MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation72362910.1155/2011/723629723629Research ArticleNumerical Integration and Synchronization for the 3-Dimensional Metriplectic Volterra SystemIvanGheorghe1IvanMihai1PopCamelia2MessiasMarcelo1Seminarul de Geometrie şi TopologieWest University of Timişoara4, B-dul V. Pârvan 300223 TimişoaraRomaniauvt.ro2Math DepartmentThe “Politehnica” University of TimişoaraPiaţa Victoriei nr. 2300006 TimişoaraRomaniaupt.ro201119072010201121012010290620102011Copyright © 2011 Gheorghe Ivan et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The main purpose of this paper is to study the metriplectic system associated to 3-dimensional Volterra model. For this system we investigate the stability problem and numerical integration via Kahan's integrator. Finally, the synchronization problem for two coupled metriplectic Volterra systems is discussed.

1. Introduction

To give a unification of the conservative and dissipative dynamics, Kaufman  has introduced the concept of metriplectic system.

Let (x1,x2,,xn) be a local coordinate system on Rn. We considerẋ(t)=P(x(t))·H(x(t)) be a Hamilton-Poisson system on Rn with Hamiltonian HC(Rn), where x(t)=(x1(t),,xn(t))T and H(x)=(H/x1,,H/xn)T.

We add to the Hamilton-Poisson system (1.1) a dissipation term of the form G(x(t))·C̃(x(t)), where G(x) is a symmetric matrix which satisfies certain compatibility conditions, and C̃(x(t))=a·C(x(t)), where aR and CC(Rn) are a Casimir function (i.e., P(x)·C(x)=0). One obtains a family of metriplectic systems of the formẋ(t)=P(x(t))·H(x(t))+G(x(t))·C̃(x(t)). This family of metriplectic systems have the same Hamiltonian H and the same Casimir function C. For each aR, the metriplectic systems (1.2) can be viewed as a perturbation of Hamilton-Poisson system (1.1).

The differential systems of the form (1.2) and their applications have been studied in connection with several dynamical systems derived from mathematical physics; see for instance, .

Another way for giving rise to a dynamical system of the form of (1.2) is based on the definition of a metriplectic structure on Rn. These systems can be expressed in terms of Leibniz bracket, see .

The paper is structured as follows. In Section 2, the metriplectic system associated to 3-dimensional Volterra model (2.8) is constructed. For this dynamical system, the stability of equilibrium states is investigated. In Section 3, we discuss the numerical integration for the system (2.8).

Synchronization problem for dynamical systems has received a great deal of interest due to their application in different fields of science; see . For this reason, Section 4 is dedicated to synchronization problem for two coupled metriplectic Volterra systems of the form of (2.8).

2. The Metriplectic System Associated to 3-Dimensional Volterra Model

Let (Rn,P,H) be a Hamilton-Poisson system given by (1.1). For this system we determine the symmetric matrix G=(Gij), whereGii(x)=-k=1,kin(Hxk)2,Gij(x)=Hxi·Hxj,forij.

If CC(Rn) is a Casimir function for the configuration (Rn,P,H), then we take C̃=a·C, where aR is a parameter.

For P,H,C̃, and the matrix G determined by relations (2.1), we write the differential system (1.2) in the following tensorial form:ẋi=PijHxj+GijC̃xj,i,j=1,n¯.

System (2.2) is a metriplectic system in Rn (see [2, 7]), that is the following conditions are satisfied:(i)G(x)·H(x)=0;(ii)(C̃(x))T·G(x)·C̃(x)0.

System (2.2) is called the metriplectic system associated to Hamilton-Poisson system (1.1) and is denoted by (Rn,P,H,G,C̃).

Let us construct a metriplectic system of the form of (2.2), starting a Hamilton-Poisson realization of the 3-dimensional Volterra model.

The phase space of the 3-dimensional Volterra model consists of variables xi, 1i3; see [13, 14]. This is described by the equationsẋ1=x1x2,ẋ2=-x1x2+x2x3,ẋ3=-x2x3.

