Physica A. Preserving synchronization under matrix product modifications

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1 Physica A 387 (2008) Contents lists available at ScienceDirect Physica A journal homepage: Preserving synchronization under matrix product modifications Dan Becker-Bessudo, G. Fernandez-Anaya, J.J. Flores-Godoy Departamento de Física y Matemáticas, Universidad Iberoamericana, Prolongación Paseo de la Reforma 880, México, D. F , Mexico a r t i c l e i n f o a b s t r a c t Article history: Received 29 May 2008 Received in revised form 11 July 2008 Available online 14 August 2008 PACS: Xt a Gg Pq In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under modification through matrix multiplication. The objective is to show that under a defined multiplicative group, hyperbolic critical points are preserved along the stable and unstable manifolds. The properties of this multiplicative group were determined through the use of simultaneous Jordan decomposition. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of different modified systems Elsevier B.V. All rights reserved. Keywords: Control Preservation of synchronization Chaotic systems Nonlinear systems Output feedback and observers 1. Introduction The study of synchronization preservation is relevant when it comes to chaos control problems. As a matter of fact, the generalized synchronization can even be derived 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 master dynamics (see Ref. [1] and references therein). This result can be seen as a timely contribution; however, in accordance to the goal of keeping intact the stability under the transformation, a new question arises: how can stability be preserved under transformations suffered by a dynamical system? An answer to this question might allow us to ensure synchronization in strictly different systems, in the sense that stability of the error dynamics is preserved under the transformation. Preservation of stability for a class of nonlinear autonomous dynamical systems has been reported in the last decades [2 4]. 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) and ensure 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 [2]. In the case of linear dynamical systems there exist several results of stability preservation, for instance in Refs. [5 7], stability is asymptotically 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 or also as Corresponding author. Tel.: ; fax: addresses: danbecker87@gmail.com (D. Becker-Bessudo), guillermo.fernandez@uia.mx (G. Fernandez-Anaya), job.flores@uia.mx (J.J. Flores-Godoy) /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.physa

2 6632 D. Becker-Bessudo et al. / Physica A 387 (2008) perturbations on the value of the physical parameters involved in the description of the model. The problem of stability preservation has been recently addressed for the case of a nonlinear systems with chaotic dynamics [8]. The results presented in this paper give an answer to the question posed above by insuring that the proposed transformation is robust, meaning that although the system has been modified it still preserves the aforementioned desired qualities (stability, synchronization, hyperbolic points). We propose to develop a methodology to extend some classic results of the dynamical systems theory by preserving the signature of the real parts of the eigenvalues of an underlying Jacobian matrix in the equilibrium points of the dynamical system. In particular, we present a simple extension of the Stable Unstable Manifold Theorem. The developed methodology is used to study the problem of preservation of synchronization in chaotic dynamical systems. It is based on the use of matrix theory tools, specifically, simultaneous Jordan decomposition, the multiplicative group structure for Jordan matrices, the closure under product of positive definite matrices and the eigenvalue sign-preservation for both real and complex matrices under matrix multiplication. The results on preservation of structure, stability and synchronization based on the extension of the stable-unstable manifold theorem show that stability and synchronization can be preserved by transforming the linear part of the synchronization system. We present a series of modified dynamical systems to show the preservation of the synchronization manifold. 2. Structure preservation In this section we present the necessary definitions and results that will allow us to prove the main propositions of this paper. The results stem from the use of matrix products of simultaneous Jordan decomposable matrices and are focused on the task of preserving hyperbolic points under a transformation over the linear part of a vector field of a nonlinear autonomous system. The results will be used in Section 4 where we will present some examples on preservation of synchronization in dynamical systems. Simultaneous Jordan decompositions are defined as follows Definition 2.1. The group of matrices A 1, A 2,..., A n is said to be Jordan simultaneously decomposable if there exists and invertible similarity matrix P such that A 1 = PJ 1 P 1, A 2 = PJ 2 P 1,..., A n = PJ n P 1 where J i is the Jordan canonical form of the matrix A i. For the following discussion consider the dynamical system described by ẋ = f (x) where x R n and f : R n R n is a continuous differentiable function of its argument. Let A = f be the Jacobian matrix x 0 associated with f evaluated at an equilibrium point x 0. We will use the fact that every square matrix can be decomposed as a sum of two matrices, one diagonalizable and one nilpotent, i.e., let A be a square matrix of order n, then it can be decomposed as A = S + N, with S a diagonalizable matrix and N a nilpotent matrix such that N k = 0 for 1 < k n. An interesting property of this decomposition is that the S and N matrices commute, i.e., SN = NS, [9], which is an ideal property for the purposes of this paper. We introduce the following lemma in order to establish structure preservation under matrix multiplication of simultaneous Jordan decomposable matrices. Lemma 2.2. The modifying matrix M, which is a simultaneous Jordan decomposition of our dynamical system s associated Jacobian matrix A, will maintain the system s structure. Proof. After further decomposing the J A matrix as a sum of diagonal and nilpotent matrices (see Ref. [9]) we arrive at PJ A P 1 = P(S + N)P 1. We propose a positive definite modifying matrix M that is a simultaneous decomposition to our A matrix and further decompose it into diagonal and nilpotent matrices M = PJ M P 1 = P(Ŝ + ˆN)P 1 then our modified system will now become MA = PJ M P 1 PJ A P 1 = P(Ŝ + ˆN)(S + N)P 1 carrying out the matrix products MA = P(ŜS + ŜN + ˆNS + ˆNN)P 1. Considering the characteristics of diagonal and strictly triangular matrices, we know that these structures may be redefined as S = ŜS, a diagonal eigenvalue matrix and N = ŜN + ˆNS + ˆNN, a nilpotent strictly upper triangular matrix. x

