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

+is work is dedicated to the study of an approach that allows the generation of hidden attractors based on a class of piecewiselinear (PWL) systems. +e systems produced with the approach present the coexistence of self-excited attractors and hidden attractors such that hidden attractors surround the self-excited attractors. +e first part of the approach consists of the generation of self-excited attractors based on pairs of equilibria with heteroclinic orbits.+en, 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 selfexcited attractors.


Introduction
ere 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. e 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. erefore, 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]. is 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]. e 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 fourdimensional 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][16][17][18][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. e 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 selfexcited attractors.
An approach that allows the generation of hidden attractors based on a kind of piecewise-linear (PWL) system is studied in this work. e 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.
e 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]. us, 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. e structure of the article is as follows: In Section 2, a class of piecewise-linear systems with double-scroll selfexcited 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.

Heteroclinic Chaos
To introduce the approach, let us first consider a partition P of the metric space X ⊂ R 3 , endowed with the Euclidean metric d. Let P � P 1 , . . . , P η (η > 1) be a finite partition of X, that is, X � ∪ 1≤i≤η P i , and P i ∩ P j � ∅ 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, P i have a saddle equilibrium point, then within each atom there is a stable manifold and also an unstable manifold. ese stable manifolds W s and unstable manifolds W u are necessary for the mechanism of expansion and contraction present in chaotic dynamics.
Let T: X ⟶ X, with X ⊂ R 3 , be a piecewise-linear dynamical system whose dynamics is given by a family of subsystems of the form T is a constant vector, and f is a functional. e vector f(x)B is a constant vector in each atom P i such that the equilibria are given by x * eq i � (x * Oscillations of the flow around the equilibria x * eq i are desired. Let us assign a negative real eigenvalue λ 1 � c to the complexification of the operator A(A 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 . Additionally, we restrict b/a ≥ 10. us the stable and unstable manifolds are given by W s 2 and v 3 are given as follows: e matrix of the linear operator A is defined as follows: In this work, we denote the local stable and unstable manifolds of an equilibrium point x * eq as W s x * eq and W u x * eq , 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 set P i as cl(P i ). us, for each pair of atoms P i and P j , i ≠ j, if cl(P i ) ∩ cl(P j ) ≠ ∅, then these atoms are adjacent and the switching surface between them is given by the intersection, i.e., SW ij � cl(P i ) ∩ cl(P j ).
Each SW ij has associated an equation of the form is the normal vector. en the atoms P i , i � 1, 2 are defined as follows:

Remark 1.
e 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 a P 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. erefore, consider a partition with two atoms P � P 1 , P 2 , the constant vector B ∈ R 3 is defined as follows: and the functional f is given by with α > 0. So the equilibria are at x * eq 1 � (− α, 0, 0) T ∈ P 1 and x * eq 2 � (α, 0, 0) T ∈ P 2 , and the stable and the unstable manifolds are given by Proposition 1 (see [28,29]). " e hyperbolic system given by (1), (3), (5), and (6) generates a pair of heteroclinic orbits if the switching surface between the atoms P 1 and P 2 is given by the plane SW 12 � x ∈ R 3 : 2x 1 − x 3 � 0 ." e points where the stable and unstable manifolds intersect at SW are given by ese points x in 1 and x in 2 belong to SW 12 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.
us, the heteroclinic orbits are defined as follows:
us, 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 P 1 is as follows: and for P 2 with k ∈ Z + . Example 1. Consider the system (1), (3), (5), and (6) with SW 12 � x ∈ R 3 : 2x 1 − x 3 � 0 and the parameters a � 0.2, b � 5, c � − 3, α � 1. e above-defined system fulfills Proposition 1, so it presents a heteroclinic orbit. From (10) and (11), two initial conditions, x 02 � (0.9999976751050959, 0, 2.3248949041393315e − 6) T , (12) 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 x 0 � (0, 0, 0) T . e unstable manifolds W u e intersection points are given by cl(W s Proposition 2 (see [29]). "If the partition P contains more than two atoms P 1 , P 2 , . . . , P k , with 2 < k ∈ Z + , and each atom is a hyperbolic set defined as above. Furthermore, the atoms by pairs P i and P i+1 fulfill Proposition 1. en, the system generates 2(k − 1) heteroclinic orbits." Proof. A direct consequence of Proposition 1.

