A simple chaotic snap circuit based on a single transistor is presented with tunable damping. It suggests, at present, the simplest chaotic snap circuit in the sense that it requires only 9 devices, which offer the minimum number of devices for a chaotic snap circuit. It also suggests the first and simplest circuit realization of either a four-dimensional (4D) chaotic system or a 4th-order (snap) chaotic system that demonstrates a maximized attractor dimension (D L) of a parameter set, or of the entire parameter space of the system, at minimized damping. The tendency of an increase in D L until its peak is illustrated by a decrease in damping. It offers the highest attractor dimension in a category of unit-damping snap chaos. As an initial report, a Clapp oscillator is able to exhibit 4D chaos but does not allow snap chaos. The proposed snap circuit embeds two simple mechanisms: (i) a Clapp oscillator as a simple core engine of oscillations avoiding a need for op-amps, and (ii) a single resistor as a remarkably simple realization of adjustable damping for snap chaos. A current-tunable equilibrium exhibits one of the 4 different types, two of which are of an (unstable) spiral saddle equilibrium, whereas the others are of a spiral stable equilibrium. They reveal the first report on either saddle-equilibrium or stable-equilibrium snap chaos based on a single transistor. Multistability and hidden attractors are demonstrated. The simple circuit offers a novel damping-tunable single-transistorbased approach to such rich dynamics of snap flows through various types of self-excited and hidden attractors. INDEX TERMS Hidden attractor, maximized attractor dimension, minimized damping, multistability, simplest chaotic snap circuit, single transistor, stable equilibrium. BANLUE SRISUCHINWONG received the B.Eng. degree (Hons.) from King Mongkut's Institute of Technology Ladkrabang, Bangkok, Thailand, in 1985, and the M.Sc. and Ph.D. degrees in electronics from The University of Manchester,
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.
Since the invention of Chua’s circuit, numerous generalizations based on the substitution of the nonlinear function have been reported. One of the generalizations is obtained by substituting cubic nonlinearity for piece-wise linear (PWL) nonlinearity. Although hidden chaotic attractors with a PWL nonlinearity have been discovered in the classical Chua’s circuit, chaotic attractors with a smooth cubic nonlinearity have long been known as self-excited attractors. Through a systematically exhaustive computer search, this paper identifies coexisting hidden attractors in the smooth cubic Chua’s circuit. Either self-excited or coexisting hidden attractors are now possible in the smooth cubic Chua’s circuit with algebraically elegant values of both initial points and system parameters. The newly found coexisting attractors exhibit an inversion symmetry. Both initial points and system parameters are equally required to localize hidden attractors. Basins of attraction of individual equilibria are illustrated and clearly show critical areas of multistability where a tiny drift in an initial point potentially induces jumps among different basins of attraction and coexisting states. Such multistability poses potential threats to engineering applications. The dynamical regions of hidden and self-excited attractors, and areas of stability of equilibria, are illustrated against two parameter spaces. Both illustrations reveal that two nonzero equilibrium points of the smooth cubic Chua’s circuit have a transition from unstable to stable equilibrium points, leading to generations of self-excited and hidden attractors simultaneously.
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