Numerical Integration and Synchronization for the 3-Dimensional Metriplectic Volterra System

be a Hamilton-Poisson system on R with Hamiltonian H ∈ C∞ R , 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 · ∇ ̃ C x t , where G x is a symmetric matrix which satisfies certain compatibility conditions, and ̃ C x t a · C x t , where a ∈ R and C ∈ C∞ R are a Casimir function i.e., P x · ∇C x 0 . One obtains a family of metriplectic systems of the form


Introduction
To give a unification of the conservative and dissipative dynamics, Kaufman 1 has introduced the concept of metriplectic system.
Let x 1 , x 2 , . . ., x n be a local coordinate system on R n .We consider ẋ t P x t • ∇H x t 1.1 be a Hamilton-Poisson system on R n with Hamiltonian H ∈ C ∞ R n , where x t x 1 t , . . ., x n t T and ∇H x ∂H/∂x 1 , . . ., ∂H/∂x n T .
We add to the Hamilton-Poisson system 1.1 a dissipation term of the form G x t • ∇ C x t , where G x is a symmetric matrix which satisfies certain compatibility conditions, and C x t a • C x t , where a ∈ R and C ∈ C ∞ R n are a Casimir function i.e., P x • ∇C x 0 .One obtains a family of metriplectic systems of the form

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This family of metriplectic systems have the same Hamiltonian H and the same Casimir function C. For each a ∈ R, 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, 1-4 .
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 R n .These systems can be expressed in terms of Leibniz bracket, see 5-8 .
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 9-12 .For this reason, Section 4 is dedicated to synchronization problem for two coupled metriplectic Volterra systems of the form of 2.8 .

Volterra Model
Let R n , P, H be a Hamilton-Poisson system given by 1.1 .For this system we determine the symmetric matrix G G ij , where Casimir function for the configuration R n , P, H , then we take C a • C, where a ∈ R is a parameter.
For P, H, C, and the matrix G determined by relations 2.1 , we write the differential system 1.2 in the following tensorial form: System 2.2 is a metriplectic system in R n see 2, 7 , that is the following conditions are satisfied: System 2.2 is called the metriplectic system associated to Hamilton-Poisson system 1.1 and is denoted by R n , P, H, G, C .
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 x i , 1 ≤ i ≤ 3; see 13, 14 .This is described by the equations It is well known that system 2.4 has a Hamilton-Poisson realization R 3 , P V , H V with the Casimir C V ∈ C ∞ R 3 , R see 14 , where We apply now relations 2.1 to the function We take H H V and C C V given by 2.6 , the skew-symmetric matrix P P V given by 2.5 and the symmetric matrix G V given by 2.7 .For the function C V aC V with a ∈ R, system 2.2 becomes

2.8
Proposition 2.1.The dynamical system R 3 , P V , H V , G V , C V given by 2.8 is a metriplectic system on R 3 .
Proof.We have ∂C V /∂x 1 x 3 , and ∂C V /∂x 2 0, ∂C V /∂x 3 x 1 .Then We prove that conditions i and ii from 2.3 are verified.

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We have

2.10
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 f i x , i 1, 3, where

Proposition 2.2. i
The function H V 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 C V decreases along the solutions of system 2.8 .
ii The derivative of C V along the solutions of system 2.8 verifies the condition is not a constant of motion for the metriplectic system 2.8 .Proof.The equilibria are the solutions of system f i x 0, i 1, 3.

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Proposition 2.5.The equilibrium states e M 2 , M ∈ R are unstable.
Proof.Let A be the matrix of the linear part of the system 2.8 , that is, The characteristic polynomial of the matrix

Numerical Integration of the Metriplectic Volterra System 2.8
For 2.8 , Kahan's integrator see for details 16 can be written in the following form:

3.1
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 H V of the dynamics of 2.8 .
Proof.Indeed, adding all equations 3.1 we obtain For the initial conditions x 1 0 1, x 2 0 2, and x 3 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 Ox 1 x 2 x 3 in Figure 1.
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 Ox 1 x 2 x 3 in Figure 2. Remark 3.3.Using Runge-Kutta 4 steps integrator, we obtain almost the same result; see Figure 3.

The Synchronization of Two Metriplectic Volterra Systems
In this section we apply Pecora and Carroll method for constructing the drive-response configuration see 12 .
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 x i with y i and adding u i for i 1, 3 .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 by where u 1 t , u 2 t , and u 3 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 :

4.4
We define the active control inputs u 1 t , u 2 t , and u 3 t as follows:

4.7
If we choose
Proof.An easy computation shows that the all conditions of Lyapunov-Malkin theorem 17 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 u 1 t , u 2 t , u 3 t given by 4.9 .
The initial values of the drive system 4.1 and response system 4.2 are x 1 0 1, x 2 0 2, x 3 0 1 and y 1 0 1, y 2 0 2, y 3 0 3.These choices result in initial errors of e 1 0 0.001, e 2 0 0.01, and e 3 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.
According to numerical simulations, by a good choice of parameters the synchronization error states e 1 t , e 2 t , e 3 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 .

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 18 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 R 3 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.

Proposition 2 . 4 .
i If a ∈ R * , then the equilibrium states of the dynamics of 2.8 are e M 2 0, M, 0 for all M ∈ R. ii) For a 0, the equilibrium states of the dynamics of 2.8 are e MN 1 M, 0, N , and e M 2 0, M, 0 , e M 3 0, 0, M for all M, N ∈ R.

e 3 t y 3 tFigure 7 :
Figure 7: Synchronization of systems 4.1 and 4.2 for a 1.The solutions x 3 t , y 3 t and the evolution of error e 3 t .
Then the assertion follows via Lyapunov's theorem 15 .N < 0 and spectrally stable if M − N > 0; e M 2 is unstable, and e M 3 is unstable if M ≥ 0 and spectrally stable if M < 0.
are real functions which depend on x 1 t , x 2 t , and x 3 t .Then the differential system of errors 4.7 is given by ė1 e 1 e 2 x 2 a k 11 e 1 x 1 k 12 e 2 k 13 − 2a e 3 , ė2 −e 1 e 2 e 2 e 3 a − x 2 k 21 e 1 x 3 − x 1 k 22 e 2 a x 2 k 23 e 3 , ė3 −e 2 e 3 k 31 − 2a e 1 k 23 − x 3 e 2 a − x 2 k 33 e 3 .