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

Abstract

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 [1] has introduced the concept of metriplectic system.

Let (𝑥₁,𝑥₂,…,𝑥_𝑛) be a local coordinate system on 𝐑^𝑛. We consider𝑥̇(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)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇 and ∇𝐻(𝑥)=(𝜕𝐻/𝜕𝑥₁,…,𝜕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 form𝑥̇(𝑡)=𝑃(𝑥(𝑡))⋅∇𝐻(𝑥(𝑡))+𝐺(𝑥(𝑡))⋅∇𝐶(𝑥(𝑡)). (1.2) 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).

2. The Metriplectic System Associated to 3-Dimensional 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), where𝐺^𝑖𝑖(𝑥)=−^𝑛_{𝑘=1,𝑘≠𝑖}(𝜕𝐻𝜕𝑥_𝑘)²,𝐺^𝑖𝑗(𝑥)=𝜕𝐻𝜕𝑥_𝑖⋅𝜕𝐻𝜕𝑥_𝑗,for𝑖≠𝑗. (2.1)

If 𝐶∈𝐶^∞(𝐑^𝑛) is a 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:𝑥̇_𝑖=𝑃^𝑖𝑗𝜕𝐻𝜕𝑥_𝑗+𝐺^𝑖𝑗𝜕𝐶𝜕𝑥_𝑗,𝑖,𝑗=1,𝑛. (2.2)

System (2.2) is a metriplectic system in 𝐑^𝑛 (see [2, 7]), that is the following conditions are satisfied:(i)𝐺(𝑥)⋅∇𝐻(𝑥)=0;(ii)(∇𝐶(𝑥))^𝑇⋅𝐺(𝑥)⋅∇𝐶(𝑥)≤0. (2.3)

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 equations𝑥̇₁=𝑥₁𝑥₂,𝑥̇₂=−𝑥₁𝑥₂+𝑥₂𝑥₃,𝑥̇₃=−𝑥₂𝑥₃. (2.4)

It is well known that system (2.4) has a Hamilton-Poisson realization (𝐑³,𝑃_𝑉,𝐻_𝑉) with the Casimir 𝐶_𝑉∈𝐶^∞(𝐑³,𝐑) (see [14]), where𝑃_𝑉=0𝑥₁𝑥₂0−𝑥₁𝑥₂0𝑥₂𝑥₃0−𝑥₂𝑥₃0, (2.5)𝐻_𝑉(𝑥₁,𝑥₂,𝑥₃)=𝑥₁+𝑥₂+𝑥₃,𝐶_𝑉(𝑥₁,𝑥₂,𝑥₃)=𝑥₁𝑥₃. (2.6)

We apply now relations (2.1) to the function 𝐻=𝐻_𝑉∈𝐶^∞(𝐑³,𝐑) given by (2.6). Then the symmetric matrix 𝐺∶=𝐺_𝑉=(𝐺_𝑉^𝑖𝑗) is𝐺_𝑉=−2111−2111−2. (2.7)

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) becomes𝑥̇₁=𝑥₁𝑥₂+𝑎(𝑥₁−2𝑥₃𝑥̇),₂=−𝑥₁𝑥₂+𝑥₂𝑥₃+𝑎(𝑥₁+𝑥₃𝑥̇),₃=−𝑥₂𝑥₃+𝑎(−2𝑥₁+𝑥₃). (2.8)

Proposition 2.1. The dynamical system (𝐑³,𝑃_𝑉,𝐻_𝑉,𝐺_𝑉,𝐶_𝑉) given by (2.8) is a metriplectic system on 𝐑³.

