Multistable Systems with Hidden and Self-Excited Scroll Attractors Generated via Piecewise Linear Systems

In this work, we present an approach to design a multistable system with one-directional (1D), two-directional (2D), and threedirectional (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.


Introduction
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. Some works have extended this system in order to obtain a greater number of scrolls or different geometries. According to [1], an attractor with three or more scrolls in the attractor is considered a multiscroll attractor. Recently in [2], the generation of scroll attractors via multistable systems have been observed.
Multistability can be considered undesirable for some applications, so some works focus on how to avoid this behavior. For example, in [28], a method that allows to transform a periodic or chaotic multistable system into a monostable was studied, and some experiments were carried out with a fiber laser doped with erbium. However, for some applications it may be considered desirable to be able to switch from monostable to bistable behavior, for example, in [29], a parameterized method to design multivibrator circuits with stable, monostable, and bistable regimes was proposed.
Some works deal with multistable systems with infinite number of equilibria and their electronic realization [30,31].
A study on the widening of the basin of attraction of a class of piecewise linear (PWL) systems was recently performed in [2]. In this work, a bifurcation from a bistable system with two self-excited double-scroll attractors to a multistable system with two self-excited attractors and one hidden attractor was reported. Other study on the emergence of hidden double-scroll attractors in a class of PWL systems is reported in [32].
Based on the observations made in previous works, a question of whether or not it is possible to generate a hidden multiscroll attractor with scrolls along more than one direction emerges. Depending on the number of directions in which the scrolls in the attractor extend, they are usually referred to as one-directional (1D), two-directional (2D), and three-directional (3D) grid scroll attractors.
Here, we introduce an approach for the construction of multistable PWL systems that exhibit hidden multiscroll attractors with 1D, 2D, and 3D grid arrangements. In Section 2, a system with a chaotic double-scroll self-excited attractor is introduced. In Section 3, the construction is extended to 1D grid scroll self-excited attractor; then, the equilibria are separated by pairs to generate multistable systems with hidden and self-excited attractors. In Section 4, the construction is generalized to 1D grid scroll hidden attractor. In Section 5, the construction is generalized to 2D and 3D grid scroll hidden attractors; in Section 6, conclusions are given.

Heteroclinic Chaos
Let P � P 1 , . . . , P η (η > 1) be a finite partition of X ⊂ R 3 , that is, X � ∪ 1≤i≤η P i , and P i ∩ P j � ∅ for i ≠ j. e approach to generate hidden attractors is based on the existence of self-excited attractors; thus, the class of systems considered in this work are those that present a saddle equilibrium point in each element of the partition P that is called an atom.
We denote the closure of a set P i as cl(P i ). For each pair of adjacent atoms P i and P j , i ≠ j, SW i,j � cl(P i ) ∩ cl(P j ) is the switching surface. Consider a dynamical system T : X ⟶ X whose dynamic is given by

is the state vector and
A � α ij ∈ R 3×3 is a linear operator whose matrix is as follows: where a, b ∈ R + , and c ∈ R − . us, the linear operator A has a negative real eigenvalue λ 1 � c with the corresponding eigenvector v 1 and a pair of complex conjugate eigenvalues with positive real part, λ 2 � a + ib and λ 3 � a − ib, with the corresponding eigenvectors v 2 and v 3 , respectively. e eigenvectors are given by B � (β 1 , β 2 , β 3 ) T ∈ R 3 is a constant vector, and f : X ⟶ R is a functional such that f(x)B is a constant vector in each atom P i , and there exists an equilibrium point . . , η, in each atom P i . us, in each atom P i there exists a saddle equilibrium point with a local stable manifold of dimension one given by W s 3 . We begin to explain the generation of chaotic attractors by first considering a partition with two atoms P � P 1 , P 2 and the constant vector B ∈ R 3 given by and the functional f given by with 0 < α ∈ R. Please note that the vector − B is the first column of the linear operator A, and thus, system (1) can be rewritten as en, the functional f(x) determines the location of the equilibria along the x 1 -axis. So, Proposition 1. If the PWL system (1) is given by (2), (4), and (5), then the functional f(x) determines the location of the equilibria along the x 1 -axis.
and then Now, in order to find the equilibria, we equate the vector field to zero: Since A ≠ 0, it follows that us, the equilibria is determined by f(x) given by (5) along the x 1 -axis.
According to f(x), the first components of the equilibria fulfill that x * e switching plane SW has associated an equation en, the atoms P i , i � 1, 2, are defined as follows: □ Assumption 1. e switching plane SW intersects the x 1 -axis at the midpoint between the equilibrium points x * eq 1 and x * eq 2 .
Proof. We want to prove that if a system is given by (1), (2), (4), and (5), then ∃ x * eq 1 ∈ P 1 and ∃ x * eq 2 ∈ P 2 . From (9) and (5), the two equilibrium points are From Assumption 1, the parameter D can be defined as ) is the intersection of SW with the x 1 -axis: us, , the stable and unstable manifolds of the PWL system (1) given by (2), (4), and (5) intersect at two points given by Proof. e equilibrium points are located at x * eq 1 � (− α, 0, 0) T and x * eq 2 � (α, 0, 0) T . In each atom P i , the equilibrium point x * eq i has a local stable manifold (14) of dimension one given by W s span v 1 and a two-dimensional local unstable manifold given by us, the local manifolds are given as follows: W u According to (18), the stable and unstable manifolds, and the intersection points are given by ese points belong to SW. □ Assumption 2. e parameters a and b control the oscillation around the equilibrium point x * eq i , and we consider b/a > 10.

