Detecting invariant manifolds, attractors, and generalized KAM tori in aperiodically forced mechanical systems

TL;DR

This paper introduces a method using geodesic transport barriers to identify invariant manifolds, attractors, and KAM tori in aperiodically forced mechanical systems, enabling visualization without Poincare maps.

Contribution

It applies geodesic transport barrier theory to forced mechanical systems, providing a new way to visualize invariant sets in finite time without relying on Poincare maps.

Findings

Successfully visualized invariant manifolds and attractors in Duffing oscillators

Demonstrated the method's applicability to both conservative and dissipative systems

Enabled finite-time analysis of complex invariant structures

Abstract

We show how the recently developed theory of geodesic transport barriers for fluid flows can be used to uncover key invariant manifolds in externally forced, one-degree-of-freedom mechanical systems. Specifically, invariant sets in such systems turn out to be shadowed by least-stretching geodesics of the Cauchy-Green strain tensor computed from the flow map of the forced mechanical system. This approach enables the finite-time visualization of generalized stable and unstable manifolds, attractors and generalized KAM curves under arbitrary forcing, when Poincare maps are not available. We illustrate these results by detailed visualizations of the key finite-time invariant sets of conservatively and dissipatively forced Duffing oscillators.