1 Introduction
Systems of differential and difference equations with various types of exponential nonlinearities are very popular both in theoretical research and in different applications (see, for instance, [2, 4, 5, 7–9, 15, 18–22, 24, 26–29, 33–41] and references therein).
Here, similarly to [38], 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, 17, 31, 32] and the method of linear matrix inequalities (LMIs) [3, 6, 10–12, 14, 23, 25, 30]. However, unlike, for instance, [28, 37–39], 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 numerical simulations of solutions of the considered system of stochastic differential equations. Numerical analysis of the considered LMI is carried out using MATLAB.
Consider the system of two nonlinear delay differential equations
with positive parameters and positive initial conditions.
(1.1)
\[ \begin{array}{c}\displaystyle {\dot{x}_{1}}(t)+{a_{1}}{x_{1}}(t)+{b_{1}}{x_{1}}(t-h)={c_{1}}{x_{1}}(t){e^{-{p_{1}}{x_{1}}(t)-{q_{1}}{x_{2}}(t)}},\\ {} \displaystyle {\dot{x}_{2}}(t)+{a_{2}}{x_{2}}(t)+{b_{2}}{x_{2}}(t-h)={c_{2}}{x_{2}}(t){e^{-{p_{2}}{x_{1}}(t)-{q_{2}}{x_{2}}(t)}},\\ {} \displaystyle {x_{i}}(s)={\phi _{i}}(s),\hspace{0.2778em}\hspace{0.2778em}s\in [-h,0],\hspace{0.2778em}\hspace{0.2778em}i=1,2,\hspace{0.2778em}\hspace{0.2778em}t\ge 0,\end{array}\]It is clear that the system (1.1) has the zero equilibrium ${E_{0}}=(0,0)$, and the nonzero equilibrium ${E^{\ast }}=({x_{1}^{\ast }},{x_{2}^{\ast }})$ of the system (1.1) is defined by the system of two algebraic equations
that can be presented in the form of the system of two linear equations
with the solution
(1.2)
\[ \begin{array}{c}\displaystyle {a_{1}}+{b_{1}}={c_{1}}{e^{-{p_{1}}{x_{1}}-{q_{1}}{x_{2}}}},\\ {} \displaystyle {a_{2}}+{b_{2}}={c_{2}}{e^{-{p_{2}}{x_{1}}-{q_{2}}{x_{2}}}},\end{array}\](1.3)
\[ \begin{array}{c}\displaystyle {p_{1}}{x_{1}}+{q_{1}}{x_{2}}={r_{1}},\\ {} \displaystyle {p_{2}}{x_{1}}+{q_{2}}{x_{2}}={r_{2}},\\ {} \displaystyle {r_{i}}=\ln \big({c_{i}}/({a_{i}}+{b_{i}})\big),\hspace{0.2222em}\hspace{0.2778em}i=1,2,\end{array}\](1.4)
\[ \displaystyle {x_{i}^{\ast }}=\frac{{\Delta _{i}}}{\Delta },\hspace{1em}i=1,2,\hspace{1em}\Delta =\left|\begin{array}{c@{\hskip10.0pt}c}{p_{1}}& {q_{1}}\\ {} {p_{2}}& {q_{2}}\end{array}\right|,\hspace{1em}{\Delta _{1}}=\left|\begin{array}{c@{\hskip10.0pt}c}{r_{1}}& {q_{1}}\\ {} {r_{2}}& {q_{2}}\end{array}\right|,\hspace{1em}{\Delta _{2}}=\left|\begin{array}{c@{\hskip10.0pt}c}{p_{1}}& {r_{1}}\\ {} {p_{2}}& {r_{2}}\end{array}\right|.\]Note that via (1.3) for an equilibrium ${E^{\ast }}=({x_{1}^{\ast }},{x_{2}^{\ast }})$ with both positive coordinates the condition ${r_{i}}\gt 0$, i.e. ${c_{i}}\gt {a_{i}}+{b_{i}}$, $i=1,2$, must be satisfied. It is clear that for ${c_{i}}\lt {a_{i}}+{b_{i}}$ we get a negative equilibrium, while for ${c_{i}}={a_{i}}+{b_{i}}$ there is the zero equilibrium. However, the inequality ${c_{i}}\gt {a_{i}}+{b_{i}}$ is only a necessary condition for the solution of the system (1.3) to be positive, but not a sufficient one. Indeed, for example, the system $2{x_{1}}+{x_{2}}=4$, ${x_{1}}+{x_{2}}=1$ with ${r_{1}}=4$, ${r_{2}}=1$ has the solution ${x_{1}}=3$, ${x_{2}}=-2$, the system $2{x_{1}}+{x_{2}}=4$, ${x_{1}}+{x_{2}}=2$ with ${r_{1}}=4$, ${r_{2}}=2$ has the solution ${x_{1}}=2$, ${x_{2}}=0$.
