Almost Sure Asymptotic Stability of Scalar Stochastic Delay Equations: Finite State Markov Process

Abstract

In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of the stochastic system.

Perturbations of linear delay differential equations at the verge of instability

“…Using singular perturbation methods and Furstenberg-Khasminskii formula, the following theorem for scalar processes is proved in [26] and [27]. Theorem 6.2.…”

“…In the case when F is constant and G, G q are Lipschitz; the probability distribution of H ε until any finite time T > 0, converges as ε → 0, to the probability distribution of a processȟ which is the solution of the SDEdȟ(t) = (b H + b q,(1) H + b q,(2) H )(ȟ(t))dt + σ H (ȟ(t))dW (t),ȟ(0) = h(ϕ),where b H and σ H are same as in(26) and(27)and b q,(k) H for k = 1, 2 are given by b q,(k) H…”

Exponentially-ergodic Markovian noise perturbations of delay differential equations at Hopf bifurcation

“…Here z(t) is R 2 -valued. Then, using (11), equation ( 9) can be replaced by the following system of equations (15) ż(t) = Bz(t), d dt y t = Ay t with initial values z(0) and y 0 (•) given by Π…”

“…If the zero fixed point of the limit ODE is exponentially stable, then it is proven that x is also exponentially stable in the moments. [15,16] considers equations of the form ẋ(t) = L 0 (Π t x) + εσ(ξ(t))L 1 (Π t x) with σ a mean zero function of the noise process ξ and L 1 is a bounded linear operator on C. Define the exponential growth rate λ ε := lim sup t→∞ 1 t log |x ε (t)| and expand it as λ ε = λ 0 +ελ 1 +ε 2 λ 2 +. .…”

“….. Using perturbation methods and Furstenberg-Khasminskii representation, [15,16] show that λ 0 = λ 1 = 0 and give explicit expression for λ 2 .…”