Preservation of stability and synchronization in nonlinear systems

Physics Letters A 371 (2007) 205–212 www.elsevier.com/locate/pla

Preservation of stability and synchronization in nonlinear systems G. Fernández-Anaya a , J.J. Flores-Godoy a,∗ , R. Femat b , J.J. Álvarez-Ramírez c a Departamento de Física y Matemáticas, Universidad Iberoamericana, Prol. Paseo de la Reforma 880, Lomas de Santa Fe, México, D.F. 01210, Mexico b División de Matemáticas Aplicadas y Sistemas Computacionales, IPICyT, Camino a la Presa San José 2055, Col. Lomas 4a. sección,

San Luis Potosí, San Luis Potosí 78216, Mexico c Ingeniería de Procesos e Hidráulica, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México, D.F. 09340, Mexico

Received 24 October 2006; received in revised form 26 April 2007; accepted 1 June 2007 Available online 12 June 2007 Communicated by C.R. Doering

Abstract Preservation of stability in the presence of structural and/or parametric changes is an important issue in the study of dynamical systems. A specific case is the synchronization of chaos in complex networks where synchronization should be preserved in spite of changes in the network parameters and connectivity. In this work, a methodology to establish conditions for preservation of stability in a class of dynamical system is given in terms of Lyapunov methods. The idea is to construct a group of dynamical transformations under which stability is retained along certain manifolds. Some synchronization examples illustrate the results. © 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Gg Keywords: Stability preservation; Chaotic synchronization

1. Introduction Several feedback schemes for the chaos suppression and synchronization have been widely studied in last two decades (see the reviews in [1–3]). More recently, the complex networks have opened new challenges for the stabilization of the chaotic dynamics [4,5]. In this direction, a new problem is to study the conditions under which the stabilization is kept intact during transformation induced by the (feedback or feedforward) interconnections of dynamical systems in a network; i.e., the stability preservation of complex networks. The study of the stability preservation makes sense in the chaos control problems. As matter of fact, the generalized synchronization can be derived even for different systems by finding a diffeomorphic transformation such that the states of the slave system can be written as a function of the states of the * Corresponding author.

E-mail addresses: [email protected] (G. Fernández-Anaya), [email protected] (J.J. Flores-Godoy), [email protected] (R. Femat), [email protected] (J.J. Álvarez-Ramírez). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.06.017

master dynamics (see [6] and references therein). This result can be seen as a timely contribution; however, accordingly to the goal of keeping intact the stability under the transformation, a new question arises: how can the stability be preserved under transformations suffered by a dynamical system? An answer to this question might allow us to ensure the synchronization in strictly different systems, in the sense that stability of the error is preserved under the transformation. The preservation of stability for a class of nonlinear autonomous dynamical system has been reported in last decades [7–9]. The underlying idea is to preserve the stability properties under transformation of finite-dimensional dynamical systems. Thus, for example, by using a change of variables (e.g., using a diffeomorphism in the neighborhood of an equilibrium point), a feedback can be designed such that the original system is stable (or asymptotically stable) as the transformed system is stable (or asymptotically stable). Some results on stability preservation have been reported as successful by computing the multiplication of the vector field in the nonlinear dynamical system by a continuously differentiable function [7]. In the case of linear dynamical system there exist several results of preservation of stability, for

