Multistability and chaotic dynamics of a simple Jerk system with a smoothly tuneable symmetry and nonlinearity

Abstract

A novel autonomous chaotic Jerk system with a nonlinearity in the form $$ \phi_{k} \left( x \right) = 0.5\left( {\exp \left( {kx} \right) - \exp \left( { - x} \right)} \right) $$ which for $$ k = 1 $$ reduces to the hyperbolic sine is proposed. Two general purpose semiconductor diodes connected in antiparallel are utilized to synthesize the tuneable nonlinearity. The new system presents three rest points among which two unstable ones. Thus, the dynamics is organized around the zero equilibrium point. Correspondingly, the model develops only mono-scroll strange attractors. The numerical analysis of the system reveals some interesting phenomena such as period doubling cascades to chaos, period-doubling reversals, periodic windows, hysteresis, and coexisting bifurcations as well. The presence of parallel bifurcation branches justifies the occurrence of multiple coexisting attractors in some ranges of parameters. Several basins of attraction with extremely complex structures are provided to illustrate the magnetization of the state space due to the presence of numerous competing attractors. The practical implication of this phenomenon is that it might be very difficult to predict the dynamical state of the system. Multistability in the symmetry boundary is studied by using $$ k $$ as control parameter. A very good agreement is observed between practical experiments and the theoretical predictions.