Research Article Multistable Systems with Hidden and Self-Excited Scroll Attractors Generated Via Piecewise Linear Systems

Hindawi Complexity Volume 2020, Article ID 7832489, 12 pages https://doi.org/10.1155/2020/7832489 Research Article Multistable Systems with Hidden and Self-Excited Scroll Attractors Generated via Piecewise Linear Systems R. J. Escalante-Gonza´lez and Eric Campos Division of Applied Mathematics, Institute for Scientific and Technological Research of San Luis Potos´ı, Camino a la Presa San José 2055, Lomas 4 Sección 78216, San Luis Potos´ı, Mexico Correspondence should be addressed to Eric Campos; [email protected] Received 4 January 2020; Revised 26 February 2020; Accepted 23 March 2020; Published 14 April 2020 Guest Editor: Sundarapandian Vaidyanathan Copyright © 2020 R. J. Escalante-González and Eric Campos. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. ,e 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. ,e 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. 1. Introduction [7–19]. ,ere are some works focused on control systems with hidden attractors, as in references [20, 21]. Most works Piecewise linear systems that display scroll attractors have related to multiscroll attractors are based on the first class. been studied since the publication of the well-known Chua’s Multiscroll attractors have been reported in circuit. ,e attractor exhibited by Chua’s circuit is an ex- [12, 13, 18, 19, 22–26]. In [27], a system with a multiscroll ample of chaotic attractor whose chaotic nature has been chaotic sea was introduced. explained through the Shilnikov method. Some works have Multistability can be considered undesirable for some extended this system in order to obtain a greater number of applications, so some works focus on how to avoid this scrolls or different geometries. According to [1], an attractor behavior. For example, in [28], a method that allows to with three or more scrolls in the attractor is considered a transform a periodic or chaotic multistable system into a multiscroll attractor. Recently in [2], the generation of scroll monostable was studied, and some experiments were carried attractors via multistable systems have been observed. out with a fiber laser doped with erbium. However, for some According to [3], there are two classes of attractors, one applications it may be considered desirable to be able to of them is a class called self-excited attractors that includes switch from monostable to bistable behavior, for example, in all the attractors excited by unstable equilibria, i.e., the basin [29], a parameterized method to design multivibrator cir- of attraction intersects with an arbitrarily small open cuits with stable, monostable, and bistable regimes was neighborhood of equilibria [4]. Examples of this class are the proposed. well-known Lorenz attractor [5] and the scroll attractor of Some works deal with multistable systems with infinite Chua’s circuit [6]. ,e other class is called hidden attractors number of equilibria and their electronic realization [30, 31]. and their basins of attractions do not contain neighborhoods A study on the widening of the basin of attraction of a of equilibria. Some hidden attractors have been studied in class of piecewise linear (PWL) systems was recently 2 Complexity performed in [2]. In this work, a bifurcation from a bistable corresponding eigenvectors v2 and v3, respectively. ,e ei- system with two self-excited double-scroll attractors to a genvectors are given by multistable system with two self-excited attractors and one 1 hidden attractor was reported. Other study on the emer- B0 C1 B C gence of hidden double-scroll attractors in a class of PWL B C B C systems is reported in [32]. v �B 0 C; 1 B C Based on the observations made in previous works, a B C B C question of whether or not it is possible to generate a hidden @ 1 A multiscroll attractor with scrolls along more than one di- 2 rection emerges. Depending on the number of directions in which the scrolls in the attractor extend, they are usually 