# Emergence of Hidden Attractors through the Rupture of Heteroclinic-Like Orbits of Switched Systems with Self-Excited Attractors

Authors: R. J. Escalante-González; Eric Campos
Journal: Complexity
 (2021)

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## Abstract

This work is dedicated to the study of an approach that allows the generation of hidden attractors based on a class of piecewise-linear (PWL) systems. The systems produced with the approach present the coexistence of self-excited attractors and hidden attractors such that hidden attractors surround the self-excited attractors. The first part of the approach consists of the generation of self-excited attractors based on pairs of equilibria with heteroclinic orbits. Then, additional equilibria are added to the system to obtain a bistable system with a second self-excited attractor with the same characteristics. It is conjectured that a necessary condition for the existence of the hidden attractor in this class of systems is the rupture of the trajectories that resemble heteroclinic orbits that join the two regions of space that surround the pairs of equilibria; these regions resemble equilibria when seen on a larger scale. With the appearance of a hidden attractor, the system presents a multistable behavior with hidden and self-excited attractors.


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## Body

## 1. Introduction

There are two classes of attractors according to [1], which are defined as follows: the first class is given by those classical attractors excited from unstable equilibria called self-excited attractors whose basin of attraction intersects at least a neighborhood of an equilibrium point [2], and they are not difficult to find via numerical methods, and the second class is called hidden attractors whose basin of attraction does not contain neighborhoods of equilibria. The localization of this last class represents a more difficult task which has led to interesting approaches as the analytical-numerical algorithm suggested in [1] for the localization of hidden attractors of Chua’s circuit.Definition 1.
(see [2]).“An attractor is called a self-excited attractor if its basin of attraction intersects with any open neighborhood of an unstable fixed point. Otherwise, it is called a hidden attractor.”A hidden attractor is commonly observed in systems without equilibria or systems with a stable equilibrium point. Therefore, these classes of systems could serve as a starting point in the search for hidden attractors. However, according to the definition, a hidden attractor could be found in a system with any type and number of equilibria as well as any kind of attractors. Multistability is usually related to the existence of more than one attractor. Different scenarios of multistability are reported in [3].Arnold Sommerfeld worked with one of the first dynamical systems with oscillating behavior but no equilibria [4]. In 1994, a conservative system without equilibria that presents a chaotic flow was reported in [5]. This system, known as Sprott case A, presents two quadratic nonlinearities, and it is a particular case of the Nose-Hoover system [6]. After this work, several three-dimensional systems without equilibria with chaotic attractors have been reported, like the one in [7] with two quadratic nonlinearities based on the Sprott system case D, the one in [8] with three quadratic nonlinearities, or the piecewise-linear system reported in [9]. In [10], three methods are used to produce seventeen three-dimensional systems without equilibria with chaotic flows, which present only quadratic nonlinearities.Four-dimensional systems without equilibria with chaotic or hyperchaotic attractors have also been reported. For instance, systems with quadratic and cubic nonlinearities with hyperchaotic attractors are reported in [11, 12]. The first piecewise-linear system without equilibria that exhibits a hyperchaotic attractor is reported in [13]. It is the result of the approximation made to the quadratic nonlinearities of an extended diffusionless Lorenz system. In [14], a four-dimensional system without equilibria with chaotic multiwing butterfly attractors is presented.Since the double-scroll attractor in Chua’s circuit, there exists an interest to generate double-scroll and multiscroll attractors. In circuits based on Chua’s circuit, the implementation of piecewise-linear resistors with multiple segments is not an easy task due to their irregular breakpoints and slopes. Some approaches for self-excited scroll attractors have been reported in [15–19]. Recently, in [20], an approach for the generation of multiscroll hidden attractors with any number of scrolls in a system without equilibria was introduced. In [21], two systems with multiscroll hidden attractors are constructed by introducing nonlinear functions into Sprott system case A. In [22], a no-equilibrium system with a multiscroll hidden chaotic sea is introduced. In [23], a memristive system with chaotic attractors is presented. The multiscroll hidden attractors and multiwing hidden attractors exhibited by the system are sensitive to the transient simulation.In [24], the widening of the basins of attraction of a class of piecewise-linear systems is studied. Also, a system with a double-scroll hidden attractor along with two double-scroll self-excited attractors is introduced. Based on this result, it is natural to think about the possibility of generating hidden attractors via multistable systems with double-scroll self-excited attractors.An approach that allows the generation of hidden attractors based on a kind of piecewise-linear (PWL) system is studied in this work. The study reveals a relationship between the emergence of a hidden attractor and the existence of trajectories that, when are seen on a larger scale, resemble heteroclinic orbits joining the self-excited attractors.The study performed in this work suggests that some classes of systems with a multistable behavior could be designed geometrically to exhibit hidden and self-excited attractors. Chaotic scroll attractors have been widely studied and have been found useful in the design of pseudorandom number generation [25]. It has been demonstrated that the number of scrolls on some classes of systems affects the properties of the generated sequences determining if they fulfill the statistical test of the NIST and affecting the stream ciphering of images [26]. Some chaotic systems can be restored by reconstructing the attractor, which is not desirable in an encryption algorithm since it would reduce the security [27]. In a hidden scroll attractor, the restoration of the system is harder [27]. Thus, the class of systems discussed in this work could lead to the development of new cryptographic algorithms with more complex multistable systems with self-excited and hidden scroll attractors.The structure of the article is as follows: In Section2, a class of piecewise-linear systems with double-scroll self-excited chaotic attractors is introduced. In Section 3, additional equilibria are considered to generate two self-excited attractors. In Section 4, the transitory behavior of the trajectories surrounding the self-excited attractors of the system is studied. In Section 5, the relation between the emergence of a hidden attractor and the existence of trajectories that, when are seen on a larger scale, resemble heteroclinic orbits joining the self-excited attractors is discussed. Finally, conclusions are given in Section 6.


