Abstract
In this work, we present an approach to design a multistable system with one-directional (1D), two-directional (2D), and three-directional (3D) hidden multiscroll attractor by defining a vector field on ℝ3 with an even number of equilibria. The design of multistable systems with hidden attractors remains a challenging task. Current design approaches are not as flexible as those that focus on self-excited attractors. To facilitate a design of hidden multiscroll attractors, we propose an approach that is based on the existence of self-excited double-scroll attractors and switching surfaces whose relationship with the local manifolds associated to the equilibria lead to the appearance of the hidden attractor. The multistable systems produced by the approach could be explored for potential applications in cryptography, since the number of attractors can be increased by design in multiple directions while preserving the hidden attractor allowing a bigger key space.
Highlights
Piecewise linear systems that display scroll attractors have been studied since the publication of the well-known Chua’s circuit. e attractor exhibited by Chua’s circuit is an example of chaotic attractor whose chaotic nature has been explained through the Shilnikov method
It was found that the separation between self-excited double-scroll attractors and the switching plane between these self-exited attractors lead to the emergence of a hidden attractor
A generalized construction was proposed for the generation of multistable systems with self-excited double-scroll chaotic attractors and a hidden multiscroll/grid attractor. e coexistence of selfexcited attractors and a hidden attractor is presented via piecewise linear (PWL) systems and the approach considers for each scroll in the hidden attractors a self-excited attractor inside the scroll
Summary
Piecewise linear systems that display scroll attractors have been studied since the publication of the well-known Chua’s circuit. e attractor exhibited by Chua’s circuit is an example of chaotic attractor whose chaotic nature has been explained through the Shilnikov method. According to [3], there are two classes of attractors, one of them is a class called self-excited attractors that includes all the attractors excited by unstable equilibria, i.e., the basin of attraction intersects with an arbitrarily small open neighborhood of equilibria [4] Examples of this class are the well-known Lorenz attractor [5] and the scroll attractor of Chua’s circuit [6]. We introduce an approach for the construction of multistable PWL systems that exhibit hidden multiscroll attractors with 1D, 2D, and 3D grid arrangements.
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