Proof. We have 𝜕𝐶_𝑉/𝜕𝑥₁=𝑥₃, and 𝜕𝐶_𝑉/𝜕𝑥₂=0, 𝜕𝐶_𝑉/𝜕𝑥₃=𝑥₁. Then 𝑃_𝑉⋅∇𝐶_𝑉=0𝑥₁𝑥₂0−𝑥₁𝑥₂0𝑥₂𝑥₃0−𝑥₂𝑥₃0𝑥₃0𝑥₁=000, (2.9) that is, 𝐶_𝑉 is a Casimir of Hamilton-Poisson system (𝐑³,𝑃_𝑉,𝐻_𝑉).
We prove that conditions (i) and (ii) from (2.3) are verified.
We have 𝐶_𝑉(𝑥)=𝑎𝑥₁𝑥₃, 𝜕𝐻_𝑉/𝜕𝑥₁=𝜕𝐻_𝑉/𝜕𝑥₂=𝜕𝐻_𝑉/𝜕𝑥₃=1 and  𝜕𝐶_𝑉/𝜕𝑥₁=𝑎𝑥₃, 𝜕𝐶_𝑉/𝜕𝑥₂=0, 𝜕𝐶_𝑉/𝜕𝑥₃=𝑎𝑥₁. Then 𝐺_𝑉⋅∇𝐻_𝑉=111=000,−2111−2111−2(∇𝐶(𝑥))^𝑇⋅𝐺(𝑥)⋅∇𝐶(𝑥)=(𝑎𝑥₃,0,𝑎𝑥₁)−2111−2111−2𝑎𝑥₃0𝑎𝑥₁=−2𝑎²(𝑥²₁−𝑥₁𝑥₃+𝑥²₃)≤0. (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 𝑥̇_𝑖=𝑓_𝑖(𝑥), 𝑖=1,3, where 𝑓₁(𝑥)=𝑥₁𝑥₂+𝑎(𝑥₁−2𝑥₃),𝑓₂(𝑥)=−𝑥₁𝑥₂+𝑥₂𝑥₃+𝑎(𝑥₁+𝑥₃𝑓),₃(𝑥)=−𝑥₂𝑥₃+𝑎(−2𝑥₁+𝑥₃). (2.11)

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).

Proof. (i) We have 𝑑𝐻_𝑉/𝑑𝑡=𝑥̇₁+𝑥̇₂+𝑥̇₃=𝑓₁(𝑥)+𝑓₂(𝑥)+𝑓₃(𝑥)=0.
(ii) The derivative of 𝐶_𝑉 along the solutions of system (2.8) verifies the condition 𝑑𝐶_𝑉/𝑑𝑡≤0. Indeed, 𝑑𝐶_𝑉/𝑑𝑡=𝑎𝑥̇₁𝑥₃+𝑎𝑥₁𝑥̇₃=𝑎(𝑥₃𝑓₁(𝑥)+𝑥₁𝑓₃(𝑥))=−2𝑎²(𝑥²₁−𝑥₁𝑥₃+𝑥²₃)≤0.

Remark 2.3. If 𝑎≠0, then 𝐶_𝑉=𝑎𝐶_𝑉 is not a constant of motion for the metriplectic system (2.8).

Proposition 2.4. (i) If 𝑎∈𝐑^∗, then the equilibrium states of the dynamics of (2.8) are 𝑒^𝑀₂=(0,𝑀,0) for all 𝑀∈𝐑.
(ii) For 𝑎=0, the equilibrium states of the dynamics of (2.8) are 𝑒₁^𝑀𝑁=(𝑀,0,𝑁), and  𝑒^𝑀₂=(0,𝑀,0), 𝑒^𝑀₃=(0,0,𝑀) for all 𝑀,𝑁∈𝐑.

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

Proposition 2.5. The equilibrium states 𝑒^𝑀₂, 𝑀∈𝐑 are unstable.

