A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems

Chaos, Solitons & Fractals 71 (2015) 100–106

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems B. Aguirre-Hernández a, E. Campos-Cantón b,⇑, J.A. López-Renteria c, E.C. Díaz González a a Departamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 México, D.F., Mexico b División de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica A.C., Camino a la Presa San José 2055 col. Lomas 4a Sección, 78216 San Luis Potosí, SLP, Mexico c Departamento de Matemáticas, División de Ciencias Exactas y Naturales, Universidad de Sonora, Boulevard Rosales y Luis Encinas s/n, Col. Centro, 83000 Hermosillo, Sonora, Mexico

a r t i c l e

i n f o

Article history: Received 2 August 2014 Accepted 18 December 2014 Available online 10 January 2015

a b s t r a c t In this paper, we consider characteristic polynomials of n-dimensional systems that determine a segment of polynomials. One parameter is used to characterize this segment of polynomials in order to determine the maximal interval of dissipativity and unstability. Then we apply this result to the generation of a family of attractors based on a class of unstable dissipative systems (UDS) of type affine linear systems. This class of systems is comprised of switched linear systems yielding strange attractors. A family of these chaotic switched systems is determined by the maximal interval of perturbation of the matrix that governs the dynamics for still having scroll attractors. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Let us to consider a linear system capable of displaying multiscroll attractors with chaotic behavior when is driven by the suitable control law:

x_ ¼ Ax

ð1Þ

where x 2 Rn is the state vector and A 2 Rnn determines the dynamics of the system. A great deal of qualitative information about the behaviors of its solutions is determined by the linear operator A of (1) and specifically its eigenvalues K ¼ fk1 ; . . . ; kn g. An equilibrium point x of (1) is called saddle if it is a hyperbolic equilibrium point in the sense that A has at least one eigenvalue with positive real part and at least one with negative real part but no eigenvalues having zero real parts. Saddle equilibria, which ⇑ Corresponding author. E-mail addresses: [email protected] (B. Aguirre-Hernández), [email protected] (E. Campos-Cantón), [email protected] (J.A. López-Renteria), [email protected] (E.C. Díaz González). http://dx.doi.org/10.1016/j.chaos.2014.12.012 0960-0779/Ó 2015 Elsevier Ltd. All rights reserved.

connect the stable and unstable manifolds, W s and W u , respectively, are responsible for successive stretching and folding therefore play an important role in generating chaos, particularly saddle-focus equilibria. The stretching causes the system trajectories to exhibit sensitive dependence on initial conditions whereas the folding creates the complicated microstructure [1]. The saddlefocus equilibria of a chaotic systems in R3 can be characterized  into two types according to their eigenvalues K ¼ ki ; kj ; kk  C: (i) the saddle-focus equilibria that is stable in one of its components but unstable and oscillatory in the other two ones [2]. That is, the stable component corresponds to a negative real eigenvalue (Refki g < 0 and Imfki g ¼ 0), whereas the unstable components are related to two complex conjugate eigenvalues (Refkk g > 0 and Imfkk g – 0). (ii) The saddle-focus equilibria that are stable in two of its components but unstable in the other one [3]. That is, the dissipative components are oscillatory: Imfkk g – 0 and Refkk g < 0, while the unstable component corresponds to the real positive eigenvalue Refkk g > 0 and Imfkk g ¼ 0.

