Open access peer-reviewed chapter

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

By Guillermo Fernadez‐Anaya, Luis Alberto Quezada‐Téllez, Jorge Antonio López‐Rentería, Oscar A. Rosas‐Jaimes, Rodrigo Muñoz‐ Vega, Guillermo Manuel Mallen‐Fullerton and José Job Flores‐ Godoy

Submitted: May 23rd 2016Reviewed: November 16th 2016Published: March 15th 2017

DOI: 10.5772/66957

Downloaded: 825

Abstract

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.

Keywords

  • preservation
  • synchronization
  • Tracy‐Singh product
  • chaotic dynamical system

1. 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 master‐slave 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.

2. 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 Ebe an open subset of Rn

containing the origin. Let fC1(E)and ϕtbe the flow of the nonlinear system of the form x˙=f(x). Suppose that f(0)=0and that Df(0)are the Jacobian matrix, which has keigenvalues with negative real part and nkeigenvalues with positive real part.

  1. (Stable manifold) Then, there exists a k‐dimensional differentiable manifold Stangent to the stable subspace ESof the linear system x˙=A(x)at x0such that for all t0, ϕt(S)Sand for all x0S, limtϕt(x0)=0.

  2. (Unstable manifold) Also there exists an nkdimensional differentiable manifold Wtangent to the unstable subspace EWof x˙=A(x)at x0such that for all t0, ϕt(W)Wand for all x0W, limtϕt(x0)=0.

It should be noted that the manifolds Sand 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 Ebe an open subset of Rncontaining the origin and r1. Let fCr(E), that is, fis a continuously differentiable function on Eof order r. Now we suppose that f(0)=0and that Df(0)have keigenvalues with negative real part, jeigenvalues with positive real part and l=nkjeigenvalues with zero real part. Therefore, there exists an l

‐dimensional center manifold WC(0)of class Crtangent to the center subspace ECof x˙=A(x)at 0which is invariant under the flow ϕtof x˙=f(x).

By what it is established in Theorem 2.2, the center manifold WC(0)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 Ebe an open subset of Rncontaining the origin, let ϕtbe the flow of the nonlinear system x˙=f(x). Now, we assume that f(0)=0, that is, the origin is an equilibrium point of the dynamical system; also the Jacobian matrix evaluated at the origin, A=Df(0). If His an homeomorphism of an open set Wonto an open set Vsuch that for each x0W, it exists an open interval I0Rsuch that for all x0Wand tI0

Hφt(x0)=eAtH(x0);E1

that is, Hmaps trajectories of the nonlinear system x˙=f(x)near the origin onto trajectories of x˙=Axnear the origin and preserves the parametrization.

From the following argument, it is show that for any matrix A=UTTAU, there exists an homeomorphism H^=UHsuch that for an open set Wcontaining the origin onto an open set Valso containing the origin such that for each x0Wand there is an open interval I0Rcontaining zero such that for all x0Wand tI0

H^φt(x0)=eTAtH^(x0); E2

This last equality is a consequence of the Hartman‐Grobman theorem and of the fact of UeAt=eTAtU, that is, H^maps trajectories of the nonlinear system x˙=f(x)near the origin onto trajectories of x˙=TAxnear the origin and preserves the parametrization.

On the other hand, some classical definitions are now included. A linear system of the form x˙=Axwhere xRn, Ais a n×nmatrix and x˙=dxdt. It is shown that the solution of the linear system together with the initial condition x(0)=x0is given by x(t)=eAtx0. The mapping eAt:RnRnis called the flow of the linear system.

Definition 2.1. For all eigenvalues of a matrix A(n×n)have nonzero real part, then the flow eAtis called a hyperbolic flow and therefore, x˙=Axis called a hyperbolic linear system [9].

Definition 2.2. A subspace ERnis said to be invariant with respect to the flow eAt:RnRnif eAtEfor all tR[9].

Lemma 2.1. Let ARn×n. If Rn=EsEuEcwhere Es, Euand Ecare the stable, unstable and center subspaces of the linear system x˙=Ax. By the above, Es, Euand Ecare invariant with respect to the flow eAt, respectively [9].

Definition 2.3. Let Ebe an open subset of Rnand let fC1(E), that is, fis a continuous differentiable function defined on E. For x0E, let ϕ(t,x0)be the solution of the initial value problem x˙=f(x), x(0)=x0defined on its maximal interval of existence I(x0). Then for tI(x0), the mapping ϕt:EEdefined by ϕt(x0)=ϕt(t, x0)is called the flow of the differential equation [9].