## 2. Heteroclinic Chaos

To introduce the approach, let us first consider a partitionP of the metric space X⊂ℝ3, endowed with the Euclidean metric d. Let P=P1,…,Pηη>1 be a finite partition of X, that is, X=∪1≤i≤ηPi, and Pi∩Pj=∅ for i≠j. Each element of the set P is called an atom and each atom contains a saddle equilibrium point. Due to these atoms, Pi have a saddle equilibrium point, then within each atom there is a stable manifold and also an unstable manifold. These stable manifolds Ws and unstable manifolds Wu are necessary for the mechanism of expansion and contraction present in chaotic dynamics.LetT:X⟶X, with X⊂ℝ3, be a piecewise-linear dynamical system whose dynamics is given by a family of subsystems of the form(1)x˙=Ax+fxB,where x=x1,x2,x3T∈ℝ3 is the state vector, and A=αij∈ℝ3×3 is a linear operator, B=β1,β2,β3T is a constant vector, and f is a functional. The vector fxB is a constant vector in each atom Pi such that the equilibria are given by xeqi∗=x1eqi∗,x2eqi∗,x3eqi∗T=−fxA−1B∈Pi, with i=1,…,η.Oscillations of the flow around the equilibriaxeqi∗ are desired. Let us assign a negative real eigenvalue λ1=c to the complexification of the operator AAC 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. Additionally, we restrict b/a≥10. Thus the stable and unstable manifolds are given by Wxeqi∗s=x+xeqi∗:x∈spanv1 and Wxeqi∗u=x+xeqi∗:x∈spanv2,v3, where v1, v2 and v3 are given as follows:(2)v1=1012,v2=0−10,v3=−101.The matrix of the linear operatorA is defined as follows:(3)A=a3+2c3b2c3−2a3−b3a2b3c3−a3−b2a3+c3.In this work, we denote the local stable and unstable manifolds of an equilibrium pointxeq∗ as Wxeq∗s and Wxeq∗u, respectively, and they are responsible for connecting the equilibria of a dynamical system. Recall that a heteroclinic orbit is a path that joins two equilibrium points in the phase space. Similarly, a homoclinic orbit is a path that starts and ends at the same equilibrium point.We also denoted the closure of a setPi as clPi. Thus, for each pair of atoms Pi and Pj, i≠j, if clPi∩clPj≠∅, then these atoms are adjacent and the switching surface between them is given by the intersection, i.e., SWij=clPi∩clPj.EachSWij has associated an equation of the form A^x1+B^x2+C^x3+D=N12⋅xT+D=0, with A^>0 where N12=A^,B^,C^ is the normal vector. Then the atoms Pi, i=1,2 are defined as follows:(4)P1=x∈ℝ3:x3>0,N12⋅xT≤−D∪x∈ℝ3:x3≤0,N12⋅xT<−D,P2=x∈ℝ3:x3>0,N12⋅xT>−D∪x∈ℝ3:x3≤0,N12⋅xT≥−D.Remark 1.
The divergence of the PWL system (1) considering the linear operator A given by (3) is ∇=2a+c, so the system is dissipative in each atom of the partition P if 2a<c.With the atoms of aP partition containing a saddle equilibrium point in each of them as defined above, it is possible to generate heteroclinic orbits. To generate a heteroclinic orbit, at least two equilibria are required. Therefore, consider a partition with two atoms P=P1,P2, the constant vector B∈ℝ3 is defined as follows:(5)B=−a3−2c3b3a3−c3,and the functional f is given by(6)fx=−α,x∈P1,α,x∈P2,with α>0. So the equilibria are at xeq1∗=−α,0,0T∈P1 and xeq2∗=α,0,0T∈P2, and the stable and the unstable manifolds are given by(7)Wxeq1∗s=x∈ℝ3:|x1+α=2x3,x2=0,Wxeq1∗u=x∈ℝ3:|x1+x3=−α,Wxeq2∗s=x∈ℝ3:|x1−α=2x3,x2=0,Wxeq2∗u=x∈ℝ3:|x1+x3=α.Proposition 1.
(see [28, 29]). “The hyperbolic system given by (1), (3), (5), and (6) generates a pair of heteroclinic orbits if the switching surface between the atoms P1 and P2 is given by the plane SW12=x∈ℝ3:2x1−x3=0.”The points where the stable and unstable manifolds intersect atSW are given by(8)xin1=clWxeq1∗s∩clWxeq2∗u=α3,0,2α3T,xin2=clWxeq2∗s∩clWxeq1∗u=−α3,0,−2α3T.These pointsxin1 and xin2 belong to SW12 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. Thus, the heteroclinic orbits are defined as follows:(9)HO1=x∈φxin1,t:t∈−∞,∞,HO2=x∈φxin2,t:t∈−∞,∞.For the system given by (1), (3), (5), and (6), it is possible to find several points x0∈HOi such that xeqi−x0<ε with ε arbitrarily small and i=1,2. Thus, one can find initial conditions for the simulation of the heteroclinic orbits as close to the equilibria as desired. One example of the initial condition formula for P1 is as follows:(10)x01=23αe−2kaπ/b−α0−23αe−2kaπ/b,and for P2(11)x02=−23αe−2kaπ/b+α023αe−2kaπ/b,with k∈ℤ+.Example 1.
Consider the system (1), (3), (5), and (6) with SW12=x∈ℝ3:2x1−x3=0 and the parameters a=0.2,b=5,c=−3,α=1.The above-defined system fulfills Proposition1, so it presents a heteroclinic orbit. From (10) and (11), two initial conditions,(12)x01=−0.9999976751050959,0,−2.3248949041393315e−6T,x02=0.9999976751050959,0,2.3248949041393315e−6T,are chosen with k=50 to simulate the two heteroclinic orbits shown in Figure 1(a). A double-scroll attractor with heteroclinic chaos is generated, and it is shown in Figure 1(a) for the initial condition x0=0,0,0T.Figure 1
In (a) the heteroclinic loop of the system (1), (3), (5), and (6) with the switching surface x∈ℝ3:2x1−x3=0, the parameters a=0.2,b=5,c=−3,α=1, and the initial conditions x01=−0.9999976751050959,0,−2.3248949041393315e−6T (red) and x02=0.9999976751050959,0,2.3248949041393315e−6T (blue), and in (b) a double-scroll attractor that emerges from a heteroclinic orbit using the following initial condition x0=0,0,0T and the same parameters.
(a)(b)The unstable manifoldsWxeq1∗u=x∈ℝ3:x1+x3+1=0 and Wxeq2∗u=x∈ℝ3:x1+x3−1=0 and the stable manifolds Wxeq1∗s=x∈ℝ3:x1+1/2=x3;x2=0 and Wxeq2∗s=x∈ℝ3:x1−1/2=x3;x2=0. The intersection points are given by clWxeq2∗s∩clWxeq1∗u=−1/3,0,−2/3T, clWxeq1∗s∩clWxeq2∗u=1/3,0,2/3T.Proposition 2.
(see [29]). “If the partition P contains more than two atoms P1,P2,…,Pk, with 2<k∈ℤ+, and each atom is a hyperbolic set defined as above. Furthermore, the atoms by pairs Pi and Pi+1 fulfill Proposition 1. Then, the system generates 2k−1 heteroclinic orbits.”Proof.
A direct consequence of Proposition1.


