Abstract

A system of two difference equations with exponential nonlinearity in each equation is studied under stochastic perturbations. Conditions of the stability in probability of a positive equilibrium are studied by virtue of the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The obtained results are illustrated via examples and figures with numerical simulation of the solution of the system of stochastic difference equations. The proposed research method can be applied to nonlinear systems of higher dimension with an order of nonlinearity higher than one, both for stochastic difference equations and for stochastic differential equations with delay in various important applications, for example, in quantum physics, in population models and others.

Keywords:Nonlinear difference equations; positive equilibrium; stochastic perturbations; asymptotic mean square stability; stability in probability; linear matrix inequality (LMI); numerical simulations; MATLAB

Root Pressure

Systems of both difference and differential equations with different forms of exponential nonlinearities are very popular in research and various applications (see, for instance, [1–17] and references therein), in particular, the model from quantum physics [5], the model of Nicholson’s blowflies [2] or Mosquito population equation [12].

Here, similarly to [14], the stability of the positive equilibrium of a system with exponential nonlinearity is investigated under stochastic perturbations via the general method of Lyapunov functionals construction [16,18– 20] and the method of linear matrix inequalities (LMIs) [21–29]. However, unlike, for instance, [7,14], where the exponential nonlinearity in each equation depends on only one variable, here each equation exponentially depends on all variables of the system under consideration. The obtained results are illustrated via examples and figures with the equilibrium and numerical simulation of the solution of the considered system of difference equations. Numerical analysis of the considered LMIs is carried out using MATLAB.

Consider the system of two nonlinear difference equations.

with positive parameters, bi < 1, and positive initial conditions

Equilibrium

It is clear that the equilibrium (x₁, x₂) of the system (1) is defined by the system of two algebraic equations

Presenting the first equation (2) in the form

and calculating the logarithm, we get

Similarly, from the second equation (2) we have

It is clear that the function x₂ = f₁(x₁) given by (4) is defined and positive if x₁ ∈ (x_1min, x_1max), where unique root of the equation

which follows from (3) by x₂ = 0. Calculating the derivative in (4)

it is easy to see that x₂ = f₁(x₁) is strictly decreasing function. Moreover, and f₁ (x_1max) = 0 . Calculating the derivative of the function x₂ = f₂(x₁) , defined implicitly by (6), we have

i.e x'₂ > 0 It means that x= f₂(x₁) is strictly decreasing function for x₁ >0 . Moreover and x₂= f₂(0) is a unique root of the equation

which follows from (5) by x1 = 0. It is easy to see that the root x2 of this equation satisfies the condition . It is clear that two strictly decreasing functions x₂ = f₁ x₁ and x₂ = f₂ x₁ have (see Figure1 and 2) one common point, which is a solution of the system (2) and is the unique equilibrium (x*₁, x*₂) of the system (1).

where x_1max and x_2max are roots of the equations (7) and (8) respectively. Example 1.1 Consider the system (1) with

Then the solution of the system (2) is from (9) and (7), (8) it follows that

. In Figures 1 and 2 the graphs of the functions (red) and the equilibrium (x*₁, x*₂) are shown.

In Fig.1 also the asymptotes and of the functions f₁(x₁) and f₂(x₁) respectively are shown.

Let {Ω,ℑ, P} be a basic probability space, , be a nondecreasing family of sub--algebras of , i.e.,

ℑ_n1 ⊂ ℑ_n2 for n_n1 < n_n2 , E be the mathematical expectation with respect to the measure P, ξ₁ (n) and ξ₁ (n) , n∈N be two mutually independent sequences with adapted mutually independent random values such that [16]

Let us assume that the system (1) is exposed to stochastic perturbations that are directly proportional to the deviation of the system state (x1(n); x2(n)) from the equilibrium (x*₁, x*₂) . Then the system (1) takes the form

Remark 2.1 Note that the such type of stochastic perturbations was firstly proposed in [30] for a system of Ito’s stochastic delay differential equations and was later used in many other works for both differential and difference equations (see, for instance, [16,20] and references therein). With this type of stochastic perturbations, the equilibrium of the original deterministic system remains also a solution of the stochastically perturbed system.

Using (2) with (x₁, x₂) = (x*₁, x*₂), from the first equation (12) we have Similarly, for the second equation (12) we get

As a result, we obtain the nonlinear system with the zero solution

Remark 2.2 Note that stability of the zero solution of the system (13) is equivalent to stability of the equilibrium of the system (11).

