Some Generalization of the Method of Stability Investigation for Nonlinear Stochastic Delay Differential Equations

Abstract

It is known that the method of Lyapunov functionals is a powerful method of stability investigation for functional differential equations. Here, it is shown how the previously proposed method of stability investigation for nonlinear stochastic differential equations with delay and a high order of nonlinearity can be extended to nonlinear mathematical models of a much more general form. An important feature is the combination of the method of Lyapunov functionals with the method of Linear Matrix Inequalities (LMIs). Some examples of applications of the proposed method of stability research to known mathematical models are given.

Stability of the Exponential Type System of Stochastic Difference Equations

“…Note that the equilibrium E * (x * 1 , x * 2 , x * 3 ) of the deterministic system (1) is also the solution of the stochastic system (11). Stochastic perturbations of such a type were first proposed in [31], and they were successfully used later by many other researchers for different mathematical models with continuous and discrete time applications (see, for instance, [10][11][12][13][28][29][30] and the references therein). By substituting into system (11) x i (n) = y i (n) + x * i , i = 1, 2, 3, and using (3), we centralize system (11) around the equilibrium E * (x * 1 , x * 2 , x * 3 )…”

“…The method of studying stability used here is of a fairly general nature and can be applied to nonlinear systems that are described by both difference and differentials equations [28][29][30]. To demonstrate that this method can be used for systems of higher dimension and for systems with other types of nonlinearity, besides of system (1), a special system with exponential and fractional nonlinearities was also investigated.…”

About Stability of Nonlinear Stochastic Differential Equations with State-Dependent Delay

“…Substituting (11) into system (8) and using (9), we obtain the following system of nonlinear Ito's stochastic differential equations…”

“…Among different types of delay differential equations, in particular, stochastic delay differential equations, equations with a delay, that depends on the state of the system under consideration, play a special role and are very popular in research (see, for instance, [1][2][3][4][5][6][7][8][9] and the references therein). Here, the method of stability investigation described in [10,11] for nonlinear stochastic differential equations with usual delay is used for investigation of the following stage-structured single population model with a state-dependent delay [8]. ẋ(t) = αy(t) − γx(t) − α[1 − τ (z(t)) ż(t)]y(t − τ(z(t)))e −γτ(z(t)) , ẏ(t) = α[1 − τ (z(t)) ż(t)]y(t − τ(z(t)))e −γτ(z(t)) − βy 2 (t), where z(t) = x(t) + y(t), x(s) = φ(s) ≥ 0, y(s) = ψ(s) ≥ 0, s ∈ [−τ M , 0].…”

"On Stability of One Mathematical Model of the Epidemic Spread Under Stochastic Perturbations"

“…In [1] some properties of stability of the system (1.1) equilibria are studied. Below, following the method from [4], stability in probability of two equilibria of the system (1.1) is investigated by the assumption that the system (1.1) is exposed to stochastic perturbations that are of the type of the white noise and are directly proportional to the deviation of a system state from an appropriate equilibrium.…”