Hyperchaotic attractors through coupling of systems without equilibria

Abstract

Systems without equilibria with chaotic flows have been the focus of recent works. Since there are no equilibria to start a local analysis the study of this class of systems is still a challenging task. Some systems have already been found by means of numerical searches that they consider different classes of three-dimensional nonlinear ordinary differential equations, usually with quadratic nonlinearities. Some works present construction approaches for the generation of chaotic attractors via piecewise linear systems (PWL) without equilibria in ℝ³. There are few works on the generation of chaotic attractors through systems without equilibria with differentiable nonlinearities or mechanisms to generate higher dimensional systems with chaotic or hyperchaotic attractors. Here we report a class of systems without equilibria which exhibit a scroll attractor and whose vector field is differentiable. The system construction presents great flexibility for the selection of the number of scrolls exhibited by the attractor. We also report a special coupling for this class of systems which allows the coupling without introducing new equilibria in the system. The coupling is illustrated with a nine-dimensional system which was numerically studied through Lyapunov exponents, Kaplan-Yorke dimension and Poincaré maps. The proposed coupling approach presents flexibility that can be further studied.