Below, the system (1.1) is investigated under stochastic perturbations, the conditions of stability for both the zero and one positive equilibria are studied, the obtained results are illustrated by numerical simulations of solutions of the system under consideration.
2 Stochastic perturbations, centralization and linearization
Let $({x_{1}^{\ast }},{x_{2}^{\ast }})$ be one of the equilibria of the system (1.1). Let us suppose that the system (1.1) is exposed to stochastic perturbations of the white noise type, that are directly proportional to the deviation of the system (1.1) state $({x_{1}}(t),{x_{2}}(t))$ from the equilibrium $({x_{1}^{\ast }},{x_{2}^{\ast }})$. As a result, we obtain the system of Ito’s stochastic delay differential equations [13]
(2.1)
\[ \begin{aligned}{}d{x_{1}}(t)& =\big(-{a_{1}}{x_{1}}(t)-{b_{1}}{x_{1}}(t-h)+{c_{1}}{x_{1}}(t){e^{-{p_{1}}{x_{1}}(t)-{q_{1}}{x_{2}}(t)}}\big)dt\\ {} & \hspace{1em}+{\sigma _{1}}\big({x_{1}}(t)-{x_{1}^{\ast }}\big)d{w_{1}}(t),\\ {} d{x_{2}}(t)& =\big(-{a_{2}}{x_{2}}(t)-{b_{2}}{x_{2}}(t-h)+{c_{2}}{x_{2}}(t){e^{-{p_{2}}{x_{1}}(t)-{q_{2}}{x_{2}}(t)}}\big)dt\\ {} & \hspace{1em}+{\sigma _{2}}\big({x_{2}}(t)-{x_{2}^{\ast }}\big)d{w_{2}}(t).\end{aligned}\]Note that stochastic perturbations of the type of (2.1) were first used in [1] and later in many other works (see, for instance, [31, 32] and references therein). In this case, the equilibrium $({x_{1}^{\ast }},{x_{2}^{\ast }})$ of the deterministic system (1.1) is also the solution of the stochastic system (2.1).
2.1 Nonzero equilibrium
Let $({x_{1}^{\ast }},{x_{2}^{\ast }})$ be the nonzero equilibrium of the system (1.1). Using the new variables ${y_{i}}(t)={x_{i}}(t)-{x_{i}^{\ast }}$, $i=1,2$, and (1.2), transform the first equation of the system (2.1) by the following way:
\[\begin{aligned}{}d{y_{1}}(t)& =\big[-{a_{1}}\big({y_{1}}(t)+{x_{1}^{\ast }}\big)-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\\ {} & \hspace{1em}+{c_{1}}\big({y_{1}}(t)+{x_{1}^{\ast }}\big){e^{-{p_{1}}{y_{1}}(t)-{q_{1}}{y_{2}}(t)}}{e^{-{p_{1}}{x_{1}^{\ast }}-{q_{1}}{x_{2}^{\ast }}}}\big]dt+{\sigma _{1}}{y_{1}}(t)d{w_{1}}(t)\\ {} & =\big[-{a_{1}}\big({y_{1}}(t)+{x_{1}^{\ast }}\big)-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\\ {} & \hspace{1em}+({a_{1}}+{b_{1}})\big({y_{1}}(t)+{x_{1}^{\ast }}\big){e^{-{p_{1}}{y_{1}}(t)-{q_{1}}{y_{2}}(t)}}\big]dt+{\sigma _{1}}{y_{1}}(t)d{w_{1}}(t)\\ {} & =\big[\big({y_{1}}(t)+{x_{1}^{\ast }}\big)\big(({a_{1}}+{b_{1}}){e^{-{p_{1}}{y_{1}}(t)-{q_{1}}{y_{2}}(t)}}-{a_{1}}\big)\\ {} & \hspace{1em}-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\big]dt+{\sigma _{1}}{y_{1}}(t)d{w_{1}}(t).\end{aligned}\]