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instance in [10–12] asymptotically stability is preserved using transformations on rational functions in the frequency domain. Some of these transformations can be interpreted as a special class of noise present in the system also as perturbation on the value of the physical parameters involved in the description of the model. Nevertheless, the problem of the stability preservation has not been addressed for the case of a nonlinear systems with chaotic dynamics. The following problem, particularly interesting, is how a given collective dynamics (for instance, a synchronous motion) can be preserved when important changes occur in the dynamical system. This issue is important for the case of networking systems. We know that most of real world networks are not stationary, in the sense that they are growing, with new nodes continuously being added to the graph (WWW, Internet, Science Citation index, regulatory networks, are just a few of such examples). Therefore it comes out a natural question on how these networks can preserve a given collective dynamics or functioning, while the process of their growth is taking place in time. The stability preservation is studied in this contribution for the chaotic synchronization problem. The results show that stability can be preserved by transforming the linear part of the synchronization system which, in complex networks, is related to the connectivity of nodes. The transformation can be performed in time or Laplace domain. Thus, the results can be also used in the chaos suppression problem. The results include very relaxed conditions to preserve the stability. Specifically two extensions are studied: an extension of the Lyapunov indirect method and a result of preservation for diagonalizable systems in their linear part. Different examples are provided and the results are applied to the chaotic synchronization problem. As we shall see, the results depart from the hypothesis of the existence of a constant state feedback as nominal synchronizing force. The Letter is organized as follows. Section 2 includes the problem statement and the main results: Also illustrative examples are shown. Then, in Section 3, the stability preservation is discussed in terms of the chaotic synchronization of Chua’s oscillators. The numerical experiments on the stability preserved for the chaotic synchronization are shown in Section 4. Finally, conclusions are in Section 5. 2. Preservation of stability The main results of this work are presented in this section. These results depart from the Lyapunov’s indirect method. The idea is to preserve the stability in the linear part of the vector field associated with a nonlinear autonomous dynamic system. The following theorem is known as the Lyapunov’s indirect method [7]. Theorem 1. Let x = 0 be an equilibrium point for the nonlinear system x˙ = f (x), where f : D → Rn is continuously differentiable function and D ⊂ Rn is a neighborhood of the origin. Let   ∂f (x) . A= ∂x x=0

Then: (1) The origin is asymptotically stable, if the real part of all the eigenvalues of A is negative and g(x)2 →0 x2

as x2 → 0,

with g : D → Rn continuously differentiable. (2) The origin is unstable, if the real part of one or more of the eigenvalues of A is positive. Let us consider the following notions of stability for polynomials and square matrices. A polynomial p(s) is asymptotically stable if all its roots have strictly negative real part. On the other hand, a square matrix M is asymptotically stable if all its eigenvalues have strictly negative real part. Let Pn (s) denote the set of real polynomials of fixed degree n. Also, let Rn×n denote the set of square matrix of order n with real entries. A transformation that preserves stability of polynomials is a function Ξ : Pn (s) → Pm (s), such that if p(s) ∈ Pn (s) is asymptotically stable, then Ξ (p(s)) ∈ Pm (s) is also asymptotically stable. In the same way, a transformation that preserves stability of matrices is a function Φ : Rn×n → Rm×m such that if M ∈ Rn×n is asymptotically stable, then Φ(M) ∈ Rm×m is asymptotically stable. The following proposition is an extension to Lyapunov’s indirect method. Proposition 2. Let x = 0 be an equilibrium point for the nonlinear systems x˙ = f1 (x) and x˙ = f2 (x), where f1 : D1 → Rn and f2 : D2 → Rm are continuously differentiable functions and D1 ⊂ Rn and D2 ⊂ Rm are neighborhoods of the origin. Let   ∂f1  ∂f2  A1 = A2 = (x) , (x) , ∂x ∂x x=0 x=0 with A1 ∈ Rn×n and A2 ∈ Rm×m ; and consider the following decomposition x˙ = f1 (x) = A1 x + g1 (x), x˙ = f2 (x) = A2 x + g2 (x), with g1 : D1 → Rn and g2 : D2 → Rm continuously differentiable. Let Pn (s) and Pm (s) be the characteristic polynomials of A1 and A2 , respectively. (1) Suppose that there exists a transformation Ξ : Pn (s) → Pm (s) such that preserves stability of polynomials and Ξ (det(sI − A1 )) = det(sI − A2 ). Then the following statements are equivalent: (a) The origin of the nonlinear system x˙ = f2 (x) is asymptotically stable, if the real part of all the eigenvalues of A1 is negative and for all g2 continuously differentiable function such that g2 (x)2 →0 x2

as x2 → 0.

