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

Authors: R. J. Escalante-González; Eric Campos
Journal: Complexity
 (2020)

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## 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 onℝ3 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.


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## Body

## 1. Introduction

Piecewise linear systems that display scroll attractors have been studied since the publication of the well-known Chua’s circuit. The 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.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]. The other class is called hidden attractors and their basins of attractions do not contain neighborhoods of equilibria. Some hidden attractors have been studied in [7–19]. There are some works focused on control systems with hidden attractors, as in references [20, 21]. Most works related to multiscroll attractors are based on the first class. Multiscroll attractors have been reported in [12, 13, 18, 19, 22–26]. In [27], a system with a multiscroll chaotic sea was introduced.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 Section2, 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.


## 2. Heteroclinic Chaos

LetP=P1,…,Pηη>1 be a finite partition of X⊂ℝ3, that is, X=∪1≤i≤ηPi, and Pi∩Pj=∅ for i≠j. The 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 setPi as clPi. For each pair of adjacent atoms Pi and Pj, i≠j, SWi,j=clPi∩clPj is the switching surface.Consider a dynamical systemT:X⟶X whose dynamic is given by(1)x˙=Ax+fxB,where x=x1,x2,x3T∈ℝ3 is the state vector and A=αij∈ℝ3×3 is a linear operator whose matrix is as follows:(2)A=a3+2c3b2c3−2a3−b3a2b3c3−a3−b2a3+c3,where a,b∈ℝ+, and c∈ℝ−. Thus, the linear operator A has a negative real eigenvalue λ1=c with the corresponding eigenvector v1 and a pair of complex conjugate eigenvalues with positive real part, λ2=a+ib and λ3=a−ib, with the corresponding eigenvectors v2 and v3, respectively. The eigenvectors are given by(3)v1=1012,v2=0−10,v3=−101.B
=
β
1
,
β
2
,
β
3
T
∈
ℝ
3 is a constant vector, and f:X⟶ℝ is a functional such that fxB is a constant vector in each atom Pi, and there exists an equilibrium point xeqi∗=x1eqi∗,x2eqi∗,x3eqi∗T=−fxA−1B, with i=1,…,η, in each atom Pi. Thus, in each atom Pi there exists a saddle equilibrium point with a local stable manifold of dimension one given by Wxeqi∗s=x+xeqi∗:x∈spanv1. A two-dimensional local unstable manifold is given by Wxeqi∗u=x+xeqi∗:x∈spanv2,v3.We begin to explain the generation of chaotic attractors by first considering a partition with two atomsP=P1,P2 and the constant vector B∈ℝ3 given by(4)B=−a3−2c3b3a3−c3,and the functional f given by(5)fx=−α,x∈P1,α,x∈P2,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 x˙=Ax1−fx,x2,x3T. Then, the functional fx determines the location of the equilibria along the x1-axis. So, xeqi∗=x1eqi∗,x2eqi∗,x3eqi∗T∈Pi, i=1,2.Proposition 1.
