Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix

Preservation is related to local asymptotic stability in nonlinear systems by using dynamical systems tools. It is known that a system, which is stable, asymptotically stable, or unstable at origin, through a transformation can remain stable, asymptotically stable, or unstable. Some systems permit partition of its nonlinear equation in a linear and nonlinear part. Some authors have stated that such systems preserve their local asymptotic stability through the transformations on their linear part. The preservation of synchronization is a typical application of these types of tools and it is considered an interesting topic by scientific community. This chapter is devoted to extend the methodology of the dynamical systems through a partition in the linear part and the nonlinear part, transforming the linear part using the Tracy-Singh product in the Jacobian matrix. This methodology preserves the structure of signs through the real part of eigenvalues of the Jacobian matrix of the dynamical systems in their equilibrium points. The principal part of this methodology is that it permits to extend the fundamental theorems of the dynamical systems, given a linear transformation. The results allow us to infer the hyperbolicity, the stability and the synchronization of transformed systems of higher dimension.


Introduction
In nonlinear autonomous dynamical systems, the study of synchronization is not new. We can see several papers about these themes from different approaches. Some examples show the use of change of variables, that is, through a diffeomorphism of the origin. From this, it is possible to say if a system is stable, asymptotically stable, or unstable. Some results are also obtained by the product in a vector field in the nonlinear dynamical system by a continuously differentiable function at the origin [1]. On one hand, there are studies showing the use of statistical properties to characterize the synchronization [2]. The eigenvalues of a system determine a system dynamics, but they are not derivable from the statistical features of such a system. One way to observe the stability is through a linear part of a dynamical system. But the problem to preserve stability by the transformation of its linear part in a nonlinear autonomous system has just been analyzed recently.
In [3], it is presented a methodology under which stability and synchronization of a dynamical master-slave system configuration are preserved under a modification through matrix multiplication. The conservation of stability is important for chaos control. A generalized synchronization can also be derived for different systems by finding a diffeomorphic transformation such as the slave system written as a function of the master system. One example of preservation for asymptotic stability is the use of transformations on rational functions in the frequency domain [4,5].
This class of transformation can be interpreted as noise in the system or as a simple disturbance on the value of the physical parameters of the model. The chaotic synchronization problem studied in [6] is mainly related to preservation of the stability of the master-slave system presented in it. Results included therein show that stability is preserved by transforming the linear part of system. The same results can also be used in the chaos suppression problem. In [7], the authors show the viability of preserving the hyperbolicity of a masterslave pair of chaotic systems under different types of nonlinear modifications to its Jacobian matrix.
In [8], the developed methodology is used to study the problem of preservation of synchronization in chaotic dynamical systems, in particular the case of dynamical networks. Given a chaotic system, its transformed version is also a chaotic system. By means of a master-slave scheme obtained a controller for the system using a linear-quadratic regulator, preserving the stability even after the master-slave controller is transformed. This chapter is inspired by the same objective, that is, to preserve the stability in a master-slave system even through a transformation is performed over it. One way to achieve it is by extending some of the results in [8], particularly those of the local stable-unstable manifold theorem and extension of the center manifold theorem based in the preservation of the linear part of the vector field in nonlinear dynamical systems. As we will see, these results depart from the hypothesis of the existence of a constant state feedback as anominal synchronization force. In this work, we elaborate another approach to the problem of preservation of synchronization. We focus particularly on autonomous nonlinear dynamical systems, extending the previous results already mentioned. This chapter is organized as follows: First, in Section 2, we will give basic concepts of dynamical systems. The fundamental theorem for linear systems, the local stable-unstable manifold theorem, the center manifold theorem, the Hartman-Grobman theorem and the concept of group action are introduced. In Section 3, we present some definitions about matrices and Tracy-Singh product of matrices. Also in this section, the main result is presented as a generalization of Proposition 4 in [6]. In Section 4, we will show that it is possible to preserve synchronization under a class of transformations defined under a certain method. Numerical experiments on the stability preservation for chaotic synchronization are shown in Section 5. Finally, a set of concluding remarks is given in Section 6.