3 D. Becker-Bessudo et al. / Physica A 387 (2008) We may now write our modified system as MA = P( S + N)P 1. Thus preserving the system s structure. This small proof enables us to pursue the definition of a multiplicative group under the product of simultaneous Jordan decomposable matrices. What follows now is the explicit definition of the multiplicative group and its properties which allow for preservation of stability. Using this group we present an extension of the Stable Unstable Manifold theorem (Proposition 3.1). 3. Local Stable Unstable theorem extension The following proposition is a simple extension of the Local Stable Unstable Manifold Theorem for the action of group Ω P on the matrix A and the vector field f (x) where A, the system s linear coefficients matrix, may be decomposed as A = PJ A P 1 with J A a Jordan canonical form, PP 1 = P 1 P = I. We define pd as the set of matrices in their Jordan canonical and whose real coefficients are all positive and the group Ω P as Ω P = M M = PJ M P 1 Rn n with J M the Jordan canonical form of M and J M. pd This proposition is an alternative result to proposition 4.2 presented in Ref. [10] using simultaneous Jordan decomposition and standard matrix product. Proposition 3.1. Let E be an open subset of R n containing the origin, let f C 1 (E), and let φ t be the flow of the nonlinear system ẋ = f (x) = Ax + g(x). Suppose that f (0) = 0 and that A = Df (0) has k eigenvalues with negative real part and n k eigenvalues with positive real part, i.e., the origin is an hyperbolic fixed point. Then for each matrix M Ω P, as previously defined, there exists a k-dimensional differentiable manifold S M tangent to the stable subspace E S M of the linear system ẋ = MAx at 0 such that for all t 0, φ M,t (S M ) S M and for all x 0 S M, lim φ M,t(x 0 ) = 0, t where φ M,t be the flow of the nonlinear system ẋ = MAx + g(x); and there exists an n k dimensional differentiable manifold W M tangent to the unstable subspace E W M of ẋ = MAx at 0 such that for all t 0, φ M,t(W M ) W M and for all x 0 W M, lim φ M,t(x 0 ) = 0. t An interesting property is that Proposition 3.1 is valid for each g C 1 (E) such that ẋ = f (x) = Ax + g(x) and g(x) 2 x 2 0 as x 2 0. In consequence, the set of matrices Ω P generate the action of the group Ω P on the set of hyperbolic nonlinear systems (formally on the set of hyperbolic vector fields f C 1 (E)) ẋ = f (x) = Ax + g(x) with g C 1 (E) and { } Λ A R n n A = PJ A P 1 with J A the Jordan decomposition of A satisfying the last condition, where P is a fixed similarity matrix, this action is faithful and free. The former action is generated by the action of the group Ω P on the set Λ. The action preserves hyperbolic nonlinear systems and dimension of the stable and unstable manifolds, i.e., an hyperbolic nonlinear system (ẋ = Ax + g(x)) is mapped in an hyperbolic nonlinear system (ẋ = MAx + g(x)), and dim S = dim S M and dim W = dim W M. The following remarks are necessary for our proof Remark 3.2. For a complex eigenvalue a k + ib k, belonging to A where a k < 0; it is required that the eigenvalues of our complex modifying matrix M, ā k +i bk, fulfill ā k > 0 and ±b k ± bk to insure that adding the product of complex coefficients will preserve the original real coefficient s sign values in our modified matrix. Remark 3.3. For a complex eigenvalue a k + ib k, belonging to A where a k > 0; it is required that the eigenvalues of our complex modifying matrix M, ā k + i bk, fulfill ā k > 0 and ±b k bk for the same reason mentioned above. Proof. (1) Real coefficients Consider a matrix A of order n n with decomposition A = PJ A P 1, where J A is a Jordan canonical form with k negative real eigenvalues and n k positive real eigenvalues and P is a similarity n n matrix, and the decomposition M = PJ M P 1, M Ω P. By our proposition we carry out the matrix product MA = PJ M P 1 PJ A P 1 = PJ M J A P 1. Looking at the product of