Complexity 3
From (2), the linear operator A can be expressed as where e intersection points x in 1 and x in 2 belong to SW and x in 1 ∈ P 1 and x in 2 � P 2 . Because these points x in 1 and x in 2 belong to the stable manifolds W s , respectively, they are points whose trajectories remain in atoms P 1 and P 2 , respectively.
By definition, x * ; then, the x 2 -axis belongs to the plane SW. e sets cl(W u x * eq i ) ∩ SW, for i � 1, 2, can be written as follows: Consider the following changes of coordinates en, the vector field in z (i) coordinates for the space given by the atom P i is given by the sets given by (22) in z (i) coordinates are given as follows: where 1 -axis that corresponds to the transformation of the intersection points When t < 0, φ(x in 1 , t) leaves the atom P 1 and enters to atom P 2 , the transformation of In a similar way, when t < 0, φ(x in 2 , t) leaves the atom P 2 and enters to atom P 1 , the transformation of x in 2 under With the uncoupled system in z (i) coordinates, we can analyze the flow on the plane z (i) 2 − z (i) 3 and see how the flow converges at the equilibrium point z * (i) eq j when t ⟶ − ∞: If r � r 0 e at .

Multiple Self-Excited Attractors
A study on the widening of the basins of attraction of multistable switching dynamical system with symmetrical equilibria is performed in [2]. In this study, the increment of distance between the equilibria of the self-excited doublescroll attractors increases the basin of attraction of both attractors. e study also reveals that, for the system under study, there is a distance at which a hidden double-scroll attractor emerges.
Based on the idea of generating hidden scroll attractors from sufficiently separated self-excited attractors, we consider now a partition with more atoms P � P 1 , P 2 , P 3 , P 4 along with the PWL system (1), with A and B given by (2) and (4), respectively. For this partition, the functional f(x) is defined in the four atoms as follows:

Complexity
where 0 < α, c ∈ R and c > α. Now, there are four atoms and each atom contains an equilibrium point, e first components of the equilibrium points fulfill that x * , i � 1, 2, 3. ere are three switching planes and each switching plane SW i,i+1 crosses the x 1 -axis and is located between the equilibrium points x * eq i and x * eq i+1 , i � 1, 2, 3. e switching plane SW ij with i, j ∈ 1, . . . , η has associated an equation is the normal vector. en, in order to know if a point x belongs to a P i the following conditions are considered: If x ∈ P k for a k ≥ i and x ∈ P k for a k ≤ i, it follows that x ∈ P i . e equilibria are located on the x 1 -axis at so x * eq 1 ∈ P 1 , x * eq 2 ∈ P 2 , x * eq 3 ∈ P 3 , and x * eq 4 ∈ P 4 .

Assumption 3.
e distance between the self-excited attractors should be big enough to allow the existence of a hidden double-scroll attractor, so we consider (c/α) ≥ 10.
Consider three switching surfaces with the following orientation: Complexity 5 which fulfill that e switching surfaces given in (33) along with the condition given in (34) ensure the existence of the four heteroclinic orbits. Moreover, cl(W u As example, consider the system with parameters a � 0.2, b � 5, c � − 3, α � 1, and c � 10, the system presents two self-excited attractors; however, the hidden double-scroll attractor is not present and a transitory double-scroll oscillation is exhibited instead. Figure 2(a) shows in blue the trajectory for the initial condition x 0 � (0, 0, 0) T for a time t ∈ [0, 150] a.u. (arbitrary units). us, an enough separation of the self-excited double-scroll attractors is not sufficient to produce a hidden double-scroll attractor. e absence of a hidden attractor is related to the switching plane SW 2,3 and how the trajectories of the transitory double-scroll cross it near the stable manifolds. Consider now the following switching surfaces: With the change in SW 2,3 , the intersections cl(W u ∩ SW 2,3 and by symmetry from cl(W u ∩ SW 2,3 allow the trajectories to cross SW 2,3 far from the stable manifolds and allowing the existence of the hidden double-scroll attractor. As example, consider the system given by (1), (2), (4), and (30) with a � 0.2, b � 5, c � − 3, α � 1, and c � 10 and the new switching surfaces given by (35). Figure 2(b) shows in blue the trajectory for the initial condition x 0 � (0, 0, 0) T for a time t ∈ [50000, 51000] a.u. e trajectory reaches the hidden doublescroll instead of converging to one of the self-excited attractors. A trajectory has been simulated for a time t > 1000000 to verify that the double-scroll oscillation is not a transitory behavior and that the trajectory does not converge to a self-excited attractor. e construction can be further extended to the number of scrolls desired just by adding two atoms for each scroll; for instance, for a triple scroll attractor the partition is P � P 1 , . . . , P 6 } and a possible set of switching surfaces is given by en, the equilibria are located at the x 1 -axis with