Classical concepts of dynamical systems
We introduce theorems and classical definitions on properties of dynamical systems in this section. The fundamental theorem for linear systems, the local stable-unstable manifold theorem and the center manifold theorem are those important propositions mainly needed to develop analyses in this chapter. We will combine them with the Hartman-Grobman theorem in order to achieve a necessary generalization for those particular results of this chapter.
Theorem 2.1. (The local stable-unstable manifold theorem [9]). Let E be an open subset of R n containing the origin. Let f ∈ C 1 ðEÞ and φ t be the flow of the nonlinear system of the form _ x ¼ f ðxÞ. Suppose that f ð0Þ ¼ 0 and that Df ð0Þ are the Jacobian matrix, which has k eigenvalues with negative real part and n−k eigenvalues with positive real part.
1. (Stable manifold) Then, there exists a k-dimensional differentiable manifold S tangent to the stable subspace E S of the linear system _ x ¼ AðxÞ at x 0 such that for all t ≥ 0, φ t ðSÞ⊂S and for all x 0 ∈S, lim t!∞ φ t ðx 0 Þ ¼ 0.
2. (Unstable manifold) Also there exists an n−k dimensional differentiable manifold W tangent to the unstable subspace E W of _ x ¼ AðxÞ at x 0 such that for all t ≤ 0, φ t ðWÞ⊂W and for all x 0 ∈W, lim t!−∞ φ t ðx 0 Þ ¼ 0.
It should be noted that the manifolds S and W mentioned in Theorem 2.1 are unique. We define now the central manifold theorem in the following.
Theorem 2.2. (The center manifold theorem [9]). Let E be an open subset of R n containing the origin and r ≥ 1. Let f ∈ C r ðEÞ, that is, f is a continuously differentiable function on E of order r. Now we suppose that f ð0Þ ¼ 0 and that Df ð0Þ have k eigenvalues with negative real part, j eigenvalues with positive real part and l ¼ n−k−j eigenvalues with zero real part. Therefore, there exists an l -dimensional center manifold W C ð0Þ of class C r tangent to the center subspace E C of _ x ¼ AðxÞ at 0 which is invariant under the flow φ t of _ x By what it is established in Theorem 2.2, the center manifold W C ð0Þ is not unique, which is an important difference for the stable character of the systems to be studied. Theorem 2.3. (The Hartman-Grobman theorem [9]). Let E be an open subset of R n containing the origin, let φ t be the flow of the nonlinear system _ x ¼ f ðxÞ. Now, we assume that f ð0Þ ¼ 0, that is, the origin is an equilibrium point of the dynamical system; also the Jacobian matrix evaluated at the origin, A ¼ Df ð0Þ. If H is an homeomorphism of an open set W onto an open set V such that for each x 0 ∈W, it exists an open interval I 0 ⊂ R such that for all x 0 ∈ W and t ∈ I 0 that is, H maps trajectories of the nonlinear system _ x ¼ f ðxÞ near the origin onto trajectories of _ x ¼ Ax near the origin and preserves the parametrization.
This last equality is a consequence of the Hartman-Grobman theorem and of the fact of Ue At ¼ e T A t U, that is,Ĥ maps trajectories of the nonlinear system _ x ¼ f ðxÞ near the origin onto trajectories of _ x ¼ T A x near the origin and preserves the parametrization.
On the other hand, some classical definitions are now included. A linear system of the form _ x ¼ Ax where x ∈ R n , A is a n · n matrix and _ x ¼ dx dt . It is shown that the solution of the linear system together with the initial condition xð0Þ ¼ x 0 is given by xðtÞ ¼ e At x 0 . The mapping e At : R n ! R n is called the flow of the linear system. Definition 2.1. For all eigenvalues of a matrix Aðn · nÞ have nonzero real part, then the flow e At is called a hyperbolic flow and therefore, _ x ¼ Ax is called a hyperbolic linear system [9].
