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

Escalante-González, R. J.; Campos, Eric

doi:https://doi.org/10.1155/2020/7832489

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Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability

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Research Article | Open Access

Volume 2020 | Article ID 7832489 | https://doi.org/10.1155/2020/7832489

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

R. J. Escalante-González¹and Eric Campos¹

Guest Editor: Sundarapandian Vaidyanathan

Received04 Jan 2020

Revised26 Feb 2020

Accepted23 Mar 2020

Published14 Apr 2020

Abstract

In this work, we present an approach to design a multistable system with one-directional (1D), two-directional (2D), and three-directional (3D) hidden multiscroll attractor by defining a vector field on with an even number of equilibria. The design of multistable systems with hidden attractors remains a challenging task. Current design approaches are not as flexible as those that focus on self-excited attractors. To facilitate a design of hidden multiscroll attractors, we propose an approach that is based on the existence of self-excited double-scroll attractors and switching surfaces whose relationship with the local manifolds associated to the equilibria lead to the appearance of the hidden attractor. The multistable systems produced by the approach could be explored for potential applications in cryptography, since the number of attractors can be increased by design in multiple directions while preserving the hidden attractor allowing a bigger key space.

1. Introduction

Piecewise linear systems that display scroll attractors have been studied since the publication of the well-known Chua’s circuit. The attractor exhibited by Chua’s circuit is an example of chaotic attractor whose chaotic nature has been explained through the Shilnikov method. Some works have extended this system in order to obtain a greater number of scrolls or different geometries. According to [1], an attractor with three or more scrolls in the attractor is considered a multiscroll attractor. Recently in [2], the generation of scroll attractors via multistable systems have been observed.

According to [3], there are two classes of attractors, one of them is a class called self-excited attractors that includes all the attractors excited by unstable equilibria, i.e., the basin of attraction intersects with an arbitrarily small open neighborhood of equilibria [4]. Examples of this class are the well-known Lorenz attractor [5] and the scroll attractor of Chua’s circuit [6]. The other class is called hidden attractors and their basins of attractions do not contain neighborhoods of equilibria. Some hidden attractors have been studied in [7–19]. There are some works focused on control systems with hidden attractors, as in references [20, 21]. Most works related to multiscroll attractors are based on the first class. Multiscroll attractors have been reported in [12, 13, 18, 19, 22–26]. In [27], a system with a multiscroll chaotic sea was introduced.

Multistability can be considered undesirable for some applications, so some works focus on how to avoid this behavior. For example, in [28], a method that allows to transform a periodic or chaotic multistable system into a monostable was studied, and some experiments were carried out with a fiber laser doped with erbium. However, for some applications it may be considered desirable to be able to switch from monostable to bistable behavior, for example, in [29], a parameterized method to design multivibrator circuits with stable, monostable, and bistable regimes was proposed.

Some works deal with multistable systems with infinite number of equilibria and their electronic realization [30, 31].

A study on the widening of the basin of attraction of a class of piecewise linear (PWL) systems was recently performed in [2]. In this work, a bifurcation from a bistable system with two self-excited double-scroll attractors to a multistable system with two self-excited attractors and one hidden attractor was reported. Other study on the emergence of hidden double-scroll attractors in a class of PWL systems is reported in [32].

Based on the observations made in previous works, a question of whether or not it is possible to generate a hidden multiscroll attractor with scrolls along more than one direction emerges. Depending on the number of directions in which the scrolls in the attractor extend, they are usually referred to as one-directional (1D), two-directional (2D), and three-directional (3D) grid scroll attractors.

Here, we introduce an approach for the construction of multistable PWL systems that exhibit hidden multiscroll attractors with 1D, 2D, and 3D grid arrangements. In Section 2, a system with a chaotic double-scroll self-excited attractor is introduced. In Section 3, the construction is extended to 1D grid scroll self-excited attractor; then, the equilibria are separated by pairs to generate multistable systems with hidden and self-excited attractors. In Section 4, the construction is generalized to 1D grid scroll hidden attractor. In Section 5, the construction is generalized to 2D and 3D grid scroll hidden attractors; in Section 6, conclusions are given.

