Abstract
Symmetry is an important property found in a large number of nonlinear systems. The study of chaotic systems with symmetry is well documented. However, the literature is unfortunately very poor concerning the dynamics of such systems when their symmetry is altered or broken. In this paper, we investigate the dynamics of a simple jerk system with hyperbolic tangent nonlinearity (Kengne et al., Chaos Solitons, and Fractals, 2017) whose symmetry is broken by adding a constant term modeling an external excitation force. We demonstrate that the modified system experiences several unusual and striking nonlinear phenomena including coexisting bifurcation branches, hysteretic dynamics, coexisting asymmetric bubbles, critical transitions, and multiple (i.e., up to six) coexisting asymmetric attractors for some suitable ranges of system parameters. These features are highlighted by exploiting common nonlinear analysis tools such as graphs of largest Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The control of multistability is investigated by using the method of linear augmentation. We demonstrate that the multistable system can be converted to a monostable state by smoothly adjusting the coupling parameter. The theoretical results are confirmed by performing a series of PSpice simulations based on an electronic analogue of the system.
Highlights
A particular attention has been paid to the study of nonlinear and chaotic dynamic systems. is is due to the rapid development of increasingly powerful computers on the one hand and on the other hand to the many potential applications in several fields of science and engineering.ese systems are capable of several forms of complexity such as chaos, hyperchaos, multirhythmicity, bifurcations, intermittency, hysteresis, and multistability [1,2,3]
To the best of the authors’ knowledge, the literature is very poor concerning the behavior of these systems when their symmetry is altered or broken. e symmetry break purposefully induced in a nonlinear dynamical system may be adjusted to discover many complex nonlinear phenomena as previously discussed in several nonlinear systems [11,12,13,14,15,16,17]
Motivated by previous results on jerk dynamical systems, this paper focuses on the effects engendered by symmetry break in a simple autonomous jerk system with hyperbolic tangent nonlinearity previously analyzed in [18]. us, the novel chaotic flow is smoothly tuned to behave either symmetrically or to develop no symmetry property using a single parameter
Summary
A particular attention has been paid to the study of nonlinear and chaotic dynamic systems. is is due to the rapid development of increasingly powerful computers on the one hand and on the other hand to the many potential applications in several fields of science and engineering. Ese systems are capable of several forms of complexity such as chaos, hyperchaos, multirhythmicity, bifurcations, intermittency, hysteresis, and multistability [1,2,3] Concerning the latter feature, it should be noted that a multistable dynamic system is capable of displaying two or more attractors for the same set of parameters. Multistability in simple jerk dynamic systems has recently drained tremendous research interest in varied fields of science and technology resulting in several publications. On this line, Kengne and colleagues reported the coexistence of four self-excited mutually symmetric attractors in a jerk system possessing a cubic nonlinearity [23] based on both numerical and experimental methods.
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