Similarly, for the second equation of the system (2.1) we have
\[\begin{aligned}{}d{y_{2}}(t)& =\big[-{a_{2}}\big({y_{2}}(t)+{x_{2}^{\ast }}\big)-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\\ {} & \hspace{1em}+{c_{2}}\big({y_{2}}(t)+{x_{2}^{\ast }}\big){e^{-{p_{2}}{y_{1}}(t)-{q_{2}}{y_{2}}(t)}}{e^{-{p_{2}}{x_{1}^{\ast }}-{q_{2}}{x_{2}^{\ast }}}}\big]dt+{\sigma _{2}}{y_{2}}(t)d{w_{2}}(t)\\ {} & =\big[-{a_{2}}\big({y_{2}}(t)+{x_{2}^{\ast }}\big)-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\\ {} & \hspace{1em}+({a_{2}}+{b_{2}})\big({y_{2}}(t)+{x_{2}^{\ast }}\big){e^{-{p_{2}}{y_{1}}(t)-{q_{2}}{y_{2}}(t)}}\big]dt+{\sigma _{2}}{y_{2}}(t)d{w_{2}}(t)\\ {} & =\big[\big({y_{2}}(t)+{x_{2}^{\ast }}\big)\big(({a_{2}}+{b_{2}}){e^{-{p_{2}}{y_{1}}(t)-{q_{2}}{y_{2}}(t)}}-{a_{2}}\big)\\ {} & \hspace{1em}-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\big]dt+{\sigma _{2}}{y_{2}}(t)d{w_{2}}(t).\end{aligned}\]
As a result, we obtain the system of two nonlinear Ito’s stochastic delay differential equations [13] with the zero solution:
(2.2)
\[ \begin{aligned}{}d{y_{1}}(t)& =\big[\big({y_{1}}(t)+{x_{1}^{\ast }}\big)\big(({a_{1}}+{b_{1}}){e^{-{p_{1}}{y_{1}}(t)-{q_{1}}{y_{2}}(t)}}-{a_{1}}\big)\\ {} & \hspace{1em}-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\big]dt+{\sigma _{1}}{y_{1}}(t)d{w_{1}}(t),\\ {} d{y_{2}}(t)& =\big[\big({y_{2}}(t)+{x_{2}^{\ast }}\big)\big(({a_{2}}+{b_{2}}){e^{-{p_{2}}{y_{1}}(t)-{q_{2}}{y_{2}}(t)}}-{a_{2}}\big)\\ {} & \hspace{1em}-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\big]dt+{\sigma _{2}}{y_{2}}(t)d{w_{2}}(t).\end{aligned}\]Using the representation ${e^{-x}}=1-x+o(x)$, where ${\lim \nolimits_{x\to 0}}\frac{o(x)}{x}=0$, we obtain the linear part of the system (2.2). Really, neglecting the nonlinear terms in the square brackets of the first equation (2.2), we have
where
\[\begin{aligned}{}& \big({y_{1}}(t)+{x_{1}^{\ast }}\big)\big(({a_{1}}+{b_{1}}){e^{-{p_{1}}{y_{1}}(t)-{q_{1}}{y_{2}}(t)}}-{a_{1}}\big)-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\\ {} & \hspace{1em}=\big({y_{1}}(t)+{x_{1}^{\ast }}\big)\big(({a_{1}}+{b_{1}})\big(1-{p_{1}}{y_{1}}(t)-{q_{1}}{y_{2}}(t)\big)-{a_{1}}\big)-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\\ {} & \hspace{1em}=\big({y_{1}}(t)+{x_{1}^{\ast }}\big)\big({b_{1}}-({a_{1}}+{b_{1}})\big({p_{1}}{y_{1}}(t)+{q_{1}}{y_{2}}(t)\big)\big)-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\\ {} & \hspace{1em}={b_{1}}\big({y_{1}}(t)+{x_{1}^{\ast }}\big)-({a_{1}}+{b_{1}}){x_{1}^{\ast }}\big({p_{1}}{y_{1}}(t)+{q_{1}}{y_{2}}(t)\big)-{b_{1}}\big({y_{1}}(t-h)+{x_{1}^{\ast }}\big)\\ {} & \hspace{1em}=\big({b_{1}}-({a_{1}}+{b_{1}}){x_{1}^{\ast }}{p_{1}}\big){y_{1}}(t)-({a_{1}}+{b_{1}}){x_{1}^{\ast }}{q_{1}}{y_{2}}(t)-{b_{1}}{y_{1}}(t-h).\end{aligned}\]