If the PWL system (1) is given by (2), (4), and (5), then the functional fx determines the location of the equilibria along the x1-axis.Proof.
LetA=A1,A2,A3 be the linear operator so that each Ai, i=1,2,3, is a column vector. Since B=−A1, then x.=Ax+fxB can be rewritten as(6)x˙=x1−fxA1+x2A2+x3A3,and then(7)x˙=Ax1−fx,x2,x3T.Now, in order to find the equilibria, we equate the vector field to zero:(8)0=Ax1−fx,x2,x3T.
SinceA≠0, it follows that(9)xeq∗=fx,0,0T.Thus, the equilibria is determined by fx given by (5) along the x1-axis.
According tofx, the first components of the equilibria fulfill that x1eq1∗<x1eq2∗.
The switching planeSW has associated an equation A^x1+B^x2+C^x3+D=N⋅xT+D=0, with A^>0, and N=A^,B^,C^ is the normal vector. Then, the atoms Pi, i=1,2, are defined as follows:(10)P1=x∈ℝ3:x3>0,N⋅xT≤−D∪x∈ℝ3:x3≤0,N⋅xT<−D,P2=x∈ℝ3:x3>0,N⋅xT>−D∪x∈ℝ3:x3≤0,N⋅xT≥−D.Assumption 1.
The switching planeSW intersects the x1-axis at the midpoint between the equilibrium points xeq1∗ and xeq2∗.Proposition 2.
Under Assumption1, each atom Pi given by (10), i=1,2, contains an equilibrium point of the PWL system (1) given by (2), (4), and (5).Proof.
We want to prove that if a system is given by (1), (2), (4), and (5), then ∃xeq1∗∈P1 and ∃xeq2∗∈P2. From (9) and (5), the two equilibrium points are(11)xeq1∗=−α,0,0Tandxeq2∗=α,0,0T.
From Assumption1, the parameter D can be defined as D=−xsw1A^, where xsw1∈x1eq1∗,x1eq2∗ is the intersection of SW with the x1-axis:(12)N⋅xeq1∗=−αA^,N⋅xeq2∗=αA^.
Thus,(13)N⋅xeq1∗<−D,N⋅xeq2∗>−D,from (10) xeq1∗∈P1 and xeq2∗∈P2.Proposition 3.
IfSW=clP1∩clP2=x∈ℝ3:2x1−x3=0, the stable and unstable manifolds of the PWL system (1) given by (2), (4), and (5) intersect at two points given by xin1=α/3,0,2α/3T and xin2=−α/3,0,−2α/3T.Proof.
The equilibrium points are located atxeq1∗=−α,0,0T and xeq2∗=α,0,0T. In each atom Pi, the equilibrium point xeqi∗ has a local stable manifold (14) of dimension one given by Wxeqi∗s=x+xeqi∗:x∈spanv1 and a two-dimensional local unstable manifold given by Wxeqi∗u=x+xeqi∗:x∈spanv2,v3. Thus, the local manifolds are given as follows:(14)Wxeq1∗s=x∈P1:x1+α=2x3,x2=0,(15)Wxeq1∗u=x∈P1:x1+x3=−α,(16)Wxeq2∗s=x∈P2:x1−α=2x3,x2=0,(17)Wxeq2∗u=x∈P2:x1+x3=α.
According to (18), the stable and unstable manifolds, and the intersection points are given by(18)xin1=clWxeq1∗s∩clWxeq2∗u=α/3,0,2α/3T,(19)xin2=clWxeq2∗s∩clWxeq1∗u=−α/3,0,2α/3T.
These points belong toSW.Assumption 2.
The parametersa and b control the oscillation around the equilibrium point xeqi∗, and we consider b/a>10.Proposition 4.
The hyperbolic system given by (1), (2), (4), and (5) generates a pair of heteroclinic orbits if the switching surface between the atoms P1 and P2 is given by the plane SW=clP1∩clP2=x∈ℝ3:2x1−x3=0.Proof.
We want to show that there exist initial conditionsx01,x02∈SW, such that two solution curves φx01,t and φx02,t of the hyperbolic system given by (1), (2), (4), and (5) fulfill that φx01,t⟶xeq1∗ and φx02,t⟶xeq2∗ as t⟶∞ and φx01,t⟶xeq2∗ and φx02,t⟶xeq1∗ as t⟶−∞; in particular, these initial conditions correspond to the intersection points clWxeq1∗s∩clWxeq2∗ and clWxeq2∗s∩clWxeq1∗u.