4 6634 D. Becker-Bessudo et al. / Physica A 387 (2008) Jordan matrices J M and J A whose eigenvalues are σ M = {λ i } and σ A = {µ i }, respectively. It is simple to observe that the resulting matrix s eigenvalues are precisely the individual products of the original matrices eigenvalues; σ MA = {λ i µ i }. Since M Ω P is strictly positive, then the matrix MA has k eigenvalues with negative real part and n k eigenvalues with positive real part. Since the dimensions of each manifold have not changed, the result is a consequence of the Stable Unstable Manifold Theorem and Lemma 2.2. (2) Complex coefficients Consider a matrix A of order n n with decompositions A = PJ A P 1, where J A is a Jordan canonical form made up of complex values with k eigenvalues with negative real part and n k eigenvalues with positive real part and P is a similarity n n matrix, and the decomposition M = PJ M P 1, M Ω P. Then the matrix product yields MA = PJ M J A P 1. Seeing that the product of matrices J M and J A results in a matrix of complex eigenvalues we must adhere to what was previously established in Remarks 3.2 and 3.3. For the case of the eigenvalues related to matrix A with negative real part a k < 0 (Remark 3.2) we have the following product of eigenvalues (a k ±ib k )(ā k ±i bk ) = a k ā k bk b k ±i(a k bk +ā k b k ), since ā k > 0 it follows that a k ā k < 0 and subtracting bk b k will keep the real part of the new eigenvalue negative. Similarly in the case that a k > 0 (Remark 3.3) we have (a k ± ib k )(ā k i bk ) = a k ā k + bk b k ± i(a k bk ā k b k ), since ā k > 0 it follows that a k ā k > 0 and adding bk b k will keep the real part of the new eigenvalue positive. Then the matrix MA has k eigenvalues with negative real part and n k eigenvalues with positive real part. Again the dimension of each manifold is the same as the original system s, therefore the result is again a consequence of the Stable Unstable Manifold Theorem and Lemma 2.2. The relevance of this proposition resides on the fact that critical hyperbolic points are preserved. As a consequence of this we established properties which allow us to preserve the signature of the associated Jacobian matrix. In Sections 4 and 5 it will be shown, through various examples, that stability and synchronization are preserved under modifications performed on dynamical systems following the methodology of Proposition 3.1. This proposition on the extension of the Stable Unstable Manifold Theorem is different to other approaches for stability and synchronization preservation such as Ref. [8] where Lyapunov s indirect method was employed. Notice that given a particular nonlinear system, the stable and unstable manifolds S and W are unique, then for each matrix M Ω P there exist an unique pair of manifolds (S M, W M ) in such way that it is possible to define a pair of functions in the following form Γ : Ω P Man S Man S Γ (M, S) = S M Υ : Ω P Man W Man W Υ (M, W) = W M where Man S is the set of stable manifolds and Man W is the set of unstable manifolds for autonomous nonlinear systems. Remark 3.4 (See Ref. [10]). 1. The group Ω P satisfies the group axioms since the product of simultaneously decomposable positive definite Jordan matrices has no effect on the signature of the system s real coefficient s eigenvalues. 2. The action is faithful since no two different modifying matrices acting on the same set will yield the same manifold. 3. The action is free since our stabilizer, for both real and complex cases, is trivially an identity matrix of proper order. Notice that if A = Df (0) is a stable matrix, i.e., A has all the n eigenvalues with negative real part, then the origin of the nonlinear system ẋ = MAx + g(x) is asymptotically stable; and if A = Df (0) is an unstable matrix, i.e., A has at least one eigenvalue with positive real part, then the origin of the nonlinear system ẋ = MAx + g(x) is unstable. 4. Preservation of synchronization in modified systems In this section we show how it is possible to preserve synchronization after a system s eigenvalues have been modified under the action of a class of transformation on the linear part of the nonlinear system. Consider following n-dimensional systems in a master-slave configuration, where the master system is given by ẋ = Ax + g(x) and the slave system is ẏ = Ay + f (y) + u(t) where A R n n is a constant matrix, f, g : R n R n are continuous nonlinear functions and u R n is the control input. The problem of synchronization considered in this section is the complete-state exact synchronization. That is, the master system and the slave system are synchronized by designing an appropriate nonlinear state feedback control u(t) which is attached to the slave system such that lim y(t) x(t) 0 t where is the Euclidean norm of a vector.