## 3. Emergence of Multiscroll Attractors through Multiple Heteroclinic Orbits

According to the Proposition2, it is possible to generate multiscroll attractors based on multiple heteroclinic orbits. So in this Section, we consider more than two hyperbolic sets in the partition with the aim of studying the existence of heterocyclic cycles and the attractors exhibited by the system when varying the location of the equilibria.Consider the partitionP=P1,P2,P3,P4 along with the piecewise-linear dynamical system (1), with A and B given by (3) and (5), respectively. Thus the function fx is defined in the four atoms as follows:(13)fx=−α−γ,x∈P1,α−γ,x∈P2,−α+γ,x∈P3,α+γ,x∈P4,where α,γ>0. The equilibria are at(14)xeq1∗=−γ+α00,xeq2∗=−γ−α00,xeq3∗=γ−α00,xeq4∗=γ+α00,so xeq1∗∈P1, xeq2∗∈P2, xeq3∗∈P3 and xeq4∗∈P4. The location of the equilibria according to the parameters 0<α and 0<γ is as follows:(i)
The equilibria are on thex1 axis and for α=γ the system only have three equilibria. Otherwise, it has four equilibria.(ii)
Forα<γ the distance of the equilibria xeq1∗ and xeq4∗ to the origin O=0,0,0T are the same dxeq1∗,O=dxeq4∗,O and also for dxeq2∗,O=dxeq3∗,O.(iii)
Forγ=2α, all equilibria are at the same distance dxeq1∗,xeq2∗=dxeq2∗,xeq3∗=dxeq3∗,xeq4∗=2α.(iv)
The other case is whenγ≠2α, and dxeq1∗,xeq2∗=dxeq3∗,xeq4∗=2α, but dxeq2∗,xeq3∗≠2α.In this section, we are especially interested in the case ofγ≠2α such that γ>α with switching surfaces given by(15)SW12=clP1∩clP2=x∈ℝ3:2x1−x3=−2γ,SW23=clP2∩clP3=x∈ℝ3:2x1−x3=0,SW34=clP3∩clP4=x∈ℝ3:2x1−x3=2γ,which fulfill that(16)SWii+1∩x∈ℝ3:x3>0∈Pi,SWii+1∩x∈ℝ3:x3≤0∈Pi+1.This way of defining the switching surfaces provokes that the intersections between them and the stable manifolds contain a point, and the intersections between them and the unstable manifolds are the empty set, i.e.,Wxeq1∗u∩SW12=∅ and Wxeq1∗s∩SW12≠∅.Let us define two points,pb=Wxeq1∗s∩SW12 and pa=Wxeq3∗s∩SW23 as shown in Figure 2. Then pa and pb are given as follows:(17)pa=−γ−α30−2γ−α3,pb=α3−γ02α3.Figure 2
Projection of the stable and unstable manifolds and switching planes onto thex1−x3 plane. The diagram shows the location of the unstable manifold marked with red lines, the stable manifold marked with blue lines, and switching planes marked with green lines.The setclWxeq2∗u∩SW12 can be written as follows:(18)x∈ℝ3:x=0,ε,0T+pb,ε∈ℝ,and the set clWxeq2∗u∩SW23 can be written as follows:(19)x∈ℝ3:x=0,ε,0T+pa,ε∈ℝ.Consider the transformationz2=Q−1x−xeq2∗, since Q−10,ε,0T=0,−ε,0T, so z2=0,ε,0T+Q−1pb−xeq2∗, where Q−1pb−xeq2∗ is a point on the plane z22−z32. And z2=0,ε,0T+Q−1pa−xeq2∗, where Q−1pa−xeq2∗ is also a point on the plane z22−z32. Then, the set (18) in z2 coordinates is given by(20)z2∈ℝ3:z2=0,ε,2α3T,ε∈ℝ,and the set (19) in z2 coordinates is given by(21)z2∈ℝ3:z2=0,ε,2α−γ3T,ε∈ℝ.Thus, the sets (20) and (21) are orthogonal lines to the z32 axis. The points pa and pb in z2 coordinates will be denoted as follows:(22)paz=002α−γ3,pbz=002α3.With the uncoupled system inz2 coordinates, we can analyze the flow on the plane z22−z32 close to zeq2∗2.(23)z˙22=az22−bz32,z˙3=bz22+az32,(24)rr˙=z22z2.2+z32z3.2=ar2,(25)r˙=ar,(26)r=r0eat.It follows from (22) that if α=γ−α, then the points paz and pbz are at the same distance from zeq2∗2=0,0,0T. Thus, from (26), it follows that the trajectories with initial conditions paz and pbz remain in P2 for all t<0.Our case study isγ−α≠α, such that γ>α. Let us consider the case γ−α>α, and it can be seen from (22) that pbz is closer to zeq22 than paz, this is, dpbz,zeq22<dpaz,zeq22. Then, if γ is sufficiently big with respect to α, the trajectory with the initial condition paz will eventually reach the set given by (20) for t<0; i.e., the trajectory of the initial condition pa∈SW23 reaches the switching plane SW12 and not the equilibrium point xeq2∗. This means that in x coordinates, the heteroclinic orbit from xeq2∗ to xeq3∗ does not exist. Similarly, when pbz is further than paz from zeq2∗2, this is, dpaz,zeq22<dpbz,zeq22, for γ sufficiently small, the trajectory with the initial condition pbz will eventually reach the set given by (21) for t<0, i.e., the trajectory of the initial condition pb∈SW12 reaches the switching plane SW23 and not the equilibrium point xeq2∗. This means that in x coordinates, the heteroclinic orbit from xeq2∗ to xeq1∗ does not exist.The next proposition warranty the existence of heteroclinic orbits whenγ belongs to an interval of real numbers where the case γ−α≠α is considered, such that γ>α.Proposition 3.
The hyperbolic system given by (1), (3), (5), and (13) with the switching surfaces given in (15) generates six heteroclinic orbits if(27)αe−aτcosbτ−1e−aτcosbτ>γ>α1−e−aτcosbτ,where(28)τ=arctanb/a+π/2b.Proof.
To find the values ofγ for which these heteroclinic orbits exist, let us assume pa is a point of the heteroclinic orbit joining xeq2∗ and xeq3∗, i.e.,(29)limt⟶−∞φpa,t=xeq2∗,limt⟶∞φpa,t=xeq3∗.
Becausepa∈Wxeq3∗s then limt⟶∞φpa,t=xeq3∗. For the other part of the heteroclinic orbit, we analyze the system in z2 coordinates, we have paz, pbz, zeq2∗2, and the orbit is given by z2t. We assume that z20=paz, so we want that z2t remains in P2 for all t<0. Thus, we need to find the first maximum in the component z32 of the trajectory whose initial condition is paz for t<0. According to (22), the third component of paz and pbz are 2α−γ/3<0 and 0<2α/3, respectively. This maximum gives us the intersection point between the trajectory z2t and the axis z32. Then we can compare the third component of the trajectory z2t and the point pbz, in terms of α and γ to ensure that z2t remains in P2 for all t<0.
The trajectoryz2t for the initial condition z02=z102,z202,z302T is as follows:(30)z12t=z102e−ct,z22t=z202eatcosbt−z302eatsinbt,z32t=z202eatsinbt+z302eatcosbt.This set of equations is analyzed for t<0. The same analysis can be done for 0<t by using the following set of equations:(31)z12t=z102ect,(32)z22t=z202e−atcosbt+z302e−atsinbt,(33)z32t=−z202e−atsinbt+z302e−atcosbt.
Since we are looking for the first maximum inz32t for 0<t.
Then from (33) with the initial condition paz given in (22),(34)z32t=2α−γ3e−atcosbt,(35)z˙32t=−2α−γ3e−atbsinbt+acosbt,(36)z˙32t=−2α−γ3e−ata2+b2cosbt−arctanba,to find the maximum, we equate to zero(37)0=−2α−γ3e−ata2+b2cosbt−arctanba.Thus, it turns out that(38)bt−arctanba=π2+nπ,withn∈ℤ,t=arctanb/ab+π2b+nπb,withn∈ℤ.
We will calltmax the time for the first maximum. Thus, it follows that(39)tmax=arctanb/a+π/2bthen from (34),(40)z32tmax=2α−γ3e−atmaxcosbtmax.This maximum z32 must be part of P2, since pbz belongs to P1 it follows from (22) that(41)2α3>2α−γ3e−atmaxcosbtmax,αe−atmaxcosbtmax−1e−atmaxcosbtmax>γ.
Now, let us assumepb is a point of the heteroclinic orbit joining xeq2∗ and xeq1∗, i.e.,(42)limt⟶−∞φpb,t=xeq2∗,limt⟶∞φpb,t=xeq1∗.
Becausepb∈Wxeq1∗s then limt⟶∞φpb,t=xeq1∗.
Following the same procedure described above but looking for a minimum, due to the third component ofpbz is 0<2α/3. It is found that(43)tmin=arctanb/a+π/2b.Then from (33) and the point pbz given in (22),(44)z32tmin=2α3e−atmincosbtmin.This minimum z32 must be part of P2, since paz belongs to P3 it follows from (22) that(45)2α−γ3<2α3e−atmincosbtmin,γ>α1−e−atmincosbtmin.
Then definingτ=tmax=tmin(46)αe−aτcosbτ−1e−aτcosbτ>γ>α1−e−aτcosbτ.
The same conclusion applies to the pointxeq3∗ due to the symmetry of the system. Finally, the heteroclinic orbit from xeq1∗ to xeq2∗ and the one from xeq4∗ to xeq3∗ are always present in the system as there are no more switching surfaces.□