Emergence of Multiscroll Attractors through Multiple Heteroclinic Orbits
According to the Proposition 2, 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 partition P � P 1 , P 2 , P 3 , P 4 along with the piecewise-linear dynamical system (1), with A and B given by (3) and (5), respectively. us the function f(x) is defined in the four atoms as follows: where α, c > 0. e equilibria are at so x * eq 1 ∈ P 1 , x * eq 2 ∈ P 2 , x * eq 3 ∈ P 3 and x * eq 4 ∈ P 4 . e location of the equilibria according to the parameters 0 < α and 0 < c is as follows: (i) e equilibria are on the x 1 axis and for α � c the system only have three equilibria. Otherwise, it has four equilibria. (ii) For α < c the distance of the equilibria x * eq 1 and x * eq 4 to the origin O � (0, 0, 0) T are the same d( iv) e other case is when c ≠ 2α, and d(x * eq 1 , x * eq 2 ) � d(x * eq 3 , x * eq 4 ) � 2α, but d(x * eq 2 , x * eq 3 ) ≠ 2α. In this section, we are especially interested in the case of c ≠ 2α such that c > α with switching surfaces given by 4 Complexity which fulfill that is 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., W u ∩ SW 23 as shown in Figure 2. en pa and pb are given as follows: e set cl(W u ) ∩ SW 12 can be written as follows: and the set cl(W u ) ∩ SW 23 can be written as follows: Consider the transformation z (2) is also a point on the plane z (2) 2 − z (2) 3 . en, the set (18) in z (2) coordinates is given by x 02 � (0.9999976751050959, 0, 2.3248949041393315e − 6) T (blue), and in (b) a double-scroll attractor that emerges from a heteroclinic orbit using the following initial condition x 0 � (0, 0, 0) T and the same parameters.
SW 12 Figure 2: Projection of the stable and unstable manifolds and switching planes onto the x 1 − x 3 plane. e 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.
and the set (19) in z (2) coordinates is given by us, the sets (20) and (21) are orthogonal lines to the z (2) 3 axis. e points pa and pb in z (2) coordinates will be denoted as follows: With the uncoupled system in z (2) coordinates, we can analyze the flow on the plane z ( It follows from (22) that if α � c − α, then the points pa z and pb z are at the same distance from z * (2) eq 2 � (0, 0, 0) T . us, from (26), it follows that the trajectories with initial conditions pa z and pb z remain in P 2 for all t < 0.
Our case study is c − α ≠ α, such that c > α. Let us consider the case c − α > α, and it can be seen from (22) that pb z is closer to z (2) eq 2 than pa z , this is, d(pb z , z (2) eq 2 ) < d(pa z , z (2) eq 2 ). en, if c is sufficiently big with respect to α, the trajectory with the initial condition pa z will eventually reach the set given by (20) for t < 0; i.e., the trajectory of the initial condition pa ∈ SW 23 reaches the switching plane SW 12 and not the equilibrium point x * eq 2 . is means that in x coordinates, the heteroclinic orbit from x * eq 2 to x * eq 3 does not exist. Similarly, when pb z is further than pa z from z * (2) eq 2 , this is, d(pa z , z (2) eq 2 ) < d(pb z , z (2) eq 2 ), for c sufficiently small, the trajectory with the initial condition pb z will eventually reach the set given by (21) for t < 0, i.e., the trajectory of the initial condition pb ∈ SW 12 reaches the switching plane SW 23 and not the equilibrium point x * eq 2 . is means that in x coordinates, the heteroclinic orbit from x * eq 2 to x * eq 1 does not exist. e next proposition warranty the existence of heteroclinic orbits when c belongs to an interval of real numbers where the case c − α ≠ α is considered, such that c > α. Proposition 3.