From (2), the linear operator A can be expressed as(20)A=QEQ−1,where Q=v1v2v3 and(21)E=c000a−b0ba.
The intersection pointsxin1 and xin2 belong to SW and xin1∈P1 and xin2=P2. Because these points xin1 and xin2 belong to the stable manifolds Wxeq1∗s and Wxeq2∗s, respectively, they are points whose trajectories remain in atoms P1 and P2, respectively.
By definition,x1eq1∗=−x1eq2∗; then, the x2-axis belongs to the plane SW. The sets clWxeqi∗u∩SW, for i=1,2, can be written as follows:(22)0,ϵ,0T+xini:ϵ∈ℝ,fori=1,2.
Consider the following changes of coordinateszi=Q−1x−xeqi∗, for i=1,2. Then, the vector field in zi coordinates for the space given by the atom Pi is given by z˙i=Ezi, with i=1,2.
SinceQ−10,ϵ,0T=0,−ϵ,0T, the sets given by (22) in zi coordinates are given as follows:(23)0,ϵ,0T+Q−1xini−xeqi∗:ϵ∈ℝ,fori=1,2,where zinii=Q−1xini−xeqi∗=−1i+14α/3,0,0T is a point on the z1i-axis that corresponds to the transformation of the intersection points xini∈Pi to zinii∈Q−1x−xeqi∗:x∈Pi, for i=1,2.
Whent>0, φxin1,t remains in the atom P1, the transformation of xin1 under Q−1xin1−xeq1∗ is zin11=4α/3,0,0T. In a similar way, when t>0, φxin2,t, remains in the atom P2, the transformation of xin2 under Q−1xin2−xeq2∗ is zin22=−4α/3,0,0T. So, zinii belongs to the stable manifold Wzeqi∗s, for i=1,2; then, the trajectories zit=eEtzinii⟶0 when t⟶∞. This implies that(24)limt⟶∞φxin1,t=xeq1∗,limt⟶∞φxin2,t=xeq2∗.
Whent<0, φxin1,t leaves the atom P1 and enters to atom P2, the transformation of xin1 under Q−1xin1−xeq2∗ to zin12∈Q−1x−xeq2∗n:x∈clP2 is zin12=0,0,2α/3T. In a similar way, when t<0, φxin2,t leaves the atom P2 and enters to atom P1, the transformation of xin2 under Q−1xin2−xeq1∗ to zin21∈Q−1x−xeq1∗:x∈clP1 is zin21=0,0,−2α/3T. Thus, zinij=0,0,−1j2α/3T is a point on the axis z3j and belongs to clWzeqj∗u for i,j∈1,2 and i≠j .
With the uncoupled system inzi coordinates, we can analyze the flow on the plane z2i−z3i and see how the flow converges at the equilibrium point zeqj∗i when t⟶−∞:(25)z˙2i=az2i−bz3i,z˙3i=bz2i+az3i,z2iz˙2i+z3iz˙3i=az2i2+z3i2.
Ifr2=z2i2+z3i2, then rr˙=ar2(26)r˙=ar,(27)r=r0eat.
As0<a∈ℝ, so r⟶0 when t⟶−∞. Then, the trajectories zit=eEtzinii⟶0 when t⟶−∞. This implies that(28)limt⟶−∞φxin1,t=xeq2∗,andlimt⟶−∞φxin2,t=xeq1∗,
Thus, the heteroclinic orbits are defined as(29)HO1=x∈φxin1,t:t∈−∞,∞,HO2=x∈φxin2,t:t∈−∞,∞.
As example, consider systems (1), (2), (4), and (5) with the parameters a=0.2,b=5,c=−3,α=1, which fulfills (10) with the switching surface SW=clP1∩clP2=x∈ℝ3:2x1−x3=0. Then, heteroclinic chaos emerges from this system, a double-scroll attractor is exhibited, as it is shown in Figure 1(b), for the initial condition x0=0,0,0T.Figure 1
In (a) the heteroclinic loop of system (1), (2), (4), and (5) with the switching surface x∈ℝ3:2x1−x3=0 and the parameters a=0.2,b=5,c=−3,α=1 and in (b) a double-scroll attractor that emerges from an heteroclinic loop using the following initial condition x0=0,0,0T.
(a)
(b)