To illustrate the effect of the parametersγ, α, a, and b on the existence of heteroclinic orbits of the system given by (1), (3), (5), (13), and (15), we use Proposition 3 to determine the open interval of real values for γ given by(47)Γ=α1−e−aτcosbτ,αe−aτcosbτ−1e−aτcosbτ,with τ=arctanb/a+π/2/b. So, six initial conditions were calculated as in (10) and (11) with k=50 for the parameters a=0.2, b=5, c=−3, and α=1. Four cases of different values of γ are analyzed. The first two correspond to γ1,2∈Γ and the last two correspond to γL,U∉Γ:.(1)
For this caseγ1∈Γ, with γ1=α1−e−aτcosbτ+0.00001, so there exist six heteroclinic orbits, as shown in Figure 3(a).(2)
Forγ2=αe−aτcosbτ−1/e−aτcosbτ−0.00001∈Γ, in this case, there exist also six heteroclinic orbits as shown in Figure 3(b).(3)
In this caseγL=α1−e−aτcosbτ∉Γ, then there exist four heteroclinic orbits, as shown in Figure 3(c). The green orbit stating close to xeq2∗ cannot reach xeq1∗ and goes to P3. In the same way, the yellow orbit starting close to xeq3∗ cannot reach xeq4∗ and goes to P2. Then there is no heteroclinic orbits from xeq2∗ to xeq1∗ and from xeq3∗ to xeq4∗.(4)
ForγU=αe−aτcosbτ−1/e−aτcosbτ∉Γ, there also exist four heteroclinic orbits as shown in Figure 3(d). The red orbit stating close to xeq2∗ cannot reach xeq3∗ and goes to P1. In the same way, the blue orbit starting close to xeq3∗ cannot reach xeq2∗ and goes to P4. Then there is no heteroclinic orbit from xeq2∗ to xeq3∗, nor vice versa.
The open intervalΓ is given as Γ=γL,γU, where(48)γL=α1−e−aτcosbτ≈1.8826170015164836,γU=αe−aτcosbτ−1e−aτcosbτ≈2.1329942639693464.
The four cases mentioned generate three types of systems determined byγ and Γ. For instance, for γ=2∈Γ and α=1 corresponds to the above first and second cases. Then the system presents six heteroclinic orbits which comprise three heteroclinic loops between equilibria: xeq1∗ and xeq2∗; xeq2∗ and xeq3∗; xeq3∗ and xeq4∗. For γ=1.5<γL, then γ∉Γ, and this case corresponds to the above third case. So there are four heteroclinic orbits, and two of them comprise a heteroclinic loop between equilibria xeq2∗ and xeq3∗. For γU<γ=3, then γ∉Γ, and this case corresponds to the above fourth case. So there are four heteroclinic orbits that comprise two heteroclinic loops, but now between equilibria: xeq1∗ and xeq2∗; xeq3∗ and xeq4∗. The above three cases generate self-excited attractors as shown below:.(1)
Forγ=2∈Γ, the system presents a self-excited attractor with four scrolls which are shown in Figure 4(a), and its corresponding three heteroclinic loops are shown in Figure 4(d). According to [30], a scroll attractor can be considered a multiscroll attractor when it has at least three scrolls. Thus the attractor shown in Figure 4(a) is a multiscroll attractor. The scrolls are generated around each equilibrium point of the system xeqi∗, with i=1,2,3,4.(2)
Forγ=3, γ>γU, then γ∉Γ. The system presents bistability; the two double-scroll self-excited attractors are shown in Figure 4(b). In this case, two heteroclinic orbits are lost, the system exhibits four heteroclinic orbits, i.e., two heteroclinic loops, as shown in Figure 4(e). One double-scroll self-excited attractor oscillates around equilibria xeq1∗ and xeq2∗, while the other self-excited attractor oscillates around equilibria xeq3∗ and xeq4∗. The basin of attraction of each self-excited attractor has surrounded both attractors.(3)
Forγ=1.5, γ<γL, then γ∉Γ. The system presents only one double-scroll self-excited attractor, shown in Figure 4(c). In this case, two heteroclinic orbits are also lost, but only a heteroclinic loop is exhibited. The heteroclinic orbits are shown in Figure 4(f). The double-scroll self-excited attractor oscillates around equilibria xeq2∗ and xeq3∗.
Based on the results reported in [24] about the relation between the location of the symmetric equilibria and the size of the basin of attraction, we could ponder the possible existence of a hidden attractor for the case γ>γU because there are oscillations surrounding the two self-excited attractors as a hidden attractor exists. However, the simulations of these systems let us know that hidden attractors are not present. For example, if the γ value is increased, then also the distance between the two self-excited attractors increases. This provokes that some initial conditions in the basins of attraction of both attractors generate transitory oscillations resembling a double-scroll attractor. However, after some time, these transitory oscillations converge to one of the double-scroll self-excited attractors.
We analyze the trajectory for the initial conditionx0=0,0,0T and different values of γ fulfilling γU<γ. The first case is γ=5 and t∈0,40 in arbitrary units (a.u). The equilibria are at(49)xeq1∗=−600,xeq2∗=−400,xeq3∗=400,xeq4∗=600.
Figure5(a) shows the trajectory which consists of the transitory behavior resembling a double-scroll attractor and after a short time reaches a double-scroll self-excited attractor around equilibria xeq1∗ and xeq2∗.
Increasing the value ofγ to 15, the equilibria are located at(50)xeq1∗=−1600,xeq2∗=−1400,xeq3∗=1400,xeq4∗=1600.
And the transitory time to reaches the self-excited attractor is increased. In Figure5(b), the trajectory is shown for t∈0,60a.u. Now, for γ=100 the equilibria are located at(51)xeq1∗=−10100,xeq2∗=−9900,xeq3∗=9900,xeq4∗=10100.
The transitory time lasts longer for the same initial condition. Figure5(c) shows the transitory oscillations of the trajectory for t∈0,300a.u. After a long time, the trajectory reaches a double-scroll self-excited attractor around equilibria xeq1∗ and xeq2∗; see Figure 5(d) for t∈356.6,400a.u. Continuing to increase the value to γ=1000, then this sets the equilibria at(52)xeq1∗=−100100,xeq2∗=−99900,xeq3∗=99900,xeq4∗=100100.
Figure5(e) shows the transitory oscillation of the trajectory when t∈0,300a.u., again, transitory time increases, and after this long time, the trajectory again reaches a double-scroll self-excited attractor around equilibria xeq1∗ and xeq2∗, see 5f for t∈3091,3200a.u.
In brief, forγ=5 it took the trajectory around 35a.u. To converge to a self-excited attractor, for γ=15 around 50a.u. To converge, for γ=100 around 350a.u. And for γ=1000 around 3090a.u. Thus, transitory time seems to increase for some initial conditions when γ increases. In all the cases, the trajectories reach a self-excited attractor.Figure 3
Heteroclinic orbits of the system given by (1), (3), (5), (13), and (15) for the parameters a=0.2, b=5, c=−3, α=1 and different values of γ. There are six heteroclinic orbits for: (a) γ1=α1−e−aτcosbτ+.00001, and (b) γ2=αe−aτcosbτ−1/e−aτcosbτ−.00001. Four heteroclinic orbits for (c) γL=α1−e−aτcosbτ, and (d) γU=αe−aτcosbτ−1/e−aτcosbτ.
(a)(b)(c)(d)Figure 4
Attractors and heteroclinic orbits of the system are given by (1), (3), (5), (13), and (15) for α=1 and different values of γ. For γ=2 the system is exhibited in (a) a quad-scroll attractor, and in (d) six heteroclinic orbits. For γ=3 the system is exhibited in (b) two double-scroll attractors, and in (e) four heteroclinic orbits. For γ=1.5 the system is exhibited in (c) a double-scroll attractor and in (f) four heteroclinic orbits.
(a)(b)(c)(d)(e)(f)Figure 5
Trajectory of the system given by (1), (3), (5), (13), and (15) for the initial condition x=0,0,0T, a=0.2, b=5, c=−7, α=1 and different values of γ: (a) γ=5, the transitory oscillation of double-scroll exhibited and after some time converge to one of the double-scroll self-excited attractors, t∈0,40a.u.; (b) γ=15, the transitory oscillation of double-scroll exhibited and after some time converge to one of the double-scroll self-excited attractors, t∈0,60a.u.; (c) γ=100, the transitory oscillation of double-scroll exhibited for t∈0,300a.u.; (d) γ=100, double-scroll self-excited attractor for t=356.6,400a.u.; (e) γ=1000, the transitory oscillation of double-scroll exhibited for t∈0,300a.u.; (f) γ=1000, the double-scroll self-excited attractor for t=3091,3200a.u.
(a)(b)(c)(d)(e)(f)


