Chaotic Behavior of a Generalized Sprott E Differential System

Abstract

A chaotic system with only one equilibrium, a stable node-focus, was introduced by Wang and Chen [2012]. This system was found by adding a nonzero constant [Formula: see text] to the Sprott E system [Sprott, 1994]. The coexistence of three types of attractors in this autonomous system was also considered by Braga and Mello [2013]. Adding a second parameter to the Sprott E differential system, we get the autonomous system [Formula: see text] where [Formula: see text] are parameters and [Formula: see text]. In this paper, we consider theoretically some global dynamical aspects of this system called here the generalized Sprott E differential system. This polynomial differential system is relevant because it is the first polynomial differential system in [Formula: see text] with two parameters exhibiting, besides the point attractor and chaotic attractor, coexisting stable limit cycles, demonstrating that this system is truly complicated and interesting. More precisely, we show that for [Formula: see text] sufficiently small this system can exhibit two limit cycles emerging from the classical Hopf bifurcation at the equilibrium point [Formula: see text]. We also give a complete description of its dynamics on the Poincaré sphere at infinity by using the Poincaré compactification of a polynomial vector field in [Formula: see text], and we show that it has no first integrals in the class of Darboux functions.