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

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

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

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

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

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

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

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

9. 2. Historrcal Background
Poincare ( )
“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 ( )
Existence of Many Periodic Solutions
Double Pendulum

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

11. 2. Historical Background
Balthasar van der Pol ( )
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

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

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

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

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

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

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

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

19. 2. Historical Background
Time Almost Periodic Perturbations
Scheurle
Palmer
Meyer and Sell
Stoffer
Yagaski
Others
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.
Transversal homoclinic point
Analytic Shadowing
Horseshoes

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

21. 3. Chaotic Behavior Driven by Brwonian Motion
Assume

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

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

24. 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

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

26. 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.

27. 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.

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

29. 4. Chaotic Behavior Driven by Nonautonomous forcing
Assume

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

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

32. 4. Chaotic Behavior Driven by Nonautonomous forcing
Let

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

50. 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

51. Muchas gracias