Proof. Let 𝐴 be the matrix of the linear part of the system (2.8), that is, 𝑥𝐴=²+𝑎𝑥₁−2𝑎−𝑥₂+𝑎−𝑥₁+𝑥₃𝑥₂+𝑎−2𝑎−𝑥₃−𝑥₂+𝑎. (2.12) The characteristic polynomial of the matrix 𝐴(𝑒^𝑀₂) is 𝑝(𝜆)=𝜆(𝜆²−2𝑎𝜆−3𝑎²−𝑀²) with the roots 𝜆₁=0, 𝜆_2,3=𝑎±4𝑎²+𝑀². Then the assertion follows via Lyapunov's theorem [15].

Remark 2.6. (i) The dynamics of (2.4) and (2.8) have not the same equilibria.
(ii) For 𝑎=0, 𝑒₁^𝑀𝑁,𝑒^𝑀₂,𝑒^𝑀₃ have the following behaviors (see [14]): 𝑒₁^𝑀𝑁 is unstable if 𝑀−𝑁<0 and spectrally stable if 𝑀−𝑁>0; 𝑒^𝑀₂ is unstable, and 𝑒^𝑀₃ is unstable if 𝑀≥0 and spectrally stable if 𝑀<0.

3. 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:𝑥₁^𝑛+1−𝑥^𝑛₁=ℎ2(𝑥₁^𝑛+1𝑥^𝑛₂+𝑥₂^𝑛+1𝑥^𝑛₁)+𝑎ℎ(𝑥^𝑛₁+𝑥₁^𝑛+1−2𝑥^𝑛₃−2𝑥₃^𝑛+1𝑥),_2𝑛+1−𝑥^𝑛₂=ℎ2(−𝑥₁^𝑛+1𝑥^𝑛₂−𝑥₂^𝑛+1𝑥^𝑛₁+𝑥₃^𝑛+1𝑥^𝑛₂+𝑥₂^𝑛+1𝑥^𝑛₃)+𝑎ℎ(𝑥^𝑛₁+𝑥₁^𝑛+1+𝑥^𝑛₃+𝑥₃^𝑛+1𝑥),_3𝑛+1−𝑥^𝑛₃ℎ=−2(𝑥₂^𝑛+1𝑥^𝑛₃+𝑥₃^𝑛+1𝑥^𝑛₂)+𝑎ℎ(𝑥^𝑛₃+𝑥₃^𝑛+1−2𝑥^𝑛₁−2𝑥₁^𝑛+1). (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 𝑥₁^𝑛+1+𝑥₂^𝑛+1+𝑥₃^𝑛+1=𝑥^𝑛₁+𝑥^𝑛₂+𝑥^𝑛₃. (3.2) Hence 𝐻_𝑉(𝑥₁^𝑛+1,𝑥₂^𝑛+1,𝑥₃^𝑛+1)=𝐻_𝑉(𝑥^𝑛₁,𝑥^𝑛₂,𝑥^𝑛₃).

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.

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.

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.