5 D. Becker-Bessudo et al. / Physica A 387 (2008) Considering the error state vector e = y x R n, f (y) g(x) = L(x, y) and an error dynamics equation ė = Ae + L(x, y) + u(t). Based in the active control approach [11], to eliminate the nonlinear part of the error dynamics, and choosing u(t) = Bv(t) L(x, y), where B is a constant gain matrix which is selected such that (A, B) be controllable, we obtain ė = Ae + Bv(t). Notice that the original synchronization problem is equivalent to the problem of stabilizing the zero-input solution of the last system by a suitable choice of the state feedback control. Since the pair (A, B) is controllable one such suitable choice for state feedback is a linear-quadratic state-feedback regulator [12], which minimizes the quadratic cost function ( ) J(u(t)) = e(t) Qe(t) + v(t) Rv(t) dt 0 where Q and R are positive semi-definite and a positive definite weighting matrices, respectively. The state-feedback law is given by v = Ke with K = R 1 B S and S the solution to the Riccati equation A S + SA SBR 1 B S + Q = 0. This state-feedback law renders the error equation to ė = (A BK)e, with (A BK) a Hurwitz matrix. 1 The linear quadratic regulator (LQR) is a well-known design technique that provides practical feedback gains [12]. An interesting property of (LQR) is robustness. Now consider M Ω P, and suppose that the following two n-dimensional systems are chaotic ẋ = (MA) x + g(x) ẏ = (MA) y + f (y) + û(t) for some f, g : R n R n continuous nonlinear functions and û R n is the control input. We have that û(t) = θ(t) L(x, y) stabilizes the zero solution of the error dynamics system, where θ(t) = (MBK) e, i.e., the resultant system ė = (MA) e + θ(t) ė = (MA MBK) e is asymptotically stable. Notice that using K = R 1 B S, we obtain ė = (M (A BK)) e ė = ( M ( A BR 1 B S )) e. The original control u(t) = BKe L(x, y) is preserved in its linear part by the matrix product M ( ) and the new control is given by û(t) = (MBK) e L(x, y). Therefore, we can interpret the last procedure as one in which the controller u(t) which achieves the synchronization in the two original systems is preserved under the transformation M ( ) so that û(t) achieves the synchronization in the two resultant systems after the transformation. A similar procedure is possible if we consider the transformation ( ) M. In general, under the transformations (A, g) (MA, g), and under the hypothesis of the existence of a constant state feedback U = Kx which achieves synchronization of the original chaotic systems, and also that the transformed systems are chaotic, synchronization can be preserved. The main contribution in this section does not deal with a better synchronization methodology, rather it deals with the fact that synchronization is preserved when the underlying nonlinear chaotic dynamical system is altered in such a way as to change its dynamical behavior yet preserving the topological structure near the origin. 5. Synchronization of several dynamical systems 5.1. The Rössler attractor The original dynamical system of what is know as the Rössler attractor is defined by ẋ 1 = x 2 x 3 ẋ 2 = x x 2 ẋ 3 = x 3(x 1 14) 1 A Hurwitz matrix is a matrix for which all its eigenvalues are such that their real part is strictly less than zero.