Similarly, for the second equation (2.2) we obtain
\[\begin{aligned}{}& \big({y_{2}}(t)+{x_{2}^{\ast }}\big)\big(({a_{2}}+{b_{2}}){e^{-{p_{2}}{y_{1}}(t)-{q_{2}}{y_{2}}(t)}}-{a_{2}}\big)-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\\ {} & \hspace{1em}=\big({y_{2}}(t)+{x_{2}^{\ast }}\big)\big(({a_{2}}+{b_{2}})\big(1-{p_{2}}{y_{1}}(t)-{q_{2}}{y_{2}}(t)\big)-{a_{2}}\big)-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\\ {} & \hspace{1em}=\big({y_{2}}(t)+{x_{2}^{\ast }}\big)\big({b_{2}}-({a_{2}}+{b_{2}})\big({p_{2}}{y_{1}}(t)+{q_{2}}{y_{2}}(t)\big)\big)-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\\ {} & \hspace{1em}={b_{2}}\big({y_{2}}(t)+{x_{2}^{\ast }}\big)-({a_{2}}+{b_{2}}){x_{2}^{\ast }}\big({p_{2}}{y_{1}}(t)+{q_{2}}{y_{2}}(t)\big)-{b_{2}}\big({y_{2}}(t-h)+{x_{2}^{\ast }}\big)\\ {} & \hspace{1em}=-({a_{2}}+{b_{2}}){x_{2}^{\ast }}{p_{2}}{y_{1}}(t)+\big({b_{2}}-({a_{2}}+{b_{2}}){x_{2}^{\ast }}{q_{2}}\big){y_{2}}(t)-{b_{2}}{y_{2}}(t-h).\end{aligned}\]
As a result, we obtain the linear part of the system (2.2)
\[\begin{aligned}{}d{z_{1}}(t)& =\big[\big({b_{1}}-({a_{1}}+{b_{1}}){x_{1}^{\ast }}{p_{1}}\big){z_{1}}(t)-({a_{1}}+{b_{1}}){x_{1}^{\ast }}{q_{1}}{z_{2}}(t)-{b_{1}}{z_{1}}(t-h)\big]dt\\ {} & \hspace{1em}+{\sigma _{1}}{z_{1}}(t)d{w_{1}}(t),\\ {} d{z_{2}}(t)& =\big[-({a_{2}}+{b_{2}}){x_{2}^{\ast }}{p_{2}}{z_{1}}(t)+\big({b_{2}}-({a_{2}}+{b_{2}}){x_{2}^{\ast }}{q_{2}}\big){z_{2}}(t)-{b_{2}}{z_{2}}(t-h)\big]dt\\ {} & \hspace{1em}+{\sigma _{2}}{z_{2}}(t)d{w_{2}}(t),\end{aligned}\]
or in the matrix form
(2.3)
\[ \displaystyle dz(t)=\big(Az(t)-Bz(t-h)\big)dt+{\sum \limits_{i=1}^{2}}{C_{i}}z(t)d{w_{i}}(t),\](2.4)
\[ \begin{array}{c}\displaystyle z(t)=\left[\begin{array}{c}{z_{1}}(t)\\ {} {z_{2}}(t)\end{array}\right],\hspace{1em}A=\left[\begin{array}{c@{\hskip10.0pt}c}{b_{1}}-({a_{1}}+{b_{1}}){x_{1}^{\ast }}{p_{1}}& -({a_{1}}+{b_{1}}){x_{1}^{\ast }}{q_{1}}\\ {} -({a_{2}}+{b_{2}}){x_{2}^{\ast }}{p_{2}}& {b_{2}}-({a_{2}}+{b_{2}}){x_{2}^{\ast }}{q_{2}}\end{array}\right],\\ {} \displaystyle B=\left[\begin{array}{c@{\hskip10.0pt}c}{b_{1}}& 0\\ {} 0& {b_{2}}\end{array}\right],\hspace{1em}{C_{1}}=\left[\begin{array}{c@{\hskip10.0pt}c}{\sigma _{1}}& 0\\ {} 0& 0\end{array}\right],\hspace{1em}{C_{2}}=\left[\begin{array}{c@{\hskip10.0pt}c}0& 0\\ {} 0& {\sigma _{2}}\end{array}\right].\end{array}\]2.2 Zero equilibrium
Let us linearize the system (2.1) with the zero equilibrium. Neglecting the nonlinear terms, transform the expression in brackets of the first equation (2.1) by the following way:
