Non-perturbative features of driven scattering systems

TL;DR

This paper explores the complex scattering behavior of driven oscillators, revealing hierarchical singularity structures, the role of invariant manifolds, and power-law decay in survival probabilities, highlighting non-perturbative dynamics.

Contribution

It uncovers the hierarchical structure of scattering singularities and links them to invariant manifold intersections and phase space 'stickiness' in driven oscillators.

Findings

Hierarchical singularity patterns in periodic forcing cases.

Power-law decay of survival probabilities due to phase space 'stickiness'.

Connection between scattering function singularities and invariant manifold intersections.

Abstract

We investigate the scattering properties of one-dimensional, periodically and non-periodically forced oscillators. The pattern of singularities of the scattering function, in the periodic case, shows a characteristic hierarchical structure where the number Nc of zeros of the solutions plays the role of an order parameter marking the level of the observed self-similar structure. The behavior is understood both in terms of the return map and of the intersections pattern of the invariant manifolds of the outermost fixed points. In the non-periodic case the scattering function does not provide a complete development of the hierarchical structure. The singularities pattern of the outgoing energy as a function of the driver amplitude is connected to the arrangement of gaps in the fundamental regions. The survival probability distribution of temporarily bound orbits is shown to decay asymptotically as a power law. The "stickiness" of regular regions of phase space, given by KAM surfaces and remnant of KAM curves, is responsible for this observation.