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(b) The origin of the nonlinear system x˙ = f1 (x) is unstable, if the real part of one or more of the eigenvalues of A2 is positive. (2) Similarly, suppose that there exists a transformation Φ : Rn×n → Rm×m such that preserves stability of matrices and Φ(A1 ) = A2 . Then the following statements are equivalent, (a) The origin of the nonlinear system x˙ = f2 (x) is asymptotically stable, if the real part of all the eigenvalues of A1 is negative and for all g2 continuously differentiable function such that g2 (x)2 → 0 as x2 → 0. x2 (b) The origin of the nonlinear system x˙ = f1 (x) is unstable, if the real part of one or more of the eigenvalues of A2 is positive. Proof. We proceed to prove each item as follows (1) (a) By Theorem 1, the origin of the systems x˙ = f1 (x) and x˙ = f2 (x) are asymptotically stable if the real part of all the eigenvalues of A1 and A2 is negative. If there exists a transformation Ξ : Pn (s) → Pm (s) such that preserves stability of polynomials and Ξ [det(sI − A1 )] = det(sI − A2 ). Then, if the real part of all the eigenvalues of A1 is negative implies that the real part of all the eigenvalues of A2 is negative. Now, by Theorem 1 the origin of the nonlinear system x˙ = f2 (x) is asymptotically stable. (b) This item is equivalent to the subitem (1a), and a well known tautological formula (A ⇒ B ≡ (∼ B) ⇒ (∼ A)), A implies B is equivalent to not B implies not A. (2) The proof for item (2) subitem (2a) and subitem (2b) is similar to item (1) subitem (1a) and subitem (1b), respectively. 2 Notice that the propose transformation just actuates on the linear part of the vector fields f1 and f2 . The higher order terms g1 and g2 are free as long as they satisfy their respective restrictions. Proposition 3. Let x = 0 be an equilibrium point for the nonlinear system x˙ = f (x) where f : D → Rn is continuously differentiable function and D is a neighborhood of the origin. Let   ∂f A= (x) ∂x x=0

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such that MA = AM. Then, the origin of the nonlinear systems ˆ x˙ = fM (x) = MAx + g(x) is asymptotically stable, for all M ∈ ΛM and for all gˆ : D → Rn ˆ 2 continuously differentiable function such that g(x) x2 → 0 as x2 → 0. Proof. First note that by Theorem 12.4.1 in [13], if A = N DA N −1 is diagonalizable, then for all M ∈ ΛM such that MA = AM. Hence, we have M = N DM N −1 where DM is diagonal matrix with strictly positive elements only. Second observe that the set   ΛM = M: M = N DM N −1 , ∀DM , (DM )jj > 0 is a commutative group with the multiplication of matrices. In consequence, if MA is Hurwitz stable for some M ∈ ΛM , then MA is Hurwitz stable for all M ∈ ΛM . Now taking M as the identity matrix In and by Theorem 1, if Re λi < 0 for all eigenvalues of A, the matrices MA are Hurwitz stable and the origin of the nonlinear systems x˙ = fM (x) is asymptotically stable for all M ∈ ΛM and for all gˆ : D → Rn continuously differentiable ˆ 2 function such that g(x) x2 → 0 as x2 → 0. 2 Proposition 3 is a statement on a commutative group of transformations that generates an action on stable linear systems. This action preserves stability among linear systems. With respect to the nonlinear system since it can be decomposed into a linear part and higher order terms, this result applied to the linear part preserves local stability of the original nonlinear system and at the same time allows for perturbations on the higher order terms that satisfy the restrictions imposed, which are not worst than the initial hypothesis in the proposition. In what follows, we present some examples to show the wide range of possible transformations that can be made on the linear part of dynamical systems vector fields which preserves local stability. 2.1. Examples In the following example we show how a scalar function of certain type transforms the linear part of a dynamical system vector field preserving stability. Example 4. Let the matrix A ∈ Rn×n with distinct eigenvalues λ1 , . . . , λs be such that the minimal polynomial of A with degree m = m1 + · · · + ms is given by m(λ) = (λ − λ1 )m1 (λ − λ2 )m2 · · · (λ − λs )ms .

and consider the following decomposition

Given a function h : R → R, we say that h(λ) is defined on the spectrum of A if there exist the numbers

x˙ = f (x) = Ax + g(x).