Definition 2.4. For any x0Rn, let ϕt(x0)be the flow of the differential equation through x0.(i) The local stable set Scorresponding to a neighborhood Vof x0is defined by S=S(0)={x0Rn:ϕt(x0)V,t0and ϕt(x0)0as t}. (ii) The local unstable set Wof x0corresponding to a neighborhood Vof x0is defined by W=W(0)={x0Rn:ϕt(x0)V,t0and ϕt(x0)0as t}. Then, these stable and unstable local sets are submanifolds of Rnin a sufficiently small neighborhood Vof x0 [9].

Definition 2.5. If Gis a group and Xis a set, then a (left) group action of Gon Xis a binary function G×XX, denoted by [9]

(g,x)gxE3

which satisfies the following two axioms:

  1. (gh)x=g(hx)for all g,hGand xX;

  2.  ex=xfor every xX(where edenotes the identity element of G).

The action is faithful (or effective) if for any two different g,hG, there exists an xXsuch that gxhx; or equivalently, if for any gein G, there exists an xXsuch that gxx.

The action is free or semiregular if for any two different g,hGand all xX,we have gxhx; or equivalently, if gx=xfor some ximplies g =e.

For every xX, we define the stabilizer subgroup of x(also called the isotropy group or little group) as the set of all elements in Gthat fix x:

Gx={gG:gx=x}E4

This is a subgroup of G, though typically not a normal one. The action of Gon Xis free if and only if all stabilizers are trivial.

3. 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×nmatrix A of multiplicity mn. Then for k=1,, m, any nonzero solution wof [9]

(AλI)kw=0E5

is called a generalized eigenvector of A.

In this case, let wj=uj+vjbe a generalized eigenvector of the matrix Acorresponding to an eigenvalue λj=aj+ibj(note that if bj=0then vj=0). Then, let B={u1,v1,, uk,vk, , um,vm} 

be a basis of Rn(with n=2mkas 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=(aij)and C=(cij)of order m×nand B=(bkl)of order p×q. Let A=(Aij)be partitioned with Aijof order mi×njas the (i,j)th block submatrix and B=(Bkl)of order pk×qlas the (k,l)th block submatrix (mi=m, nj=n,pk=p, ql=q). Then, the definitions of the matrix products or sums of Aand Bare given as follows [10].

Tracy‐Singh product

AB=(AijB)ij=((AijBkl)kl)ijE6

where AijBklis of order mipk×njql, AijBis a Kronecker product of order mip×njq, and ABis of order mp×nq.

Tracy‐Singh sum

AB=AIp+ImBE7

where A=(Aij)and B=(Bkl)are square matrices of respective order m×mand p×pwith Aijof order mi×mjand Bklof order pk×pl; Ipand Imare compatibly partitioned identity matrices.

Theorem 3.1. Let A,B,C,D, E, and Fbe compatibly partitioned matrices, then [10]

  1. (AB)(CD)=(AC)(BD).

  2. ABBA.

  3. (CB=BC)where C=(cij)and cijis a scalar.

  4. (AB)'=AB'.

  5. (A+D)(B+E)=AB+AE+DB+DE.

  6. (AB)F=A(BF)

The next proposition presents some extensions to the local stable‐unstable manifold theorem and to the center manifold theorem.

Proposition 3.1. Let Ebe an open subset of Rncontaining the origin, let fC1(E)and ϕtbe the flow of the nonlinear system x˙=f(x)=Ax+g(x). Suppose that f(0)=0and that A=Df(0)have keigenvalues with negative real part and nkeigenvalues with positive real part, that is, the origin is an hyperbolic fixed point. Then for each matrix MΛU, there exists a k

‐dimensional differentiable manifold SMtangent to the stable subspace EMSof the linear system x˙=MAxat 0such that for all t0, ϕM,t(SM)SMand for all x0SM[8],

limtφM,t(x0)=0,E8

where ϕM,tis the flow of the nonlinear system x˙=MAx+g(x)and there exists an nkdimensional differentiable manifold WMtangent to the unstable subspace EMWof x˙=MAxat 0such that for all t0, ϕM,t(WM)WMand for all x0WM,

limtφM,t(x0)=0.E9

An interesting property is that Proposition 4.1 is valid for each g¯C1(E)such that x˙=f¯(x)=Ax+g¯(x)and

g¯(x)2x20 as x20.E10

In consequence, the set of matrices ΛUgenerates the action of the group ΛUon the set of the hyperbolic nonlinear systems, formally on the set of the hyperbolic vector fields fC1(E), x˙=f¯(x)=Ax+g¯(x)with g¯C1(E)and

AΩU{PRn×n: P=UTTPU with TP any upper triangular matrix}E11

Satisfying the last condition, where Uis a fixed unitary matrix, the action is generated by the action of the group ΛUon the set ΩU. 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˙=Ax+g¯(x))is mapped in a hyperbolic nonlinear systems (x˙=MAx+g¯(x))and dimS=dimSMand dimW=dimWM.