## 3. 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 double-scroll attractors increases the basin of attraction of both attractors. The 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 atomsP=P1,P2,P3,P4 along with the PWL system (1), with A and B given by (2) and (4), respectively. For this partition, the functional fx is defined in the four atoms as follows:(30)fx=−α−γ,x∈P1,α−γ,x∈P2,−α+γ,x∈P3,α+γ,x∈P4,where 0<α,γ∈ℝ and γ>α. Now, there are four atoms and each atom contains an equilibrium point, xeqi∗=x1eqi∗,x2eqi∗,x3eqi∗T∈Pi, i=1,2,3,4. The first components of the equilibrium points fulfill that x1eqi∗<x1eqi+1∗, i=1,2,3.There are three switching planes and each switching planeSWi,i+1 crosses the x1-axis and is located between the equilibrium points xeqi∗ and xeqi+1∗, i=1,2,3.The switching planeSWij with i,j∈1,…,η has associated an equation Aijx1+Bijx2+Cijx3+Dij=Nij⋅xT+Dij=0, where Aij>0 and Nij=Aij,Bij,Cij is the normal vector. Then, in order to know if a point x belongs to a Pi the following conditions are considered:(31)Ifx∈x∈ℝ3:x3>0,Nij⋅xT≤−Dij,thenx∈Pkfor ak≤i,Ifx∈x∈ℝ3:x3>0,Nij⋅xT>−Dij,thenx∈Pkfor ak≥j,Ifx∈x∈ℝ3:x3≤0,Nij⋅xT≥−Dij,thenx∈Pkfor ak≥j,Ifx∈x∈ℝ3:x3≤0,Nij⋅xT<−Dij,thenx∈Pkfor ak≤i.Ifx∈Pk for a k≥i and x∈Pk for a k≤i, it follows that x∈Pi.The equilibria are located on thex1-axis at(32)xeq1∗=−γ+α00,xeq2∗=−γ−α00,xeq3∗=γ−α00,xeq4∗=γ+α00,so xeq1∗∈P1, xeq2∗∈P2, xeq3∗∈P3, and xeq4∗∈P4.Assumption 3.
The distance between the self-excited attractors should be big enough to allow the existence of a hidden double-scroll attractor, so we considerγ/α≥10.
Consider three switching surfaces with the following orientation:(33)SW1,2=clP1∩clP2=x∈ℝ3:2x1−x3=−2γ,SW2,3=clP2∩clP3=x∈ℝ3:2x1−x3=0,SW3,4=clP3∩clP4=x∈ℝ3:2x1−x3=2γ,which fulfill that(34)SWi,i+1∩x∈ℝ3:x3>0∈Pi,SWi,i+1∩x∈ℝ3:x3≤0∈Pi+1.
The switching surfaces given in (33) along with the condition given in (34) ensure the existence of the four heteroclinic orbits. Moreover, clWxeq2∗u∩Wxeq3∗s≠∅ and clWxeq3∗u∩Wxeq2∗s≠∅.
As example, consider the system with parametersa=0.2,b=5,c=−3,α=1, and γ=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 x0=0,0,0T for a time t∈0,150 a.u. (arbitrary units). Thus, an enough separation of the self-excited double-scroll attractors is not sufficient to produce a hidden double-scroll attractor.Figure 2
Simulation of the system given by (1), (2), (4), and (30) with a=0.2,b=5,c=−3,α=1,γ=10, and α=1 for the initial condition x=0,0,0T in blue. In (a) the surfaces are defined by (33) and (34), the simulation is shown for t∈0,300a⋅u. (arbitrary units), and the trajectory converges to one of the self-excited attractors in red. In (b) the surfaces are defined by (35) and the simulation is shown for t∈50000,51000a⋅u., and the hidden attractor emerges.
(a)
(b)The absence of a hidden attractor is related to the switching planeSW2,3 and how the trajectories of the transitory double-scroll cross it near the stable manifolds. Consider now the following switching surfaces:(35)SW1,2=clP1∩clP2=x∈ℝ3:2x1−x3=−2γ,x3<0,SW2,3=clP2∩clP3=x∈ℝ3:x1=0,SW3,4=clP3∩clP4=x∈ℝ3:2x1−x3=2γ,x3>0.With the change inSW2,3, the intersections clWxeq2∗u∩Wxeq3∗s=∅ as well as clWxeq3∗u∩Wxeq2∗s=∅. Furthermore, the distance from clWxeq2∗u∩SW2,3 to Wxeq3∗s∩SW2,3 and by symmetry from clWxeq3∗u∩SW2,3 to Wxeq2∗s∩SW2,3 allow the trajectories to cross SW2,3 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 γ=10 and the new switching surfaces given by (35). Figure 2(b) shows in blue the trajectory for the initial condition x0=0,0,0T for a time t∈50000,51000 a.u. The trajectory reaches the hidden double-scroll 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.The 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 isP=P1,…,P6 and a possible set of switching surfaces is given by(36)SW1,2=clP1∩clP2=x∈ℝ3:2x1−x3=−4γ,x3<0,SW2,3=clP2∩clP3=x∈ℝ3:x1=−γ,SW3,4=clP3∩clP4=x∈ℝ3:2x1−x3=0,x3>0,SW4,5=clP4∩clP5=x∈ℝ3:x1=γ,SW5,6=clP5∩clP6=x∈ℝ3:2x1−x3=4γ,x3>0,with(37)fx=−α−2γ,x∈P1,α−2γ,x∈P2,−α,x∈P3,α,x∈P4,−α+2γ,x∈P3,α+2γ,x∈P4.Then, the equilibria are located at the x1-axis with x1eq1∗=−2γ−α, x1eq2∗=−2γ+α, x1eq3∗=−α, x1eq4∗=α, x1eq5∗=2γ−α, and x1eq6∗=2γ+α.


