IROORZLQJ SURSHUW\ IRU DOPRVW DOO ω ∈ Ω LW KROGV WKDW Vt (ω) LV DEVROXWHO\ FRQWLQXRXV RQ I0 ZLWK V˙ t (ω) ≤ 0 IRU DOPRVW DOO t ∈ I0  7KHQ WKH QXOOVROXWLRQ RI  LV SDWKZLVH HTXLVWDEOH ,I DGGLWLRQDOO\ LW KROGV IRU DOPRVW DOO ω ∈ Ω WKDW V˙ t (ω) ≤ −cVt (ω)

IRU DOPRVW DOO t ∈ I ,

!DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   Ŕ /@QS ( +X@OTMNU %TMBSHNMR @MC h 2S@AHKHSX 0 6HSG SGD @HC NE @ RTHS@AKX BGNRDM +X@OTMNU ETMBSHNM FHUD RTƧBHDMS BNM CHSHNMR ENQ h RS@AHKHSX NE SGD MTKK RNKTSHNM NE X˙ = f (Xt , t, ω) 0 2JDSBG SGD OQNNE SGD @RRDQSHNM XNT TRDC HM 0 0 6GHBG BNMCHSHNMR @CCHSHNM@K SN SGNRD FHUDM HM 0 @QD QDPTHQDC ENQ H @RXLOSNSHB @MC HH DWONMDMSH@K h RS@AHKHSX 0 +DS SGD ENKKNVHMF Q@MCNL CHƤDQDMSH@K DPT@SHNM AD FHUDM  −1 0 Xt + g(Xt , t, ω) . X˙ t = 0 −1 6GHBG QDPTHQDLDMSR G@UD SN AD HLONRDC NM SGD Q@MCNL ETMBSHNM g @MC SGD HMHSH@K BNMCHSHNMR RTBG SG@S SGD MTKK RNKTSHNM NE SGHR DPT@SHNM HR DWON MDMSH@KKX h RS@AKD HM RNLD RDMRD 0 "NMRHCDQ SGD KHMD@Q GNLNFDMDNTR Q@MCNL CHƤDQDMSH@K DPT@SHNM  −1 0 ˙ Xt = W t Xt , 1 −1 · DWO(−t) VGDQD Wt HR SGD  CHLDMRHNM@K RS@MC@QC 6HDMDQ OQNBDRR BBNQCHMF SN 3GDNQDL   LD@M RPT@QD RS@AHKHSX V Q S M NE SGD MTKK RNKTSHNM B@M AD RGNVM HE SGDQD HR @ RTHS@AKD PT@CQ@SHB L@SQHW B 6GHBG BNMCHSHNMR G@UD SN AD HLONRDC NM B "NMRSQTBS @ RTHS@AKD L@SQHW B ENQ SGD FHUDM DPT@SHNM

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3NAH@R 1HFNHO  %KNQH@M 5XSS ZKHUH c LV D SRVLWLYH FRQVWDQW WKHQ WKH QXOOVROXWLRQ RI  LV SDWKZLVH DV\PSWRWLFDOO\ HTXLVWDEOH 0RUHRYHU LI IRU SRVLWLYH FRQVWDQWV a DQG b axr

d ×I RK 0

≥

v(x, t, ω)

d ×I RK 0

≥

bxr

(r > 0)

KROGV WRJHWKHU ZLWK V˙ t (ω) ≤ −cVt (ω)