The proof of this Proposition 3.1 can be revised in Ref. [8].

Given a particular nonlinear system, the stable and unstable manifolds Sand Ware unique; then for each matrix MΛU, there exists an unique pair of manifolds (SM, WM)in such a way that it is possible to define a pair of functions in the following form

Θ:ΛU×ManSManSΘ(M,S)=SMΦ:ΛU×ManWManWΦ(M,W)=WME12

Where ManSis the set of stable manifolds and ManWis the set of unstable manifold for autonomous nonlinear systems.

Therefore, we can say that if A=Df(0)is an stable matrix Ahas all the neigenvalues with negative real part, then the origin of the nonlinear system x˙=MAx+g¯(x)is asymptotically stable; but if A=Df(0)is an unstable matrix Ahas nk(with n>k)eigenvalues with positive real part, then the origin of the nonlinear system x˙=MAx+g¯(x)is unstable.

As an extension of the local stable‐unstable manifold theorem in terms of Tracy‐Singh product of matrices in ΛNand the matrix Aof the vector field f(x), we present the following proposition.

Proposition 3.2.

1. Let Ebe an open subset of Rncontaining the origin, let fC1(E)and let ϕtbe the flow of the nonlinear system x˙=f(x)=Ax+g(x). We suppose that f(0)=0and that A=Df(0)have a keigenvalues with negative real part and nkeigenvalues with positive real part; thus, the origin is a hyperbolic fixed point. Now, take a fixed continuously differentiable function

F : C1(E)C1(E¯)E13

such that F(g)=g^where g^ : E¯RmnRmnis a fixed continuously differentiable function with domain all C1(E); moreover, g^C1(E¯)with E¯an open subset of Rncontaining the origin such that

g^(x)2x20asx20.E14

Then, for each matrix MΛUof m×m, there exists a mkdimensional differentiable manifold SMAtangent to the stable subspace EMASof the linear system x˙=(MA)xat 0such that for all t0, ϕMA,t(SMA)SMAand for all x0SMA,

limtφMA,t(x0)=0,E15

where ϕMA,tbe the flow of the nonlinear system x˙=(MA)x+g^(x)and there exists an m(nk)dimensional differentiable manifold WMAtangent to the unstable subspace EMAWof x˙=(MA)xat 0such that for all t0, ϕMA,t(WMA)WMAand for all x0WMA,

limtφMA,t(x0)=0.E16

2. Also, there exists a function of the group ΛNand the set of all the autonomous hyperbolic nonlinear systems of dimension n(hyperbolic vector fields of dimension n) denoted by Γn, to the set Γmnof all the autonomous hyperbolic nonlinear systems of dimension mn(hyperbolic vector fields of dimension mn); this function (which is similar to an action of the group ΛNon the set Γn) is defined as follows

ϑ : ΛN×ΓnΓmnϑ(M,Ax+g(x))=(MA)x+g^(x)E17

and the new nonlinear system is

x˙=ϑ(M,Ax+g(x))x˙=(MA)x+g^(x))E18

which satisfies the following two axioms:

  1. (gh)z=g(hz)for all g,hΛNand zΓn;

  2. For every zΓn, there exists an unique z^Γmnsuch that ez=z^and hz^=hz(edenotes the identity element of ΛN, that is, is the identity matrix Imof m×m).

Where zis associated with Ax+g(x)(denoted by zAx+g(x)); hzmeans (MhA)x+g^(x)(denoted by hz(MhA)x+g^(x)); ghis associated with the usual product of matrices Mg,Mh, that is, ghMgMhand ezmeans (ImA)x+g^(x), that is, (ez(ImA)x+g^(x))and g(hz)means (MgIn)(MhA)x+g^(x)(denoted by g(hz)(MgIn)(MhA)x+g^(x)).

Proof.

1. Consider a matrix Awith eigenvalues λifor i=1,2,,nand the matrix Mwith eigenvalues μjfor j=1,2,,m. Then, the eigenvalues of the matrix MAare the mnnumbers λiμjand taking account that μj>0for each j=1,2,,m. Therefore, the matrix MAhas mkeigenvalues with negative real part and m(nk)eigenvalues with positive real part. For this, the result is a consequence of the stable‐unstable manifold theorem.