It is well known that system (2.4) has a Hamilton-Poisson realization (R3,PV,HV) with the Casimir CVC(R3,R) (see ), wherePV=(0x1x20-x1x20x2x30-x2x30),HV(x1,x2,x3)=x1+x2+x3,CV(x1,x2,x3)=x1x3.

We apply now relations (2.1) to the function H=HVC(R3,R) given by (2.6). Then the symmetric matrix G:=GV=(GVij) isGV=(-2111-2111-2).

We take H=HV and C=CV given by (2.6), the skew-symmetric matrix P=PV given by (2.5) and the symmetric matrix GV given by (2.7). For the function C̃V=aCV with aR, system (2.2) becomesẋ1=x1x2+a(x1-2x3),ẋ2=-x1x2+x2x3+a(x1+x3),ẋ3=-x2x3+a(-2x1+x3).

Proposition 2.1.

The dynamical system (R3,PV,HV,GV,C̃V) given by (2.8) is a metriplectic system on R3.

Proof.

We have CV/x1=x3, and CV/x2=0, CV/x3=x1. Then PV·CV=(0x1x20-x1x20x2x30-x2x30)(x30x1)=(000), that is, CV is a Casimir of Hamilton-Poisson system (R3,PV,HV).

We prove that conditions (i) and (ii) from (2.3) are verified.

We have C̃V(x)=ax1x3, HV/x1=HV/x2=HV/x3=1 and  C̃V/x1=ax3, C̃V/x2=0, C̃V/x3=ax1. Then GV·HV=(-2111-2111-2)(111)=(000),(C̃(x))T·G(x)·C̃(x)=(ax3,0,ax1)(-2111-2111-2)(ax30ax1)=-2a2(x12-x1x3+x32)0. Hence (2.8) is a metriplectic system.

System (2.8) is called the 3-dimensional metriplectic Volterra system. For a=0, it is reduced to Volterra model (2.4).

System (2.8) can be written in the form ẋi=fi(x), i=1,3¯, where f1(x)=x1x2+a(x1-2x3),f2(x)=-x1x2+x2x3+a(x1+x3),f3(x)=-x2x3+a(-2x1+x3).

Proposition 2.2.

( i) The function HV given by (2.6) is a constant of the motion for the metriplectic Volterra system, that is, it is conserved along the solutions of the dynamics (2.8).

( ii) The function C̃V decreases along the solutions of system (2.8).

Proof.

(i) We have dHV/dt=ẋ1+ẋ2+ẋ3=f1(x)+f2(x)+f3(x)=0.

(ii) The derivative of C̃V along the solutions of system (2.8) verifies the condition dC̃V/dt0. Indeed, dC̃V/dt=aẋ1x3+ax1ẋ3=a(x3f1(x)+x1f3(x))=-2a2(x12-x1x3+x32)0.

Remark 2.3.

If a0, then C̃V=aCV is not a constant of motion for the metriplectic system (2.8).

Proposition 2.4.

( i) If aR*, then the equilibrium states of the dynamics of (2.8) are e2M=(0,M,0) for all MR.

( ii) For a=0, the equilibrium states of the dynamics of (2.8) are e1MN=(M,0,N), and  e2M=(0,M,0), e3M=(0,0,M) for all M,NR.

Proof.

The equilibria are the solutions of system fi(x)=0, i=1,3¯.

Proposition 2.5.

The equilibrium states e2M, MR are unstable.

Proof.

Let A be the matrix of the linear part of the system (2.8), that is, A=(x2+ax1-2a-x2+a-x1+x3x2+a-2a-x3-x2+a). The characteristic polynomial of the matrix A(e2M) is p(λ)=λ(λ2-2aλ-3a2-M2) with the roots λ1=0, λ2,3=a±4a2+M2. Then the assertion follows via Lyapunov's theorem .

Remark 2.6.

