Chaos in Differential Equations Driven By Brownian Motions

Chaos in Differential Equations Driven By Brownian Motions Qiudong Wang, University of Arizona (Joint With Kening Lu, BYU) 1

1. Equation Driven by Brownian Motion q Random Force by Brownian Motion. Ø Wiener probability space Open compact topology Ø Wiener shift: Ø Brownian motion 2

3. Chaotic Behavior Driven by Brwonian Motion Ø Random forcing 3

1. Basic Problem Mathematical Model Driven by a Random Forcing Question: What is the change of dynamics? 4

1. Basic Problem q Random Forcing: Ø Probability space: Ø Stochastic process: Let Real noise 5

1. Basic Problem q Random Forcing Driven by Brownian Motion. Ø Brownian motion: Ø Stochastic process given by Wiener shift: • Stationary process with a normal distribution • Discrete version of “white noise” • Unbounded almost surely. 6

1. Basic Problem q Problems: Ø Study the dynamical behavior of DE driven by a sample path, leading to nonautonomous DE Ø Study the almost sure dynamical behavior, i. e. , A property holds almost surely 7

1. Basic Problem q Poincare’s Problem: Assume has a homoclinic orbit. Complicated behavior of solutions 8

2. Historrcal Background Poincare (1854 -1912) “it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible. . . ". George Birkhoff (1884 -1944) Existence of Many Periodic Solutions Double Pendulum 9

2. Historical Background Ø Homoclinic Orbit Ø Perturbation Poincare Map F Stable Manifold Unstable Manifold Transversal Homoclinic Point q 10

2. Historical Background Balthasar van der Pol (1889 -1959) Dutch electrical engineer Experiments on electrical circuits, 1927 reports on the "irregular noise" heard in a telephone earpiece attached to an electronic tube circuit. Eric W. Weisstein 11

2. Historical Background John Littlewood (1885 -1977) Mary Cartwright (1900 -1998) Nonlinear differential equations arising in radio research. Forced oscillations in nonlinear systems of second order Forced van der Pol’s equation 12

2. Historical Background Norman Levinson (1912 -1975) Forced periodic oscillations of the Van der Pol oscillator Ø Ø Infinite many periodic solutions Periodic solutions of singularly perturbed differential Ø Ø equations Invariant Torus Integral Manifolds 13

2. Historical Background Smale (1930 -) Ø Horseshoe maps. Ø Birkhoff-Smale Transversal Homoclinic Theorem Existence of a transversal homoclinic point implies the existence of a horseshoe 14

2. Historical Background Ø Time Periodic Perturbations q Applications and extension of Birkhoff-Smale Theorem § Time-periodic map § Melnikov function Alekseev, Sitnikov, Melnikov Shilnikov, Holmes, Marsden, Guckenheimer, … 15

2. Historical Background Ø Time Periodic Perturbations § Time-periodic map § Lyapunov-Schmidt Chow, Hale, and Mallet-Paret Existence of subharmonic solutions 16

2. Historical Background Ø Time Periodic Perturbations q Analytic Shadowing Approach Palmer Hale, Lin others 17

2. Historical Background Scheurle Palmer Meyer and Sell Stoffer Yagaski Transversal homoclinic point Others Shadowing Horseshoes L and Wang Strange Attractor 18

2. Historical Background Ø Time Almost Periodic Perturbations Scheurle Palmer Meyer and Sell Stoffer Transversal homoclinic point Yagaski Analytic Shadowing Horseshoes Others Ø Time Dependent Perturbations Lerman and Shilnikov Assume that there exists a set of homoclinic solutions satisfying exponential dichotomy and certain uniform properties. There are solutions associated with Bernoulli shift. 19

3. Chaotic Behavior Driven by Brwonian Motion q Unforced Equations: 20

3. Chaotic Behavior Driven by Brwonian Motion q Assume 21

3. Chaotic Behavior Driven by Brwonian Motion q Equation Driven by Random Force: where Multiplicative noise, singular, unbounded. 22

3. Chaotic Behavior Driven by Brwonian Motion q Random Poincare Return Maps in Extended Space Poincare Return Map 23

3. Chaotic Behavior Driven by Brwonian Motion Theorem. (Chaos almost surely) has a topological horseshoe of infinitely many branches almost surely. • Sensitive dependence on initial time. • Sensitive dependence on initial position 24

3. Chaotic Behavior Driven by Brwonian Motion q Topological Horseshoe: 25

3. Chaotic Behavior Driven by Brwonian Motion Corollary A. (Duffing equation) the randomly forced Duffing equation has a topological horseshoe of infinitely many branches almost surely. 26

3. Chaotic Behavior Driven by Brwonian Motion Corollary B. (Pendulum equation) the randomly forced pendulum equation has a topological horseshoe of infinitely many branches almost surely. 27

4. Chaotic Behavior Driven by Nonautonomous forcing q Unforced Equations: 28

4. Chaotic Behavior Driven by Nonautonomous forcing q Assume 29

4. Chaotic Behavior Driven by Nonautonomous forcing q Equation Driven by Time Dependent Force: 30

4. Chaotic Behavior Driven by Nonautonomous forcing q Characteristic function Let be the homoclininc orbit. Let 31

4. Chaotic Behavior Driven by Nonautonomous forcing Let 32

4. Chaotic Behavior Driven by Nonautonomous forcing q Characteristic Function – Generalized Melnikov Function Let 33

4. Chaotic Behavior Driven by Nonautonomous forcing q Poincare Return Maps in Extended Space Poincare Return Map 34

4. Chaotic Behavior Driven by Nonautonomous forcing q Invariant Sets: 35

4. Chaotic Behavior Driven by Nonautonomous forcing q Topological Horseshoe: 36

4. Chaotic Behavior Driven by Nonautonomous forcing Theorem A. (Intersections of Stable and Unstable Manifolds) 37

4. Chaotic Behavior Driven by Nonautonomous forcing Theorem B. (Integral Manifolds) 38

4. Chaotic Behavior Driven by Nonautonomous forcing Theorem C. (Intersection and Full Horseshoe) 39

4. Chaotic Behavior Driven by Nonautonomous forcing Theorem D. (Intersection and Half Horseshoe) 40

4. Chaotic Behavior Driven by Nonautonomous forcing Theorem F. (Trivial Dynamics) 41

4. Chaotic Behavior Driven by Nonautonomous forcing Theorem E. (Non-intersection and a Half Horseshoe) 42

4. Chaotic Behavior Driven by Nonautonomous forcing Theorem G. (Non-intersection and Full Horseshoe) 43

4. Chaotic Behavior Driven by Nonautonomous forcing q Application: Forced Duffing’s Equations 44

4. Chaotic Behavior Driven by Nonautonomous forcing q Application: Forced Duffing’s Equations Theorem H. All above phenomena mentioned in Theorem A-G appear by adjusting the parameters. 45

5. Chaotic Behavior Driven by Bounded Random forcing q Equation Driven by Bounded Random Forcing: 46

5. Chaotic Behavior Driven by Bounded Random forcing q Random Melnikov Function: Theorem I. 47

5. Chaotic Behavior Driven by Bounded Random forcing q Expectation and Variance: Proposition. The condition of Theorem I holds if 48

5. Chaotic Behavior Driven by Bounded Random forcing q Application. Randomly Forced Duffing Equation Ø Quasiperiodic Force. Ø Random Force Driven by Wiener Shift: 49

6. Idea of Proof: • Invariant stable and unstable segments, • Random Poincare return map • Random Melnikov function • Birkhoff Ergodic Theorem • Random partially Linearization • Topological horseshore • Nonautonomous Linearization 50

Muchas gracias 51