## 4. Route to a Self-Excited Attractor

In this section, the transitory behavior presented in the previous section is studied in order to visualize the route of the transitory double scroll to a self-excited attractor. The idea is to estimate two regionsR1,R2⊂SW23, such that any trajectory φx0, with x0∈R1∪R2, will eventually go to the self-excited attractor Aself1 or Aself2. These regions are symmetric with respect to the origin and are crossed by the trajectories of the transitory double scroll.Consider the pointpa and its symmetric point pc=−pa:(53)pa=−γ−α30−2γ−α3,pc=γ−α302γ−α3.These points are the intersections of the local manifolds inSW23 and are shown in Figure 6.Figure 6
Projection of the manifolds on (a)x1−x3 plane and (b)z12−z32 plane. The stable and unstable manifolds are marked with blue and red solid lines, respectively, the switching surfaces with green lines.
(a)(b)The trajectories with initial conditions in the pointspa,pc∈SW23 converge to equilibria xeq3 and xeq2, respectively. So, the transient oscillation of the trajectory that resembles a double-scroll attractor interferes when the trajectory reaches neighborhoods Npa⊂SW23 and Npc⊂SW23 around pa or pc, respectively, because each trajectory with initial condition in Npa or Npc is led to one of the self-excited attractors Aself1 or Aself2, respectively. So, the aim is to visualize the route to a self-excited attractor when Npa∩R1≠∅ and Npc∩R2≠∅.Thus, the study in this section has the following structure: First, two regionsR1 and R2 are estimated. The regions are then evaluated numerically to verify their validity. Finally, based on the geometry of the system and the observation of the simulations, conjecture about the necessary conditions for the existence of the hidden attractor.To simplify the study, some assumptions are made to restrict the systems to a subset of the class.Assumption 1.
The parameter values fulfill the following relations:b/a≥25, 2≥c/b≥7/5 and γ/α≥10.To start the analysis, let us find the points inSW23 where the vector fields of P2 and P3 are tangent to the plane SW23. These points will be called tangent points and can be found from the following equation:(54)2,0,−1a3+2c3b2c3−2a3−b3a2b3c3−a3−b2a3+c3x32−x1eqix2x3=−a+cx1eqi+3c−a2x3+3bx2=0.It follows that(55)x2=a+c3bx1eqi−c−a2bx3,withi=1,2.For the vector field ofP2, x1eq2=−γ+α, then(56)x2=a+c3b−γ+α−c−a2bx3,while for the vector field of P3, x1eq3=γ−α:(57)x2=a+c3bγ−α−c−a2bx3.According to (16), if x3>0 then SW23 belongs to P2, and the tangent points to consider in SW23 for x3>0 are given by (56). And if x3≤0, then SW23 belongs to P3 and the tangent points are given by (57). An illustration of the tangent points in SW23 is shown in Figure 7, where the points for P2 are indicated by a dotted line, while for P3 are drawn as a continuous line.Figure 7
Illustration of the local manifold of the system. Stable manifolds are in blue, unstable manifold in red, and switching surfaces in green. The points where the vector field ofP2 is tangent to SW23 are in a red dotted line, while the points where the vector field of P3 are tangent to SW23 are in a continuous black line. The symmetric regions R1 and R2 are in white.As a starting point to propose the regionR1 defined by four points p1,…,p4, consider the point in clWxeq2u∩SW23 given by (19) that fulfills (56):(58)pt1=−γ−α3−2aγ−α3b−2γ−α3,and in z2 coordinates(59)pt1z2=02aγ−α3b−2γ−α3.The points inz2 coordinate system are denoted with the suffix z2. If we evaluate the trajectory with an initial condition in x0=pt1z2, under the vector field of P2 and ignoring the effect of the vector field of P1 and P3, reaches the point pt2z2∈SW23. The flow φ could go from P2 to P3 through the segment pt1z2pt2z2¯. Thus, trajectories with initial condition are close to Aself1 but not in the attractor cross SW23 close to the segment pt1z2pt2z2¯ then R1 should include this segment. However, when the vector field of all atoms is considered, trajectories with initial conditions close to pt1z2 could reach SW23 in points whose second component in z2 coordinates are further from 0 than the second component in z2 coordinates of pt2z2. Thus, let us propose the region R1 based on a larger segment pa1z2pa2z2¯ such that pt1z2pt2z2¯⊂pa1z2pa2z2¯. Consider the initial condition paz2 given in z2 coordinates by(60)paz2=00−2γ−α3,then, the radius with respect to zeq2∗ would be 2γ−α/3. Remember that only the vector field of P2 is considered and the trajectory rotates around the axis z12. Let us think in an imaginary and impossible case when a trajectory with an initial condition in paz2 rotates around the axis z12 and reaches SW23, but instead of the normal increment of radius, let us imagine that the increment in radius corresponds to an evolution time t=2π/b (360°). Thus, the z22 component of this point is further from 0 than the z22 component of pt2z2. Then, we could take pa2z2=pt1z2 and find the z22 component of pa1z2 from(61)2γ−α3ea2π/b2−2γ−α32.By using Assumption1,(62)2γ−α3ea2π/b2−2γ−α32≤γ23ea2π/b2−232≤0.5388γ<3γ5.Remember thatz22=−x2, then, the points pa1 and pa2 are given by(63)pa1=−γ−α33γ5−2γ−α3,pa2=−γ−α3−2aγ−α3b−2γ−α3,where −2aγ−α/3b is the tangent coordinate given by (56) for x3=−2γ−α/3.So, let us propose the regionR1 delimited by the following four points:(64)p1=pa1+γ10,0,γ5T,p2=pa1−γ10,0,γ5T,p3=pa2+−γ10,−c−a−γ/52b,−γ5T,p4=pa2+γ10,−c−aγ/52b,γ5T.BecauseR2 is symmetric to R1 with respect to the origin, then the symmetric region R2 is delimited by the points:(65)q1=−p1,q2=−p2,q3=−p3,q4=−p4.These regionsR1 and R2 have been proposed taking into consideration that points pd and pe shown in Figure 6(b) are part of the regions. In Figure 8, R1 and R2 are shown in z2 coordinates.Figure 8
RegionsR1 and R2 on the projection z22−z32.Now, let us analyze some scenarios on these regions to see if the proposed regions are good candidates, at least in an estimated manner. In order to simplify the scenarios, in the following, it is considered that trajectories inP1∪P2 rotate only around the stable manifold of xeq2, i.e., if only the vector field of P2 is considered.Let us define the setR1b as follows:(66)R1b=z2∈ℝ3:z12∈−γ5,γ5.First, let us verify that the points inR2 go to R1b.The evaluation of the vector field inpc1z2 tells us that the spin is counterclockwise in z2 coordinates. From Figures 8 and 6(b), it is not hard to see that the points below the segment pcz2pc1z2¯ produce trajectories that can perform a turn of π around the z12 axis without reaching SW23 again. The time that corresponds to a turn of 2π is T=2π/b. The point pc1z2 is given by(67)pc1z2=4γ−α33γ50.Consider the trajectory with an initial condition inpc1z2, and an evolution time that corresponds to a turn of π around the z12 axis. After this time, the first component of the state vector can be found from(68)z12=4γ−α3ecπ/b.Ifz12≤γ/5 