Abstract
Background: Since the invention of Chua’s circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua’s circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system’s coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua’s circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.
Highlights
Multistability of a dynamical system is usually taken to mean that there are coexisting attractors, each with a basin of attraction that depends crucially on the system’s initial values
In the present paper we investigated the coexisting bifurcations and coexisting attractors in a Chua’s system with adjustable symmetry and adjustable nonlinearity
The system is solved under three different initial conditions (a) (±1, 0, 0) ; (b) (20, 0, 0) leading to coexisting bifurcations (Figure 2)
Summary
Multistability of a dynamical system is usually taken to mean that there are coexisting attractors, each with a basin of attraction that depends crucially on the system’s initial values. Cheng et al reported a memristive bridge-based canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive bridge diode [29] They reported the complex nonlinear phenomena of coexisting bifurcations and coexisting attractors though the number of coexisting attractors depend crucially on the symmetry of the system. Symmetry always plays an important role in physical system This property is found in a variety of system including nonlinear and chaotic systems [30,31,32,33,34,35,36,37,38,39,40]. The electronic analog circuit corresponding to the system under investigation is very simple This is due to the fact that no analog multiplier is used to build the quadratic and the cubic polynomials
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