e hyperbolic system given by (1), (3), (5), and (13) with the switching surfaces given in (15) generates six heteroclinic orbits if where Proof. To find the values of c for which these heteroclinic orbits exist, let us assume pa is a point of the heteroclinic orbit joining x * eq 2 and x * eq 3 , i.e., lim Because pa ∈ W s For the other part of the heteroclinic orbit, we analyze the system in z (2) coordinates, we have pa z , pb z , z * (2) eq 2 , and the orbit is given by z (2) (t). We assume that z (2) (0) � pa z , so we want that z (2) (t) remains in P 2 for all t < 0. us, we need to find the first maximum in the component z (2) 3 of the trajectory whose initial condition is pa z for t < 0. According to (22), the third component of pa z and pb z are (2(α − c)/3) < 0 and 0 < (2α/3), respectively. is maximum gives us the intersection point between the trajectory z (2) (t) and the axis z (2) 3 . en we can compare the third component of the trajectory z (2) (t) and the point pb z , in terms of α and c to ensure that z (2) (t) remains in P 2 for all t < 0. e trajectory z (2) (t) for the initial condition z (2) 0 � (z (2) 10 , z (2) 20 , z (2) 30 ) T is as follows: is set of equations is analyzed for t < 0. e same analysis can be done for 0 < t by using the following set of equations: Since we are looking for the first maximum in z (2) en from (33) with the initial condition pa z given in (22), 6 Complexity to find the maximum, we equate to zero (37) us, it turns out that We will call t max the time for the first maximum. us, it follows that then from (34), is maximum z (2) 3 must be part of P 2 , since pb z belongs to P 1 it follows from (22) that (41) Now, let us assume pb is a point of the heteroclinic orbit joining x * eq 2 and x * eq 1 , i.e., Because pb ∈ W s x * eq 1 then lim t⟶∞ φ(pb, t) � x * eq 1 . Following the same procedure described above but looking for a minimum, due to the third component of pb z is 0 < (2α/3). It is found that en from (33) and the point pb z given in (22), is minimum z (2) 3 must be part of P 2 , since pa z belongs to P 3 it follows from (22) e same conclusion applies to the point x * eq 3 due to the symmetry of the system. Finally, the heteroclinic orbit from x * eq 1 to x * eq 2 and the one from x * eq 4 to x * eq 3 are always present in the system as there are no more switching surfaces.
□ To illustrate the effect of the parameters c, α, a, and b on the existence of heteroclinic orbits of the system given by (1) with τ � (arctan(b/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 c are analyzed. e first two correspond to c 1,2 ∈ Γ and the last two correspond to c L,U ∉ Γ:.
(3) In this case c L � α(1 − e − aτ cos(bτ)) ∉ Γ, then there exist four heteroclinic orbits, as shown in Figure 3(c). e green orbit stating close to x * eq 2 cannot reach x * eq 1 and goes to P 3 . In the same way, the yellow orbit starting close to x * eq 3 cannot reach x * eq 4 and goes to P 2 . en there is no heteroclinic orbits from x * eq 2 to x * eq 1 and from x * eq 3 to x * eq 4 . (4) For c U � (α(e − aτ cos(bτ) − 1)/e − aτ cos(bτ)) ∉ Γ, there also exist four heteroclinic orbits as shown in Figure 3(d). e red orbit stating close to x * eq 2 cannot reach x * eq 3 and goes to P 1 . In the same way, the blue orbit starting close to x * eq 3 cannot reach x * eq 2 and goes to P 4 . en there is no heteroclinic orbit from x * eq 2 to x * eq 3 , nor vice versa. e open interval Γ is given as e four cases mentioned generate three types of systems determined by c and Γ. For instance, for c � 2 ∈ Γ and α � 1 Complexity 7 corresponds to the above first and second cases. en the system presents six heteroclinic orbits which comprise three heteroclinic loops between equilibria: x * eq 1 and x * eq 2 ; x * eq 2 and x * eq 3 ; x * eq 3 and x * eq 4 . For c � 1.5 < c L , then c ∉ 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 x * eq 2 and x * eq 3 . For c U < c � 3, then c ∉ 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: x * eq 1 and x * eq 2 ; x * eq 3 and x * eq 4 . e above three cases generate self-excited attractors as shown below:.
