A new method to probe the boundary where KAM tori persist by square matrix

TL;DR

This paper introduces an iterative method using square matrices to accurately identify action-angle variables near resonance, enabling better analysis of stability boundaries and KAM tori in nonlinear dynamical systems.

Contribution

The paper presents a novel iterative approach to find action-angle variables near resonance, improving stability analysis beyond traditional perturbation methods.

Findings

Iterative method yields closer agreement with forward integration.

Method converges near stability boundary where conventional methods fail.

Action-angle variables approximate KAM invariants more accurately.

Abstract

The nonlinear dynamics of a system can be analyzed using a square matrix. If off resonance, the lead vector of a Jordan chain in a left eigenspace of the square matrix is an accurate action- angle variable for sufficiently high power order. The deviation from constancy of the action-angle variable provides a measure of the stability of a trajectory. However, near resonance or the stability boundary, the fluctuation increases rapidly and the lead vector no longer represents an accurate action-angle variable. In this paper we show that near resonance or stability boundary, it is possible to find a set of linear combinations of the vectors in the degenerate Jordan chains as the action-angle variables by an iteration procedure so that the fluctuation is minimized. Using the Henon-Heiles problem as an example on resonance, we show that when compared with conventional canonical perturbation theory, the iteration leads to result in much more close agreement with the forward integration, and the iteration is convergent even very close to the stability boundary. It is further shown that the action-angle variables found in the iteration can be used to find another action-angle variables even much more closer to rigid rotations (KAM invariants). The fast convergent result is not in the form of polynomials, it is an exponential function with a rational function in the exponent more similar to a Laurent series than a Taylor series. Hence the method provides a new way to probe the boundary of the region where KAM tori exist.