Scenario to chaos and multistability in a modified Coullet system: effects of broken symmetry

Abstract

Recently, the study of nonlinear systems with multiple attractors has drained the attention of researchers worldwide owing to its fundamental and technological relevance. In this work, we consider a modified Coullet system with a cubic polynomial nonlinearity in the form $$ \varphi_{m} \left( x \right) = x - mx^{2} - x^{3} $$ where $$ m $$ is a control parameter. The new system bridges the gap between chaotic systems with symmetry and those without symmetry. In fact for $$ m = 0 $$ , the system displays a perfect symmetry which is reflected on the location of equilibria, the type of attractors, and the shape of basins of attraction as well. In this special case, multistability involves a pair of mutually symmetric periodic or chaotic attractors, a pair of chaotic attractors with a pair of limit cycles, two pairs of limit cycles with a pair of chaotic attractors, and so on. For $$ m \ne 0 $$ , the system is non symmetric and more complex nonlinear phenomena are observed such as parallel bifurcation branches, coexisting multiple asymmetric attractors, and hysteresis. For instance, the coexistence of a stable fixed point with a limit cycle, two different non symmetric limit cycles or chaotic attractors, three or four different non symmetric periodic and chaotic attractors are reported when monitoring the system parameters and initial conditions as well. Basins of attraction with non-trivial basin boundaries structures are computed to visualize the magnetization of the state space in the presence of multiple attractors. Experimental results based on a suitable electronic implementation of the proposed jerk system are included.