## 4. Generalization

The 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 systemT:X⟶X whose dynamic is given by(38)x˙=Ax+BFx,where x=x1,x2,x3T∈ℝ3 is the state vector, A=αij∈ℝ3×3 is the linear operator given by (2), and F is a functional such that BFx is a constant vector in each atom Pi. The saddle equilibrium point of each atom is given by xeqi∗=x1eqi∗,x2eqi∗,x3eqi∗T=Fx,0,0T, with i=1,…,η. Fx is defined as follows:(39)Fx=αg2x1−f1x1−x3,x3+f1x1,where(40)f1x1=∑j=1Nx1γux1+2γj−1−γNx1−1,with Nx1∈ℤ+. The parameters α and γ fulfill Assumption 3. The function uy is the Heaviside step function:(41)uy=1,ify≥0,−1,ify<0,and g is a step function defined as follows:(42)gy,z=1,ify>0andz≥0,−1,ify≤0andz≥0,1,ify≥0andz<0,−1,ify<0andz<0.Note thatgy,z is equal to uy, when z<0; while, for z≥0, it is similar to uy 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:(43)x˙=Ax−ABx,where B:X⟶X is a vector valued function defined as follows:(44)Bx=B1B2B3=αg2x1−f1x1−x3,x3+f1x100,such that ABx is a constant vector in each atom Pi. The saddle equilibrium point of each atom is now given by xeqi∗=x1eqi∗,x2eqi∗,x3eqi∗T=Bx, with i=1,…,η.To understand the form ofBx, it is useful to separate B1 and analyze the effect of each term of the sum, i.e., the effect of the term αg2x1−f1x1−x3,x3 and the effect of f1x1.First, consider the functionf1x1 whose plot resembles a stair centered at the origin whose plateaus are of 2γ in height and width. Two examples for Nx1=3 and Nx1=6 are shown in Figure 3.Figure 3
The plot of the functionf1x1 for (a) Nx1=3 and (b) Nx1=6.
(a)
(b)Thus,f1x1 generates Nx1 switching planes of the form x∈ℝ3:x1=ϵ∈ℝ, which are parallel to the plane x2−x3. Then, f1x1 generates a partition R=R1,…,RNx1+1 of X.Now, consider the termαg2x1−f1x1−x3,x3, and this term generates a switching plane x∈Ri:2x1−f1x1−x3=0 for i=1,…,Nx+1. Since f1x1 takes Nx1+1 values, then αg2x1−f1x1−x3,x3 generates Nx1+1 switching planes, one for each element of the partition R. Thus, the elements Ri are split and the partition P=P1,…,P2Nx1+2 is generated.In this way,Bx locates two equilibrium points with a separation of 2α in the middle of each element Ri (along x1) for i=1,…,Nx1+1. In the partition P, Bx locates an equilibrium point in each Pi for i=1,…,2Nx1+2.Thus, the equilibria along thex1-axis is located in Nx1 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 cpμ with μ=1,…,Nx1+1. Then, the distance from cpi to cpi+1 is 2γ. The 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. This larger scroll attractor is indeed the hidden scroll attractor.The equilibria are located along thex1-axis as follows:(45)x1eq1∗=02γ−γNx1−α,x1eq3∗=12γ−γNx1−α,x1eq5∗=22γ−γNx1−α,⋮x1eq2Nx1−1∗=Nx1−12γ−γNx1−α,x1eq2Nx1+1∗=Nx12γ−γNx1−α,x1eq2∗=02γ−γNx1+α,x1eq4∗=12γ−γNx1+α,x1eq6∗=22γ−γNx1+α,⋮x1eq2Nx1∗=Nx1−12γ−γNx1+α,x1eq2Nx1+2∗=Nx12γ−γNx1+α.The switching planes located in the middle of the self-excited attractors are given as follows:(46)SW1,2=x∈ℝ3:2x1−x1eq1∗+α−x3=0,x1<x1eq2∗+γ−α,SW3,4=x∈ℝ3:2x1−x1eq3∗+α−x3=0,x1eq3∗−γ−α≤x1<x1eq4∗+γ−α,⋮SW2Nx1+1,2Nx1+2=x∈ℝ3:2x1−x1eqNx1+1∗+α−x3=0,xeq2Nx1+1∗−γ−α≤x1.The rest of the switching surfaces are(47)SW2,3=x∈ℝ3:x=x1eq2∗+γ−α,SW4,5=x∈ℝ3:x=x1eq4∗+γ−α,⋮SW2Nx1−2,2Nx1−1=x∈ℝ3:x=x1eq2Nx1−2∗+γ−α,SW2Nx1,2Nx1+1=x∈ℝ3:x=x1eq2Nx1∗+γ−α.To illustrate the construction, consider the parametersa=0.2,b=5,c=−7, and Nx1=1, the system presents two self-excited attractors and a hidden double-scroll attractor, which is shown in Figure 4(a).Figure 4
Simulation of the hidden attractor (in blue) and the self-excited attractors (in red) exhibited by system (43) with parameters a=0.2, b=5, and c=−7. In (a) and (b) Bx is given by (44) with Nx1=1 and Nx1=4, respectively. In (c) Bx is given by (48) with E2=0,E3=1,Nx1=2, and Nx3=2. In (d) Bx is given by (48) with E2=1,E3=1,Nx1=2, Nx2=2, Nx3=2, and w=0.2.
(a)
(b)
(c)
(d)According to the definition, the attraction basin of a hidden attractor does not intersect neighborhoods of equilibria. Figure5 shows the cut of the numerically evaluated basins of attraction given by the plane x3=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.Figure 5
Cut of basins of attraction of system (43) with parameters a=0.2, b=5, c=−7, and Nx1=1 at x3=0. Red and green dots belong to the basins of attraction of self-excited attractors, and blue dots belong to the basin of attraction of the hidden attractor. In (a) grid of 0.5 and in (b) and (c) grid of 0.1.
(a)
(b)
(c)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 Figure6, it is shown that the simulation for the initial condition x0=0,0,0T with t∈1000000,10001000. This last approach requires less computing time, so it was the approach used in all the examples in the manuscript.Figure 6
Simulation of the hidden attractor exhibited by system (43) with parameters a=0.2, b=5, c=−7, and Nx1=1 for t∈1000000,10001000.For a second example, considerNx1=4; then, the system presents five self-excited attractors and a hidden 5-scroll attractor shown in Figure 4(b). Thus, the number of scrolls is equal to the number of self-excited attractors, which is Nx1+1.


