Chaotic Jumping Near Resonances: Infinite-Dimensional Systems

Abstract

In this chapter we study the global effect of resonances in near-integrable evolution equations. As in Chapter 2, the unperturbed equation is assumed to have a set of resonant states, which appear as fixed points in a frame moving with the forcing. The resonant states are connected through a heteroclinic network of orbits before perturbation and admit infinitely many neutrally stable directions. These directions correspond to infinitely many Fourier modes exhibiting fast oscillations around the finitely many slowly varying modes. As earlier, we seek fast multipulse transitions between the resonant states after perturbation. These recurrent jumping solutions are again expected to cause irregular or chaotic dynamics, but their exact implications are mostly unknown as of yet. While we discuss some partial results in this direction, future developments in infinite-dimensional chaotic dynamics will still have a lot to offer to this theory.