h(λk ), h (λk ), . . . , h(mk −1) (λk ),

Suppose that A is diagonalizable (i.e., there exists a matrix N such that A = N DA N −1 with DA a diagonal matrix) also all eigenvalues of A have negative real part, Re λi < 0. Consider the set ΛM of positive definite matrices M (i.e., M = M T > 0)

where h

(mk −1)

 d mk −1 h(x)  (λk ) = . dx mk −1 x=λk

k = 1, 2, . . . , s,

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Now, by Theorem 9.4.6 in [13] and Proposition 2 item (2a), if h(·) maps the left-half complex plane into itself, i.e., h(C− ) ⊂ C− , and A is Hurwitz and the dynamical system x˙ = Ax + g(x) has a stable equilibrium point at x = 0. Then the origin of x˙ = fh (x) = h(A)x + g(x) ¯ is asymptotically stable for all g¯ : D¯ ⊂ Rn → Rn continuously ¯ 2 differentiable function such that g(x) x2 → 0 as x2 → 0. Notice that the action of h(·) is only applied to the linear part of the dynamical system, g(x) ¯ may be any continuously differ¯ 2 entiable function such that limx→0 g(x) x2 = 0. Depending on the context this example might be understood as a perturbation that preserves dimension in the sense that the number of variables is the same. As a particular case of Example 4 we have the following one. Example 5. Consider the function h : R → R, defined by h(s) =

b + s

n  k=1

bk , s + ak

 where b  0, bk  0 and ak  0, but b + nk=1 bk = 0 and  n − − k=1 ak = 0, then h(C ) ⊂ C . In particular, we take the function 2 5 7 + , h1 (x) = + x x −2 x −3 and consider the nonlinear system x˙ = Ax + g(x) given by x˙1 = −x1 + 12x2 + x22 , x˙2 = 8x1 − 2x2 + x1 x22 .

(1)

In this case the matrix A is given by   −1 12 A= , −8 −2 which is Hurwitz, i.e., all its eigenvalues are in C− . Note that h1 (x) is defined on the spectrum of A as in Example 4. Therefore the nonlinear system x˙ = h1 (A)x + g(x) given by 80989 19498 x1 − x2 + g¯ 1 (x1 , x2 ), 153468 12789 38996 20497 x˙2 = x1 − x2 + g¯ 2 (x1 , x2 ), 38367 51156 where x˙1 = −

h1 (A) = 2A−1 + 5(A − 2I )−1 + 7(A − 3I )−1  80989  − 153468 − 19498 12789 = 38996 − 20497 38367 51156 is asymptotically stable for all g¯ = [g¯ 1 (x1 , x2 ) g¯ 2 (x1 , x2 )], g(x) ¯ : D¯ ⊂ R2 → R2 continuously differentiable function such g(x) that ¯x2 2 → 0 as x2 → 0, by Example 4. In particular the nonlinear system 80989 19498 x1 − x2 + x1 x22 , x˙1 = − 153468 12789 38996 20497 x˙2 = x1 − x2 + x15 x22 38367 51156

(2)

is asymptotically stable. Notice that in this example it is not obvious to infer the stability of (2) from the original system given by (1). In the following example as far as we know it is the first time that transformation defined in the frequency domain and applied to a nonlinear systems is performed such that local stability is preserved. Example 6. Consider the nonlinear system x˙1 = −x1 + x22 , x˙2 = −x2 . In this case the matrix A is given by   −1 0 A= , 0 −1 with characteristic polynomial: PA (s) = (s + 1)2 , where the s variable is usually interpreted as a frequency domain variable due to the Laplace transform. Now, if we consider the substituz+η , then tion of the variable s by z+μ



(μ + η)2 z+η 2 2 = 4 z + (μ + η)z + (z + μ) PA z+μ 4 and taking the monic polynomial z2 + (μ + η)z + its canonical realization, we obtain the new matrix   0 2 1 . −(μ + η) − (μ+η) 4

