Abstract
This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.
Highlights
It is believed that a wide variety of natural phenomena are chaotic, including fluid flow, heartbeat irregularities, weather, and climate [1]
According to Leonov et al [9], the attractors in dynamical systems are categorized as self-excited attractors and hidden attractors
The remaining sections are devoted to the two-parameter family
Summary
It is believed that a wide variety of natural phenomena are chaotic, including fluid flow, heartbeat irregularities, weather, and climate [1]. Since the discovery of a chaotic system by Lorenz in 1963 [10], many other chaotic systems have been found and studied, such as the Rössler system [11], the Chua circuit [12,13], chaotic jerk circuit [14], the Chen system [15], the Lü system [16], and the Sprott systems [17] These examples have one or more saddle-points and the associated attractors in these papers are all self-excited. Some jerk systems and hyperjerk systems with multistability and chaotic dynamics have been found: self-excited chaos [63,70,71], hidden chaos [56,72].
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