6 6636 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 1. Original Rössler attractor showing synchronization between master and slave systems (initial conditions x 1 = 20, x 2 = 16, x 3 = 12 and y 1 = 17, y 2 = 13, y 3 = 5, respectively). Fig. 2. Magnitude of error e = y x between original Rössler master and slave systems. which has a chaotic attractor. In order to observe synchronization behavior we present two Rössler attractors arranged as a master/slave configuration. The master and the slave systems are almost identical, the only difference is that the slave system includes an extra term (the control) which is used for the purpose of synchronization with the master system. The initial conditions for the two systems are different. The master system is given by the aforementioned equations and the slave system is a copy of the master system plus a control function u(t) to be determined in order to synchronize the two systems. ẏ 1 = y 2 y 3 + u 1 (t) ẏ 2 = y y 2 + u 2 (t) ẏ 3 = y 3(y 1 14) + u 3 (t). Considering the errors e 1 = y 1 x 1, e 2 = y 2 x 2, e 3 = y 3 x 3, then the error dynamics equations may be written as ė 1 = e 2 e 3 + u 1 (t) ė 2 = e e 2 + u 2 (t) ė 3 = y 3y 1 x 3 x 1 14(e 3 ) + u 3 (t).

7 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 3. Master and slave system (initial conditions x 1 = 80, x 2 = 70, x 3 = 50 and y 1 = 60, y 2 = 50, y 3 = 40, respectively) synchronization of modified Rössler attractor (with M 1 Ω P comprised of purely real eigenvalues). Fig. 4. Magnitude of error e = y x between the modified Rössler master and slave systems (real eigenvalues). Introducing the Jacobian (A) and nonlinear terms (L) matrices A = 1 0, L(x, y) = y 3y 1 x 3 x 1, u = ( ) u1 (t) u 2 (t) u 3 (t) and selecting the matrix B such that (A, B) is controllable: B = I. Now the LQR controller is obtained by using weighting matrices Q = I and R = B B = I. The state feedback matrix is given by ( ) K = In Fig. 1 the trajectories for the solution of the master system and slave system are shown. In Fig. 2 the absolute value for the errors between the master and slave systems are shown in a semi-logarithmic plot to emphasize the fact that the error converges to zero and therefore the synchronization between the master and slave systems is successfully achieved The modified Rössler attractor The following examples show the modifications performed on the Rössler attractor with both real and complex eigenvalue matrices. The general equation for the modified Rössler master and slave systems linear and nonlinear parts