(μ+η)2 4

and

Then, by Proposition 2 item (1a), if η > 0 and μ > 0 the nonlinear system x˙1 = x2 + g¯ 1 (x1 , x2 ), (μ + η)2 x1 − (μ + η)x2 + g¯ 2 (x1 , x2 ) x˙2 = − 4 ¯ = is asymptotically stable for all g¯ : D¯ ⊂ R2 → R2 , i.e., g(x) [g¯ 1 (x) g¯ 2 (x)]top , continuously differentiable function such that g(x) ¯ 2 x2 → 0 as x2 → 0. In Example 6 the transformation is on the linear part of the dynamical system’s vector field and it was defined in the frequency domain. Notice that the dimension of the system is not changed. In the next example, the transformation induces an increment in the number of variables. This also implies that the higher order terms change in the sense that the vector field increases in dimension. Example 7. As in Example 6 consider, the substitution of the 2 , then variable s by zz2 +βz+η +δz+μ 2

2 z + βz + η z2 + δz + μ PA 2 z + δz + μ

 (β + δ)2 2 4 3 z = 4 z + (β + δ)z + η + μ + 4



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1 (η + μ)2 + (β + δ)(η + μ)z + 2 4



and taking the monic polynomial z4 + a1 z3 + a2 z2 + a3 z + a4 with a1 = β + δ, a2 = η + μ +

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changes by the action of a class of transformation on the linear part of the chaotic nonlinear dynamical system. Consider the following two n-dimensional chaotic systems: x˙ = Ax + g(x), y˙ = Ay + f (y) + u,

(β

+ δ)2

, 4 (β + δ)(η + μ) , a3 = 2 (η + μ)2 a4 = 4 and its canonical realization, we obtain the matrix ⎛ ⎞ 0 1 0 0 0 1 0 ⎟ ⎜ 0 ⎝ ⎠. 0 0 0 1 −a4 −a3 −a2 −a1

Then by Proposition 2 item (1a), if η + μ > 0 and β + δ > 0 the nonlinear system x˙1 = x2 + g¯ 1 (x1 , x2 , x3 , x4 ), x˙2 = x3 + g¯ 2 (x1 , x2 , x3 , x4 ), x˙3 = x4 + g¯ 3 (x1 , x2 , x3 , x4 ), x˙4 = −a4 x1 − a3 x2 − a2 x3 − a1 x4 g¯ 4 (x1 , x2 , x3 , x4 ) ¯ = is asymptotically stable for all g¯ : D¯ ⊂ R4 → R4 , i.e., g(x) [g¯ 1 g¯ 2 g¯ 3 g¯ 4 ] , continuously differentiable function such that g(x) ¯ 2 x2 → 0 as x2 → 0. In general, if we consider the substitution of the variable s by zn + an−1 zn−1 + · · · + a0 , α(z) = n z + bn−1 zn−1 + · · · + b0 where the rational function α(z) is a strictly positive real function of zero relative degree1 [10,14], and an asymptotically stable nonlinear system x˙ = Ax + g(x). By Theorem 1 in [10], the characteristic polynomial PA (s) of the matrix A, has the following property: PA (α(z)) is Hurwitz stable for all strictly positive real function α(z). Then considering the realization Aα of the polynomial PA (α(z)), we have that the nonlinear system x˙ = Aα x + g(x) ¯ is asymptotically stable for all g¯ : D¯ ⊂ Rn → Rn ¯ 2 continuously differentiable function such that g(x) x2 → 0 as x2 → 0. 3. Stability preservation for chaotic synchronization In this section we apply the techniques developed in Section 2 to a known problem for which solutions have been proposed in the literature. We will show that it is possible to preserve synchronization even thought the dimension of the system 1 A strictly positive real function of zero relative degree is a rational function with degrees of the numerator and denominator polynomial equal, analytical on the right-half complex plane, and the real part of the function evaluated on the imaginary axis is strictly positive.

