Chaos in the Damped and Forced Duffing Oscillator

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Authors: Javier Montenegro Joo
Page Range: 53-66
Published in: Chaos and Complexity Letters, 16#1 (2022)
ISSN: 1556-3995

ISBN: N/A Category:

Table of Contents

ABSTRACT

The chaotic behavior in the Duffing oscillator has been studied by means of computer simulations based on numerical solution of its differential equation along up to 30 million time-steps. By extracting longitudinal and transversal Poincaré’s tomographies of the state space of a chaotic event selected at random, it is found that there seems to be more than a single family of attractors, which is verified by counting the number of time-steps needed to close an orbit in state space. Those orbits in state space closing with a number of time-steps which are multiple of the simplest one of them are considered members of the same family of attractors. In this study two families of attractors have been detected. It has been encountered that the transition to chaos in the Duffing oscillator is via intermittency, this is, stages of chaos intercalated by very well defined oscillation periods, though not necessarily of a single period, the mentioned stages of chaos show sometimes short period bifurcations, but never display a complete cascade of bifurcations. It has also been verified that the chaotic oscillation in the Duffing equation have a symmetric distribution of peaks and valleys. It has as well been found that there is no gradual shrinking of periodic stages along a chaotic event, which would wind up in a continuous chaotic regime.

Keywords: chaos, duffing oscillator, state space, attractors, Maps of Return, Lyapunov exponents