Figure 3 

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

Figure 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 [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 by𝑥̇₁=𝑥₁𝑥₂+𝑎(𝑥₁−2𝑥₃𝑥̇),₂=−𝑥₁𝑥₂+𝑥₂𝑥₃+𝑎(𝑥₁+𝑥₃𝑥̇),₃=−𝑥₂𝑥₃+𝑎(−2𝑥₁+𝑥₃), (4.1) respectively𝑦̇₁=𝑦₁𝑦₂+𝑎(𝑦₁−2𝑦₃)+𝑢₁,𝑦̇₂=−𝑦₁𝑦₂+𝑦₂𝑦₃+𝑎(𝑦₁+𝑦₃)+𝑢₂,𝑦̇₃=−𝑦₂𝑦₃+𝑎(−2𝑦₁+𝑦₃)+𝑢₃, (4.2) where 𝑢₁(𝑡),𝑢₂(𝑡),and𝑢₃(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)−𝑥₁𝑒(𝑡),₂(𝑡)=𝑦₂(𝑡)−𝑥₂𝑒(𝑡),₃(𝑡)=𝑦₃(𝑡)−𝑥₃(𝑡). (4.3) By subtracting (4.2) from (4.1) and using notation (4.3) we can get𝑒₁̇=𝑒₁𝑒₂+(𝑥₂+𝑎)𝑒₁+𝑥₁𝑒₂−2𝑎𝑒₃+𝑢₁,𝑒₂̇=−𝑒₁𝑒₂+𝑒₂𝑒₃+(𝑎−𝑥₂)𝑒₁+(𝑥₃−𝑥₁)𝑒₂+(𝑎+𝑥₂)𝑒₃+𝑢₂,𝑒₃̇=−𝑒₂𝑒₃−2𝑎𝑒₁−𝑥₃𝑒₂+(𝑎−𝑥₂)𝑒₃+𝑢₃. (4.4) We define the active control inputs 𝑢₁(𝑡),𝑢₂(𝑡),and𝑢₃(𝑡) as follows:𝑢₁𝑢(𝑡)₂𝑢(𝑡)₃𝑒(𝑡)=𝐾⋅𝑒¹(𝑡)₂𝑒(𝑡)₃+(𝑡)−𝑒₁(𝑡)𝑒₂𝑒(𝑡)₁(𝑡)𝑒₂(𝑡)−𝑒₂(𝑡)𝑒₃𝑒(𝑡)₂(𝑡)𝑒₃(𝑡), (4.5) where 𝑘𝐾=¹¹𝑘₁₂𝑘₁₃𝑘₂₁𝑘₂₂𝑘₂₃𝑘₃₁𝑘₃₂𝑘₃₃ (4.6) and 𝑘_𝑖𝑗, 𝑖,𝑗=1,3 are real functions which depend on 𝑥₁(𝑡),𝑥₂(𝑡),and𝑥₃(𝑡). Then the differential system of errors (4.7) is given by𝑒₁̇=𝑒₁𝑒₂+(𝑥₂+𝑎+𝑘₁₁)𝑒₁+(𝑥₁+𝑘₁₂)𝑒₂+(𝑘₁₃−2𝑎)𝑒₃,𝑒₂̇=−𝑒₁𝑒₂+𝑒₂𝑒₃+(𝑎−𝑥₂+𝑘₂₁)𝑒₁+(𝑥₃−𝑥₁+𝑘₂₂)𝑒₂+(𝑎+𝑥₂+𝑘₂₃)𝑒₃,𝑒₃̇=−𝑒₂𝑒₃+(𝑘₃₁−2𝑎)𝑒₁+(𝑘₂₃−𝑥₃)𝑒₂+(𝑎−𝑥₂+𝑘₃₃)𝑒₃. (4.7) If we choose𝐾=−𝑎−𝑥₂−𝑥²₂𝑥00₂−𝑎𝑥₁−𝑥₃−𝑥²₁02𝑎𝑥₃𝑥₂−𝑎−𝑥²₃, (4.8) then the active controls defined by (4.5) become𝑢₁=−𝑒₁𝑒₂−(𝑎+𝑥₂+𝑥²₂)𝑒₁,𝑢₂=𝑒₁𝑒₂−𝑒₂𝑒₃+(𝑥₂−𝑎)𝑒₁+(𝑥₁−𝑥₃−𝑥²₁)𝑒₂,𝑢₃=𝑒₂𝑒₃+2𝑎𝑒₁+𝑥₃𝑒₂+(−𝑎+𝑥₂−𝑥²₃)𝑒₃. (4.9) Using (4.8), the system of errors (4.7) becomes𝑒₁̇=−𝑥²₂𝑒₁+𝑥₁𝑒₂−2𝑎𝑒₃,𝑒₂̇=−𝑥²₁𝑒₂+(𝑎+𝑥₂)𝑒₃,𝑒₃̇=−𝑥²₃𝑒₃. (4.10)

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 [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 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.

(a)

(b)

(c)

Figure 5 

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

(a)

(b)

(c)

Figure 6 

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

(a)

(b)

(c)

Figure 7 

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 [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 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.