\[\begin{aligned}{}& -{a_{1}}{x_{1}}(t)-{b_{1}}{x_{1}}(t-h)+{c_{1}}{x_{1}}(t){e^{-{p_{1}}{x_{1}}(t)-{q_{1}}{x_{2}}(t)}}\\ {} & \hspace{1em}=-{a_{1}}{x_{1}}(t)-{b_{1}}{x_{1}}(t-h)+{c_{1}}{x_{1}}(t)\big(1-{p_{1}}{x_{1}}(t)-{q_{1}}{x_{2}}(t)\big)\\ {} & \hspace{1em}=({c_{1}}-{a_{1}}){x_{1}}(t)-{b_{1}}{x_{1}}(t-h).\end{aligned}\]
Similarly, for the second equation (2.1) we have
\[\begin{aligned}{}& -{a_{2}}{x_{2}}(t)-{b_{2}}{x_{2}}(t-h)+{c_{2}}{x_{2}}(t){e^{-{p_{2}}{x_{1}}(t)-{q_{2}}{x_{2}}(t)}}\\ {} & \hspace{1em}=-{a_{2}}{x_{2}}(t)-{b_{2}}{x_{2}}(t-h)+{c_{2}}{x_{2}}(t)\big(1-{p_{2}}{x_{1}}(t)-{q_{2}}{x_{2}}(t)\big)\\ {} & \hspace{1em}=({c_{2}}-{a_{2}}){x_{2}}(t)-{b_{2}}{x_{2}}(t-h).\end{aligned}\]
As a result, we obtain that the linearized system decomposes into two unrelated equations of the same type
3 Stability
3.1 Some necessary definitions and statements
Let $\{\Omega ,\mathfrak{F},\mathbf{P}\}$ be a complete probability space, ${\{{\mathfrak{F}_{t}}\}_{t\ge 0}}$ be a nondecreasing fami-ly of sub-σ-algebras of $\mathfrak{F}$, i.e. ${\mathfrak{F}_{s}}\subset {\mathfrak{F}_{t}}$ for $s\lt t$, $\mathbf{P}\{\cdot \}$ be the probability of an event enclosed in the braces, E be the mathematical expectation, ${H_{2}}$ be the space of ${\mathfrak{F}_{0}}$-adapted stochastic processes $\varphi (s)=({\varphi _{1}}(s),{\varphi _{2}}(s))$, $s\in [-h,0]$, $\| \varphi {\| _{0}}={\sup _{s\in [-h,0]}}|\varphi (s)|$, $\| \varphi {\| _{1}^{2}}={\sup _{s\in [-h,0]}}\mathbf{E}|\varphi (s){|^{2}}$, $|\varphi (s)|=\sqrt{{\varphi _{1}^{2}}(s)+{\varphi _{2}^{2}}(s)}$, $|\varphi (s){|^{2}}={\varphi _{1}^{2}}(s)+{\varphi _{2}^{2}}(s)$.
Definition 3.1.
The zero solution of the system (2.2) with the initial condition $y(s)=({y_{1}}(s),{y_{2}}(s))=\phi (s)$, $s\in [-h,0]$, is called stable in probability if for any ${\varepsilon _{1}}\gt 0$ and ${\varepsilon _{2}}\gt 0$ there exists $\delta \gt 0$ such that the solution $y(t,\phi )$ of the system (2.2) satisfies the condition $\mathbf{P}\{{\sup _{t\ge 0}}|y(t,\phi )|\gt {\varepsilon _{1}}\}\lt {\varepsilon _{2}}$ for any initial function ϕ such that $\mathbf{P}\{\| \phi {\| _{0}}\lt \delta \}=1$.
Definition 3.2.
The zero solution of the equation (2.3) with the initial condition $z(s)=\phi (s)$, $s\in [-h,0]$, is called:
-
• mean square stable if for each $\varepsilon \gt 0$ there exists a $\delta \gt 0$ such that $\mathbf{E}|z(t,\phi ){|^{2}}\lt \varepsilon $, $t\ge 0$, provided that $\| \phi {\| _{1}^{2}}\lt \delta $;
-
• asymptotically mean square stable if it is mean square stable and ${\lim \nolimits_{t\to \infty }}\mathbf{E}|z(t,\phi ){|^{2}}=0$ for each initial function ϕ.
Remark 3.1.
