Attractors generated from switching unstable dissipative systems

Abstract

In this paper, we present a class of 3-D unstable dissipative systems, which are stable in two components but unstable in the other one. This class of systems is motivated by whirls, comprised of switching subsystems, which yield strange attractors from the combination of two unstable "one-spiral" trajectories by means of a switching rule. Each one of these trajectories moves around two hyperbolic saddle equilibrium points. Both theoretical and numerical results are provided for verification and demonstration.

A novel approach to generate attractors with a high number of scrolls

“…The switched nonlinear systems are mostly associated to the generation of chaotic behaviors with the presence of multiple scrolls in their phase spaces [1,2,3]. The study of this kind of systems has presented a great interest in the last three decades of scientific development, due to the endless number of possible applications that these systems can have in different areas of science.…”

Stability and Multiscroll Attractors of Control Systems via the Abscissa

“…Thence, the choice of 's in the definition of the step function S will determine the commutation regions D 's that enclose each equilibrium X * . The commuting system given by (29) induces in phase space R the flow ( ), ∈ R, such that each forward trajectory of the initial point X 0 = X( = 0) is the set {X( ) = (X 0 ) : ≥ 0}. Furthermore, these systems have a dissipative bounded region Ω ⊂ R named basin of attraction, such that the flow (Ω) ⊂ Ω for every ≥ 0.…”

“…, and ≥ 2. We say that system (29) can generate multiscroll attractors with the minimum of equilibrium points, if for any initial condition 0 ∈ B ⊂ R in the basin of attraction the orbit ( 0 ) generates an attractor A ⊂ R with oscillations around each X * .…”

“…. , , and each entry of the switching system is considered in order to preserve bounded trajectories of system (29). Thence, the choice of 's in the definition of the step function S will determine the commutation regions D 's that enclose each equilibrium X * .…”

“…One of the different chaotic behaviors is the presence of multiscroll attractor. Good references where the generation of multiscrolls has been studied are the works [28][29][30][31][32][33][34][35]. In this paper we use the abscissa of stability of Hurwitz polynomials to study the stability of systems in order to generate multiscroll attractors.…”