B. Aguirre-Hernández et al. / Chaos, Solitons & Fractals 71 (2015) 100–106

If system (1) has a saddle-focus equilibrium which is responsible for stable and unstable manifolds and the sum of its eigenvalues is negative, then the system is called an unstable dissipative system (UDS). According to the above discussion, it is possible to define two types of UDS, and two types of corresponding equilibria in R3 . Definition 1.1. Consider the system (1) in R3 with eigenP values ki ; i ¼ 1; 2; 3 such that 3i¼1 ki < 0. Then the system is said to be an UDS of type I (UDS-I) if one of its eigenvalues is negative real and the other two are complex conjugate with positive real part; and it is said to be of type II (UDS-II) if one of its eigenvalues is positive real and the other two are complex conjugate with negative real part. For the corresponding equilibria, their two types are defined accordingly. The above definitions imply that the UDS-I is dissipative in one of its components but unstable in the other two, which are oscillatory. The converse is the UDS-II, which are dissipative and oscillatory in two of its components but unstable in the other one. Some hyperbolic chaotic dynamical systems in R3 may be related to these two types of UDS around equilibria; for instance, Chua’s system [4] and another piecewise linear system [5] and nonlinear systems [6–9]. Switched systems have been widely used in many different areas in science. Some recent analysis have been made regarding their stability (see [10–12], and the references therein). There is interest in generating chaotic or hyperchaotic attractors with multiple scroll based on UDS-I and synchronization phenomena have been studied between them, for example, in [13] it has been studied preservation of scrolls via chaotic synchronization. Since the work reported by [14] about n-Double scroll from the Chua’s system [15,16], there have been many different approaches to yield multi-scroll attractors in the last coupled decades. These approaches may be ranged from modifying the Chua’s system by replacing the nonlinear part with different nonlinear functions [14,17–19], to the use of nonsmooth nonlinear functions such as, hysteresis [20,21], saturation [22,23], saw-tooth [24], threshold and step functions [25–28]. Multiscroll attractors based on the Jerk equation are given in [29]. Recently, it has been reported the generation of multi-scroll attractors in fractional order chaotic systems [30,31], and based on nonlinear systems [32]. On the other hand, even though there are a lot of approaches to generate multiscroll chaotic attractors, there are few works to generate families of multiscroll chaotic attractors. For example in [27] a family of multiscroll attractors was presented based on a system given by (1) and applying different controls signals. Therein families into three categories called 1D-, 2D- and 3D-grid multiscroll attractors were generated and the families consist on the possibility to increase the number of scrolls in all state variable directions. In [5] a family of multiscroll attractors was presented as a monoparametric family due to with only one parameter the system generates a single, double o tripled scrolls attractors. Recently in [33,34] a hyperchaotic family of multiscroll attractors was introduced defining a multiscroll system in R3 and this is taken

101

as a subsystem for multiscroll systems in Rn , with 4 6 n 2 N. Other interesting works have developed a systematic methodology for constructing continuous-time autonomous hyperchaotic systems with multiple positive Lyapunov exponents [35,36]. Now in this work, we are introducing a family of multiscroll attractors based on a family of polynomials. That is, consider a system as (1) with characteristic polynomial p0 and UDS-I, so the interest is to generate a polynomial family Pðt; kÞ ¼ p0 þ kp1 such that support a new system given by (1) but remaining UDS-I. The background of the theory of polynomial families Pðt; kÞ can be studied in [37–39] and references therein. The aim is to find the maximal unstability interval, i.e., the maximal robust dynamics interval (MDI) that ensures saddle equilibrium of system (1) in order to connect the stable and unstable manifolds, W s and W u , respectively. Thus these manifolds are responsible of successive stretching and folding to generate chaos. From the application point of view, chaos has been successfully employed towards encryption algorithms [40,41] and communication systems [42]. Communication systems based on parametric modulation of chaotic systems which consists of modulating the information signal in a parameter of the chaotic system without destroying the chaotic dynamics. This kind of communication systems needs an interval of values for the parameter to be modulated with the information. Thus, one of the applications of a monoparametric family of chaotic attractors via switched linear systems may be in the area of communication systems. Also several kinds of chaotic synchronization phenomena have been reported [13,43,44]. For example, in [45] a time-varying complex dynamical network model is introduced and synchronization phenomenon is investigated. In this work, we contribute to the generation of a family of UDS by constructing a monoparametric family of chaotic attractors based on a class of dynamical systems. This class of dynamical systems results of the combination of two unstable ‘‘one-spiral’’ trajectories. The article is organized as follows: Section 2 contains the preliminaries of families of chaotic attractors generated by switched linear systems. Section 3 describes the theory to generate families of polynomials in order to find the maximal robust dynamics interval (MDI). Section 4 contains the mechanism to generate a monoparametric family of multiscrolls attractors based on unstable dissipative systems (UDS); finally conclusions are drawn in Section 5. 2. Statement of the problem Consider a control system as follows:

x_ ¼ Ax þ Bu;

