What is the simplest dissipative chaotic jerk equation which is parity invariant?

Abstract

We investigate the chaotic dynamics of an autonomous scalar third-order differential equation. This system seems to be the algebraically simplest and previously unknown example of a dissipative chaotic jerky flow which is parity invariant. It displays chaotic behaviours in two distinct ranges of its control parameter. Deterministic chaos is principally observed from a symmetric limit cycle which after a symmetry-breaking bifurcation gives rise to two cascades of flip bifurcation. Then two coexisting asymmetric chaotic attractors are observed, and after a symmetry-restoring crisis a symmetric chaotic attractor is created. Chaotic attractors also coexist in another very narrow range of control parameter as results of two period doubling cascades of bifurcation from a pair of mutually symmetric coexisting limit cycles.