Nonintegrability and fluctuation by symmetry violation in perturbed harmonic oscillator systems

TL;DR

This paper investigates how symmetry violations in phase-space dynamics relate to chaos in perturbed harmonic oscillators, proposing a correlation coefficient as a new measure of nonintegrability that may outperform Lyapunov exponents.

Contribution

It introduces a correlation coefficient to quantify symmetry violation, offering a novel approach to detect nonintegrability and chaos in harmonic oscillator systems.

Findings

Correlation coefficient effectively indicates transition from integrability to chaos.

Symmetry violation correlates with nonintegrability in perturbed oscillators.

Proposed measure may have advantages over Lyapunov exponents.

Abstract

The violation of symmetry between the time series of elongation and contraction rates of phase-space point spacings is studied to examine the chaos in perturbed harmonic oscillator systems. A transition from integrability to nonintegrability is quantitatively evaluated by introducing a correlation coefficient for the degree of symmetry violation. A feature of the correlation coefficient is discussed, suggesting a possible advantage over the Lyapunov exponent. The relationship between a fluctuation property based on symmetry violation and nonintegrability is also discussed.