Nonlinear behavior of a novel chaotic jerk system: antimonotonicity, crises, and multiple coexisting attractors

Abstract

In this contribution, a novel Jerk system with a smooth piecewise quadratic nonlinearity is introduced. The new nonlinearity provides a similar smoothness as the cubic polynomial function, but a faster response and a simpler circuitry. The basic dynamical properties of the model are discussed in terms of its parameters by using standard nonlinear analysis tools including phase space trajectory plots, frequency spectra, bifurcation diagrams and Lyapunov exponent plots. The bifurcation analysis yields very rich and interesting scenarios such as period-doubling bifurcations, antimonotonicity (i.e. the concurrent creation and annihilation of periodic orbits), periodic windows, and symmetry recovering crises. One of the main findings of this work is the presence of a window in the parameter space in which the novel jerk system experiences the unusual and striking feature of multiple coexisting attractors (i.e. coexistence of four or six disconnected periodic and chaotic attractors) for the same parameters’ setting. Correspondingly, basins of attraction of various competing attractors display extremely complex basin boundaries. Compared to some lower dimensional systems (e.g. Leipnik–Newton system, modified Sprott B system) capable of displaying such type of behavior reported to date, the jerk system introduced in this work represents the simplest and the most ‘elegant’ paradigm. An electronic circuit for allowing an illustration of the theoretical model is proposed and implemented in PSpice. The results obtained in this work let us conjecture that there exist some regions in its parameter space (that need to be uncovered) in which the universal Chua’s circuit experiences six disconnected non static attractors similar to those presented in this work.