ð2Þ nm

where B 2 R and the pair ðA; BÞ is in canonical controllable form and u 2 Rm is the control signal. We consider that the system (2) is a UDS-I for u ¼ 0 and its characteristic polynomial is p0 . It is possible to generate a characteristic polynomial of a family of linear systems as follows Pðt; kÞ ¼ p0 ðtÞ þ kp1 ðtÞ, where k is a real parameter, p0 ðtÞ is the characteristic polynomial of A and p1 ðtÞ ¼ cn t n þ cn1 tn1 þ    þ c1 is an arbitrary polynomial. Therefore, the dynamics of a family of linear systems given by (2),

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for u ¼ 0, is governed by the polynomial family Pðt; kÞ. If k ¼ 0 then the behavior of the solutions for the system is given by its characteristic polynomial p0 ðtÞ. Thus, we can perturb the variable k around zero to keep the same stability and unstability that the original dynamics. Following this concept, a question arises about how can perturb the k parameter to preserve the dynamics of the system? i.e., we are interested in knowing the maximum range of the k parameter such that the system remains UDS-I. The case for p0 ðtÞ stable is called the problem of finding the maximal stability interval and was studied by Bialas around 1985 [37,46] and recently by López-Renteria et al. in [47]. Using these ideas, we generate and characterize the dynamics of systems according to their equilibria. The description of the characterization is given as follows: Suppose that the monoparametric family of systems

x_ ¼ Fk ðA; B; p1 ; x; uÞ

ð3Þ

DS1: In open loop is dissipative and unstable, that is, the origin is a saddle hyperbolic point. DS2: Is completely controllable. Respect to the hyperbolicity, it is required to be such that Að0Þ has n1 eigenvalues in C and n  n1 eigenvalues in Cþ with non pure imaginary eigenvalues. 3. The maximal UDS interval The aim is to establish a result for a family of polynomials Pðt; kÞ ¼ p0 ðtÞ þ kp1 ðtÞ for which p0 ðtÞ has n1 roots in C and n  n1 roots in Cþ for all k in the maximal robust  þ dynamics interval, ðkmin ; kmax Þ. Suppose that the family Pðt; kÞ satisfies the following assumptions. Assumption 3.1. Pðt; kÞ is a family of polynomials of

for k ¼ k0 has n1 eigenvalue in C and n  n1 eigenvalues in Cþ . By continuity of the coefficients of the polynomial p0 ðtÞ, the perturbation of k around k0 gives us the following definition.

1. fixed degree n, 2. coefficients are continuous with respect to k 2 I ¼ ½a; b.

Definition 2.1. We shall call a robust dynamics interval of the monoparametric family (3) to an interval ½a; b around k0 if the characteristic polynomial pk ðtÞ has n1 roots in C and n  n1 roots in Cþ for all k 2 ½a; b.

Therein we compute the maximal unstable and dissipative intervals and consequently, the maximal UDS interval ðk; kÞ. Finally, we study a family of attractors based on a class of unstable dissipative systems. Thus we start by giving some important theorems that help us to establish the maximal unstable dissipative interval.

Thus, the following task is the finding of the maximal robust dynamics interval, that is, find an interval for k around some k0 where the system keeps its dynamics. Under the above scheme, another important related problem is the control design in order to generate chaotic multiscroll attractors by unstable dissipative switching systems of type linear affine with multiple equilibria x ¼ A1 Bu 2 X  Rn which they are saddle hyperbolic points with stable and unstable manifolds W sx and W ux , respectively. The dynamics of the linear system (2) is characterized by the set of eigenvalues K : specðAÞ. Thus, the characteristic polynomial p0 ðtÞ of A is restrained to take values that satisfies hypothesis for systems of either type I or II given in Definition 1.1. We are interested in providing a family of piecewise affine continuous system in the following way: We shall design a monoparametric family of systems x_ ¼ AðkÞx þ Bu; k 2 R, governed by a switching law

8 u1 > > > > > . > > . > > <. u ¼ uj > > > > .. > > . > > > : ur

in Dx1 ;

in Dxj ;

