Melnikov processes and chaos in randomly perturbed dynamical systems

作者：Yagasaki Kazuyuki^*

来源：Nonlinearity, 2018, 31(7): 3057-3085.

DOI：10.1088/1361-6544/aab89f

• random perturbation
• chaos
• Melnikov method
• Gaussian stationary process
• stable and unstable manifolds
• transverse homoclinic orbit
• horseshoe sequence

We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely under some nondegenerate condition, no matter how small the random forcing terms are. This result is very contrasting to the deterministic forcing case, in which chaotic orbits exist only if the influence of the forcing terms overcomes that of the other terms in the perturbations. To obtain the result, we extend Melnikov's method and prove that the corresponding Melnikov functions, which we call the Melnikov processes, have infinitely many zeros, so that infinitely many transverse homoclinic orbits exist. In addition, a theorem on the existence and smoothness of stable and unstable manifolds is given and the Smale-Birkhoff homoclinic theorem is extended in an appropriate form for randomly perturbed systems. We illustrate our theory for the Duffing oscillator subjected to the Ornstein-Uhlenbeck process parametrically.

• 出版日期2018-7