means that the trajectory with an initial condition in pc1z2 reaches R1b. Consider Assumption 1 for a big value of z12 (when γ is too big):(69)4γ−α3ecπ/b≤4γ3ecπ/b≤0.0164γ<γ5.Thus the setz2∈R2:z22≥0,z32≤0 reaches the set R1b.Now consider the pointq2z2 given by(70)q2z2=23γ15−4α33γ5γ10.The angle produced by the radius from the pointq2z2 to the z12 axis and the plane z12−z22 is given by(71)arctanγ/103γ/5=arctan16=0.1654.Let us consider that the trajectory with the initial condition inq2z2 evolves for a duration time that corresponds to π−20.1654=2.8113. The first component of the state vector after this duration is given by(72)z12=23γ15−4α3ec2.8113/b.As before, ifz12≤γ/5 means that the trajectory with an initial condition in q2z2 reaches R1b. Consider again Assumption 1 for a big value of z12:(73)23γ15−4α3ec2.8113/b≤23γ15ec2.8113/b≤0.0299γ<γ5.Thus,q2z2 reaches the region R1b. Moreover, since the points in the segment q2z2pc1z2¯ produce a radius whose angle with the plane z12−z22 is between 0 and 2.8113, the trajectories starting in this segment also reach the set R1b.Now consider the pointpcuz2 given by(74)pcuz2=23γ15−4α30γ10.The trajectories with the initial condition in the segmentpcz2pcuz2¯ can turn π/2 without reaching SW23. Thus, let us consider that the trajectory with the initial condition in pcuz2 evolves for a duration time that corresponds to π/2. The first component of the state vector after this duration is given by(75)z12=23γ15−4α3ecπ/2b.Ifz12≤γ/5 means that the trajectory with an initial condition in pcuz2 reaches R1b, consider Assumption 1 for a big value of z12:(76)23γ15−4α3ecπ/2b≤23γ15ecπ/2b≤0.17γ<γ5.Thus the trajectories with the initial condition in the setz2∈R2:z22≥0,z32≥0 converge to the set R1b.Now consider the pointq3z2(77)q3z2=23γ15−4α33cγ−23aγ30b+2aα3bγ10.To estimate if that the trajectory starting inq3z2 is not going to reach SW23 when the radius form an angle of 3π/2 with the plane z12−z22, let us consider the following exaggerated scenario: The radius size corresponds to a duration equivalent to 3π/2 , but the z12 component corresponds to a duration equivalent to π/2 of oscillation, i.e., when the radius form an angle of 3π/2 with the plane z12−z22, a smaller radius than the real one is considered; also, a larger value of z12 than the real value is considered.Then to obtain the radius:(78)r=ea3π/2b3cγ−23aγ30b+2aα3b2+γ102.Under Assumption1 for the largest radius:(79)ea3π/2b3cγ−23aγ30b+2aα3b2+γ102≤ea3π/2b3cγ−23aγ30b2+γ102,ea3π/2b3cγ30b−23aγ30b2+γ102≤ea3π/2b−1γ5−23γ7502+γ102,ea3π/2b−1γ5−23γ7502+γ102=γea3π/2b−1737502+1102,γea3π/2b−1737502+1102≤0.3036γ.The value ofz12 after a duration equivalent to 90° is given by(80)z12=23γ15−4α3ecπ/2b.Under Assumption1 for the biggest z12(81)23γ15−4α3ecπ/2b≤23γ15ecπ/2b,23γ15ecπ/2b≤0.17γ<γ5.Inz2 coordinates, a specific value of z12SW23 fulfills z32=−2γ−α/3+z12/2. At this angle of 3π/2 the radius is r=−z32. Then, if the values found for this scenario fulfill the following inequality, it can be concluded that the trajectory with an initial condition in q3z2 does not reach SW23 after a duration that corresponds to an oscillation of π/2:(82)−0.3036γ>−2γ−α3+0.085γ,under Assumption 1 for the worst case(83)−0.3036γ>−18γ30+0.085γ=−0.515γ.Then, the trajectory remains for the duration that corresponds toπ/2. Also, the trajectory does not reach SW23 when the radius is at an angle of 3π/2 with the plane z12−z22 even when the radius growth is exaggerated. It can be concluded that the trajectory with the initial condition in q3z2 could reach SW23 until the second time it approaches SW23 and reaches R1b.Sinceq3z2 is the point in the set z2∈R2:z22≤0 that produces the largest radius of that set, the trajectories with the initial condition in this set also reach R1b. Then the trajectories starting at R2 reach R1b.Now to estimate if the trajectories that start inR2 reach R1⊂R1b is enough to verify the trajectories starting in the segment q1z2q2z2¯ since these produce the largest radius in R1b.Consider the pointsq1z2 and q2z2(84)q1z2=17γ15−4α33γ5−γ10,q2z2=23γ15−4α33γ5γ10,both points produce the same radius with a different angle. However, more oscillation time before reaching SW23 is expected from q1z2. Thus, consider the trajectory with the initial condition in q1z2 and the evolution time that corresponds to 3π/2+0.1651=4.8775, which is an exaggerated angle since SW23 is reached before that.(85)r=ea4.8775/b3γ52+γ102=γea4.8775/b37100.Under Assumption1,(86)γ37100ea4.8775/b≤0.7393γ.Consider the pointsp1z2, pa1z2 and p2z2(87)p1z2=γ5−3γ52α3−17γ30,pa1z2=0−3γ5−2γ−α3,p2z2=−γ5−3γ52α3−23γ30.The minimum radius in the segmentpa1z2,pa2z2¯ is as follows:(88)r=3γ52+2α3−17γ302.Under Assumption1, the radius from the segment pa1z2,pa2z2¯ is as follows:(89)3γ52+2α3−17γ302≥3γ52+−γ22,γ352+−122=0.781γ.Since0.7393γ<0.781γ even when the increment of radius was exaggerated, it is expected that the trajectories with an initial condition in R2 reach R1 or a self-excited attractor. In the same way, the trajectories with an initial condition in R1 reach R2 or go to a self-excited attractor.To verify the region for the parametersa=0.2, b=5, c=−7 and α=1, seven trajectories have been simulated and are shown in Figure 9(a) for γ=10 and γ=100 in Figure 9(b).Figure 9
Seven trajectories of the system given by (1), (3), (5), (13), and (15) starting in R1 with a=0.2, b=5, c=−7, α=1 and different values of γ: (a) γ=10 and (b) γ=100.
(a)(b)Now consider two sets of initial conditions inSW23, I1 and I2, such that subsets Npa and Npc of these sets produce trajectories that end in one of the self-excited attractors. These sets are drawn by circles in Figure 7.It is easy to see that ifγ increases, then the regions R1 and R2 grow, but the subsets of initial conditions in I1 and I2 that reach a self-excited attractor without reaching SW23 again are reduced.Let us look at the system inz2 coordinates, as γ grows, SW23 and pcz2 are further from the zeq2∗ and then it takes more time for the trajectories close to pcz2 to travel along the z12 direction to get close to zeq2∗; however, the expansion along z22 and z32 remains the same; then the subsets of initial conditions that reach the self-excited attractors without reaching SW23 again shrink in I1 and I2 but never disappear. As pc and pa belong to R1 and R2, respectively, then there will always be an intersection of these regions R1 and R2 with the subsets of initial conditions that reaches the self-excited attractors in the regions I1 and I2.This explains why asγ is increased, it is easy to find initial conditions such that the transitory lasts long. Then we come to the conjecture that a necessary condition for the existence of a hidden attractor is that the intersection of regions R1 and R2 with those sets given by I1 and I2 must be empty, i.e., Npa∩R1=∅ and Npc∩R2=∅.