ð4Þ

Theorem 3.2 (Boundary crossing theorem). Under Assumption 3.1, if Pðt; aÞ has all its roots in S  C whereas Pðt; bÞ has at least one root in U ¼ C  S. Then there exists at least one q in ða; b such that  Pðt; qÞ has all its roots in S [ @S,  Pðt; qÞ has at least one root in @S. where @S denotes the boundary of S. For a specific polynomial Pðt; k0 Þ, with k0 2 ½a; b, its coefficients depend continuously on the parameter vector ^cðk0 Þ 2 Rn , then the parameter vectors ^cðkÞ associated to the family of polynomials Pðt; kÞ vary in a set X 2 Rn . If Pðt  ; k0 Þ ¼ 0 then t is a root, thus the set of roots Kk0 ðtÞ :¼ ft jPðt; k0 Þ ¼ 0g is comprised of the eigenvalues of the system. Theorem 3.3 (Zero exclusion principle). Under Assumption 3.1, the polynomial family Pðt; kÞ contains at least one stable polynomial, and X is pathwise connected. Then the entire family is stable if and only if

0 R Pðt  ; kÞ;

8 t 2 @S:

ð5Þ

As well as its generalization presented in [47].

in Dxr :

that generate multi-scroll attractors, with ui on a domain Di  Rm , i ¼ 1; 2; . . . ; r, with \ri¼1 Di ¼ ; and [ri¼1 Di ¼ Rm . Moreover, we shall determine the maximal dissipative interval and robust dynamics ðk; kÞ around k ¼ 0, that is, the maximal interval of perturbation of the matrix AðkÞ by mean of the parameter k for still having scrolls attractors. To achieve it we assume that the system (2):

Theorem 3.4 (Generalization of zero exclusion principle). Consider the polynomial family Pðk; tÞ with constant degree where k 2 X and X  Rl is a pathwise connected set. Suppose there exists an element of the family with n1 roots in C and n  n1 roots in Cþ . Then the entire family still having n1 roots in C and n  n1 roots in Cþ if and only if Pðk; ixÞ – 0 for all k 2 X and for all x 2 Rl .

B. Aguirre-Hernández et al. / Chaos, Solitons & Fractals 71 (2015) 100–106

103

The aforementioned system is simultaneously satisfied if Gðxl Þ þ kFðxl Þ ¼ 0, where xl 2 RðHÞRþ [ f0g. Thence we

In [47] the maximal robust stability interval is computed by using the before generalized zero exclusion principle, which can be seen as a particular case in our new context of robust dynamics (Definition 2.1 for n1 ¼ n). Now, in a more general case, we consider the problem of finding the maximal interval of robust dynamics with a polynomial approach. So we are interested in the  þ maximum interval ðkmin ; kmax Þ such that Pðt; kÞ has n1 roots  þ  in C and n  n1 roots in Cþ for all k in ðkmin ; kmax Þ and  n > deg p1 ðtÞ. In the same spirit that [47], we calculate kmin þ 2 2 and kmax . If p0 ðixÞ ¼ Pðx Þ  ixQ ðx Þ and p1 ðixÞ ¼ pðx2 Þ þ ixqðx2 Þ, then

  Gðxl Þ  kmin ¼ max  : Fðxl Þ > 0; xl 2 RðHÞRþ [ f0g Fðxl Þ   Gðxl Þ : Fðxl Þ 2 K þ ¼ max  Fðxl Þ   Gðxl Þ þ : Fðxl Þ < 0; xl 2 RðHÞRþ [ f0g kmax ¼ min  Fðxl Þ   Gðxl Þ : Fðxl Þ 2 K  ¼ min  Fðxl Þ

Pðix; kÞp0 ðixÞ ¼ GðxÞ þ kFðxÞ þ ikxHðxÞ;

as we claim.

xl Þ . Therefore, since GðxÞ > 0 for all x 2 R, the get k ¼  Gð Fðx Þ l

desire values are given by

h

where Remark 3.6. In the proof we just consider positive real

FðxÞ ¼ pðx2 ÞPðx2 Þ þ x2 qðx2 ÞQ ðx2 Þ;

xl 2 RðHÞRþ due to the symmetry of HðxÞ and the realness

GðxÞ ¼ P2 ðx2 Þ þ x2 Q 2 ðx2 Þ;

of values of k. Moreover, it is not necessary to consider cases when Fðxl Þ ¼ 0, since otherwise the system 6,7 we would have Gðxl Þ ¼ 0, but