(1) For c � 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. us the attractor shown in Figure 4(a) is a multiscroll attractor. e scrolls are generated around each equilibrium point of the system x * eq i , with i � 1, 2, 3, 4.
(2) For c � 3, c > c U , then c ∉ Γ. e 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 x * eq 1 and x * eq 2 , while the other self-excited attractor oscillates around equilibria x * eq 3 and x * eq 4 . e basin of attraction of each self-excited attractor has surrounded both attractors.
(3) For c � 1.5, c < c L , then c ∉ Γ. e 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. e heteroclinic orbits are shown in Figure 4(f ). e double-scroll self-excited attractor oscillates around equilibria x * eq 2 and x * eq 3 .
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 c > c U 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 c value is increased, then also the distance between the two self-excited attractors increases. is provokes that some initial conditions in the basins of attraction of both attractors generate transitory oscillations resembling a double-scroll attractor. However, e equilibria are at Figure 5(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 x * eq 1 and x * eq 2 . Increasing the value of c to 15, the equilibria are located at And the transitory time to reaches the self-excited attractor is increased. In Figure 5(b), the trajectory is shown for t ∈ [0, 60]a.u. Now, for c � 100 the equilibria are located at e transitory time lasts longer for the same initial condition. Figure 5(c) shows the transitory oscillations of the trajectory for t ∈ [0, 300]a.u. After a long time, the trajectory reaches a double-scroll self-excited attractor around equilibria x * eq 1 and x * eq 2 ; see Figure 5(d) for t ∈ [356.6, 400]a.u. Continuing to increase the value to c � 1000, then this sets the equilibria at Figure 5(e) shows the transitory oscillation of the trajectory when t ∈ [0, 300]a.u., again, transitory time increases, and after this long time, the trajectory again reaches a double-scroll self-excited attractor around equilibria x * eq 1 and x * eq 2 , see 5f for t ∈ [3091, 3200]a.u. In brief, for c � 5 it took the trajectory around 35 a.u. To converge to a self-excited attractor, for c � 15 around 50 a.u. To converge, for c � 100 around 350 a.u. And for c � 1000 around 3090 a.u. us, transitory time seems to increase for some initial conditions when c increases. In all the cases, the trajectories reach a self-excited attractor.

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. e idea is to estimate two regions R 1 , R 2 ⊂ SW 23 , such that any trajectory φ(x 0 ), with x 0 ∈ R 1 ∪ R 2 , will eventually go to the self-excited attractor A self1 or A self2 . ese regions are symmetric with respect to the origin and are crossed by the trajectories of the transitory double scroll.
Consider the point pa and its symmetric point pc � − pa: ese points are the intersections of the local manifolds in SW 23 and are shown in Figure 6. e trajectories with initial conditions in the points pa, pc ∈ SW 23 converge to equilibria x eq 3 and x eq 2 , respectively. So, the transient oscillation of the trajectory that resembles a double-scroll attractor interferes when the trajectory reaches neighborhoods N(pa) ⊂ SW 23 and N(pc) ⊂ SW 23 around pa or pc, respectively, because each trajectory with initial condition in N(pa) or N(pc) is led to one of the self-excited attractors A self1 or A self2 , respectively. So, the aim is to visualize the route to a self-excited attractor when N(pa) ∩ R 1 ≠ ∅ and N(pc) ∩ R 2 ≠ ∅.
us, the study in this section has the following structure: First, two regions R 1 and R 2 are estimated. e 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.