Note that the level of nonlinearity of the system (2.2) is more than one. It is known [32] that in this case sufficient conditions for asymptotic mean square stability of the zero solution of the linear part of this system, i.e. of the equation (2.3), are also sufficient conditions for stability in probability of the zero solution of the nonlinear system (2.2) and therefore are sufficient conditions for stability in probability of the equilibrium $({x_{1}^{\ast }},{x_{2}^{\ast }})$ of the system (2.1).
Let $z(t)$ be a value of the solution of the equation (2.3) in the time moment t and ${z_{t}}=z(t+s)$, $s\in [-h,0]$, be a trajectory of the solution of the equation (2.3) until the time moment t.
Consider a functional $V(t,\varphi ):[0,\infty )\times {H_{2}}\to {\mathbf{R}_{+}}$ that can be represented in the form $V(t,\varphi )=V(t,\varphi (0),\varphi (s))$, $s\in [-h,0)$, and for $\varphi ={z_{t}}$ put
Let D be the set of the functionals, for which the function ${V_{\varphi }}(t,z)$ defined by (3.1) has a continuous derivative with respect to t and two continuous derivatives with respect to z. The generator L of the equation (2.3) is defined on the functionals from D and has the form [13, 32] (here and everywhere below ′ is the sign of transposition)
(3.2)
\[ \begin{aligned}{}LV(t,{z_{t}})& =\frac{\partial }{\partial t}{V_{\varphi }}\big(t,z(t)\big)+\nabla {V^{\prime }_{\varphi }}\big(t,z(t)\big)\big(Az(t)-Bx(t-h)\big)\\ {} & \hspace{1em}+\frac{1}{2}{\sum \limits_{i=1}^{2}}{z^{\prime }}(t){C^{\prime }_{i}}{\nabla ^{2}}{V_{\varphi }}\big(t,z(t)\big){C_{i}}z(t).\end{aligned}\]Theorem 3.1 ([32]).
Let there exist a functional $V(t,\varphi )\in D$, positive constants ${c_{1}}$, ${c_{2}}$, ${c_{3}}$, such that the following conditions hold:
\[ \mathbf{E}V(t,{x_{t}})\ge {c_{1}}\mathbf{E}|x(t){|^{2}},\hspace{1em}\mathbf{E}V(0,\phi )\le {c_{2}}\| \phi {\| ^{2}},\hspace{1em}\mathbf{E}LV(t,{x_{t}})\le -{c_{3}}\mathbf{E}|x(t){|^{2}}.\]
Then the zero solution of the equation (2.3) is asymptotically mean square stable.
3.2 Nonzero equilibrium
Theorem 3.2.
Let there exist positive definite matrices P and R, such that the linear matrix inequality (LMI)
holds, where via (2.4)
(3.3)
\[ \Phi =\left[\begin{array}{c@{\hskip10.0pt}c}PA+{A^{\prime }}P+{S_{0}}+R& -PB\\ {} -{B^{\prime }}P& -R\end{array}\right]\lt 0\]
\[ {S_{0}}={\sum \limits_{i=1}^{2}}{C^{\prime }_{i}}P{C_{i}}=\left[\begin{array}{c@{\hskip10.0pt}c}{\sigma _{1}^{2}}{p_{11}}& 0\\ {} 0& {\sigma _{2}^{2}}{p_{22}}\end{array}\right]\]
and ${p_{11}}$, ${p_{22}}$ are diagonal elements of the matrix P. Then the equilibrium $({x_{1}^{\ast }},{x_{2}^{\ast }})$ of the system (2.1) is stable in probability.
Proof.