IRU DOPRVW DOO t ∈ I ,

IRU DOPRVW DOO ω ∈ Ω WKHQ WKH QXOOVROXWLRQ RI  LV SDWKZLVH H[SRQHQWLDOO\ HTXLVWDEOH 3URRI %NKKNVHMF :< O  KDS ε > 0 VHSG ε < K AD FHUDM 6D BGNNRD δ > 0 ˆ RTBG SG@S RTO x s 3GDQDENQD SGD KHLHSHMF RSQTBSTQD NE U (0, −t)x = ϕ(t, θ−t ω)x VGDM t → ∞ B@M AD HMSDQOQDSDC @R SGD RS@SD NE NTQ RXRSDL VGHBG VD NARDQUD MNV t = 0 OQNUHCDC HS V@R HM RS@SD x HM SGD HMƥMHSDKX CHRS@MS O@RS t = −∞ 3GTR SGD TMHNM NE @KK SGDRD KHLHSR OQNUHCDR TR VHSG SGD QD@K OHBSTQD NE SGD OQDRDMS RS@SD NE SGD RXRSDL Ş 3GD @RXLOSNSHB ADG@UHNQ NE ϕ(t, θ−t ω)x OQNUHCDR TR VHSG RNLD HMENQL@ SHNM @ANTS SGD KNMF SHLD ETSTQD 2HMBD θt V@R @RRTLDC SN AD DQFNCHB H D  LD@RTQD OQDRDQUHMF VD G@UD SG@S P (ω ∈ Ω : ϕ(t, ω)x ∈ D) = P (ω ∈ Ω : ϕ(t, θ−t ω)x ∈ D) ENQ @MX DKDLDMS x ∈ Rd NE SGD RS@SD RO@BD @MC @MX !NQDK RDS D ∈ B(Rd ) 3GDQDENQD KHL P (ω ∈ Ω : ϕ(t, ω)x ∈ D) = KHL P (ω ∈ Ω : ϕ(t, θ−t ω)x ∈ D) ,

t→∞

t→∞

HE SGD KHLHS NM SGD QHFGS G@MC RHCD DWHRSR 3GTR SGD KHLHSHMF ADG@UHNQ NE ϕ(t, θ−t ω)x ENQ @KK ω ∈ Ω CDSDQLHMDR SGD KNMF SHLD ADG@UHNQ NE ϕ(t, ω)x VHSG QDRODBS SN SGD BNMUDQFDMBD HM OQNA@AHKHSX Ş (S ENKKNVR SG@S HE NM SGD RDS NE Q@MCNL U@QH@AKDR A(ω) VHSG U@KTDR HM Rd VD CDƥMD SGD NODQ@SNQR Tt UH@ (Tt (a))(ω) = ϕ(t, θ−t ω)a(θ−t ω) ,

t ∈ R+ ,

SGDM SGD E@LHKX {Tt }t∈R+ HR @ NMD O@Q@LDSDQ RDLH FQNTO (MCDDC AX @O OKXHMF SGD BNBXBKD OQNODQSX VD G@UD SG@S (Ts ◦ Tt )(a)(ω) = ϕ(s, θs ω)(T−t a)(θ−s ω) = ϕ(s, θs ω) ◦ ϕ(t, θ−t−s ω)a(θ−t−s ω) = ϕ(t + s, θ−t−s ω)a(θ−t−s ω) = Tt+s (a)(ω) . 3GTR HS ADBNLDR ONRRHAKD SN @OOKX HCD@R EQNL SGD SGDNQX NE CDSDQLHMHR SHB @TSNMNLNTR CXM@LHB@K RXRSDLR ENQ VGHBG SGD RDLH FQNTO RSQTBSTQD NE SGD DUNKTSHNM NODQ@SNQ HR BQTBH@K ITRS SGHMJ NE SGD DWHRSDMBD NE @M @S SQ@BSNQ 3GD SGQDD NARDQU@SHNMR @ANUD LNSHU@SD SGD ENKKNVHMF 6HFWLRQ 



3NAH@R 1HFNHO  %KNQH@M 5XSS #DƥMHSHNM   $PTHKHAQHTL NE @ 1@MCNL #XM@LHB@K 2XRSDL +DS (Rd , R, ϕ) AD @ Q@MCNL CXM@LHB@K RXRSDL NUDQ SGD LDSQHB CXM@LHB@K RXRSDL (Ω, F, P, (θt )t∈R ) Q@MCNL U@QH@AKD u : Ω → Rd HR R@HC SN AD @M DPTHKHAQHTL NE SGD Q@MCNL CXM@LHB@K RXRSDL HE HS HR HMU@QH@MS TMCDQ ϕ H D  HE ϕ(t, ω)u(ω) = u(θt ω) ,

ENQ @KK t ∈ R @MC @KK ω ∈ Ω.