Definition 2.2. A subspace E⊂R n is said to be invariant with respect to the flow e At : R n ! R n if e At ⊂E for all t∈R [9].
and E c are the stable, unstable and center subspaces of the linear system _ x ¼ Ax. By the above,E s , E u and E c are invariant with respect to the flow e At , respectively [9]. Definition 2.3. Let E be an open subset of R n and let f ∈ C 1 ðEÞ, that is, f is a continuous differentiable function defined on E. For x 0 ∈ E, let φðt, x 0 Þ be the solution of the initial value problem _ x ¼ f ðxÞ, xð0Þ ¼ x 0 defined on its maximal interval of existence Iðx 0 Þ. Then for t∈Iðx 0 Þ, the mapping φ t : E ! E defined by φ t ðx 0 Þ ¼ φ t ðt, x 0 Þ is called the flow of the differential equation [9]. Definition 2.4. For any x 0 ∈R n , let φ t ðx 0 Þ be the flow of the differential equation through x 0 .(i) The local stable set S corresponding to a neighborhood V of x 0 is defined by S ¼ Sð0Þ Then, these stable and unstable local sets are submanifolds of R n in a sufficiently small neighborhood V of x 0 ½9. Definition 2.5. If G is a group and X is a set, then a (left) group action of G on X is a binary function G · X ! X, denoted by [9] ðg, xÞ↦g Á x which satisfies the following two axioms: 2. e Á x ¼ x for every x ∈ X (where e denotes the identity element of G).
The action is faithful (or effective) if for any two different g, h ∈ G, there exists an x ∈ X such that g Á x ≠ h Á x; or equivalently, if for any g ≠ e in G, there exists an x ∈ X such that g Á x ≠ x.
The action is free or semiregular if for any two different g, h ∈ G and all x ∈ X, we have For every x ∈ X, we define the stabilizer subgroup of x (also called the isotropy group or little group) as the set of all elements in G that fix x: This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial.

Tracy-Singh product and other mathematical extensions
In this third section, we show a definition and some properties of the Tracy-Singh product. We also include a simple extension of the local stable-unstable manifold theorem and the center manifold theorem, using the tools presented in Section 2. These extensions are tools that will also be used in Section 4, where we will present the results on preservation of synchronization in nonlinear dynamical systems.
Definition 3.1. Let λ be an eigenvalue of the n · n matrix A of multiplicity m≤n. Then for k ¼ 1, …, m, any nonzero solution w of [9] ðA−λIÞ k w ¼ 0 is called a generalized eigenvector of A.
In this case, let w j ¼ u j þ v j be a generalized eigenvector of the matrix A corresponding to an be a basis of R n (with n ¼ 2m−k as established by Theorems 1.7.1 and 1.7.2, see [9]). Now, we introduce the definition of Tracy-Singh product and some properties.
Definition 3.2. If taken the matrices A ¼ ða ij Þ and C ¼ ðc ij Þ of order m · n and B ¼ ðb kl Þ of order p · q. Let A ¼ ðA ij Þ be partitioned with A ij of order m i · n j as the ði, jÞ th block submatrix and B ¼ ðB kl Þ of order p k · q l as the ðk, lÞ th block submatrix ð∑m i ¼ m, ∑n j ¼ n, ∑p k ¼ p, ∑q l ¼ qÞ. Then, the definitions of the matrix products or sums of A and B are given as follows [10].
where A ij ⊗B kl is of order m i p k · n j q l , A ij ∘B is a Kronecker product of order m i p · n j q, and A∘B is of order mp · nq.
Tracy-Singh sum where A ¼ ðA ij Þ and B ¼ ðB kl Þ are square matrices of respective order m · m and p · p with A ij of order m i · m j and B kl of order p k · p l ; I p and I m are compatibly partitioned identity matrices. 1. ðA∘BÞðC∘DÞ ¼ ðACÞ∘ðBDÞ.

ðA∘BÞ∘F ¼ A∘ðB∘FÞ
The next proposition presents some extensions to the local stable-unstable manifold theorem and to the center manifold theorem.