2. Heteroclinic Chaos

Let be a finite partition of , that is, , and for . The approach to generate hidden attractors is based on the existence of self-excited attractors; thus, the class of systems considered in this work are those that present a saddle equilibrium point in each element of the partition that is called an atom.

We denote the closure of a set as . For each pair of adjacent atoms and , , is the switching surface.

Consider a dynamical system whose dynamic is given bywhere is the state vector and is a linear operator whose matrix is as follows:where , and . Thus, the linear operator has a negative real eigenvalue with the corresponding eigenvector and a pair of complex conjugate eigenvalues with positive real part, and , with the corresponding eigenvectors and , respectively. The eigenvectors are given by

is a constant vector, and is a functional such that is a constant vector in each atom , and there exists an equilibrium point , with , in each atom . Thus, in each atom there exists a saddle equilibrium point with a local stable manifold of dimension one given by . A two-dimensional local unstable manifold is given by .

We begin to explain the generation of chaotic attractors by first considering a partition with two atoms and the constant vector given byand the functional given bywith .

Please note that the vector is the first column of the linear operator , and thus, system (1) can be rewritten as . Then, the functional determines the location of the equilibria along the -axis. So, , .

Proposition 1. If the PWL system (1) is given by (2), (4), and (5), then the functional determines the location of the equilibria along the -axis.

Proof. Let be the linear operator so that each , , is a column vector. Since , then can be rewritten asand thenNow, in order to find the equilibria, we equate the vector field to zero:Since , it follows thatThus, the equilibria is determined by given by (5) along the -axis.
According to , the first components of the equilibria fulfill that .
The switching plane has associated an equation , with , and is the normal vector. Then, the atoms , , are defined as follows:

Assumption 1. The switching plane intersects the -axis at the midpoint between the equilibrium points and .

Proposition 2. Under Assumption 1, each atom given by (10), , contains an equilibrium point of the PWL system (1) given by (2), (4), and (5).

Proof. We want to prove that if a system is given by (1), (2), (4), and (5), then and . From (9) and (5), the two equilibrium points areFrom Assumption 1, the parameter can be defined as , where is the intersection of with the -axis:Thus,from (10) and .

Proposition 3. If , the stable and unstable manifolds of the PWL system (1) given by (2), (4), and (5) intersect at two points given by and .

Proof. The equilibrium points are located at and . In each atom , the equilibrium point has a local stable manifold (14) of dimension one given by and a two-dimensional local unstable manifold given by . Thus, the local manifolds are given as follows:According to (18), the stable and unstable manifolds, and the intersection points are given byThese points belong to .

Assumption 2. The parameters and control the oscillation around the equilibrium point , and we consider .

Proposition 4. The hyperbolic system given by (1), (2), (4), and (5) generates a pair of heteroclinic orbits if the switching surface between the atoms and is given by the plane .

Proof. We want to show that there exist initial conditions , such that two solution curves and of the hyperbolic system given by (1), (2), (4), and (5) fulfill that and as and and as ; in particular, these initial conditions correspond to the intersection points and .
From (2), the linear operator can be expressed aswhere andThe intersection points and belong to and and . Because these points and belong to the stable manifolds and , respectively, they are points whose trajectories remain in atoms and , respectively.
By definition, ; then, the -axis belongs to the plane . The sets , for , can be written as follows:Consider the following changes of coordinates , for . Then, the vector field in coordinates for the space given by the atom is given by , with .
Since , the sets given by (22) in coordinates are given as follows:where is a point on the -axis that corresponds to the transformation of the intersection points to , for .
When , remains in the atom , the transformation of under is . In a similar way, when , , remains in the atom , the transformation of under is . So, belongs to the stable manifold , for ; then, the trajectories when . This implies thatWhen , leaves the atom and enters to atom , the transformation of under to is . In a similar way, when , leaves the atom and enters to atom , the transformation of under to is . Thus, is a point on the axis and belongs to for and .
With the uncoupled system in coordinates, we can analyze the flow on the plane and see how the flow converges at the equilibrium point when :If , then As , so when . Then, the trajectories when . This implies thatThus, the heteroclinic orbits are defined asAs example, consider systems (1), (2), (4), and (5) with the parameters , which fulfills (10) with the switching surface . Then, heteroclinic chaos emerges from this system, a double-scroll attractor is exhibited, as it is shown in Figure 1(b), for the initial condition .

