Parametric Modulation of Nonlinear Coupling in the H\'enon Heiles System: Resonances, Chaos, and Stabilization

TL;DR

This paper explores how parametric modulation of nonlinear coupling in the Henon Heiles system affects resonance structures, chaos, and stabilization, combining theoretical analysis and simulations to reveal new control mechanisms.

Contribution

It introduces a novel approach of parametric modulation to modify resonance and chaos in the Henon Heiles system, complementing existing forcing methods.

Findings

Resonance tongues scale as √ε near commensurate frequencies.

Parametric modulation can suppress chaos through high-frequency averaging.

Transition from localized resonances to global transport observed in simulations.

Abstract

We investigate parametric modulation of the nonlinear coupling in the Henon Heiles system, which directly modifies intrinsic resonance structure in a manner complementary to additive forcing. Canonical perturbation theory in extended phase space yields normal forms predicting resonance tongues scaling as $\sqrt{\varepsilon}$ near commensurate frequencies. Melnikov analysis quantifies separatrix splitting and chaos onset, confirmed by symplectic simulations showing transition from localized resonances to global transport via overlap. High-frequency averaging reveals potential stiffening that suppresses chaos. parametric modulation of nonlinear coupling provides an alternative route for generating combination resonances and influencing chaotic dynamics