Following Remark 3.1, it is enough to prove the asymptotic mean square stability of the zero solution of the linear equation (2.3). Using the general method of Lyapunov functionals construction [16, 17, 31, 32], consider the functional $V={V_{1}}+{V_{2}}$, where ${V_{1}}(z(t))={z^{\prime }}(t)Pz(t)$, $P\gt 0$, the additional functional ${V_{2}}$ will be chosen below. Via (2.3), (3.2) and $\nabla {V_{1}}(z(t)=2Pz(t)$, ${\nabla ^{2}}{V_{1}}(z(t))=2P$, we have
\[\begin{aligned}{}L{V_{1}}\big(z(t)\big)& =2{z^{\prime }}(t)P\big(Az(t)-Bx(t-h)\big)+{z^{\prime }}(t){S_{0}}z(t)\\ {} & ={z^{\prime }}(t)\big(PA+{A^{\prime }}P+{S_{0}}\big)z(t)-2{z^{\prime }}(t)PBz(t-h).\end{aligned}\]
Putting ${V_{2}}({z_{t}})={\textstyle\int _{t-h}^{t}}{z^{\prime }}(s)Rz(s)ds$, $R\gt 0$, with $L{V_{2}}({z_{t}})={z^{\prime }}(t)Rz(t)-{z^{\prime }}(t-h)Rz(t-h)$, as a result for the functional $V={V_{1}}+{V_{2}}$, we obtain
\[\begin{aligned}{}LV\big(z(t)\big)& ={z^{\prime }}(t)\big(PA+{A^{\prime }}P+{S_{0}}+R\big)z(t)\\ {} & \hspace{1em}-2{z^{\prime }}(t)PBz(t-h)-{z^{\prime }}(t-h)Rz(t-h)\\ {} & ={\eta ^{\prime }}(t)\Phi \eta (t),\end{aligned}\]
where $\eta (t)={(z(t),z(t-h))^{\prime }}$ and the matrix Φ is defined in (3.3). From here and the LMI (3.3) it follows that $LV(z(t))\le -c|\eta (t){|^{2}}\le -c|z(t){|^{2}}$ for some $c\gt 0$. Via Theorem 3.1 it means that the zero solution of the linear equation (2.3) is asymptotically mean square stable.Via Remark 3.1 the sufficient conditions for asymptotic mean square stability of the zero solution of the linear equation (2.3) are also sufficient conditions for stability in probability of the zero solution of the nonlinear system (2.2) and therefore are sufficient conditions for stability in probability of the equilibrium $({x_{1}^{\ast }},{x_{2}^{\ast }})$ of the system (2.1). The proof is completed. □
Example 3.1.
Consider the system (2.1) with the following values of the parameters:
By that from (1.4) and (2.4) we obtain
Via MATLAB it was shown that for the parameters (3.4), (3.5) there exist the positive definite matrices
(3.4)
\[ \begin{array}{c}\displaystyle {a_{1}}=1,\hspace{1em}{b_{1}}=0.02,\hspace{1em}{c_{1}}=4,\hspace{1em}{p_{1}}=2,\hspace{1em}{q_{1}}=0.2,\hspace{1em}{\sigma _{1}}=1,\hspace{1em}h=1,\\ {} \displaystyle {a_{2}}=2,\hspace{1em}{b_{2}}=0.03,\hspace{1em}{c_{2}}=5,\hspace{1em}{p_{2}}=0.2,\hspace{1em}{q_{2}}=2,\hspace{1em}{\sigma _{2}}=1.\end{array}\](3.5)
\[ \begin{array}{c}\displaystyle {x_{1}^{\ast }}=0.6446,\hspace{1em}{x_{2}^{\ast }}=0.3862,\\ {} \displaystyle A=\left[\begin{array}{c@{\hskip10.0pt}c}-1.2950& -0.1315\\ {} -0.1568& -1.5381\end{array}\right],\hspace{0.1667em}\hspace{2.5pt}B=\left[\begin{array}{c@{\hskip10.0pt}c}0.02& 0\\ {} 0& 0.03\end{array}\right],\hspace{0.1667em}\hspace{2.5pt}{C_{1}}=\left[\begin{array}{c@{\hskip10.0pt}c}1& 0\\ {} 0& 0\end{array}\right],\hspace{0.1667em}\hspace{2.5pt}{C_{2}}=\left[\begin{array}{c@{\hskip10.0pt}c}0& 0\\ {} 0& 1\end{array}\right].\end{array}\]
\[ P=\left[\begin{array}{c@{\hskip10.0pt}c}301.6380& -23.3803\\ {} -23.3803& 252.7107\end{array}\right],\hspace{1em}R=\left[\begin{array}{c@{\hskip10.0pt}c}268.6902& 4.3515\\ {} 4.3515& 281.0097\end{array}\right],\]
for which the LMI (3.3) holds. It means that the equilibrium $({x_{1}^{\ast }},{x_{2}^{\ast }})=(0.6446,0.3862)$ of the system (2.1) is stable in probability. In Fig. 1 100 trajectories of the solution of the system (2.1) are shown, with the initial conditions ${x_{1}}(s)=0.77$, ${x_{2}}(s)=0.27$, $s\in [-h,0]$. One can see that all trajectories converge to the equilibrium.3.3 Zero equilibrium
Note that both equations of the system (2.5) have the form
where $a={c_{i}}-{a_{i}}$, $b=-{b_{i}}$, σ, $h\ge 0$ are known constants.