## 5. Emergence of Hidden Attractors

In this section, a modification is made to the previous class of systems to meet the conjecture requirement. A way to produceNpa∩R1=∅ and Npc∩R2=∅ and allow the existence of a hidden attractor is by modifying the commutation surface SW23 between the two self-excited attractors. Consider the following switching planes:(90)SW12=clP1∪clP2=x∈ℝ3:2x1−x3=−2γ,x1<0,SW23=clP2∪clP3=x∈ℝ3:x1=0,SW34=clP3∪clP4=x∈ℝ3:2x1−x3=2γ,x1>0.Note that the switching surfaceSW23 has a new location while the switching surfaces SW12 and SW34 keep their original locations. This new arrangement keeps the existence of the two heteroclinic loops and thus the two self-excited attractors. The new projections of the system in x and z2 coordinates are shown in Figure 10.Figure 10
Projection of the manifolds on (a)x1−x3 and (b) z12−z32. The stable and unstable manifolds are marked with blue and red solid lines, respectively, the switching surfaces with green lines.
(a)(b)To study the emergence of a hidden attractor, the same procedure is followed as in the previous section.Let us find the points inSW23 where the vector fields of P2 and P3 are tangent to the plane SW23. These points can be found from the following equation:(91)1,0,0a3+2c3b2c3−2a3−b3a2b3c3−a3−b2a3+c3−xeqix2x3=−xeqia+2c3+bx2+2c−2a3x3=0,x2=xeqia+2c3b−2c−2a3bx3,withi=1,2.Then for the vector field of P2 we have the following expression:(92)x2=−γ−αa+2c3b−2c−2a3bx3.For the vector field ofP3 the expression is as follows:(93)x2=γ−αa+2c3b−2c−2a3bx3.Consider the point inclWxeq2u∩SW23 that fulfills (92):(94)pt1=0−aγ−αb−γ−α,and in z2 coordinates(95)pt1z2=0aγ−αb−γ−α.If we evaluate the trajectory with initial condition inx0=pt1z2, under the vector field of P2 ignoring the effect of the vector field of P1 and P3, reaches the point pt2z2∈SW23. The flow φ could go from P2 to P3 through the segment pt1z2pt2z2¯. Thus, trajectories with initial condition close to Aself1 but not in the attractor cross SW23 close to the segment pt1z2pt2z2¯, then, R1 should include this segment. However, when the vector field of all atoms is considered, trajectories with initial conditions close to pt1z2 could reach SW23 in points whose second component in z2 coordinates is further from 0 than the second component in z2 coordinates of pt2z2. This allows us to propose the region R1 based on a larger segment pi1z2pi2z2¯ such that pt1z2pt2z2¯⊂pi1z2pi2z2¯. Consider the initial condition piz2 given in z2 coordinates by(96)piz2=00−γ−α,then, the radius with respect to zeq2∗ would be γ−α. Remember that only the vector field of P2 is considered and the trajectory rotates around the axis z12. Let us think in an imaginary and impossible case when a trajectory with an initial condition in piz2 rotates around the axis z12 and reaches SW23, but instead of the normal increment of radius, let us imagine that the increment in radius corresponds to an evolution time t=2π/b (360°). Thus, the z22 component of this point is further from 0 than the z22 component of pt2z2. Then, we could take pi2z2=pt1z2 and find the z22 component of pa1z2 from(97)ea2π/bγ−α2−γ−α2.Consider Assumption1, then,(98)ea2π/bγ−α2−γ−α2≤γea2π/b2−1≤0.80815γ≈4γ5.Remember thatz22=−x2, then, the points pi1 and pi2 are given by(99)pi1=04γ5−γ−α,pi2=0−aγ−αb−γ−α,where −aγ−α/b is the tangent coordinate given by (92) for x3=−γ−α.Let us propose a regionR1 delimited by the following four points:(100)p1=pi1+0,0,γ5T,p2=pi1−0,0,γ5T,p3=pi2+0,−2c−2a3b−γ5,−γ5T,p4=pi2+0,−2c−2a3bγ5,γ5T.Then, the symmetric regionR2 is delimited by the points:(101)q1=−p1,q2=−p2,q3=−p3,q4=−p4.The regionsR1 and R2 have been proposed taking into consideration that pd and pe are part of the regions and pa and pc are not. In Figure 11, R1 and R2 are shown in z2 coordinates.Figure 11
RegionsR1 and R2 on the projection z22−z32.Now, let us analyze some scenarios on these regions to see if the proposed regions are good candidates, at least in an estimated manner. In order to simplify the scenarios, in the following, it is considered that trajectories inP1∪P2 rotate only around the stable manifold of xeq2, i.e., if only the vector field of P2 is considered. Since p1z2−pi and piz2−p2z2 are given by(102)p1z2−pi=2γ15−4γ52γ15,pi−p2z2=2γ154γ52γ15.Let us define the set R1b as follows:(103)R1b=z2∈ℝ3:z12∈−2γ15,2γ15.First, let us verify that the points inR2 go to R1b. The evaluation of the vector field in pj1z2 tells us that the spin is counterclockwise in z2 coordinates.Consider the pointq2z2 given by(104)q2z2=22γ15−4α34γ57γ15−α3.The angle produced by the radius from the pointq2z2 to the z12 axis and the plane z12−z22 is given by(105)arctan7γ/15−α/34γ/5=arctan712−5α12γ.Under Assumption1, the angle obeys the following inequality:(106)arctan712−5α12γ≤arctan712≤0.5281.Consider then that the trajectory with an initial condition inq2z2 evolves for a duration that corresponds to π−20.5281=2.0854. If after this duration the first component of the state vector z12≤2γ/15 means that the trajectory with initial condition in q2z2 reaches R1b. After this duration, z12 is given by(107)z12=22γ15−4α3ec2.0854/b.Under Assumption1 for a big value of z12:(108)22γ15−4α3ec2.0854/b≤22γ15ec2.0854/b≤0.0791γ<2γ15.Thus,q2z2 reaches the region R1b. Moreover, since the points in the segment q2z2q1z2¯ produce a radius whose angle with the plane z12−z22 is between 0 and 0.5281, the trajectories with the initial condition in this segment also reach the set R1b.Now consider the pointq3z2(109)q3z2=22γ15−4α32cγ−17aγ15b+aαb7γ−5α15.The angle of the radius atq3z2 with the plane z12−z22 is as follows:(110)−arctan2cγ−17aγ/15b+aα/b7γ−5α/15=−arctan2cγ−17aγ+15aα7bγ−5bα.Under Assumption1, the angle should be less than(111)−arctan2cγ−17aγ+15aα7bγ−5bα≤−arctan2c6.5b−17a6.5b+1.5a6.5b,−arctan2c6.5b−17a6.5b+1.5a6.5b≤−arctan−46.5−176.525+1.56.525=0.6179.To verify that the trajectory starting inq3z2 is not going to reach SW23 when the radius form an angle of 3π/2 with the plane z12−z22, let us consider the following exaggerated scenario: The radius size corresponds to a duration equivalent to 10π/9 , but the z12 component corresponds