8 6638 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 5. Master and slave system (initial conditions x 1 = 35, x 2 = 33, x 3 = 23 and y 1 = 27, y 2 = 24, y 3 = 17, respectively) synchronization of modified Rössler attractor (with M 2 Ω P having both real and complex eigenvalues). Fig. 6. Magnitude of error e = y x between modified Rössler master and slave systems (real and complex eigenvalues). may be defined as follows [ ẋ = (MA)x [ ẏ = (MA)y ] x 3x 1, ] y 3y 1 + u(t). Considering the error vector e = y x, then the error dynamics may be written as ė = (MA)e + L(x, y) + u(t) with u = L(x, y) + v and v = (MBK)e. Defining the Jacobian and modifying matrices A = , M 1 = ( 2 0 ) , where the M 1 matrix s eigenvalues are all real positive numbers and with K the same as in Section 5.1. In Fig. 4 the absolute error for the master/slave system configuration, with a real positive definite matrix perturbation, is plotted in semi-logarithmic form to show that the error between them effectively converges to zero. We can appreciate that convergence is achieved relatively quickly (about the same period as the unaltered configuration).

9 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 7. Original Lü attractor showing synchronization between master and slave systems (initial conditions x 1 = 1, x 2 = 1, x 3 = 1 and y 1 = 3, y 2 = 3, y 3 = 3, respectively). Fig. 8. Magnitude of error e = y x between original Lü master and slave systems. Next we define a new matrix M 2 (which has both real and complex eigenvalues) ( ) M 2 = which was constructed using simultaneous Jordan decomposition and following the sign relationships established in Remark 3.3 (Since the real coefficients of the complex eigenvalues of A are positive). Once again we use K as in Section 5.1 and u = (M 2 BK)e L(x, y). In Fig. 6 we have the absolute error of the master/slave system configuration, this time with a complex matrix perturbation, and again we see that there is an effective convergence to zero. However, the amount of time needed to achieve this was considerably larger, thus showing that these types of perturbations have a much more severe effect on the dynamical system s behavior The Lü attractor The dynamical system associated with the Lü attractor, with parameters a = 35, b = 3 and c = 28, is defined by the equations ẋ 1 = 35(x 2 x 1 ) ẋ 2 = 28x 2 x 1 x 3 ẋ 3 = x 1 x 2 3x 3. Just as was done with the Rössler attractor we have a modified Lü attractor arranged as a master/slave configuration. Again the initial conditions for the two systems are different.

