Characterization in bi-parameter space of a non-ideal oscillator

TL;DR

This paper explores the complex dynamical behaviors of a non-ideal Duffing oscillator driven by a limited power supply, revealing new organized structures like Arnold tongues and shrimp-shaped regions through extensive numerical analysis.

Contribution

It provides the first detailed bi-parameter space characterization of a non-ideal Duffing oscillator with a limited power supply, uncovering novel bifurcation structures.

Findings

Identification of Arnold tongues and shrimp-shaped structures in parameter space

Discovery of self-similar and organized distribution of periodic windows

Observation of codimension-2 bifurcation points as accumulation points

Abstract

We investigate the dynamical behavior of a non-ideal Duffing oscillator, a system composed of a mass-spring-pendulum driven by a DC motor with limited power supply. To identify new features on Duffing oscillator parameter space due to the limited power supply, we provide an extensive numerical characterization in the bi-parameter space by using Lyapunov exponents. Following this procedure, we identify remarkable new periodic windows, the ones known as Arnold tongues and also shrimp-shaped structures. Such windows appear in highly organized distribution with typical self-similar structures for the shrimps, and, surprisingly, codimension-2 bifurcation as a point of accumulations for the tongues.