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Numerical Integration and Synchronization for the 3-Dimensional Metriplectic Volterra System
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  has introduced the concept of metriplectic system.
Let be a local coordinate system on . We consider be a Hamilton-Poisson system on with Hamiltonian , where and
We add to the Hamilton-Poisson system (1.1) a dissipation term of the form , where is a symmetric matrix which satisfies certain compatibility conditions, and , where and are a Casimir function (i.e., ). One obtains a family of metriplectic systems of the form This family of metriplectic systems have the same Hamiltonian and the same Casimir function . For each , the metriplectic systems (1.2) can be viewed as a perturbation of Hamilton-Poisson system (1.1).
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 . 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 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).
2. The Metriplectic System Associated to 3-Dimensional Volterra Model
Let be a Hamilton-Poisson system given by (1.1). For this system we determine the symmetric matrix , where
If is a Casimir function for the configuration , then we take , where is a parameter.
Let us construct a metriplectic system of the form of (2.2), starting a Hamilton-Poisson realization of the dimensional Volterra model.
Proposition 2.1. The dynamical system given by (2.8) is a metriplectic system on .
System (2.8) can be written in the form , , where
Proposition 2.2. The function 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).
The function decreases along the solutions of system (2.8).
Proof. We have
The derivative of along the solutions of system (2.8) verifies the condition Indeed,
Remark 2.3. If , then is not a constant of motion for the metriplectic system (2.8).
Proof. The equilibria are the solutions of system , .
Proposition 2.5. The equilibrium states , are unstable.
Remark 2.6. The dynamics of (2.4) and (2.8) have not the same equilibria.
For , have the following behaviors (see ): is unstable if and spectrally stable if ; is unstable, and is unstable if and spectrally stable if .
3. Numerical Integration of the Metriplectic Volterra System (2.8)
Proof. Indeed, adding all equations (3.1) we obtain Hence
Remark 3.3. Using Runge-Kutta 4 steps integrator, we obtain almost the same result; see Figure 3.
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 with and adding for ). 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 respectively where 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): By subtracting (4.2) from (4.1) and using notation (4.3) we can get We define the active control inputs as follows: where and , are real functions which depend on Then the differential system of errors (4.7) is given by If we choose then the active controls defined by (4.5) become Using (4.8), the system of errors (4.7) becomes
Proposition 4.1. The equilibrium state 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 is asymptotically stable.
Numerical simulations are carried out using the software MATHEMA-TICA 6. We consider the case . The fourth-order Runge-Kutta integrator is used to solve systems (4.1), (4.2), and (4.10) with the control functions given by (4.9).
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 converge to zero, and hence the synchronization between two coupled metriplectic Volterra systems is achieved.
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 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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