On the formation of nested invariant tori and hidden chaotic attractors in the Sprott A system [ Back ]

Date:
27.02.17   
Times:
15:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Marcelo Messias
University:
Universidade Estadual Paulista

Abstract

In this talk, we consider the well-known Sprott A system, which depends on a single real parameter $a$ and, for $a=1$, it was shown to present a hidden chaotic attractor. We investigate the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter $a$. We prove that, for $a=0$ the Sprott A system has a line of equilibria in the $z$--axis, the phase space is foliated by concentric invariant spheres with two equilibrium points and each one of these spheres is filled by heteroclinic orbits. For $a\neq 0$, the spheres are no longer invariant algebraic surfaces of the system and the heteroclinic orbits are destroyed. We perform a detailed numerical study for $a>0$ small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the $\alpha$-- and $\omega$--limit sets of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the value of parameter $a$ increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for $a<1$. Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincaré compactification, showing that the only orbit which escapes to infinity is the one contained in the $z$--axis, and the other orbits are homoclinics to the limit set or to the hidden chaotic attractor, depending on the parameter value.

Joint work with Alisson C. Reinol