On explosion of the chaotic attractor

TL;DR

This paper investigates the sudden and violent explosion of a strange attractor in a driven damped anharmonic oscillator, analyzing the phenomenon through Lyapunov exponents and fractal dimensions.

Contribution

It provides a phenomenological characterization of the explosive transition of strange attractors induced by small parameter changes.

Findings

Explosion associated with linear increase in Lyapunov exponent

Sudden jump in fractal dimension at explosion onset

Behavior observed in a one-dimensional driven damped anharmonic oscillator

Abstract

There are presented examples of the rather sudden and violent explosion of the strange attractor of a one-dimensional driven damped anharmonic oscillator induced by a relatively small change of the amplitude of the strongly nonperturbative periodic driving force. A phenomenologic characterization of the explosion of the strange attractor has been given in terms of the behavior of the average maximal Lyapunov exponent ${\bar \lambda}$ and that of the fractal dimension $D_{q}$ for $q=-4$. It is shown that the building up of the exploding strange attractor is accompanied by a nearly linear increase of the maximal average Lyapunov exponent ${\bar \lambda}$. A sudden jump of the fractal dimension $D_{-4}$ is detected when the explosion starts off from an attractor consisting of disjoint bunches separated by an empty phase-space region.