(a)

(b)

3. Multiple Self-Excited Attractors

A study on the widening of the basins of attraction of multistable switching dynamical system with symmetrical equilibria is performed in [2]. In this study, the increment of distance between the equilibria of the self-excited double-scroll attractors increases the basin of attraction of both attractors. The study also reveals that, for the system under study, there is a distance at which a hidden double-scroll attractor emerges.

Based on the idea of generating hidden scroll attractors from sufficiently separated self-excited attractors, we consider now a partition with more atoms along with the PWL system (1), with and given by (2) and (4), respectively. For this partition, the functional is defined in the four atoms as follows:where and . Now, there are four atoms and each atom contains an equilibrium point, , . The first components of the equilibrium points fulfill that , .

There are three switching planes and each switching plane crosses the -axis and is located between the equilibrium points and , .

The switching plane with has associated an equation , where and is the normal vector. Then, in order to know if a point belongs to a the following conditions are considered:

If for a and for a , it follows that .

The equilibria are located on the -axis atso , , , and .

Assumption 3. The distance between the self-excited attractors should be big enough to allow the existence of a hidden double-scroll attractor, so we consider .
Consider three switching surfaces with the following orientation:which fulfill thatThe switching surfaces given in (33) along with the condition given in (34) ensure the existence of the four heteroclinic orbits. Moreover, and .
As example, consider the system with parameters , and , the system presents two self-excited attractors; however, the hidden double-scroll attractor is not present and a transitory double-scroll oscillation is exhibited instead. Figure 2(a) shows in blue the trajectory for the initial condition for a time a.u. (arbitrary units). Thus, an enough separation of the self-excited double-scroll attractors is not sufficient to produce a hidden double-scroll attractor.

(a)

(b)

The absence of a hidden attractor is related to the switching plane and how the trajectories of the transitory double-scroll cross it near the stable manifolds. Consider now the following switching surfaces:

With the change in , the intersections as well as . Furthermore, the distance from to and by symmetry from to allow the trajectories to cross far from the stable manifolds and allowing the existence of the hidden double-scroll attractor.

As example, consider the system given by (1), (2), (4), and (30) with , , , , and and the new switching surfaces given by (35). Figure 2(b) shows in blue the trajectory for the initial condition for a time a.u. The trajectory reaches the hidden double-scroll instead of converging to one of the self-excited attractors. A trajectory has been simulated for a time to verify that the double-scroll oscillation is not a transitory behavior and that the trajectory does not converge to a self-excited attractor.

The construction can be further extended to the number of scrolls desired just by adding two atoms for each scroll; for instance, for a triple scroll attractor the partition is and a possible set of switching surfaces is given bywithThen, the equilibria are located at the -axis with , , , , , and .

4. Generalization

The number of scrolls for a hidden scroll attractor exhibited by the system from the previous section depends on the number of self-excited attractors. In order to simplify the description for a large number of scrolls, a generalization is introduced in this section. Consider a dynamical system whose dynamic is given bywhere is the state vector, is the linear operator given by (2), and is a functional such that is a constant vector in each atom . The saddle equilibrium point of each atom is given by , with . is defined as follows:wherewith . The parameters and fulfill Assumption 3. The function is the Heaviside step function:and is a step function defined as follows:

Note that is equal to , when ; while, for , it is similar to with the only difference that 0 is mapped to 1 instead of .

Please note that is the first column of the operator and therefore system (38) can be rewritten as follows:where is a vector valued function defined as follows:such that is a constant vector in each atom . The saddle equilibrium point of each atom is now given by , with .

To understand the form of , it is useful to separate and analyze the effect of each term of the sum, i.e., the effect of the term and the effect of .

