Symmetry Breaking, Coexisting Bubbles, Multistability, and Its Control for a Simple Jerk System with Hyperbolic Tangent Nonlinearity

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.