2

2

2

2

HðxÞ ¼ qðx ÞPðx Þ  pðx ÞQðx Þ: Now let us to define, for an arbitrary polynomial f ðtÞ, the set

Rðf Þ ¼ fn 2 Cjf ðnÞ ¼ 0g: Let Rðf ÞRþ denotes the set of positive real elements of Rðf Þ and now we define the sets

 K þ ¼ Fðxl Þ : xl 2 RðHÞRþ [ f0g;   K ¼ Fðxl Þ : xl 2 RðHÞRþ [ f0g;

 Fðxl Þ > 0 ;  Fðxl Þ < 0 :

Gðxl Þ ¼ jp0 ðixl Þj2 ¼ p0 ðixl Þp0 ðixl Þ ¼ 0; which is impossible to happens because p0 ðtÞ does not   have roots in iR. However, if it occurs either K þ ¼ 0þ or K  ¼ f0 g, then we evaluate in x ¼ 0 and depending on the sign of Fð0Þ we will get either 

kmin ¼ limþ  r!0

Gð0Þ ¼ 1; r

or

Then, with the aforementioned we give the following result for a family of polynomials.

þ

kmax ¼ lim  r!0

Theorem 3.5. (Maximal robust dynamics interval). Consider the polynomial family Pðt; kÞ ¼ p0 ðtÞ þ kp1 ðtÞ, where p0 ðtÞ is a n-degree polynomial with n1 roots in C and n  n1 roots in Cþ . Suppose the n > deg p1 ðtÞ and let FðxÞ; GðxÞ and HðxÞ be the polynomials defined above. Then Pðt; kÞ has n1 roots in C and  þ n  n1 roots in Cþ for all k 2 ðkmin ; kmax Þ, where

  Gðxl Þ  kmin ¼ max  : Fðxl Þ 2 K þ ; Fðxl Þ   Gðxl Þ þ : Fðxl Þ 2 K  : kmax ¼ min  Fðxl Þ

Gð0Þ ¼ þ1: r

Similar results but for segments and rays of Hurwitz polynomials can be seen in [48,49]. Now, we are closer of getting what we desire. If n ¼ 3 and n1 ¼ 1, we can know about the maximal interval of ‘‘saddleness’’ (MDI) and we do just need the condition for negative sum of roots (dissipativity condition). Lemma 3.7 (Dissipativity Lemma). The sum of the roots of the polynomial pðtÞ ¼ a0 t n þ a1 tn1 þ    þ an1 t þ an is negative if and only if aa10 > 0. Proof. Let z1 ; . . . ; zn be the roots of pðtÞ and

Proof. For k ¼ 0 we have that Pðt; 0Þ ¼ p0 ðtÞ has n1 roots in C and n  n1 roots in Cþ . In order to apply the generalized zero exclusion principle we need that if for all  þ k 2 ½k ; k ; Pðix; kÞp0 ðixÞ – 0 for all x 2 R, then Pðt; kÞp0 ðtÞ has n roots in C and n roots in Cþ . It is clear that if k ¼ 0 then Pðix; 0Þp0 ðixÞ ¼ p0 ðixÞp0 ðixÞ has n roots in C and n roots in Cþ for all x 2 R. Thus, we are interested in the maximum and the minimum values (around k ¼ 0) of k where it occurs a change of stability in the system.  þ That is, the maximum k and the minimum k for which occurs Pðix; kÞp0 ðixÞ ¼ GðxÞ þ kFðxÞ þ ikxHðxÞ ¼ 0. To achieve it, we must solve the system of equations

GðxÞ þ kFðxÞ ¼ 0;

ð6Þ

xHðxÞ ¼ 0:

ð7Þ

pðtÞ ¼ a0

n Y ðt  zj Þ; j¼1

its factorization. By the Viète’s formulae (see in [50]) we have that

pðtÞ ¼ a0 ðt n  s1 t n1 þ    þ ð1Þn sn Þ; P where sj ¼ 16i1 < 0. h By applying Theorem 3.5 and Lemma 3.7 to the family Pðt; kÞ ¼ p0 ðtÞ þ kp1 ðtÞ we can establish a family of multiple scrolls as is shown in the following Section.