First, consider the function whose plot resembles a stair centered at the origin whose plateaus are of in height and width. Two examples for and are shown in Figure 3.

(a)

(b)

Thus, generates switching planes of the form , which are parallel to the plane . Then, generates a partition of .

Now, consider the term , and this term generates a switching plane for . Since takes values, then generates switching planes, one for each element of the partition . Thus, the elements are split and the partition is generated.

In this way, locates two equilibrium points with a separation of in the middle of each element (along ) for . In the partition , locates an equilibrium point in each for .

Thus, the equilibria along the -axis is located in pairs, each pair of nearby equilibrium points have a separation of . Let us denote the midpoint of the line that joins a pair of nearby equilibria as with . Then, the distance from to is . The purpose of this distribution for the equilibria is to allow the existence of double-scroll self-excited attractors that are separated enough from other double-scroll self-excited attractors in a way that these resemble equilibria for the generation of a bigger scroll attractor at a larger scale. This larger scroll attractor is indeed the hidden scroll attractor.

The equilibria are located along the -axis as follows:

The switching planes located in the middle of the self-excited attractors are given as follows:

The rest of the switching surfaces are

To illustrate the construction, consider the parameters , and , the system presents two self-excited attractors and a hidden double-scroll attractor, which is shown in Figure 4(a).

(a)

(b)

(c)

(d)

According to the definition, the attraction basin of a hidden attractor does not intersect neighborhoods of equilibria. Figure 5 shows the cut of the numerically evaluated basins of attraction given by the plane . Each double-scroll self-excited attractor has its own attraction basin shown in red and green. Also, the attractor around these double-scroll self-excited attractors has its own attraction basin shown in blue and the intersection of this basin with the attraction basins of the self-excited attractor is the empty set. Because all equilibria of the system belong to the attraction basin of the self-excited attractors, the attraction basin of the attractor around the self-excited attractors does not contain an equilibrium point. So, the attractor around the self-excited attractors is a hidden attractor.

(a)

(b)

(c)

Another numerical approach to verify that it is a hidden attractor consist in performing a long-time simulation of a trajectory and make sure that the trajectory does not converge to a self-excited attractor. In Figure 6, it is shown that the simulation for the initial condition with . This last approach requires less computing time, so it was the approach used in all the examples in the manuscript.

For a second example, consider ; then, the system presents five self-excited attractors and a hidden 5-scroll attractor shown in Figure 4(b). Thus, the number of scrolls is equal to the number of self-excited attractors, which is .

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

The approach presented in Section 4 can be further extended for 2D and 3D grid scroll hidden attractors. The idea is to add switching surfaces, which are parallel to the planes and .

This requires the modification on , for simplicity, the functions are written as :wherewhere (52) , , and .

Let us denote the equilibria found in Section 4 as for . Then, the new equilibria for the new in (48) are found fromwhere and . The switching surfaces are now restricted on and and located according to the new equilibria.

As example, consider the parameters , , , and with , , , and for a 2D-grid scroll hidden attractor shown in Figure 4(c) and , , , , and for a 3D-grid scroll hidden attractor shown in Figure 4(d).

6. Conclusions

In this work, the question of whether or not it is possible to generate a hidden multiscroll attractor with an arrangement of scrolls along more than one direction from multiple self-excited attractors was addressed. It was found that the separation between self-excited double-scroll attractors and the switching plane between these self-exited attractors lead to the emergence of a hidden attractor. A generalized construction was proposed for the generation of multistable systems with self-excited double-scroll chaotic attractors and a hidden multiscroll/grid attractor. The coexistence of self-excited attractors and a hidden attractor is presented via PWL systems and the approach considers for each scroll in the hidden attractors a self-excited attractor inside the scroll. As future work, we envision working on the answer to the following question: is it possible to generate multistability with more than one hidden attractor?

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

R.J. Escalante-González is thankful to CONACYT for the scholarships granted. Eric Campos acknowledges CONACYT for the financial support through Project no. A1-S-30433.

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Copyright

Copyright © 2020 R. J. Escalante-González and Eric Campos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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