Generation of Multiscroll Attractors by Controlling the Equilibria ⋆ L. J. Onta˜ n´ on-Garc´ ıa * E. Jim´ enez-L´ opez ** E. Campos-Cant´ on *** * Instituto de Investigaci´ on en Comunicaci´ on ´ Optica, Universidad Aut´ onoma de San Luis Potos´ ı, ´ Alvaro Obreg´ on 64, Col. Centro CP. 78000 San Luis Potos´ ı, S.L.P., M´ exico.(e-mail: luisjavier.ontanon@gmail.com) ** Departamento de F´ ısico Matem´ aticas, Universidad Aut´ onoma de San Luis Potos´ ı, ´ Alvaro Obreg´ on 64, Col. Centro CP. 78000 San Luis Potos´ ı, S.L.P., M´ exico. (e-mail: jimeno@cactus.iico.uaslp.mx) *** Divisi´ on de Matem´ aticas Aplicadas, Instituto Potosino de Investigaci´ on Cient´ ıfica y Tecnol´ ogica, A.C. Camino a la Presa San Jos´ e 2055, Col. Lomas 4 secci´ on CP. 78216, San Luis Potos´ ı, S.L.P., M´ exico.(e-mail: eric.campos@ipicyt.edu.mx) Abstract: This work shows the generation of multi-scroll attractors in R 3 by controlling the equilibrium point of an unstable dissipative system. The switching control signal that governs the position of the equilibrium point changes according to the number of scrolls that is displayed in the attractor. Thus, if two systems display a different number of scrolls they have different control signals. The analysis of their Lyapunov exponents along with some bifurcation diagrams are presented. The possibility of hyper-chaos in R 3 is considered. Keywords: Chaos theory; Switching functions; System design; Piecewise linear analysis; Multi-scroll Attractors. 1. INTRODUCTION Switched systems have acquired a great deal of attention recently and they have been considered for a wide range of applications mostly in electrical engineering. These systems consist on a set of subsystems and a switching control signal which is activated or fixed at some values through some intervals of time. Among all the uses they may present, the generation of multi-scrolls and chaos has been of great interest for the scientific community. Chaos has been an extremely studied area in last decades. One of the most remarkable developments is that simpler nonlinear deterministic equations can have unpredictable (chaotic) long-term solution. Despite of the fact that there is no unique definition of chaos that all the international scientific community may adopt, there are several basis and theorems that we can seize in order to characterize the behavior of any system throughout nature. Characterization of dynamical behavior can be achieved by means of the Lyapunov exponents (LE). With the aid of their diagnostic, one can measure the average exponential rates of divergence or convergence of nearby orbits in the phase space, overall with their signs, a qualitative picture of the variety of dynamics that the systems may exhibit, ranging from fixed points via limit cycles and tori to more complex chaotic and hyperchaotic attractors. ⋆ This works was sponsored by the Doctoral scholarship of Conacyt. Whereas chaos can arise in discrete-time systems with only a single variable (which must be positive), at least three variables are required for chaos in continuous-time systems (Hirsch & Smale, 1974). Such systems are characterized by one positive (LE) in the Lyapunov spectrum. The behaviors described previously in Wolf, Swift, Swinney & Vastano (1985) can be defined with the sign of their LE as follows: • In the presence of one positive LE, one negative and, one zero (+, 0, -), the resulting attractor is “strange” or “chaotic”. • With a negative LE , and two zero (-, 0, 0), the attractor is a two-torus. • With a zero LE, and two negatives (0, -, -), the attractor is a limit cycle. • With three negative LE’s (-, -, -), is a fixed point. A natural question is the following: Is there any system with the sign of their LE (+, +, -) in R 3 ? A zero Lyapunov exponent indicates that the system is in some sort of steady state mode (Haken, 1983). A physical system with this exponent is conservative, so it is possible to construct a system that always presents stretching and folding. However, to obtain hyper-chaos, the system must be characterized by the presence of two or more positive LE’s. The reason is that the trajectory has to be nonperiodic and bounded to some finite region, and yet it cannot intersect itself because every point has a unique direction of the flow. Hyperchaos in R 4 has been reported 2012 IFAC Conference on Analysis and Control of Chaotic Systems The International Federation of Automatic Control June 20-22, 2012. Cancún, México 978-3-902823-02-1/12/$20.00 © 2012 IFAC 111 10.3182/20120620-3-MX-3012.00024