( i) The dynamics of (2.4) and (2.8) have not the same equilibria.

( ii) For a=0, e1MN,e2M,e3M have the following behaviors (see ): e1MN is unstable if M-N<0 and spectrally stable if M-N>0; e2M is unstable, and e3M is unstable if M0 and spectrally stable if M<0.

3. Numerical Integration of the Metriplectic Volterra System (<xref ref-type="disp-formula" rid="EEq2.8">2.8</xref>)

For (2.8), Kahan's integrator (see for details ) can be written in the following form:x1n+1-x1n=h2(x1n+1x2n+x2n+1x1n)+ah(x1n+x1n+1-2x3n-2x3n+1),x2n+1-x2n=h2(-x1n+1x2n-x2n+1x1n+x3n+1x2n+x2n+1x3n)+ah(x1n+x1n+1+x3n+x3n+1),x3n+1-x3n=-h2(x2n+1x3n+x3n+1x2n)+ah(x3n+x3n+1-2x1n-2x1n+1).

Remark 3.1.

Taking a=0 in relations (3.1) we obtain the numerical integration for Volterra model (2.4) via Kahan's integrator.

Proposition 3.2.

Kahan's integrator (3.1) preserves the constant of motion HV of the dynamics of (2.8).

Proof.

Indeed, adding all equations (3.1) we obtain x1n+1+x2n+1+x3n+1=x1n+x2n+x3n. Hence HV(x1n+1,x2n+1,x3n+1)=HV(x1n,x2n,x3n).

For the initial conditions x1(0)=1, x2(0)=2, and x3(0)=1, the solutions of Volterra model (2.4) (using Kahan's integrator (3.1) with a=0), are represented in the system of coordinates Ox1x2x3 in Figure 1.

Kahan's integrator for Volterra model (2.4).

For the same initial conditions, the solutions of the metriplectic Volterra system (2.8) for a=1 (using Kahan's integrator (3.1) with a=1), are represented in the system of coordinates Ox1x2x3 in Figure 2.

Kahan's integrator for the metriplectic Volterra system (2.8) with a=1.

Remark 3.3.

Using Runge-Kutta 4 steps integrator, we obtain almost the same result; see Figure 3.

Runge-Kutta's integrator for Volterra model (2.4).

Runge-Kutta's integrator for the metriplectic Volterra system (2.8) with a=1.

4. The Synchronization of Two Metriplectic Volterra Systems

In this section we apply Pecora and Carroll method for constructing the drive-response configuration (see ).

Let us build the configuration with the drive system given by the metriplectic Volterra system (2.8), and a response system (this is obtained from (2.8) by replacing xi with yi and adding ui for i=1,3̅). Suppose that these systems are coupled. More precisely, the second system is driven by the first one, but the behavior of the first system is not affected by the second one.

Therefore, the drive and response systems are given byẋ1=x1x2+a(x1-2x3),ẋ2=-x1x2+x2x3+a(x1+x3),ẋ3=-x2x3+a(-2x1+x3), respectivelyẏ1=y1y2+a(y1-2y3)+u1,ẏ2=-y1y2+y2y3+a(y1+y3)+u2,ẏ3=-y2y3+a(-2y1+y3)+u3, where u1(t),u2(t),and  u3(t) are three control functions.