!DENQD XNT BNMSHMTD L@JD RTQD SN @MRVDQ SGD ENKKNVHMF PTDRSHNMR 0THY 2DBSHNM   0 &HUD SGD CDƥMHSHNMR NE @ Q@MCNL MNQL @MC @ Q@MCNL BNLO@BS RDS 0 &HUD SGD MNSHNM NE @ RS@AKD @RXLOSNSHB@KKX RS@AKD DWONMDMSH@KKX RS@AKD QDEDQDMBD RNKTSHNM HM @ Q@MCNL CXM@LHB@K RXRSDL 0 &HUD SGD CDƥMHSHNMR @ RS@AKD BNLO@BS Q@MCNL RDS @MC @ FKNA@K Q@MCNL @SSQ@BSNQ 0 &HUD SGD CDƥMHSHNM NE @ RSNBG@RSHB +X@OTMNU ETMBSHNM 'NV B@M XNT TRD RTBG @ ETMBSHNM SN CDSDQLHMD SGD RS@AHKHSX NE @ Q@MCNL RDS 0 (KKTRSQ@SD SGD BNMBDOS NE A@BJV@QCR RS@AHKHSX @MC FHUD SGD CDƥMHSHNM NE @M DPTHKHAQHTL NE @ Q@MCNL CXM@LHB@K RXRSDL

  "G@OSDQŗR 2TLL@QX %NKKNVHMF :< @MC :< SGHR RGNQS BG@OSDQ OQNUHCDC TR VHSG @ FKHLORD HMSN SGD QDBDMS SGDNQX NE Q@MCNL CXM@LHB@K RXRSDLR %HQRS VD F@UD SGD ETMC@LDMS@K CDƥMHSHNMR NE LDSQHB LD@RTQ@AKD @MC Q@MCNL CXM@LHB@K RXRSDLR SNFDSGDQ VHSG RNLD HKKTRSQ@SHUD DW@LOKDR ,NQDNUDQ VD AQHDƦX RSTCHDC SGD MNSHNMR NE ENQV@QC @MC A@BJV@QCR RS@AHKHSX @MC SGDHQ HLOKHB@SHNMR

/QNAKDLR "K@RRHƥB@SHNM ☼ D@RX  D@RX VHSG KNMFDQ B@KBTK@SHNMR  @ KHSSKD AHS CHƧBTKS  BG@KKDMFHMF



&KDSWHU 

1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF $WDQBHRD   :☼< +HMD@Q 1@MCNL #HƤDQDMSH@K $PT@SHNMR @MC 1@MCNL #X M@LHB@K 2XRSDLR +DS A : Ω → Rd×d AD @ LD@RTQ@AKD ETMBSHNM RTBG SG@S A ∈ L1 (Ω, F, P) @MC fω (t, x) := A(θt ω)x R@SHRƥDR fω ∈ Lloc (R, C ∞ ) 2GNV SGD ENKKNVHMF @RRDQSHNMR  3GD KHMD@Q Q@MCNL CHƤDQDMSH@K DPT@SHNM x˙ t = A(θt ω)xt FDMDQ@SDR @ TMHPTD KHMD@Q C ∞ Q@MCNL CHƤDQDMSH@K DPT@SHNM ϕ R@SHREXHMF

t ϕ(t, ω) = I + A(θs ω)ϕ(s, ω)Cs 0

@MC



CDS(ϕ(t, ω)) = DWO



t 0

SQ (A(θs ω)) Cs

.

