Evolution of the Lyapunov Exponent during a Chaotic Event in the Nonlinear Damped and Forced Oscillator

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Authors: Javier Montenegro Joo
Page Range: 27-40
Published in: Chaos and Complexity Letters, 16#1 (2022)
ISSN: 1556-3995

ISBN: N/A Category:

Table of Contents

ABSTRACT

The maximum Lyapunov exponent permits to know whether or not an allegedly chaotic event is in fact chaotic and, its magnitude gives an idea of its degree of chaos. The plotting of the Lyapunov exponent during a chaotic event yields information on its behavior as time evolves, however sometimes this plotting is so dense that it results in a dark spot, impossible to understand, some other times the plotting is so free of ups and downs that it seems to be nothing distinctive is happening with the flow during the event. The solution to avoid the mentioned drawbacks is making the plotting at a pre-defined time interval. The question then is how to select a convenient plotting interval so that the graph reveals more details of the behavior of the Lyapunov exponents as the system evolves. This paper shows the results of a research dealing with the above mentioned situation and, it has been encountered that a too large interval results in a neat plotting but when using a rather smaller interval, the plotting is not neat but it reveals interesting details like a pattern that is repeated along the flow. This article makes evident that selecting the adequate time interval is important to get acquainted with the Lyapunov exponent evolution. Additionally this report shows the evolution of the Lyapunov exponent for every time higher viscosity, in this case some intuitively expected comportment of the exponent has been found.

Keywords: nonlinear oscillators, dynamics, chaos, Lyapunov, exponents, Maps of Return