Proposition 3.1. Let E be an open subset of R n containing the origin, let f ∈ C 1 ðEÞ and φ t be the flow of the nonlinear system _ x ¼ f ðxÞ ¼ Ax þ gðxÞ. Suppose that f ð0Þ ¼ 0 and that A ¼ Df ð0Þ have k eigenvalues with negative real part and n−k eigenvalues with positive real part, that is, the origin is an hyperbolic fixed point. Then for each matrix M ∈ Λ U , there exists a k where φ M, t is the flow of the nonlinear system _ x ¼ MAx þ gðxÞ and there exists an n−k dimen- An interesting property is that Proposition 4.1 is valid for each g∈C 1 ðEÞ such that _ x ¼ f ðxÞ ¼ Ax þ gðxÞ and ‖gðxÞ‖ 2 ‖x‖ 2 ! 0 as‖x‖ 2 ! 0: In consequence, the set of matrices Λ U generates the action of the group Λ U on the set of the hyperbolic nonlinear systems, formally on the set of the hyperbolic vector fields f ∈C 1 ðEÞ, _ x ¼ f ðxÞ ¼ Ax þ gðxÞ with g∈C 1 ðEÞ and A∈Ω U ≡{P∈R n · n : P ¼ U T T P U with T P any upper triangular matrix} Satisfying the last condition, where U is a fixed unitary matrix, the action is generated by the action of the group Λ U on the set Ω U . By that this first action preserves the dimension and a nonlinear systems of the stable and unstable manifolds, that is, an hyperbolic nonlinear system _ x ¼ Ax þ gðxÞ is mapped in a hyperbolic nonlinear systems _ x ¼ MAx þ gðxÞ and The proof of this Proposition 3.1 can be revised in Ref. [8].
Given a particular nonlinear system, the stable and unstable manifolds S and W are unique; then for each matrix M∈Λ U , there exists an unique pair of manifolds ðS M , W M Þ in such a way that it is possible to define a pair of functions in the following form Where Man S is the set of stable manifolds and Man W is the set of unstable manifold for autonomous nonlinear systems.
Therefore, we can say that if A ¼ Df ð0Þ is an stable matrix A has all the n eigenvalues with negative real part, then the origin of the nonlinear system _ x ¼ M∘Ax þ gðxÞ is asymptotically stable; but if A ¼ Df ð0Þ is an unstable matrix A has n−k (with n > kÞ eigenvalues with positive real part, then the origin of the nonlinear system _ x ¼ M∘Ax þ gðxÞ is unstable.
As an extension of the local stable-unstable manifold theorem in terms of Tracy-Singh product of matrices in Λ N and the matrix A of the vector field f ðxÞ, we present the following proposition.
1. Let E be an open subset of R n containing the origin, let f ∈C 1 ðEÞ and let φ t be the flow of the nonlinear system _ x ¼ f ðxÞ ¼ Ax þ gðxÞ. We suppose that f ð0Þ ¼ 0 and that A ¼ Df ð0Þ have a k eigenvalues with negative real part and n−k eigenvalues with positive real part; thus, the origin is a hyperbolic fixed point. Now, take a fixed continuously differentiable function and the new nonlinear system is which satisfies the following two axioms: 1. ðghÞ Á z ¼ g•ðh Á zÞ for all g, h∈Λ N and z∈Γ n ; 2. For every z∈Γ n , there exists an uniqueẑ∈Γ mn such that e Á z ¼ẑ and h•ẑ ¼ h Á z (e denotes the identity element of Λ N , that is, is the identity matrix I m of m · m).

Consider a matrix
A with eigenvalues λ i for i ¼ 1, 2, …, n and the matrix M with eigenvalues μ j for j ¼ 1, 2, …, m. Then, the eigenvalues of the matrix M∘A are the mn numbers λ i μ j and taking account that μ j > 0 for each j ¼ 1, 2, …, m. Therefore, the matrix M∘A has mk eigenvalues with negative real part and mðn−kÞ eigenvalues with positive real part. For this, the result is a consequence of the stable-unstable manifold theorem.
2. The function ϑ : Λ N · Γ n ! Γ mn is well defined, since F : C 1 ðEÞ ! C 1 ðEÞ is a fixed function; then given gðxÞ, the vector fieldĝðxÞ is unique; for a fixed matrix M h ∈Λ N , then M h ∘ : R n · n ! R mn · mn is a fixed function and their matrix M h ∘A is unique. From what it has been said above, we can note that if A ¼ Df ð0Þ is as stable matrix A, it has all the n eigenvalues with negative real part, then the origin of the nonlinear system _ x ¼ ðM∘AÞx þĝðxÞ is asymptotically stable; if A ¼ Df ð0Þ is an unstable matrix A, it has n−kðn > kÞ eigenvalues with positive real part, then the origin of the nonlinear system _ x ¼ ðM∘AÞx þĝðxÞ is unstable. Now the following Proposition 3.2 is an extension of the center manifold theorem, similar to Proposition 3.1.