To start the analysis, let us find the points in SW 23 where the vector fields of P 2 and P 3 are tangent to the plane SW 23 . ese points will be called tangent points and can be found from the following equation: It follows that For the vector field of P 2 , x 1eq 2 � − c + α, then while for the vector field of P 3 , x 1eq 3 � c − α: According to (16), if x 3 > 0 then SW 23 belongs to P 2 , and the tangent points to consider in SW 23 for x 3 > 0 are given by (56). And if x 3 ≤ 0, then SW 23 belongs to P 3 and the tangent points are given by (57). An illustration of the tangent points in SW 23 is shown in Figure 7, where the points for P 2 are indicated by a dotted line, while for P 3 are drawn as a continuous line.

Complexity
As a starting point to propose the region R 1 defined by four points p 1 , . . . , p 4 , consider the point in cl(W u x eq 2 ) ∩ SW 23 given by (19) that fulfills (56): and in z (2) coordinates e points in z (2) coordinate system are denoted with the suffix z 2 . If we evaluate the trajectory with an initial condition in x 0 � pt 1 z 2 , under the vector field of P 2 and ignoring the effect of the vector field of P 1 and P 3 , reaches the point pt 2 z 2 ∈ SW 23 . e flow φ could go from P 2 to P 3 through the segment pt 1 z 2 pt 2 z 2 .
us, trajectories with Figure 6: Projection of the manifolds on (a)( plane. e stable and unstable manifolds are marked with blue and red solid lines, respectively, the switching surfaces with green lines. x 1 12 Complexity initial condition are close to A self1 but not in the attractor cross SW 23 close to the segment pt 1 z 2 pt 2 z 2 then R 1 should include this segment. However, when the vector field of all atoms is considered, trajectories with initial conditions close to pt 1 z 2 could reach SW 23 in points whose second component in z (2) coordinates are further from 0 than the second component in z (2) coordinates of pt 2 z 2 . us, let us propose the region R 1 based on a larger segment pa 1 z 2 pa 2 z 2 such that pt 1 z 2 pt 2 z 2 ⊂ pa 1 z 2 pa 2 z 2 . Consider the initial condition paz 2 given in z (2) coordinates by then, the radius with respect to z * eq 2 would be (2(c − α)/3). Remember that only the vector field of P 2 is considered and the trajectory rotates around the axis z (2) 1 . Let us think in an imaginary and impossible case when a trajectory with an initial condition in paz 2 rotates around the axis z (2) 1 and reaches SW 23 , 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°). us, the z (2) 2 component of this point is further from 0 than the z (2) 2 component of pt 2 z 2 . en, we could take pa 2 z 2 � pt 1 z 2 and find the z (2) 2 component of pa 1 z 2 from ���������������������������� (61) By using Assumption 1, Remember that z (2) 2 � − x 2 , then, the points pa 1 and pa 2 are given by where − (2a(c − α)/3b) is the tangent coordinate given by (56) for x 3 � − (2(c − α)/3). So, let us propose the region R 1 delimited by the following four points: Because R 2 is symmetric to R 1 with respect to the origin, then the symmetric region R 2 is delimited by the points: (65) ese regions R 1 and R 2 have been proposed taking into consideration that points p d and p e shown in Figure 6(b) are part of the regions. In Figure 8, R 1 and R 2 are shown in z (2) coordinates. Now, let us analyze some scenarios on these regions to see if the proposed regions are good candidates, at least in an Complexity 13 estimated manner. In order to simplify the scenarios, in the following, it is considered that trajectories in P 1 ∪ P 2 rotate only around the stable manifold of x eq 2 , i.e., if only the vector field of P 2 is considered. Let us define the set R 1 b as