A plethora of coexisting strange attractors in a simple jerk system with hyperbolic tangent nonlinearity

Abstract

In the present contribution, the dynamics of a simple autonomous jerk system with hyperbolic tangent nonlinearity is considered. The system consists of a linear transformation of Model MO13 previously introduced in [Sprott, 2010]. The form of nonlinearity is interesting in the sense that with the variation of a control parameter, saturation may be approached gradually obeying hyperbolic tangent function, as in the case of magnetization in ferromagnetic system, non ideal op. amplifier, solar-wind-driven magnetosphere-ionosphere system, and activation function in neural network. The fundamental properties of the model are discussed including equilibria and stability, phase portraits, Poincaré sections, bifurcation diagrams and Lyapunov exponents’ spectrum. Period doubling bifurcation, antimonotonicity (i.e. concurrent creation an annihilation of periodic orbits), chaos, hysteresis, and coexisting bifurcations are reported. As a major outcome of this paper, a window in the parameter space is revealed in which the jerk system experiences the unusual phenomenon of multiple coexisting attractors (i.e. coexistence of two, four or six disconnected periodic and chaotic self excited attractors) resulting from the simultaneous presence of three families of parallel bifurcation branches and hysteresis. To the best of the authors’ knowledge, no example of such a simple and ‘elegant’ 3D autonomous system capable of six different strange attractors is reported in the relevant literature. Some PSpice simulations based on a physical implementation of the system are carried out to support the theoretical analysis.