Proof.
The proof is similar to proof of Proposition 3.1 and we make use of the center manifold theorem.
Also, there exists a similar functionθ to ϑ, which satisfies the axiom (i) and axiom (ii) of Proposition 3.2. However, in this case, there does not exist similar functions to Θ and Φ. due to that in general, a center manifold is not unique.
Notice that in this case, if the matrix A has l ¼ n−k−j≠0 eigenvlues with zero real part, then the origin of the nonlinear system _ x ¼ MAx þĝðxÞ and _ x ¼ ðM∘AÞx þĝðxÞ are not asymptotically stable.
Propositions 3.1 and 3.2 generalize Proposition 3 in Ref. [6] and give new tools for preservation of basic properties of dynamical systems and some of these properties are the stability and instability.

Synchronization in nonlinear dynamical system
In this section, we present that it is possible to preserve synchronization even though the dimension of the systems changes by the action of a class of transformation on the linear part to a chaotic nonlinear system. If we consider the following two n-dimensional chaotic systems, Where A∈R n · n is a constant matrix. On the other hand,u∈R n is the control input and f , g : R n ! R n are continuous nonlinear functions. Synchronization considered in this section is through the master and the slave system is synchronized by designing an appropriate nonlinear state-feedback control uðtÞ attached to slave system such that lim t!∞ xðtÞ−yðtÞ ! 0, where ‖ Á ‖ is the Euclidean norm of a vector [8]. If we consider the error state vector e ¼ y−x∈R n , f ðyÞ−f ðxÞ ¼ Lðx, yÞ and an error dynamics equation is _ e ¼ Ae þ Lðx, yÞ þ uðtÞ. Taking the active control approach [5], to eliminate the nonlinear part of the error dynamics and choosing uðtÞ ¼ BvðtÞ−Lðx, yÞ, where B is a constant gain vector which is selected such that ðA, BÞ be controllable, we obtain: We can see that the original synchronization problem is equivalent to stabilize the zero-input solution of the slave system through a suitable choice of the state-feedback control [8]. If the pair ðA, BÞ is controllable, then one such suitable choice for state feedback is a linear-quadratic regulator [5], which minimizes the quadratic cost function in the next expression, Where Q and R are positive semi-definite and positive definite weighting matrices, respectively. The state-feedback law is given by v ¼ −Ke with K ¼ R −1 B ⊺ S and S the solution to the Riccati equation This state-feedback law makes the error equation to be _ e ¼ ðA−BKÞe, with ðA−BKÞ a Hurwitz matrix. 1 The linear-quadratic regulator is a technique to obtain feedback gains [5]. It is an interesting property of (LQR) which is robustness. On the other hand, if we consider T∈R m · m be a matrix with strictly positive eigenvalues, supposing that the following two nm-dimensional systems are chaotic: for somef ,ĝ : R nm ! R nm continuous nonlinear functions andû∈R nm is the control input.
Then, for the Proposition 4.1 and the former procedure, we have thatûðtÞ ¼θðtÞ−Lðx, yÞ stabilizes the zero solution of the error dynamics system, whereθðtÞ ¼ −ðBK∘TÞe, that is, the resultant system is asymptotically stable. Then, by using Lemma 2.1 and K ¼ −R −1 B ⊺ S, we obtain that: Now, the original control uðtÞ ¼ BKe−Lðx, yÞ is preserved in its linear part by the Tracy-Singh product T∘ðÁÞ and the new control is given byûðtÞ ¼ −ðT∘BKÞe−Lðx, yÞ. Therefore, we can interpreted the last procedure as one in which the controller uðtÞ that achieves the synchronization in the two systems is preserved by the transformation T∘ðÁÞ so thatûðtÞ achieves the synchronization in the two resultant systems after the transformation. For that, a similar procedure is possible if we consider the transformation ðÁÞ∘T.