where A ∈ Rn×n is a constant matrix, f, g : Rn → Rn are continuous nonlinear functions and u ∈ Rn is the control input. The problem of synchronization considered in this section is the synchronization where the master system and the slave system are synchronized by designing an appropriate feedback function u = −Ke, where e = x − y is the synchronization error. The error dynamics are given by e˙ = (A − K)e + L(x, y), where L(x, y) = g(x) − f (y). Assume that the origin e = 0 is asymptotically stable, which implies that x(t) → y(t) as t → ∞. Now let T ∈ Rm×m be a matrix with strictly positive eigenvalues, and suppose that the following two nm-dimensional systems are chaotic: x˙ = t (A ⊗ T )x + g(x), ¯ ¯ y˙ = (A ⊗ T )y + f (y) + u(t), ¯ for some f¯, g¯ : Rnm → Rnm continuous nonlinear functions and u¯ ∈ Rnm is the control input and ⊗ denotes the Kronecker product between matrices. Then based on Proposition 2, we have that u¯ = (−K ⊗ T )e stabilizes the origin of the error dynamics system e˙ = (A ⊗ T )e + u, ¯ e˙ = (A ⊗ T − K ⊗ T )e + L(x, y). Notice that using the linear property of the Kronecker product, we obtain that:  e˙ = (A − K) ⊗ T e. The original control u(t) = −Ke is preserved in its linear part by the Kronecker product (·) ⊗ T and the new control is given by u(t) ¯ = (−K ⊗ T )e. Therefore, we can interpret the last procedure as one in which the controller u(t), that achieved the synchronization in the two original systems, is preserved by the transformation (·) ⊗ T so that u(t) ¯ achieves the synchronization in the two resultant systems after of the transformation. A similar procedure is possible if we consider the transformation T ⊗ (·) and also considering the product by the definite positive matrices M such that M ∈ ΛM where   ΛM = M: M = N DM N −1 , ∀DM , DMij > 0 and supposing that A − K = N DN −1 with D a diagonal Hurwitz matrix. In this sense the stability on the synchronization is preserved. In general, starting from a chaotic system for which a synchronization controller has been design, if either F : Pn (s) → Pm (s) or H : Rn×n → Rm×m is a linear transformation that satisfies the conditions of Proposition 2, then the synchronization

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is preserved for the transformed system given that the transformation preserves chaos. It should be stressed that, rather than dealing with the design of feedback synchronization schemes, our results show that synchronization can be preserved when the underlying chaotic dynamical system is transformed, even when the transformation lead from a lower dimension to a higher dimension. This issue is important for the case of networking system where the transformed system can be interpreted as a network in which there has been an increment in the number of nodes. 4. Simulations In this section some simulations are presented using a known chaotic system as a benchmark to illustrate the result shown in Section 3. 4.1. Synchronization of modified Chua’s circuit

Fig. 1. Modified Chua’s model (Master). Initial conditions x = [0.02, 0.05, 0.04].

In this section we present an application of the above techniques to the modified Chua’s circuit described in [15] to illustrate the result presented in the above sections. The equations that describe the modified Chua’s circuit are:

(2x13 − x1 ) , x˙1 = p x2 − (3) 7 x˙2 = x1 − x2 + x3 , (4) x˙3 = −qx2 .

(5)

The master system is given by Eqs. (3)–(5) and the slave system is a copy of the master system with a control function u(t) to be determined in order to synchronize the two systems

(2y13 − y1 ) + u1 , y˙1 = p y2 − (6) 7 y˙2 = y1 − y2 + y3 + u2 , (7) y˙3 = −qy2 + u3 .

Fig. 2. Modified Chua’s model (Slave). Initial conditions y = [0.2, 0.5, 0.4].

(8)

Considering the error vector e = y − x, then the error dynamics can be written as: e(t) ˙ = Ae(t) + M(x, y) + u, with the control law chosen as u = −Ke, where  p p 0 7 A = 1 −1 1 , 0 −q 0   3 3 M(x, y) = 2p y1 −x1 0 0 . 7

The feedback input u = −Ke = [4.5e1 6.4e2 1.9e3 ] with K = diag(4.5, 6.4, 1.9), yields complete synchronization between master and slave dynamics. The initial conditions for the master and slave systems are x = [0.02 0.05 0.04] and y = [0.2 0.5 0.4] , respectively, with p = 10 and q = 100 7 . Figs. 1 and 2 show the trajectories for the solution of the master system and slave system, respectively, and Fig. 3 shows

Fig. 3. Magnitude of the error |e| = |y − x| between the Master and the Slave systems.