We define the synchronization error system as the subtraction of the metriplectic Volterra model (4.1) from the controlled metriplectic Volterra model (4.2):e1(t)=y1(t)-x1(t),e2(t)=y2(t)-x2(t),e3(t)=y3(t)-x3(t). By subtracting (4.2) from (4.1) and using notation (4.3) we can getė1=e1e2+(x2+a)e1+x1e2-2ae3+u1,ė2=-e1e2+e2e3+(a-x2)e1+(x3-x1)e2+(a+x2)e3+u2,ė3=-e2e3-2ae1-x3e2+(a-x2)e3+u3. We define the active control inputs u1(t),u2(t),and  u3(t) as follows:(u1(t)u2(t)u3(t))=K·(e1(t)e2(t)e3(t))+(-e1(t)e2(t)e1(t)e2(t)-e2(t)e3(t)e2(t)e3(t)), where K=(k11k12k13k21k22k23k31k32k33) and kij, i,j=1,3¯ are real functions which depend on x1(t),x2(t),and  x3(t). Then the differential system of errors (4.7) is given byė1=e1e2+(x2+a+k11)e1+(x1+k12)e2+(k13-2a)e3,ė2=-e1e2+e2e3+(a-x2+k21)e1+(x3-x1+k22)e2+(a+x2+k23)e3,ė3=-e2e3+(k31-2a)e1+(k23-x3)e2+(a-x2+k33)e3. If we chooseK=(-a-x2-x2200x2-ax1-x3-x1202ax3x2-a-x32), then the active controls defined by (4.5) becomeu1=-e1e2-(a+x2+x22)e1,u2=e1e2-e2e3+(x2-a)e1+(x1-x3-x12)e2,u3=e2e3+2ae1+x3e2+(-a+x2-x32)e3. Using (4.8), the system of errors (4.7) becomesė1=-x22e1+x1e2-2ae3,ė2=-x12e2+(a+x2)e3,ė3=-x32e3.

Proposition 4.1.

The equilibrium state (0,0,0) of the differential system (4.10) is asymptotically stable.

Proof.

An easy computation shows that the all conditions of Lyapunov-Malkin theorem  are satisfied, and so we have that the equilibrium state (0,0,0) is asymptotically stable.

Numerical simulations are carried out using the software MATHEMA-TICA 6. We consider the case a=1. The fourth-order Runge-Kutta integrator is used to solve systems (4.1), (4.2), and (4.10) with the control functions u1(t),u2(t),u3(t) given by (4.9).

The initial values of the drive system (4.1) and response system (4.2) are x1(0)=1, x2(0)=2, x3(0)=1 and y1(0)=1, y2(0)=2, y3(0)=3. These choices result in initial errors of e1(0)=0.001, e2(0)=0.01, and e3(0)=0.002.

The dynamics of the metriplectic Volterra system (4.1) to be synchronized with the dynamic of (4.2) accompanied with the control functions given by (4.9) and the dynamics of synchronization errors given by (4.10) are shown in Figures 5, 6, and 7.

Synchronization of systems (4.1) and (4.2) for a=1. The solutions x1(t),y1(t) and the evolution of error e1(t).

Synchronization of systems (4.1) and (4.2) for a=1. The solutions x2(t),y2(t) and the evolution of error e2(t).

Synchronization of systems (4.1) and (4.2) for a=1. The solutions x3(t),y3(t) and the evolution of error e3(t).

According to numerical simulations, by a good choice of parameters the synchronization error states e1(t),e2(t),e3(t) converge to zero, and hence the synchronization between two coupled metriplectic Volterra systems is achieved.

Remark 4.2.

Taking a=0 in (4.1), (4.2), (4.8), and (4.9), we obtain the synchronization between two coupled Volterra models of the form of (2.4).

5. Conclusion

It is well known that many nonlinear differential systems like the Euler equations of fluid dynamics, the soliton equations can be written in the Hamiltonian form. An interesting example of nonlinear lattice equations is Volterra lattice (see ) which is a model for vibrations of the particles on lattices. Also the behavior of viscoelastic materials is an example where the dynamics is governed by Volterra equations. The metriplectic systems will be successfully used in mathematical physics, fluid mechanics, and information security; see for instance [4, 5, 10, 12].

In this paper we have build a metriplectic system on R3 associated to Volterra model. For the metriplectic Volterra system (2.8), we have presented some relevant geometrical and dynamics properties and the numerical integration. Finally, using the Pecora and Carroll method, the synchronization problem for two coupled metriplectic Volterra systems of the form of (2.8) is discussed. This technique is realized since a suitable control has been chosen to achieve synchronization.

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