Dynamic analysis of a novel jerk system with composite tanh-cubic nonlinearity: chaos, multi-scroll, and multiple coexisting attractors

Abstract

It is well known that the dynamics of very simple physical systems can be quite complex if a sufficient amount of nonlinearity is present. In this contribution, a novel jerk system with smooth composite tanh-cubic nonlinearity is proposed and investigated. Interestingly, the new nonlinearity takes advantage of the classical smooth cubic polynomial in the sense that it induces more complex and interesting dynamics (e.g. five equilibria instead of three in the case of a (traditional) cubic nonlinearity, multi-scroll, and multistability). The fundamental properties of the model are discussed including equilibria and stability, phase portraits, Poincare sections, bifurcation diagrams and Lyapunov exponent’s plots. Period doubling bifurcation, antimonotonicity, chaos, hysteresis, and coexisting bifurcations are reported. In particular, a rare phenomenon is found in which two different pairs of coexisting limit cycles born from the Hopf bifurcation follow each a different sequence of period-doubling bifurcations, then merge to form a three-scroll chaotic attractor as a parameter is smoothly changed. As another major result of this work, several windows in the parameter space are depicted in which the novel jerk system develops the striking behaviour of multiple coexisting attractors (i.e. coexistence of three, four, six, or eight disjoint periodic and chaotic attractors for the same parameters set) due to the combined effects of three families of parallel bifurcation branches and hysteresis. An electronic analogue of the new jerk system is designed and simulated in PSPICE. As far as the authors’ knowledge goes, no example of such a simple and ‘elegant’ 3D autonomous system with this combination of features is reported in the relevant literature.