2. The function ϑ : ΛN×ΓnΓmnis well defined, since F : C1(E)C1(E¯)is a fixed function; then given g(x), the vector field g^(x)is unique; for a fixed matrix MhΛN, then Mh : Rn×nRmn×mnis a fixed function and their matrix MhAis unique.

Axiom (i): Since ΛNis a multiplicative group if Mg,MhΛN, then MgMhΛN.

Then, by Theorem 3.1, we have that for all g,hΛNand zΓn

(gh)z(MgMhA)x+g^(x)=(MgIn)(MhA)x+g^(x)g(hz)E19

Axiom (ii): For every zΓn, there exists an unique z^Γmnsuch that ez(ImA)x+g^(x)=z^, then by the Theorem 2.1

hz^(MhIn)(ImA)x+g^(x)=(MhA)x+g^(x)hzE20

From what it has been said above, we can note that if A=Df(0)is as stable matrix A, it has all the neigenvalues with negative real part, then the origin of the nonlinear system x˙=(MA)x+g^(x)is asymptotically stable; if A=Df(0)is an unstable matrix A, it has nk (n>k)eigenvalues with positive real part, then the origin of the nonlinear system x˙=(MA)x+g^(x)is unstable.

Now the following Proposition 3.2 is an extension of the center manifold theorem, similar to Proposition 3.1.

Proposition 3.3. Let be fCr(E)where Eis an open subset of Rn

containing the origin and r1. Suppose that f(0)=0and that Df(0)have keigenvalues with negative real part, jeigenvalues with positive real part and l=nkjeigenvalues with zero real part. Then,

  1. For each matrix MΛU, there exists a mdimensional differentiable center manifold WMC(0)of class Crtangent to the center subspace EMCof the linear system x˙=MAx+g(x)at 0which is invariant under the flow ϕM,tof the nonlinear system x˙=MAx+g(x).

  2. If taken a fixed continuously differentiable function

    F^ : Cr(E)Cr(E¯)E21

such that F(g)=g^where g^ : E¯RmnRmnis a fixed continuously differentiable function with domain all Cr(E); moreover, g^Cr(E¯)with E¯an open subset of Rncontaining the origin such that

g^(x)2x20asx20.E22

Then for each matrix MΛNof m×m, there exists a mldimensional differentiable center manifold WMAC(0)tangent to the center subspace EMASof the linear system x˙=(MA)xat 0which is invariant under the flow ϕMA,tof the nonlinear system x˙=(MA)x+g^(x).

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 function ϑ^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 Θand Φ. due to that in general, a center manifold is not unique.

Notice that in this case, if the matrix Ahas l=nkj0eigenvlues with zero real part, then the origin of the nonlinear system x˙=MAx+g^(x)and x˙=(MA)x+g^(x)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.

4. 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,

x˙=Ax+g(x)y˙=Ay+f(y)+u(t)E23

Where ARn×nis a constant matrix. On the other hand,uRnis the control input and f, g :RnRnare 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 u(t)attached to slave system such that limtx(t)y(t)0, where is the Euclidean norm of a vector [8]. If we consider the error state vector e=yxRn,f(y)f(x)=L(x,y)and an error dynamics equation is e˙=Ae+L(x,y)+u(t). Taking the active control approach [5], to eliminate the nonlinear part of the error dynamics and choosing u(t)=Bv(t)L(x,y), where Bis a constant gain vector which is selected such that (A,B)be controllable, we obtain:

e˙=Ae+Bv(t)E24

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 (A,B)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,

J(u(t))=0(e(t)Qe(t)+v(t)Rv(t))dtE25

Where Qand Rare positive semi‐definite and positive definite weighting matrices, respectively. The state‐feedback law is given by v=Kewith K=R1BSand Sthe solution to the Riccati equation

