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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.

To give a unification of the conservative and dissipative dynamics, Kaufman [

Let

We add to the Hamilton-Poisson system (

The differential systems of the form (

Another way for giving rise to a dynamical system of the form of (

The paper is structured as follows. In Section

Synchronization problem for dynamical systems has received a great deal of interest due to their application in different fields of science; see [

Let

If

For

System (

System (

Let us construct a metriplectic system of the form of (

The phase space of the

It is well known that system (

We apply now relations (

We take

The dynamical system

We have

We prove that conditions

We have

System (

System (

If

The equilibria are the solutions of system

The equilibrium states

Let

For (

Taking

Kahan's integrator (

Indeed, adding all equations (

For the initial conditions

Kahan's integrator for Volterra model (

For the same initial conditions, the solutions of the metriplectic Volterra system (

Kahan's integrator for the metriplectic Volterra system (

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

Runge-Kutta's integrator for Volterra model (

Runge-Kutta's integrator for the metriplectic Volterra system (

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 (

Therefore, the drive and response systems are given by

We define the synchronization error system as the subtraction of the metriplectic Volterra model (

The equilibrium state

An easy computation shows that the all conditions of Lyapunov-Malkin theorem [

Numerical simulations are carried out using the software MATHEMA-TICA 6. We consider the case

The initial values of the drive system (

The dynamics of the metriplectic Volterra system (

Synchronization of systems (

Synchronization of systems (

Synchronization of systems (

According to numerical simulations, by a good choice of parameters the synchronization error states

Taking

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 [

In this paper we have build a metriplectic system on