10 6640 D. Becker-Bessudo et al. / Physica A 387 (2008) The slave system is then defined by the equations ẏ 1 = 35(y 2 y 1 ) + u 1 (t) ẏ 2 = 28y 2 y 1 y 3 + u 2 (t) ẏ 3 = y 1 y 2 3y 3 + u 3 (t) and its error dynamics equations ė 1 = 35(e 2 e 1 ) + u 1 (t) ė 2 = 28e 2 y 1 y 3 + x 1 x 3 + u 2 (t) ė 3 = y 1 y 2 x 1 x 2 3e 3 + u 3 (t). Introducing the matrices ( ) 0 ( 0 ) A = , L(x, y) = y 1 y 3 + x 1 x 3, u = y 1 y 2 x 1 x 2 The resulting state feedback matrix is ( ) K = ( ) u1 (t) u 2 (t). u 3 (t) In Fig. 7 the trajectories for the solution of the master system and slave system are shown. In Fig. 8 the absolute value for the error between the master and slave systems are shown in a semi-logarithmic plot to emphasize the fact that the error converges to zero and therefore the synchronization between the master and slave systems is achieved The modified Lü attractor Now we shall present a system showing modifications performed on the Lü attractor. The modified Lü master and slave systems linear and nonlinear parts may be defined as follows ẋ = (MA)x + [ 0 x 1 x 3 x 1 x 2 ], ẏ = (MA)y + [ 0 y 1 y 3 y 1 y 2 ] + u(t). Considering the error vector e = y x, then the error dynamics may be written as ė = (MA)e + L(x, y) + u(t) with u = L(x, y) + v and v = (MBK)e. Using the Jacobian matrix A we defined earlier and the modifying matrix ( ) 0 M = 0 4 0, where the M matrix s eigenvalues are all real positive numbers and with K the same as in Section 5.2. In Fig. 10 the absolute error for the master/slave system configuration, with a real positive definite matrix perturbation, is plotted in semi-logarithmic form to show that the error between them effectively converges to zero. However, the amount of time elapsed for this is much shorter, this may be due to the fact that we have increased the difference between the values of the eigenvalues on the x 1 and x 2 resulting in a lesser power struggle. An interesting characteristic that was found when selecting the eigenvalues used in our modifying matrix M is that when they exceed certain values, chaotic dynamics seem to be lost. This restriction is established by the original eigenvalues associated with the x 1 and x 2 elements of the Jacobian A matrix. In the case of the proposed Lü system we have σ A = { 35, 28, 3}, the absolute value of the ratio between the first and second eigenvalues is This ratio determines a minimal quota over the selection of our modifying eigenvalues. Consider the eigenvalues of our modifying matrix M as σ M = {µ 1, µ 2, µ 3 }, they must be selected so that 35µ 1 28µ If this quotient falls too far bellow this ratio it will cause the dynamic of the system to change radically. Similar behavior can be found for quotients between different pairs of eigenvalues for this system, although they are not as explicit. 2 2 Similar restrictions were found when working with the Chen and Lorentz attractors, not surprisingly since they share certain qualities and structure.

11 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 9. Master and slave system (initial conditions x 1 = 1, x 2 = 1, x 3 = 1 and y 1 = 3, y 2 = 3, y 3 = 3, respectively) synchronization of modified Lü attractor (with M 1 Ω P comprised of purely real eigenvalues) The Chua attractor Fig. 10. Magnitude of error e = y x between modified Lü master and slave systems (real eigenvalues). The system presented here is commonly known as the Modified Chua attractor, to avoid confusion we shall refer to it simply as the Chua attractor. The master system is given by the equations ( ẋ 1 = 10 x 2 2x3 1 x ) 1 7 ẋ 2 = x 1 x 2 + x 3 ẋ 3 = x 2. The slave system is then defined by ( ẏ 1 = 10 y 2 2y3 1 y ) 1 + u 1 (t) 7 ẏ 2 = y 1 y 2 + y 3 + u 2 (t) ẏ 3 = y 2 + u 3 (t) and its error dynamics equations are ė 1 = 10e e (y3 1 x3) + 1 u 1(t) ė 2 = e 1 e 2 + e 3 + u 2 (t)

12 6642 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 11. Original Chua attractor showing synchronization between master and slave systems (initial conditions x 1 = 0.01, x 2 = 0.01, x 3 = 0.01 and y 1 = 0.5, y 2 = 0.5, y 3 = 0.5, respectively). ė 3 = e 2 + u 3 (t). Introducing the matrices A = The state feedback is ( ) K = Fig. 12. Magnitude of error e = y x between original Chua master and slave systems. 20(y3 1 x3) 1, L(x, y) = 7, u = 0 0 ( ) u1 (t) u 2 (t). u 3 (t) In Fig. 11 the trajectories for the solution of the master system and slave system are shown. Fig. 12 shows the absolute value for the error between the master and slave systems in a semi-logarithmic plot to emphasize the fact that it effectively converges to zero and therefore the synchronization between the master and slave systems is achieved The modified Chua attractor The following examples show the modifications performed on the Chua attractor with both real and complex eigenvalue matrices. The general definition for the Chua master and slave systems linear and nonlinear parts may be defined as follows [ ẋ = (MA)x + 20x ],