Lemma 3.1 ([32]).
The necessary and sufficient condition for asymptotic mean square stability of the zero solution of the equation (3.6) is
where
If, in particular, $b=0$, then this necessary and sufficient stability condition takes the form $2a+{\sigma ^{2}}\lt 0$.
(3.8)
\[ \displaystyle G=\left\{\begin{array}{l@{\hskip10.0pt}l}\frac{b{q^{-1}}\sin (qh)-1}{a+b\cos (qh)},\hspace{1em}& b+|a|\lt 0,\hspace{1em}q=\sqrt{{b^{2}}-{a^{2}}},\\ {} \frac{1+|a|h}{2|a|},\hspace{1em}& b=a\lt 0,\\ {} \frac{b{q^{-1}}\sinh (qh)-1}{a+b\cosh (qh)},\hspace{1em}& a+|b|\lt 0,\hspace{1em}q=\sqrt{{a^{2}}-{b^{2}}}.\end{array}\right.\]Theorem 3.3.
Let the conditions
hold, where
Then the zero equilibrium $({x_{1}^{\ast }},{x_{2}^{\ast }})=(0,0)$ of the system (2.1) is stable in probability.
(3.9)
\[ {c_{i}}\lt {a_{i}}+{b_{i}},\hspace{1em}{G_{i}^{-1}}\gt \frac{1}{2}{\sigma _{i}^{2}},\hspace{1em}i=1,2,\](3.10)
\[ \displaystyle {G_{i}}=\left\{\begin{array}{l@{\hskip10.0pt}l}\frac{1+{b_{i}}{q_{i}^{-1}}\sin ({q_{i}}h)}{{a_{i}}-{c_{i}}+{b_{i}}\cos ({q_{i}}h)},\hspace{1em}& |{a_{i}}-{c_{i}}|\lt {b_{i}},\hspace{1em}{q_{i}}=\sqrt{{b_{i}^{2}}-{({a_{i}}-{c_{i}})^{2}}},\\ {} \frac{1+({a_{i}}-{c_{i}})h}{2({a_{i}}-{c_{i}})},\hspace{1em}& {b_{i}}={a_{i}}-{c_{i}}\gt 0,\\ {} \frac{1+{b_{i}}{q_{i}^{-1}}\sinh ({q_{i}}h)}{{a_{i}}-{c_{i}}+{b_{i}}\cosh ({q_{i}}h)},\hspace{1em}& {b_{i}}\lt {a_{i}}-{c_{i}},\hspace{1em}{q_{i}}=\sqrt{{({a_{i}}-{c_{i}})^{2}}-{b_{i}^{2}}}.\end{array}\right.\]Example 3.2.
Consider the system (2.1) with the following values of the parameters:
It is easy to see, that for the values of the parameters (3.11) the conditions (3.9) hold. Therefore, the zero equilibrium of the system (2.1) is stable in probability. In Fig. 2 100 trajectories of the solution of the system (2.1) are shown, with the initial conditions ${x_{1}}(s)=0.55$, ${x_{2}}(s)=0.35$, $s\in [-h,0]$. One can see that all trajectories converge to the zero.
(3.11)
\[ \begin{array}{c}\displaystyle {a_{1}}=2,\hspace{1em}{b_{1}}=1,\hspace{1em}{c_{1}}=1,\hspace{1em}{p_{1}}=2,\hspace{1em}{q_{1}}=0.2,\hspace{1em}{\sigma _{1}}=0.9,\hspace{1em}h=1,\\ {} \displaystyle {a_{2}}=3,\hspace{1em}{b_{2}}=1,\hspace{1em}{c_{2}}=2,\hspace{1em}{p_{2}}=0.2,\hspace{1em}{q_{2}}=2,\hspace{1em}{\sigma _{2}}=0.9.\end{array}\]4 Conclusions
A system of two differential equations with exponential nonlinearity is considered. The nonlinearity in each equation depends on both variables of the system. It is supposed that this system is exposed to stochastic perturbations of the white noise type, which are directly proportional to the deviation of the system state from one of the system’s equilibria (the zero or nonzero). Some sufficient conditions for stability in probability of the both system equilibria are obtained by virtue of the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The obtained results are illustrated by numerical simulations of solutions of the considered system. The method of stability investigation described here can be applied to systems of higher dimension and with other types of nonlinearity.