follows: First, let us verify that the points in R 2 go to R 1 b. e evaluation of the vector field in pc 1 z 2 tells us that the spin is counterclockwise in z (2) coordinates. From Figures 8  and 6(b), it is not hard to see that the points below the segment pcz 2 pc 1 z 2 produce trajectories that can perform a turn of π around the z (2) 1 axis without reaching SW 23 again. e time that corresponds to a turn of 2π is T � 2π/b. e point pc 1 z 2 is given by Consider the trajectory with an initial condition in pc 1 z 2 , and an evolution time that corresponds to a turn of π around the z (2) 1 axis. After this time, the first component of the state vector can be found from If z (2) 1 ≤ c/5 means that the trajectory with an initial condition in pc 1 z 2 reaches R 1 b. Consider Assumption 1 for a big value of z (2) 1 (when c is too big): us the set z (2) ∈ R 2 : z (2) 2 ≥ 0, z (2) 3 ≤ 0 reaches the set Now consider the point q 2 z 2 given by e angle produced by the radius from the point q 2 z 2 to the z (2) 1 axis and the plane z (2) 1 − z (2) 2 is given by Let us consider that the trajectory with the initial condition in q 2 z 2 evolves for a duration time that corresponds to π − 2(0.1654) � 2.8113. e first component of the state vector after this duration is given by As before, if z (2) 1 ≤ c/5 means that the trajectory with an initial condition in q 2 z 2 reaches R 1 b. Consider again Assumption 1 for a big value of z (2) 1 : us, q 2 z 2 reaches the region R 1 b. Moreover, since the points in the segment q 2 z 2 pc 1 z 2 produce a radius whose angle with the plane z (2) 1 − z (2) 2 is between 0 and 2.8113, the trajectories starting in this segment also reach the set R 1 b. Now consider the point pcuz 2 given by e trajectories with the initial condition in the segment pcz 2 pcuz 2 can turn π/2 without reaching SW 23 . us, let us consider that the trajectory with the initial condition in pcuz 2 evolves for a duration time that corresponds to π/2. e first component of the state vector after this duration is given by If z (2) 1 ≤ c/5 means that the trajectory with an initial condition in pcuz 2 reaches R 1 b, consider Assumption 1 for a big value of z (2) 1 : To P 2 To P 3 14 Complexity us the trajectories with the initial condition in the set z (2) ∈ R 2 : z (2) 2 ≥ 0, z (2) 3 ≥ 0 converge to the set R 1 b. Now consider the point q 3 z 2 To estimate if that the trajectory starting in q 3 z 2 is not going to reach SW 23 when the radius form an angle of 3π/2 with the plane z (2) 1 − z (2) 2 , let us consider the following exaggerated scenario: e radius size corresponds to a duration equivalent to 3π/2 , but the z (2) 1 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 z (2) 1 − z (2) 2 , a smaller radius than the real one is considered; also, a larger value of z (2) 1 than the real value is considered. en to obtain the radius: In z (2) coordinates, a specific value of z (2) 1 SW 23 fulfills z (2) 3 � − (2(c − α)/3) + (z (2) 1 /2). At this angle of 3π/2 the radius is r � − z (2) 3 . en, if the values found for this scenario fulfill the following inequality, it can be concluded that the trajectory with an initial condition in q 3 z 2 does not reach SW 23 after a duration that corresponds to an oscillation of π/2: under Assumption 1 for the worst case en, the trajectory remains for the duration that corresponds to π/2. Also, the trajectory does not reach SW 23 when the radius is at an angle of 3π/2 with the plane z (2) 1 − z (2) 2 even when the radius growth is exaggerated. It can be concluded that the trajectory with the initial condition in q 3 z 2 could reach SW 23 until the second time it approaches SW 23 and reaches R 1 b.
