A versatile class of prototype dynamical systems for complex bifurcation cascades of limit cycles

TL;DR

This paper introduces a flexible class of dynamical systems to study complex bifurcation cascades of limit cycles, including symmetry-breaking, period doubling, and chaos transitions, using systems with potential-dependent friction forces.

Contribution

It proposes a new class of prototype systems with potential-dependent friction to analyze complex bifurcation sequences in limit cycles, including symmetry-breaking and chaos.

Findings

Sequences of limit cycle bifurcations observed as energy uptake increases

Bifurcation parameters linked to zeros of friction function

Examples for 1D and 2D systems with polynomial friction

Abstract

We introduce a versatile class of prototype dynamical systems for the study of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling bifurcations and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype system consist of a $2d$-dimensional dynamical system with friction forces $f(V(\mathbf{x}))$ functionally dependent exclusively on the mechanical potential $V(\mathbf{x})$, which is typically characterized, here, by a finite number of local minima. We present examples for $d=1,2$ and simple polynomial friction forces $f(V)$, where the zeros of $f(V)$ regulate the relative importance of energy uptake and dissipation respectively, serving as bifurcation parameters. Starting from simple Hopf- and homoclinic bifurcations, complex sequences of limit cycle bifurcation are observed when energy uptake gains progressively in importance.