KRN CHƤDQDMSH@SHMF ϕ(t, ω) (ϕ(t, ω))−1 = I XHDKCR

t −1 = I+ (ϕ(s, ω))−1 A(θs ω)Cs . (ϕ(t, ω)) 0

$WDQBHRD   :☼< ƧMD 1@MCNL #HƤDQDMSH@K $PT@SHNMR @MC 1@MCNL #X M@LHB@K 2XRSDLR +DS A : Ω → Rd×d AD @ LD@RTQ@AKD ETMBSHNM RTBG SG@S A ∈ L1 (Ω, F, P) @MC fω (t, x) := A(θt ω)x R@SHRƥDR fω ∈ Lloc (R, C ∞ ) 2GNV SG@S SGD DPT@SHNM x˙ t = A(θt ω)xt + b(θt ω) ,

A, b ∈ L1 (Ω, F , P) ,

FDMDQ@SDR @ TMHPTD C ∞ Q@MCNL CXM@LHB@K RXRSDL 3GD U@QH@SHNM NE BNMRS@MSR ENQLTK@ XHDKCR

t Φ(t, ω)Φ−1 (u, ω)b(θu ω)Cu ϕ(t, ω)x = Φ(t, ω)x + 0

= Φ(t, ω)x +

0

t

Φ(t − u, θu ω)b(θu ω)Cu ,

VGDQD Φ HR SGD $WDQBHRD   :☼< 1@MCNL #HƤDQDMSH@K $PT@SHNMR VHSG /NKXMNLH@K 1HFGS '@MC 2HCD @MC 1@MCNL #XM@LHB@K 2XRSDLR k +DS f (ω, x) = N k=0 ak (ω)x AD @ ONKXMNLH@K NE CDFQDD N ≥ 2 VHSG Q@MCNL 1 BNDƧBHDMSR (E ak ∈ L (Ω, F , P) ENQ @KK k = 1, 2, . . . , N  SGDM SGD Q@MCNL CHE EDQDMSH@K DPT@SHNM N  x˙ t = ak (θt ω)xkt k=0

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3NAH@R 1HFNHO  %KNQH@M 5XSS TMHPTDKX FDMDQ@SDR @ KNB@K C ∞ Q@MCNL CXM@LHB@K RXRSDL NM R1 3GD B@RD x˙ t = a(θt ω)xt + b(θt ω)xN t B@M AD DWOKHBHSKX RNKUDC AX SQ@MRENQLHMF SGD DPT@SHNM SN @M @ƧMD DPT@SHNM UH@ y = x1−N /(1 − N ) 6QHSD CNVM DWOKHBHSKX @KK HMFQDCHDMSR NE SGD KNB@K Q@MCNL CXM@LHB@K RXRSDL 3GD B@RDR N = 2 @MC N = 3 @QD D F  SQD@SDC HM :
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8

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Base isolation with shock absorbers