Generalization
e number of scrolls for a hidden scroll attractor exhibited by the system from the previous section depends on the number of self-excited attractors. In order to simplify the description for a large number of scrolls, a generalization is introduced in this section. Consider a dynamical system T : X ⟶ X whose dynamic is given by is the linear operator given by (2), and F is a functional such that BF(x) is a constant vector in each atom P i . e saddle equilibrium point of each atom is given by is defined as follows: where with N x 1 ∈ Z + . e parameters α and c fulfill Assumption 3. e function u(y) is the Heaviside step function: and g is a step function defined as follows: Note that g(y, z) is equal to u(y), when z < 0; while, for z ≥ 0, it is similar to u(y) with the only difference that 0 is mapped to 1 instead of − 1.
Please note that − B is the first column of the operator A and therefore system (38) can be rewritten as follows: where B : X ⟶ X is a vector valued function defined as follows: such that AB(x) is a constant vector in each atom P i . e saddle equilibrium point of each atom is now given by To understand the form of B(x), it is useful to separate B 1 and analyze the effect of each term of the sum, i.e., the effect of the term αg(2(x 1 − f 1 (x 1 )) − x 3 , x 3 ) and the effect of First, consider the function f 1 (x 1 ) whose plot resembles a stair centered at the origin whose plateaus are of 2c in height and width. Two examples for N x 1 � 3 and N x 1 � 6 are shown in Figure 3.
us, f 1 (x 1 ) generates N x 1 switching planes of the form x ∈ R 3 : x 1 � ϵ ∈ R , which are parallel to the plane x 2 − x 3 . en, f 1 (x 1 ) generates a partition R � R 1 , . . . , R N x 1 +1 of X. Now, consider the term αg(2(x 1 − f 1 (x 1 )) − x 3 , x 3 ), and this term generates a switching plane x ∈ R i : 2( generates N x 1 + 1 switching planes, one for each element of the partition R.
us, the elements R i are split and the partition P � P 1 , . . . , P 2N x 1 +2 is generated.
In this way, B(x) locates two equilibrium points with a separation of 2α in the middle of each element R i (along x 1 ) for i � 1, . . . , N x 1 + 1. In the partition P, B(x) locates an equilibrium point in each P i for i � 1, . . . , 2N x 1 + 2. us, the equilibria along the x 1 -axis is located in N x 1 pairs, each pair of nearby equilibrium points have a separation of 2α. Let us denote the midpoint of the line that joins a pair μ of nearby equilibria as cp μ with μ � 1, . . . , (N x 1 + 1). en, the distance from cp i to cp i+1 is 2c. e purpose of this distribution for the equilibria is to allow the existence of double-scroll self-excited attractors that are separated enough from other double-scroll self-excited attractors in a way that these resemble equilibria for the generation of a bigger scroll attractor at a larger scale.
is larger scroll attractor is indeed the hidden scroll attractor. e equilibria are located along the x 1 -axis as follows: e switching planes located in the middle of the selfexcited attractors are given as follows: e rest of the switching surfaces are To illustrate the construction, consider the parameters a � 0.2, b � 5, c � − 7, and N x 1 � 1, the system presents two self-excited attractors and a hidden double-scroll attractor, which is shown in Figure 4(a).
According to the definition, the attraction basin of a hidden attractor does not intersect neighborhoods of equilibria. Figure 5 shows the cut of the numerically evaluated basins of attraction given by the plane x 3 � 0. Each double-scroll self-excited attractor has its own attraction basin shown in red and green. Also, the attractor around these double-scroll self-excited attractors has its own attraction basin shown in blue and the intersection of this basin with the attraction basins of the self-excited attractor is the empty set. Because all equilibria of the system belong to the attraction basin of the self-excited attractors, the attraction basin of the attractor around the self-excited attractors does not contain an equilibrium point. So, the attractor around the self-excited attractors is a hidden attractor.
Another numerical approach to verify that it is a hidden attractor consist in performing a long-time simulation of a trajectory and make sure that the trajectory does not converge to a self-excited attractor. In Figure 6, it is shown that the simulation for the initial condition x 0 � (0, 0, 0) T with t ∈ [1000000, 10001000].
is last approach requires less computing time, so it was the approach used in all the examples in the manuscript.
For a second example, consider N x 1 � 4; then, the system presents five self-excited attractors and a hidden 5-scroll attractor shown in Figure 4(b). us, the number of scrolls is equal to the number of self-excited attractors, which is N x 1 + 1.

Conclusions
In this work, the question of whether or not it is possible to generate a hidden multiscroll attractor with an arrangement of scrolls along more than one direction from multiple selfexcited attractors was addressed. 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 PWL systems and the approach considers for each scroll in the hidden attractors a self-excited attractor inside the scroll.
As future work, we envision working on the answer to the following question: is it possible to generate multistability with more than one hidden attractor?
Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.