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Fig. 4. Transformed modified Chua’s model (Master). States x1 , x2 and x3 with initial conditions 0.02, 0.05, 0.04, respectively.

Fig. 6. Transformed modified Chua’s model (Slave). States y1 , y2 and y3 with initial conditions 0.2, 0.5, 0.4, respectively.

Fig. 5. Transformed modified Chua’s model (Master). States x4 , x5 and x6 with initial conditions 0.02, 0.05, 0.04, respectively.

Fig. 7. Transformed modified Chua’s model (Slave). States y1 , y2 and y3 with initial conditions 0.2, 0.5, 0.4, respectively.

the evolution of the synchronization error (notice the semilogarithmic scale used to emphasize the fact that the error converges to zero).

M(x, y) = [ m1

4.2. Synchronization of transformed modified Chua’s circuit

We used K the same as in the previous example and initial conditions for the master system x = [0.02 0.05 0.04 0.02 0.05 0.04] and the slave system y = [0.2 0.5 0.4 0.2 0.5 0.4] . The trajectories for the Master system are shown in Figs. 4 and 5 where the chaotic attractor is observed; in Figs. 6 and 7 the trajectories for the Slave system are presented. The corresponding synchronization error is shown in Fig. 8 where the convergence to zero is observed. Notice that the transformed systems are of higher dimension than the original modified Chua’s circuit model. It is interesting to notice that despite the transformation that increases the system dimensionality, the transformed system is also chaotic with an attractor geometry quite different from the original modified Chua’s circuit model. In this form, the transformation on the

For the transformed systems we have the Master and Slave systems as   3 3 3 3 x˙ = (A ⊗ T )x + − 2x1 − 2x2 − 2x3 − 2x4 0 0 , 7 7 7  73  3 3 3 y˙ = (A ⊗ T )y + − 2y1 − 2y2 − 2y3 − 2y4 0 0 + u. 7 7 7 7 In this case, the error dynamics can be written as: e˙ = (A ⊗ T )e + M(x, y) + u, with u = −(K ⊗ T )e and p    p 0 7 2 1 , A = 1 −1 1 , T= 0 1 0 −q 0

mi = −

m2

m3

m4

0 0 ] ,

2(yi3 − xi3 ) . 7

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indirect method was used to prove that certain transformations and extensions preserve the stability of the origin. As far as we know, the result in this work becomes a first attempt to study how a given collective dynamics can be preserved when structural changes occur in system networks. We know that most of real world networks are not stationary, in the sense that they are growing, with new nodes continuously being added to the graph. This is related to the increase in dimension of the involved network. Therefore it comes out a natural question on how these networks can preserve a given collective dynamics or functioning, while the process of their growth is taking place in time. The transformed dynamical system can be interpreted as a network in which there has been an increment in the number of nodes. References Fig. 8. Magnitude of the error |e| = |y − x| between the Master and the slave systems.

vector field generated a new higher-dimensional chaotic attractor from the well-known modified Chua’s model. The methodology presented in this Letter can be useful to study how chaotic systems can be transformed into higher-dimensional ones to obtain chaotic systems with quite different characteristics. On the other hand, the transformation on the modified Chua’s system was so drastic, as in dynamical networks, that changes the interaction between variables while preserves synchronization. Finally, it should be noticed that, although the feedback controller uses full state measurements, in practice the implementation can be made on the basis of state observers. In such a case, the resulting dynamical system is higher-dimensional and the stability result still holds. Partial state feedback is possible under stabilizability assumptions (i.e., when the synchronization error converges to zero while the internal zero-dynamics is stable). In such case, given the results of Section 3, stability preservation is maintained under a rectangular input-related matrix B. 5. Conclusions In this work, we have studied the preservation of stability for a class of dynamical systems. An application of the Lyapunov’s

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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