Since q 3 z 2 is the point in the set z (2) ∈ R 2 : z (2) 2 ≤ 0 that produces the largest radius of that set, the trajectories with the initial condition in this set also reach R 1 b. en the trajectories starting at R 2 reach R 1 b. Now to estimate if the trajectories that start in R 2 reach R 1 ⊂ R 1 b is enough to verify the trajectories starting in the segment q 1 z 2 q 2 z 2 since these produce the largest radius in Consider the points q 1 z 2 and q 2 z 2 Complexity both points produce the same radius with a different angle. However, more oscillation time before reaching SW 23 is expected from q 1 z 2 . us, consider the trajectory with the initial condition in q 1 z 2 and the evolution time that corresponds to 3π/2 + 0.1651 � 4.8775, which is an exaggerated angle since SW 23 is reached before that.
Consider the points p 1 z 2 , pa 1 z 2 and p 2 z 2 e minimum radius in the segment pa 1 z 2 , pa 2 z 2 is as follows: Under Assumption 1, the radius from the segment pa 1 z 2 , pa 2 z 2 is as follows: Since 0.7393c < 0.781c even when the increment of radius was exaggerated, it is expected that the trajectories with an initial condition in R 2 reach R 1 or a self-excited attractor. In the same way, the trajectories with an initial condition in R 1 reach R 2 or go to a self-excited attractor.
To verify the region for the parameters a � 0.2, b � 5, c � − 7 and α � 1, seven trajectories have been simulated and are shown in Figure 9(a) for c � 10 and c � 100 in Figure 9(b). Now consider two sets of initial conditions in SW 23 , I 1 and I 2 , such that subsets N(pa) and N(pc) of these sets produce trajectories that end in one of the self-excited attractors. ese sets are drawn by circles in Figure 7.
It is easy to see that if c increases, then the regions R 1 and R 2 grow, but the subsets of initial conditions in I 1 and I 2 that reach a self-excited attractor without reaching SW 23 again are reduced.
Let us look at the system in z (2) coordinates, as c grows, SW 23 and pcz 2 are further from the z * eq2 and then it takes more time for the trajectories close to pcz 2 to travel along the z (2) 1 direction to get close to z * eq2 ; however, the expansion along z (2) 2 and z (2) 3 remains the same; then the subsets of initial conditions that reach the self-excited attractors without reaching SW 23 again shrink in I 1 and I 2 but never disappear. As pc and pa belong to R 1 and R 2 , respectively, then there will always be an intersection of these regions R 1 and R 2 with the subsets of initial conditions that reaches the self-excited attractors in the regions I 1 and I 2 .
is explains why as c is increased, it is easy to find initial conditions such that the transitory lasts long. en we come to the conjecture that a necessary condition for the existence of a hidden attractor is that the intersection of regions R 1 and R 2 with those sets given by I 1 and I 2 must be empty, i.e., N(pa) ∩ R 1 � ∅ and N(pc) ∩ R 2 � ∅.

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 produce N(pa) ∩ R 1 � ∅ and N(pc) ∩ R 2 � ∅ and allow the existence of a hidden attractor is by modifying the commutation surface SW 23 between the two self-excited attractors. Consider the following switching planes: 16 Complexity Note that the switching surface SW 23 has a new location while the switching surfaces SW 12 and SW 34 keep their original locations. is new arrangement keeps the existence of the two heteroclinic loops and thus the two self-excited attractors. e new projections of the system in x and z (2) coordinates are shown in Figure 10.
To study the emergence of a hidden attractor, the same procedure is followed as in the previous section.