## 5. Extension for 2D and 3D Grid Scroll Hidden Attractors

The approach presented in Section4 can be further extended for 2D and 3D grid scroll hidden attractors. The idea is to add switching surfaces, which are parallel to the planes x1−x2 and x1−x3.This requires the modification onBx, for simplicity, the functions fi⋅ are written as fi:(48)Bx=f4+f1−wf2γf2f3+wf2γ,where(49)f2=E2∑k=1Nx2γux2+2γk−1−γNx2−1,(50)f3=E3∑l=1Nx3γux3−wf2γ+2γl−1−γNx3−1,(51)f1=∑j=1Nx1γux1+wf2γ+2γj−1−γNx1−1,(52)f4=αg2x1+wf2γ−f1−x3−wf2γ−f3,x3−wf2γ−f3,where (52) w≥0, E2,E3∈0,1, and Nx1,Nx2,Nx3∈ℤ+.Let us denote the equilibria found in Section4 as xeqj∗1D=x1eqj∗1D,x2eqj∗1D,x3eqj∗1DT for j=1,…,2Nx1+2. Then, the new equilibria for the new Bx in (48) are found from(53)xjkl∗3D=xeqj∗1D+00E1l2γ−γNx3+0E2k2γ−γNx20+−wf2yγ0wf2yγ,where k=1,…,2Nx2+2 and l=1,…,2Nx3+2. The switching surfaces are now restricted on x2 and x3 and located according to the new equilibria.As example, consider the parametersa=0.2, b=5, c=−7, and w=0.2 with E2=0, E3=1, Nx1=2, and Nx3=2 for a 2D-grid scroll hidden attractor shown in Figure 4(c) and E2=1, E3=1, Nx1=2, Nx3=2, and Nx3=2 for a 3D-grid scroll hidden attractor shown in Figure 4(d).


## 6. 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 self-excited 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. The coexistence of self-excited 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?



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