Chaotic Behavior of a Generalized Sprott E Differential System

Author(s)

Oliveira, Regilene ^[1] ; Valls, Claudia ^[2]

Affiliation

^[1] Univ Sao Paulo, ICMC, Dept Matemat, Sao Paulo - Brazil

^[2] Univ Lisbon, Inst Super Tecn, Dept Matemat, P-1049001 Lisbon - Portugal

• Document type: Journal article
• Source: INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS; v. 26, n. 5 MAY 2016.
• Web of Science Citations: 0

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 b 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 (x) over dot = ayz + b, (y) over dot = x(2) - y, (z) over dot = 1 - 4x, where a, b epsilon R are parameters and a not equal 0. 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 R-3 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 b sufficiently small this system can exhibit two limit cycles emerging from the classical Hopf bifurcation at the equilibrium point p = (1/4, 1/16, 0). We also give a complete description of its dynamics on the Poincare sphere at infinity by using the Poincare compactification of a polynomial vector field in R-3, and we show that it has no first integrals in the class of Darboux functions. (AU)

• FAPESP's process: 14/00304-2 - Singularities of differentiable mappings: theory and applications
• Grantee: Maria Aparecida Soares Ruas
• Support Opportunities: Research Projects - Thematic Grants