Let us find the points in SW 23 where the vector fields of P 2 and P 3 are tangent to the plane SW 23 . ese points can be found from the following equation: en for the vector field of P 2 we have the following expression: For the vector field of P 3 the expression is as follows: Consider the point in cl(W u x eq 2 ∩ SW 23 ) that fulfills (92): and in z (2) coordinates  If we evaluate the trajectory with initial condition in x 0 � pt 1 z 2 , under the vector field of P 2 ignoring the effect of the vector field of P 1 and P 3 , reaches the point pt 2 z 2 ∈ SW 23 . e flow φ could go from P 2 to P 3 through the segment pt 1 z 2 pt 2 z 2 . us, trajectories with initial condition close to A self1 but not in the attractor cross SW 23 close to the segment pt 1 z 2 pt 2 z 2 , then, R 1 should include this segment. However, when the vector field of all atoms is considered, trajectories with initial conditions close to pt 1 z 2 could reach SW 23 in points whose second component in z (2) coordinates is further from 0 than the second component in z (2) coordinates of pt 2 z 2 . is allows us to propose the region R 1 based on a larger segment pi 1 z 2 pi 2 z 2 such that pt 1 z 2 pt 2 z 2 ⊂ pi 1 z 2 pi 2 z 2 . Consider the initial condition piz 2 given in z (2) coordinates by then, the radius with respect to z * eq 2 would be (c − α). Remember that only the vector field of P 2 is considered and the trajectory rotates around the axis z (2) 1 . Let us think in an imaginary and impossible case when a trajectory with an initial condition in piz 2 rotates around the axis z (2) 1 and reaches SW 23 , 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°). us, the z (2) 2 component of this point is further from 0 than the z (2) 2 component of pt 2 z 2 . en, we could take pi 2 z 2 � pt 1 z 2 and find the z (2) segment q 1 z 2 q 2 z 2 , since these produce the largest radius in Consider the points q 1 z 2 and q 2 z 2 e largest radius is at q 2 z 2 while the smaller angle is at q 1 z 2 , then let us consider that radius of q 2 z 2 with the angle of q 1 z 2 and the end position at 3π/2 with respect to the plane z (2) 1 − z (2) 2 , which is more than the possible rotation. e angle is given by 1.1092c ≈ 1.1081c, even when the angle of rotation was exaggerated, it is expected that the trajectories with an initial condition in R 1 reach R 2 or a self-excited attractor. In the same way, the trajectories with an initial condition in R 2 reach R 1 or go to a self-excited attractor.
Let us look at the system in z (2) coordinates, as in the previous section with the previous switching surfaces, as c grows, SW 23 and pcz 2 is further from the z * eq2 and then it take more time for the trajectories close to pcz 2 to travel along the z (2) 1 direction to get close to z * eq2 . However, the expansion along z (2) 2 and z (2) 3 remains the same; then the subsets of initial conditions that reach the self-excited attractors without reaching SW 23 again shrink in I 1 and I 2 but this time, as opposed to the previous case there exist a value of c such that the intersection disappear.
en for a sufficiently big value of c we have a region R 1 such that any trajectory starting there remains crossing R 1 for t > 0. en, 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 R 1 and R 2 .

Complexity 21
Also, as small differences in the initial conditions in R 1 could eventually produce a big separation of trajectories in SW 23 , 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 parameters a � 0.2, b � 5, c � − 7, α � 1 and c � 10 seven trajectories have been simulated and are shown in Figure 1(a). e simulations of the two particular cases coincide with the conjecture. In Figure 12(b) it is shown the hidden attractor for the parameters a � 0.2, b � 5, c � − 7, α � 1 and c � 10 and the initial condition x 0 � (0, 0, 0) T 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 x 1 − x 2 for the same parameters.

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. e approach is based on the coexistence of double-scroll selfexcited attractors surrounded the equilibria and presenting heteroclinic orbits. Increasing the distances between the double-scroll self-excited attractors generates a heterocliniclike orbit between the equilibria of two different doublescroll self-excited attractors. It is possible to generate hidden attractors surrounding the self-excited attractors by e 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. e 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.
Data Availability e data used to support the findings of this study are included within the article.

Disclosure
A preprint of this study has previously been published (https://arxiv.org/abs/1908.03789).

Conflicts of Interest
ere are no conflicts of interest regarding the publication of this paper.