to a duration equivalent to 2π/3 of oscillation, i.e., when the radius forms an angle of 3π/2 with the plane z12−z22 a smaller radius than the real one is considered; also, a larger value of z12 than the real value is considered.Then to obtain the radius,(112)r=ea10π/9b2cγ−17aγ15b+aαb2+7γ−5α152.Under Assumption1 for the largest radius,(113)ea10π/9b2cγ−17aγ15b+aαb2+7γ−5α152≤ea10π/9b2cγ−17aγ15b2+7γ152,ea10π/9bγ2c−17a15b2+7152≤0.5984γ.The value ofz12 after the duration that corresponds to 2π/3 is as follows:(114)22γ15−4α3ec2π/3b.Under Assumption1, for the biggest value of z12(115)22γ15−4α3ec2π/3b≤22γ15ec2π/3b,22γ15ec2π/3b≤0.07814γ<2γ15.The points inSW23 fulfills the following equation:(116)z32=−γ−α+z12.At this angle of3π/2 the radius is r=−z32. Then, if the found values for this scenario fulfill the following inequality, it can be concluded that the trajectory with an initial condition in q3z2 does not reach SW23 after a duration that corresponds to an oscillation of 120°:(117)−.5984γ>−γ−α+0.07814γ.Under Assumption 1 and the worst case,(118)−.5984γ>−9γ10+0.07814γ=−0.82186γ.Then, the trajectory remains for the duration that corresponds to2π/3. Furthermore, since z12<2γ/15 the trajectory reaches R1b. Since q3z2 is the point in the set z2∈R2:z22≤0 that produces the largest radius of that set, the same conclusion applies to the points in this set.For the trajectories with the initial condition in the setz2∈R2:z22>0 the duration is equivalent to more than 2π/3. Thus, the trajectories that start in R2 reach the set R1b.Now to verify that the trajectories that start inR2 reach R1⊂R1b is enough to verify the trajectories starting in the segment q1z2q2z2¯, since these produce the largest radius in R1b.Consider the pointsq1z2 and q2z2(119)q1z2=6γ5−4α34γ5γ5−α3,q2z2=22γ15−4α34γ57γ15−α3.The largest radius is atq2z2 while the smaller angle is at q1z2, then let us consider that radius of q2z2 with the angle of q1z2 and the end position at 3π/2 with respect to the plane z12−z22 , which is more than the possible rotation. The angle is given by(120)arctanγ/5−α/34γ/5=arctan14−5α12γ.Under Assumption1, the smallest angle is as follows:(121)arctan14−5α12γ≥arctan524=0.2054.Thus for 3π/2−0.2054=4.507(122)4γ52+7γ15−α32ea4.507/b.Under Assumption1 for the biggest radius,(123)4γ52+7γ15−α32ea4.507/b≤4γ52+7γ152ea4.507/b,4γ52+7γ152ea4.507/b≤γ193225ea4.507/b≤1.1091γ.Consider the pointsp1z2, pa1z2, and p2z2(124)p1z2=2γ15−4γ5α−13γ15,pa1z2=0−4γ5α−γ,p2z2=−2γ15−4γ5α−17γ15.The minimum radius in the segmentpa1z2,pa2z2¯ is as follows:(125)4γ52+α−13γ152.Under Assumption1 for the smallest radius,(126)4γ52+α−13γ152≥4γ52+23γ302=1.1081γ.1.1092γ≈1.1081γ, even when the angle of rotation was exaggerated, it is expected that the trajectories with an initial condition in R1 reach R2 or a self-excited attractor. In the same way, the trajectories with an initial condition in R2 reach R1 or go to a self-excited attractor.Let us look at the system inz2 coordinates, as in the previous section with the previous switching surfaces, as γ grows, SW23 and pcz2 is further from the zeq2∗ and then it take more time for the trajectories close to pcz2 to travel along the z12 direction to get close to zeq2∗. However, the expansion along z22 and z32 remains the same; then the subsets of initial conditions that reach the self-excited attractors without reaching SW23 again shrink in I1 and I2 but this time, as opposed to the previous case there exist a value of γ such that the intersection disappear.Then for a sufficiently big value ofγ we have a region R1 such that any trajectory starting there remains crossing R1 for t>0. Then, we could expect that exists either a periodic orbit, a hidden limit cycle, a hidden chaotic attractor, or a combination of the previous, which should go through R1 and R2.Also, as small differences in the initial conditions inR1 could eventually produce a big separation of trajectories in SW23, sensitivity to initial conditions could also be expected. However, the formal proof is out of the scope of this work.To verify the region for the parametersa=0.2, b=5, c=−7, α=1 and γ=10 seven trajectories have been simulated and are shown in Figure 1(a).The simulations of the two particular cases coincide with the conjecture. In Figure12(b) it is shown the hidden attractor for the parameters a=0.2, b=5, c=−7, α=1 and γ=10 and the initial condition x0=0,0,0T for t∈50000,50100. In Figure 1(c), it is shown the projection of the hidden attractor and the two self-excited attractors onto the plane x1−x2 for the same parameters.Figure 12
In (a), seven trajectories of the system given by (1), (3), (5), (13), and (90) starting in R1 with a=0.2, b=5, c=−7, α=1 and γ=10. In (b), hidden attractor for the same parameters and the initial condition x0=0,0,0T for t∈50000,50100. In (c), the projection of the self-excited attractors and the hidden attractor onto the plane x1−x2.
(a)(b)(c)


## 6. Conclusions

In this work, an approach for the generation of multiscroll attractors was studied based on heteroclinic orbits. Particularly, we presented a quad-scroll self-excited attractor, which is split into two double-scroll self-excited attractor, so the system bifurcates from monostability to biestability. The approach is based on the coexistence of double-scroll self-excited attractors surrounded the equilibria and presenting heteroclinic orbits. Increasing the distances between the double-scroll self-excited attractors generates a heteroclinic-like orbit between the equilibria of two different double-scroll self-excited attractors. It is possible to generate hidden attractors surrounding the self-excited attractors by breaking the heteroclinic-like orbit. The study revealed a relationship between the existence of a hidden attractor and the trajectories that, when are seen on a larger scale resemble heteroclinic orbits which join the self-excited attractors. The findings suggest that new classes of multistable systems with a different number of self-excited and hidden attractors can be designed with a geometric approach.



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