Melnikov processes and chaos in randomly perturbed dynamical systems

開催日時
2014/07/04 金 14:00 - 17:00
場所
6号館609号室
講演者
矢ヶ崎 一幸
講演者所属
京都大学大学院・情報学研究科
概要

We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely, no matter how small the random forcing terms are. This result is very contrast 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 function, which we call the Melnikov process, have infinitely many zeros, so that infinitely many transverse homoclinic orbits exist. In addition, a stable and unstable manifold theorem is given and the Smale-Birkhoff homoclinic theorem is extended in appropriate forms for randomly perturbed systems. We illustrate our theory for the randomly perturbed Duffing oscillator.