Conservative Generalized Bifurcation Diagrams

TL;DR

This paper introduces a generalized bifurcation diagram for conservative systems, revealing detailed information about periodic orbits, resonances, and tori, with applications demonstrated on the standard map and Hénon-Heiles potential.

Contribution

It presents a novel generalized bifurcation diagram that captures all relevant features of conservative systems, including higher-order resonances and irrational tori.

Findings

Contraction rates for island sizes are estimated to be approximately 3.9.

The methods effectively identify periodic orbits and characterize system dynamics.

Results are demonstrated on the standard map and Hénon-Heiles potential.

Abstract

Bifurcation cascades in conservative systems are shown to exhibit a generalized diagram, which contains all relevant informations regarding the location of periodic orbits (resonances), their width (island size), irrational tori and the infinite higher-order resonances, showing the intricate way they are born. Contraction rates for islands sizes, along period-doubling bifurcations, are estimated to be $\alpha_I\sim 3.9$. Results are demonstrated for the standard map and for the continuous H\'enon-Heiles potential. The methods used here are very suitable to find periodic orbits in conservative systems, and to characterize the regular, mixed or chaotic dynamics as the nonlinear parameter is varied.