AS+SASBR1B+Q=0E26

This state‐feedback law makes the error equation to be e˙=(ABK)e, with (ABK)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 TRm×mbe a matrix with strictly positive eigenvalues, supposing that the following two nm‐dimensional systems are chaotic:

x˙=(TA)x+g^(x)y˙=(TA)y+f^(y)+u^(t)E27

for some f^,g^:RnmRnmcontinuous nonlinear functions and u^Rnmis the control input. Then, for the Proposition 4.1 and the former procedure, we have that u^(t)=θ^(t)L^(x,y)stabilizes the zero solution of the error dynamics system, where θ^(t)=(BKT)e, that is, the resultant system

e˙=(TA)e+θ^(t)e˙=(TATBK)eE28

is asymptotically stable. Then, by using Lemma 2.1 and K=R1BS, we obtain that:

e˙=(T(A+BK))ee˙=(T(ABR1BS))eE29

Now, the original control u(t)=BKeL(x,y)is preserved in its linear part by the Tracy‐Singh product T()and the new control is given by u^(t)=(TBK)eL^(x,y). Therefore, we can interpreted the last procedure as one in which the controller u(t)that achieves the synchronization in the two systems is preserved by the transformation T()so that u^(t)achieves the synchronization in the two resultant systems after the transformation. For that, a similar procedure is possible if we consider the transformation ()T.

In general, under the transformation (A,g)(MA,g¯)or (A,g)(MA,g¯)and under the hypothesis of existence of a constant state feedback U=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.

5. 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

x˙1=a(x2x1)x˙2=cx2x1x3x˙3=x1x2bx3E30

which has a chaotic attractor when the parameters are a=35,  b=3and c=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,

x˙1=35(x2x1)x˙2=28x2x1x3x˙3=x1x23x3E31

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.

y˙1=35(y2y1)+u1(t)y˙2=28y2y1y3+u2(t)y˙3=y1y23y3+u3(t)E32

Now, we consider the errors e1=x1y1,e2=x2y2and e3=x3y3,; then, the error dynamics can be written as:

e˙1=35(e2e1)+u1(t)e˙2=28e2y1y3+x1x3+u2(t)e˙3=y1y2x1x23e3+u3(t)E33

If we introduce the matrices

A=(35350014.50003),L(x,y)=(0y1y3+x1x3y1y2x1x2),u=(u1(t)u2(t)u3(t)).E34

and selecting the matrix Bsuch that (A,B)is controllable: B=I, the LQR controller is obtained by using weighting matrices Q=Iand R=BB=I. Then, state‐feedback matrix is given by

K=(0.01430.010100.010129.05870000.1623)E35

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 (1,1,1).

Figure 1.

Master system of Lü.

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 (3,3,3). Figures 2 and 4 are phase space mappings of each system while maintaining the same initial condition.

Figure 2.

Master system of Lü.

Figure 3.

Slave system of Lü.

Figure 4.

Slave system of Lü.

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:

x˙=(TA)x+[0x1x3x1x20x4x6x4x5]y˙=(TA)y+[0y1y3y1y20y4y6y4y5]+uE36

Figure 5.

Magnitude of the error between the master and the slave systems.

Figure 6.

Synchronization of master and slave system of Lü.

Considering the error vector e=yx, then the error dynamics can be written as:

e˙=(TA)e+L(x,y)+uE37

with u=L(x,y)+vand v=(TBK)eand

A=(35350014.50003), T=(1101), B=[1 1 1 1 1 1],L(x,y)=[0y1y3+x1x3y1y2x1x20y4y6+x4x6y4y5x4x5]E38

Now, the LQR controller is obtained by using weighting matrices, B=IQ=Iand R=BB=I.So the vector L(x,y)takes these values because Tis an upper triangular matrix and the value one on the diagonal is repeated.

TA=(3535353500014.5014.5000035350000014.500000033000003)E39
K=(0.01430.010100.00710.005000.010123.305100.015111.59410000.1614000.07570.00710.015100.02140.005000.005011.594100.005034.84110000.0757000.2324)E40

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 (0.5,0.5,0.5,0.5,0.5,0.5). 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 (3,3,3,3,3,3). 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.

Figure 7.

Transformation of the master system of Lü.

Figure 8.

Phase space of the transformation of the master system of Lü.

Figure 9.

Transformation of the slave system of Lü.

Figure 10.

Phase space of the transformation of the slave system of Lü.

Figure 11.

Magnitude of the error between the transformation of master and slave systems.

Figure 12.

Synchronization of the transformation of the master and slave systems of Lü.

6. 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 linear‐quadratic 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.

Notes

  • A Hurwitz matrix is a matrix for which all its eigenvalues are such that their real part is strictly less than zero.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Guillermo Fernadez‐Anaya, Luis Alberto Quezada‐Téllez, Jorge Antonio López‐Rentería, Oscar A. Rosas‐Jaimes, Rodrigo Muñoz‐ Vega, Guillermo Manuel Mallen‐Fullerton and José Job Flores‐ Godoy (March 15th 2017). Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix, Dynamical Systems - Analytical and Computational Techniques, Mahmut Reyhanoglu, IntechOpen, DOI: 10.5772/66957. Available from:

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