Dynamics, control and symmetry-breaking aspects of a new chaotic Jerk system and its circuit implementation

Abstract

The study of chaotic systems with symmetry is very well documented. However, very little is known unfortunately about the dynamics of such systems when their symmetry is perturbed or broken. This paper introduces a new chaotic jerk system with quadratic-cubic nonlinearity $$\varphi _{k} \left( x \right) =-x+kx^{2}+x^{3}$$ where k denotes a symmetry perturbation parameter. We consider the mechanism of chaos generation both for the symmetric ($$k=0$$) and the asymmetric ($$k\ne 0$$) modes of operation. The model has three fixed points one of which is located at the origin of the state space. Oscillations are excited from the origin, giving rise to a mono-scroll chaotic attractor. We demonstrate that the presence of the quadratic terms breaks the odd symmetry of the model and engenders a plethora of new nonlinear dynamics patterns such as critical transitions, coexisting asymmetric bubbles of bifurcation, coexisting asymmetric attractors, and parallel bifurcation branches. These interesting features are rarely reported, at least in a 3D autonomous system having the simplicity of the one studied in this work. By exploiting the method of linear augmentation, a simple control strategy is developed that help to convert the multistable system from the state of six coexisting attractors to a monostable state when simply adjusting the coupling parameter. Experimental results based on an appropriate electronic realization of the system are consistent with the theoretical investigations.