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4. Generation of a monoparametric family of multiscrolls attractors

2

The control system (2) can be determined by its pair ðA; BÞ:

a12

. . . a1n

Ba B 21 A¼B B .. @ .

a22 .. .

. . . a2n C C .. C C; . A

an1

an2

. . . ann

0

b1

1

B B D¼B B @

0

0

...

0

1

0 .. .

C C C: C A

0 .. .

0 .. .

...

c0

c1

. . . cn1

0

0.2

x1

ð8Þ

0.4

0.6

0.8

5

bn

(b)

Now let define the matrix D in terms of the coefficients of the polynomial p1 ðtÞ ¼ cn1 tn1 þ cn2 t n2 þ . . . þ c0 as follows:

0

0

−2 −0.2

Bb C B 2C C B¼B B .. C: @ . A

x3

1

a11

0

−5 −0.2

0

0.2

0.4 x

ð9Þ

0.6

Theorem 4.1. The control system (2) satisfying hypothesis (DS1) and (DS2) if x_ ¼ ðA þ kDÞx þ Bu is a member of the family Fk ðA; B; p1 ; x; uÞ with characteristic polynomial Pðt; kÞ for all k 2 ðk; kÞ then it is possible to generate a monoparametric family of multiscroll attractors.

0.8

1

5

x3

0

x2

(a)

(c)

0

−5 −0.5

0

0.5

1

1.5

x2 

þ

Remark 4.2. If c1 ¼ 0 then we have that ðk; kÞ ¼ ðkmin ; kmax Þ. If c1 – 0, note that the system (2) is an UDS-I, then it satisfies that the sum of all of its eigenvalues are negative and this condition is also affected by the perturbation kc1 in the coef þ ficient a1 and therefore ðkmin ;kmax Þ \ S is not empty. To understand completely the idea, we shall develop an illustrative example. Example 4.3. Consider the control system given by (2) for

2

0

1

0

3

2

0:15

3

6 7 6 7 0 1 5 and B ¼ 4 0:3 5; A¼4 0 30 4 1 0:4 then the number of scrolls displayed in the attractor can be determined by the control signal u. For example a fourscroll attractor can be given by the following control signal

8 3; > > > < 2; u¼ > 1; > > : 0;

for 0:375 < x1 ; for 0:225 < x1 6 0:375; for x1 6 0:075:

Fig. 1 shows the projection of the attractor onto the planes: (a) ðx1 ; x2 Þ; (b) ðx1 ; x3 Þ; and (c) ðx2 ; x3 Þ. As we are interested in generating a monoparametric family of multiscroll attractors determined as follows:

ð10Þ

where k is the parameter of family given by (10). Without losing generality, the matrix D can be given as follows:

0

0 0

0

1

B C D ¼ @ 0 0 0 A: 1 0 0 2

Note that the characteristic polynomial for k ¼ 0 is p0 ðtÞ ¼ t3 þ t2 þ 4t þ 30 and from the example 4.3 and Lemma 3.7, the system (2) is UDS-I for u ¼ 0. Moreover, the characteristic polynomial of A þ kD is Pðt; kÞ ¼ t3 þ ð1  12 kÞt2 þ 4t þ 30, so we have that p1 ðtÞ ¼  12 t2 . Furthermore,

pA ðixÞ ¼ 30  x2 þ ixð4  x2 Þ; 1 p1 ðixÞ ¼ x2 ; 2 thence

Pðx2 Þ ¼ 30  x2 ; Q ðx2 Þ ¼ 4  x2 ; 1 pðx2 Þ ¼ x2 ; 2 qðx2 Þ ¼ 0;

for 0:075 < x1 6 0:225;

x_ ¼ ðA þ kDÞx þ Bu;

Fig. 1. Projections of the attractor for k ¼ 0 onto the planes: (a) ðx1 ; x2 Þ; (b) ðx1 ; x3 Þ; and (c) ðx2 ; x3 Þ.

