Weak Transient Chaos
A phenomenon of weak transient chaos is discussed that is caused by sub-exponential divergence of trajectories in the basin of a non-chaotic attractor. Such a regime is not easy to detect, because conventional characteristics such as the largest Lyapunov exponent are non-positive. Here we study how such a divergence can be exposed and detected. First, we show that weak transient chaos can be exposed if a small random perturbation is added to the system, leading to positive values of the largest Lyapunov exponent. Second, we introduce an alternative definition of the Lyapunov exponent, which allows us to detect weak transient chaos in the deterministic unperturbed system. We show that this novel characteristic becomes positive, reflecting transient chaos. We demonstrate this phenomenon and its detection using a master–slave system where the master possesses a heteroclinic cycle attractor, while the slave is the Van-der-Pol–Duffing oscillator possessing a stable limit cycle.
KeywordsWeak transient chaos Lyapunov exponent Heteroclinic cycle Random perturbation
The authors dedicate this paper to the 75th anniversary of Mikhail Rabinovich. We wish him great health, and maintenance of his remarkable enthusiasm and energy, which generated and will generate many exciting ideas and fundamental works. The authors thank T. Young for useful discussions and Ya. Pesin for allowing them to read the manuscript of his chapter  in this volume. The authors acknowledge support by the RSF grant 14-41-00044 of the Russian Science Foundation during their stay at Nizhny Novgorod University.
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