In general, under the transformation ðA, gÞ ! ðMA, gÞ or ðA, gÞ ! ðM∘A, gÞ and under the hypothesis of existence of a constant state feedback U ¼ −Kx, which achieves synchronization of the original chaotic systems and also that the transformed system is chaotic, then synchronization can be preserved [8]. The major contribution does not refer a better synchronization methodology; it deals that synchronization is preserved when a chaotic system changes from a lower dimension to a higher dimension.

Synchronization of the classical Lü system
In this section, we present the synchronization of a chaotic system. First, we propose a master and slave system. Then, from these systems, we will apply a linear transformation that allows us to preserve the synchronization. We will use the well-known Lü and Chen [11] model to show the possibility to preserve synchronization, described by which has a chaotic attractor when the parameters are a ¼ 35, b ¼ 3 and c ¼ 14:5. In order to observe synchronization behavior, we have a modified Lü attractor arranged as a master-slave configuration. The master and the slave systems are almost identical and the only difference is that the slave system includes an extra term which is used for the purpose of synchronization with the master system. The master system is defined by the following equations, (31) and the slave system is a copy of the master system with a control function uðtÞ to be determined in order to synchronize the two systems.
Now, we consider the errors e 1 ¼ x 1 −y 1 , e 2 ¼ x 2 −y 2 and e 3 ¼ x 3 −y 3 ,; then, the error dynamics can be written as: If we introduce the matrices and selecting the matrix B such that ðA, BÞ is controllable: B ¼ I, the LQR controller is obtained by using weighting matrices Q ¼ I and R ¼ B ⊺ B ¼ I. Then, state-feedback matrix is given by From the formerly said, we now present simulations made for the synchronized system of Lü and for the system also synchronized, but after the transformation of its linear part. All simulations here presented were made in Matlab software. In Figure 1, we show the trajectories of the master system of Lü. Each line represents one trajectory of the system along the time, taking an initial condition of ð1, 1, 1Þ.
For the case of Figure 3, we show the trajectories of the slave system of Lü. As it was in the first case, each line represents one trajectory of the system along the time, taking a initial condition as ð3, 3, 3Þ. Figures 2 and 4 are phase space mappings of each system while maintaining the same initial condition.
On the other hand, in Figure 5, we can see the error magnitude between master and slave systems. Phase space of synchronization of the master and slave systems in Figure 6 is presented. 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: Considering the error vector e ¼ y−x, then the error dynamics can be written as: with u ¼ −Lðx, yÞ þ v and v ¼ −ðT∘BKÞe and        After the transformation in its linear part of Lü attractor, we also have several simulations allowing us to analyze the dynamics of the transformed system. In Figure 7, we present the trajectories of the transformation of the master system of Lü. Each line represents one trajectory of the system along with the time taking an initial condition of ð0:5, 0:5, 0:5, 0:5, 0:5, 0:5Þ. For the case of Figure 9, we show the trajectories of the transformation of the slave system of Lü. Each line represents one trajectory of the system also, along the time, taking an initial condition of ð3, 3, 3, 3, 3, 3Þ. Figures 8 and 10 are the phase space mappings of each transformed system while maintaining the same initial condition. By last, in Figure 11, we can see the error magnitude of the transformation of synchronized system. A phase space mapping of the transformation of synchronized system is presented in Figure 12.

Conclusion
We have studied the preservation of stability of a chaotic dynamic system, from an extension of the stable-unstable manifold theorem and an extension of the center manifold theorem based on the preservation of the linear part in nonlinear dynamical systems. However, we can check that given a chaotic system, its transformed version is also chaotic. A scheme consisting of a master-slave system for which a controller gain is obtained using a linearquadratic regulator has been presented and synchronization is achieved and preserved even after the master-slave controller is transformed, obtaining as a consequence that the chaotic system changes to an higher dimension. It is important to note the transformation of the linear part of the chaotic system from Tracy-Singh product in which it was used to modify a Lü system, showing the effectiveness of the proposed method. The results can be extended to other techniques for feedback design, for example, adaptive control, sliding mode regulator and etcetera.