0 B0 C1 referred to as one-directional (1D), two-directional (2D), B C (3) B C and three-directional (3D) grid scroll attractors. v2 �B − 1 C; @B AC Here, we introduce an approach for the construction of multistable PWL systems that exhibit hidden multiscroll 0 attractors with 1D, 2D, and 3D grid arrangements. In Section 2, a system with a chaotic double-scroll self-excited − 1 B0 C1 B C attractor is introduced. In Section 3, the construction is B C v3 �B 0 C: extended to 1D grid scroll self-excited attractor; then, the @B AC equilibria are separated by pairs to generate multistable systems with hidden and self-excited attractors. In Section 4, 1 the construction is generalized to 1D grid scroll hidden T 3 B � (β1; β2; β3) ∈ R is a constant vector, and attractor. In Section 5, the construction is generalized to 2D f : X ⟶ R is a functional such that f(x)B is a constant and 3D grid scroll hidden attractors; in Section 6, conclu- vector in each atom Pi, and there exists an equilibrium point sions are given. ∗ ∗ ∗ ∗ T − 1 xeq � (x ; x ; x ) � − f(x)A B, with i � 1; ... ; η, in i 1eqi 2eqi 3eqi each atom Pi. ,us, in each atom Pi there exists a saddle equilibrium point with a local stable manifold of dimension one 2. Heteroclinic Chaos s ∗ given by W ∗ � nx + x : x ∈ span�v �o. A two-dimensional x eqi 1 3 eqi u Let P � nP1; ... ;Pη o(η > 1) be a finite partition of X ⊂ R , local unstable manifold is given by W ∗ � xeqi that is, X � ∪ P , and P ∩ P � ∅ for i ≠ j. ,e approach nx + x ∗ : x ∈ span�v ; v �o. 1≤i≤η i i j eqi 2 3 to generate hidden attractors is based on the existence of We begin to explain the generation of chaotic self-excited attractors; thus, the class of systems considered attractors by first considering a partition with two 3 in this work are those that present a saddle equilibrium point atoms P � �P1;P2 � and the constant vector B ∈ R given in each element of the partition P that is called an atom. by We denote the closure of a set P as cl(P ). For each pair i i a 2c of adjacent atoms Pi and Pj, i ≠ j, SWi;j � cl(Pi) ∩ cl(Pj) is B − − C B0 3 3 C1 the switching surface. B C B C Consider a dynamical system T : X ⟶ X whose dy- B C B C namic is given by B b C B �B C; (4) B C B 3 C x_ � Ax + f(x)B; (1) B C B C @B a c AC T 3 where x � (x1; x2; x3) ∈ R is the state vector and − 3×3 3 3 A � nαij o ∈ R is a linear operator whose matrix is as follows: and the functional f given by a 2c 2c 2a − α; x ∈ P1; + b − f(x) �( (5) B0 3 3 3 3 C1 B C α; x ∈ P2; B C B C B C R B b 2b C with 0 < α ∈ . A � B − a C; (2) B C Please note that the vector − B is the first column of the B 3 3 C B C linear operator A, and thus, system (1) can be rewritten as B C T B C x_ � A(x − f(x); x ; x ) . ,en, the functional f(x) de- @ c a 2a c A 1 2 3 − − b + termines the location of the equilibria along the x1-axis. So, 3 3 3 3 ∗ ∗ ∗ ∗ T xeq � (x ; x ; x ) ∈ Pi, i � 1; 2. i 1eqi 2eqi 3eqi where a; b ∈ R+, and c ∈ R− . ,us, the linear operator A has a negative real eigenvalue λ1 � c with the corresponding Proposition 1. If the PWL system (1) is given by (2), (4), and eigenvector v1 and a pair of complex conjugate eigenvalues (5), then the functional f(x) determines the location of the with positive real part, λ2 � a + ib and λ3 � a − ib, with the equilibria along the x1-axis. Complexity 3 Proof. Let A � [A1;A2;A3] be the linear operator so that Since A ≠ 0, it follows that each A , i � 1; 2; 3, is a column vector. Since B � − A , then ∗ T : i 1 x �(f(x); 0; 0) : (9) x � Ax + f(x)B can be rewritten as eq x_ � x1 − f(x) �A1 + x2A2 + x3A3; (6) ,us, the equilibria is determined by f(x) given by (5) along the x1-axis. and then According to f(x), the first components of the equilibria T fulfill that x∗ < x∗ . x_ � A x − f(x); x ; x � : (7) 1eq 1eq 1 2 3 ,e switching1 plane2 SW has associated an equation b b b T b Now, in order to find the equilibria, we equate the vector Ax1 + Bx2 + Cx3 + D � N · x + D � 0, with A > 0, and b b b field to zero: N � (A; B; C) is the normal vector.

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