and therefore

1 2 x ð30  x2 Þ; 2 GðxÞ ¼ x6  7x4  44x2 þ 900; 1 HðxÞ ¼  x2 ð4  x2 Þ: 2

FðxÞ ¼

Hence, RðHÞRþ ¼ f2g and Fð2Þ ¼ 52 > 0. Then K  ¼ f0 g þ



¼ and K þ ¼ f52g. So that, kmax ¼ þ1 and kmin ¼  Gð2Þ Fð2Þ ¼ 13. Consequently, the maximum interval of  676 52 hyperbolicity

is



þ

ðkmin ; kmax Þ ¼ ð13; þ1Þ

and

since

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B. Aguirre-Hernández et al. / Chaos, Solitons & Fractals 71 (2015) 100–106

3 0:4 6 7 B ¼ 4 0:3 5: 0:4

2

x2

a1 þ kc1 ¼ 1  12 k, then S ¼ ð1; 2Þ. Thus, if k 2 ð13; 2Þ then the system (10) is an UDS-I of which emerges a family of multi-scrolls attractor. Fig. 1 shows the projections of the attractor for k ¼ 0 and Fig. 2 shows the projections of the attractor for k ¼ 3 onto the planes: (a) ðx1 ; x2 Þ; (b) ðx1 ; x3 Þ; and (c) ðx2 ; x3 Þ. Due to the stable and unstable manifolds of the system given by Eq. 10 for k ¼ 3 change of direction it is necessary to define B as follows:

0:3

0

0.2

3

x

−5 −0.4

for 1 < x1 ; for 0:6 < x1 6 1; for 0:2 < x1 6 0:6;

1.2

−0.2

0

0.2

0.4

0.6

0.8

1

(c)

3

3

x

0

−5 −1

−0.5

0

1

1.5

2

0

0.5

1 x

1.5

2

1

5

(b)

0

0

0.5

1

x

1.5

2

1

5

Fig. 3. Projections of the attractor for k ¼ 1 onto the planes: (a) ðx1 ; x2 Þ; (b) ðx1 ; x3 Þ; and (c) ðx2 ; x3 Þ.

8 3; > > > < 2; u¼ > 1; > > : 0;

0

−5 −0.5

0.5 x2

(a)

−2 −0.5

1.2

5

2

x2

1

for x1 6 0:2:

In order to generate a four-scroll attractor, the control signal can be given as follows:

x3

0.8

x1

0:4

x3

0.6

0

6 7 B ¼ 4 0:25 5:

(c)

0

−5 −1

0.4 x1

(b)

Fig. 3 shows the projections of the attractor for k ¼ 1 onto the planes: (a) ðx1 ; x2 Þ; (b) ðx1 ; x3 Þ; and (c) ðx2 ; x3 Þ. It is necessary to define B as follows:

2

−0.2

5

In order to generate a four-scroll attractor, the control signal can be given as follows:

8 3; > > > < 2; u¼ > 1; > > : 0;

0

−2 −0.4

2

(a)

for 0:75 < x1 ; for 0:45 < x1 6 0:75; for 0:15 < x1 6 0:45; for x1 6 0:15:

5. Conclusions This paper introduced a monoparametric family of multiscrolls-generating systems based on piecewise linear systems. Particularly, it deals with UDS type I. It derives conditions to generate a family of polynomials Pðt; kÞ ¼ p0 ðtÞ þ kp1 ðtÞ for which p0 ðtÞ has n1 roots in C and n  n1 roots in Cþ for all k in the maximal robust dynamics interval, thus emerging monoparametric family of chaotic attractors. So, a way to perturb a system and still having multiscrolls attractors is derived. The attractor arises from a switching system having at least two UDS type I. This approach may be further extended to generate chaotic systems with multistabilities by adding more UDS type II. Acknowledgments

−0.5

0

0.5 x

1

1.5

2

2

Fig. 2. Projections of the attractor for k ¼ 3 onto the planes: (a) ðx1 ; x2 Þ; (b) ðx1 ; x3 Þ; and (c) ðx2 ; x3 Þ.

E.C.C. acknowledges the CONACYT financial support through project No. 181002. The fourth author wants to thank to CONACYT for its PhD scholarship support. Edgar Cristian Díaz González is doctoral fellow of the CONACYT in the Graduate Program on Science at Mathematics Department, UAM-I.

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