Using (2) and the linear approximation e^(−x) =1− x + 0(x) where we obtain the linear part of the system (13)

Representing the linear system (14) in the matrix form, we get

Stability

Some necessary definitions and statements

Let ′ be the transposition sign. Put now

Definition 3.1 ([16]). The zero solution of the system (13) is called stable in probability if for any and there exists such that the solution y(n) = y(n,φ ) of the system (13) satisfies the inequality p{sup | y(n) |>ε } <ε₁ for any initial function φ ( j) such that = where

Definition 3.2 ([16]). The zero solution of the system (14) is called mean square stable if for each there exists aδ > 0 such that E | z(n) |²<ε , n∈N for any initial function φ (j) such that asymptotically mean square stable if it is mean square stable and for each initial function φ ( j) such that ||φ ||²< ∞ the solution z(n) of the system (14) satisfies the condition

Let E_n = E {./ℑ_n} be the conditional expectation with respect to the -algebra , ℑ_n,U_ε={y:|y|≤ε},ε =0 and V (n) =V (n +1) −V (n)

Theorem 3.1 ([16]). Let for the system (13) there exists a functional V (n) = V (n, y (-1) ,…… y(n)) satisfying the Conditions.

where > 0, c0 > 0, c1 > 0. Then the zero solution of the system (13) is stable in probability.

Theorem 3.2 ([16]). Let for the system (14) there exists a nonnegative functional V (n) = V (n, z(-1) ,…. z(n)) satisfying the conditions.

where c1 > 0, c2 > 0. Then the zero solution of the system (14) is asymptotically mean square stable.

Remark 3.1 Note that the system (13) has an order of nonlinearity higher than one. It is known [16] that in this case sufficient conditions for asymptotic mean square stability of the zero solution of the linear system (14) are also sufficient conditions for stability in probability of the zero solution of the nonlinear system (13).

Stability conditions

Theorem 3.3 Let there exist positive definite 2x2-matrices P and R such that the following linear matrix inequality (LMI)

Holds, were

the matrices C₁, C₂ are defined in (16) and p₁₁; p₂₂ are diagonal elements of the m,trix P. Then the equilibrium of the system (11) is stable in probability.

Proof: Following the general method of Lyapunov functionals construction [16,18,20], consider the functional V (n) in the form V(n) = V₁(n) + V₂(n), where V₁(n) = z′ (Pz (n), P > 0, z(n) is defined in (16) and the additional functional V₂(n) will be chosen below. For the functional V₁(n) via (15) we have

From here via (10) and (20) it follows that

Using the additional functional

From (22) and the LMI (19) for some c > 0 we have EV (n) ≤| z(n) |2 , i.e., the constructed functional V (n) satisfies the conditions of Theorem 3.2. Therefore, the zero solution of the linear equation (15) is asymptotically mean square stable. Via Remarks 3.1 and 2.2 it means that the equilibrium (x*₁, x*₂) of the system (11) is stable in probability. The proof is completed.

Remark 3.2 Note that instead of the LMI (19) for definition of stability some other LMIs also can be used. Using, for instance, the additional functional V₂(n) in the form instead of the LMI (19) we obtain the LMI

If at least one from the LMIs (19) and (23) holds then the equilibrium (x*₁, x*₂) of the system (11) is stable in probability. Other ways to get appropriate LMIs are shown also in [14]. Example 3.1 Consider the system (11) with the values of the parameters given in Example 1.1. Via MATLAB the maximal values of = 0:232 and = 0:384 were obtained, by which the LMIs (19) and (23) hold respectively for the positive definite matrices

Figure 3 and

In Figure 3 50 trajectories of the solution of the system (11) are shown. All trajectories converge to the stable Equilibrium ()=(3,4).

Conclusion

Stability of a system of nonlinear difference equations under stochastic perturbations is investigated. The nonlinearity of exponential form in each equation depends on all variables of the system under consideration. The conditions of stability in probability for positive equilibrium of the considered system, obtained via the general method of Lyapunov functionals construction, are formulated in terms of linear matrix inequalities (LMIs) and are illustrated by numerical examples and figures. The method of stability investigation, used in the paper, can be applied to many other types of nonlinear systems with an order of nonlinearity higher than one for both difference and differential equations in various applications.

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