13 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 13. Master and slave system (initial conditions x 1 = 0.01, x 2 = 0.01, x 3 = 0.01 and y 1 = 0.5, y 2 = 0.5, y 3 = 0.5, respectively) synchronization of modified Chua attractor (with M 1 Ω P comprised of purely real eigenvalues). Fig. 14. Magnitude of error e = y x between modified Chua master and slave systems (real eigenvalues). Fig. 15. Master and slave system (initial conditions x 1 = 0.01, x 2 = 0.01, x 3 = 0.01 and y 1 = 0.5, y 2 = 0.5, y 3 = 0.5, respectively) synchronization of modified Chua attractor (with M 2 Ω P having both real and complex eigenvalues). [ ẏ = (MA)y + 20y ] + u(t). Considering the error vector e = y x, then the error dynamics may be written as ė = (MA)e + L(x, y) + u(t) with u = L(x, y) + v and v = (MBK)e.

14 6644 D. Becker-Bessudo et al. / Physica A 387 (2008) Fig. 16. Magnitude of error e = y x between modified Chua master and slave systems (real and complex eigenvalues). Using the modifying matrix M 1 Ω p ( ) M 1 = , where the M 1 matrix s eigenvalues are all real positive numbers and with K the same as in Section 5.3. In Fig. 14 the absolute error for the master/slave system configuration, with a real positive definite matrix perturbation, is plotted in semi-logarithmic form to show that the error between them effectively converges to zero. We can appreciate that convergence is achieved relatively quickly (about the same period as the unaltered configuration). Next we define a new matrix M 2 (which has both real and complex eigenvalues) ( ) M 2 = which was constructed using simultaneous Jordan decomposition and following the sign relationships established in Remark 3.3 (Since the real coefficients of the eigenvalues of A are negative). Once again we use K as in Section 5.3 and u = (M 2 BK)e L(x, y). In Fig. 16 we have the absolute error of the master/slave system configuration, this time with a complex matrix perturbation, and again we see that there is an effective convergence to zero. However, the amount of time needed to achieve this was considerably larger, this result concurs with what was previously seen with the Rössler attractor. Looking at the modified Figs. 3, 5, 9, 13 and 15, as far as we can see, the chaotic dynamics are preserved. Just as it was seen with the Lü attractor there exist quotas over the chosen modifying eigenvalues that would seem to cause the Rössler and Chua attractors to loose their chaotic dynamics. In the case of modifications performed with real eigenvalue matrices these quotas are less constricted. Since we deal with only two independent modifying values, results have shown that the quotient between these cannot be too large or too small, although its range does seem to be directly related to the original dimensions of the unaltered system. For modifications with complex eigenvalue matrices it was seen that our selection of complex coefficients should not alter the original system s eigenvalues significantly in order to avoid this loss of chaotic behavior. Taking into account that the control input that was applied to achieve synchronization in all the presented examples was generated through the same methodology used to modify the linear systems and that synchronization was achieved, we may infer robustness under multiplicative perturbations over the linear part of nonlinear systems. 6. Conclusions The preservation of hyperbolic behavior in chaotic synchronization is studied from an extension of the local stableunstable manifold theorem based in the preservation of the signature of the linear part of the vector fields in nonlinear dynamical systems. Furthermore, under the hypothesis that given a chaotic system, its transformed version also a chaotic system, it is shown that a scheme consisting of a master/slave pair for which a constant state feedback has achieved synchronization, it is preserved even after the master/slave/controller system has been transformed. It is an attempt to study how regular dynamics can be preserved when important changes occur in the dynamical system. From the results we may conclude that the fundamental properties of the synchronization manifold, the signature of the Jacobian matrix, hyperbolic points and the stability of the system are preserved thus showing that robustness is a consequence of this methodology. As

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