Additional dampers in the structure of the building

Water pool at the top of the building to have a counter response

Counter pendulum to generate response

%HFTQD   (KKTRSQ@SHNM NE RDUDQ@K ONRRHAHKHSHDR NE RDBTQHMF ATHKCHMFR EQNL SGD DƤDBSR NE D@QSG DWBHS@SHNMR

k1 = . . . = k4 = 25 @MC c1 = . . . = c4 = 10 !THKCHMF  RDBNMC EQNL KDES ONR RDRRDR GHFGDQ C@LOHMF U@KTDR c1 = . . . = c4 = 40 QDRTKSHMF HM KDRR CHROK@BDC RSNQDXR %NQ !THKCHMF  RDBNMC EQNL QHFGS SGD RS@MC@QC C@LOHMF ci = 10 G@R ADDM TRDC HM BNLAHM@SHNM VHSG RS@MC@QC RSHƤMDRR ki = 25 DWBDOS ENQ BNDƧBHDMS k1 = 2.5 3GD O@QSHBHO@MSR RS@QSDC VNQJHMF NM ANSG S@RJR 3 &4( @MC 3 BNMSHMTTL LDBG@MHBR LNCDK QDK@SHUDKX K@SD ATS SGDM HMUDRSDC GHFG DƤNQSR HM SQ@MREDQ QHMF SGD &4( HM O@QSHBTK@Q SN SGD CHƤDQDMS CDL@MCDC S@RJR RM@ORGNS NE SGD &4( ENQ 3 ENQ SVN CHƤDQDMS RSD@CX RS@SD RBDM@QHNR HR FHUDM HM %HF   3GD RSTCDMSR HMBKTCDC @ KNS NE U@QH@MSR SN RODBHEX ENQBDR NM SGD U@QHNTR ANTMC @QHDR NE SGD ATHKCHMF $WSDQM@K ENQBDR B@M AD @OOKHDC HM ANSG x @MC y CHQDBSHNM NMSN SGD RTQE@BD NE SGD ATHKCHMF M DW@LOKD NE @ VHMC KN@C EQNL KDES HR UHRH AKD HM SGD SNO OHBSTQD NE %HF   !DKNV @M @QSHƥBH@K SDRS B@RD NE řRPTDDYHMFŚ SGD ATHKCHMF EQNL KDES @MC QHFGS HR RGNVM 2DUDQ@K SHLD CDODMCDMS RHLTK@SHNMR TRHMF SGD -DVL@QJ RBGDLD HM BNL AHM@SHNM VHSG SGD *@M@H 3@IHLH DWBHS@SHNM G@R ADDM B@QQHDC NTS %HFTQD   RGNVR SGD QDRTKSHMF CHROK@BDLDMSR @S CHƤDQDMS SHLD RSDOR 3GD ATHKCHMF LNUDR CTD SN SGD *@M@H 3@IHLH DWBHS@SHNM RKHFGSKX SN SGD QHFGS A@BJ SN SGD BDMSQD @MC ƥM@KKX SN SGD KDES ENQ SGHR QD@KHR@SHNM NE SGD Q@MCNL HMOTS  SHLD RSDOR NE RHYD dt = 0.0049 G@UD ADDM TRDC SN QD@BG SGD ƥM@K SHLD te = 10.0 ENQ @ BN@QRD FQHC VHSG 4 × 20 CDFQDDR NE EQDDCNL #N% QDRTKSHMF HM hx = 0.05/3, hy = 0.25/19 SN CHRBQDSHRD SGD ATHKCHMF VHSG 8NTMFŗR LNCTKTR DPT@K SN E = 1e − 4 %TQSGDQLNQD SGD O@QSHBHO@MSR VDQD D@FDQ SN L@JD SGD NOSHNM@K S@RJ 3  ONRRHAKD (M %HF   SVN ATHKCHMFR FQDDM O@SBGDR NE CHƤDQDMS 8NTMFŗR LNC TKH KDES E1 = 500  QHFGS E2 = 1500 QD@BS NM @ RTQE@BD ENQBD EQNL SGD KDES

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3NAH@R 1HFNHO  %KNQH@M 5XSS

u ¨ g = -5.71

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u ¨ g = 5.52

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u ¨ g = 5.26

B

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1@MCNL #HƤDQDMSH@K $PT@SHNMR HM 2BHDMSHƥB "NLOTSHMF

%HFTQD   $W@LOKD NE SGD &4( OQNUHCDC ENQ 3 2SD@CX RS@SD RNKTSHNMR NE SGD # BNMSHMTTL LDBG@MHBR LNCDK VHSGNTS D@QSGPT@JD HMBHS@SHNM ENQ 16 × 16 MNCDR TMCDQ RODBHƥDC ENQBDR SNO OTRGHMF ENQBDR EQNL KDES VHMC KN@C D F  ANSSNL RXLLDSQHB řRPTDDYHMFŚ NE SGD ATHKCHMF EQNL KDES @MC QHFGS

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@ t = 0

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B t = 7.5

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