Periodic solutions in distribution of stochastic lattice differential equations

This paper aims to study the similarity between two lattice differential equations. Motivated by the the conjugate theory of dynamical systems, we present the definition of conjugacy and similarity for the lattice differential equations and introduce a functional to measure the degree of their similarity. We prove the maximum principle for lattice control systems. Furthermore, for two lattice differential equations, we determine the necessary conditions for the minimizer of the functional. Then we apply the definitions and results to provide the conditions that the linear minimizer satisfies.1. Introduction. Generally, ordinary differential equations (ODEs) that we are familiar with are described over continuous time [9], whereas lattice differential equations (LDEs) are described over discrete lattice points. Roughly, lattice systems are infinite dimensional systems of ordinary differential equations [4]. LDEs can be employed to depict the evolution laws of dynamical systems in discrete time or space, see [5,6,12,14] and the references therein. These types of equations are highly valuable for describing the dynamics of certain discrete systems, particularly in fields like physics, biology, economics, and engineering, see for example, [1,3,7,13]. For stochastic lattice differential equations, one can see [2,8,17,19,20] for more details.This paper is concerned with the following lattice differential equations:for all i ∈ Z, t ≥ 0, where ν, λ are positive constants, u := (u i ) i∈Z , v := (v i ) i∈Z , (f i (t)) i∈Z , (g i (t)) i∈Z ∈ l 2 are continuous in R + , and the coupled nonlinear terms (f i (u)) i∈Z , (g i (v)) i∈Z : l 2 → l 2 are smooth functions which satisfy the following Lipschitz conditions and linear growth conditions:

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Periodic solutions in distribution of stochastic lattice differential equations

Cited by 5 publications

References 42 publications

Periodicity for stochastic lattice equations with regime-switching

Periodicity for stochastic lattice equations with regime-switching

LaSalle-type stationary oscillation principle for stochastic affine periodic systems

Necessary conditions for similarity between two lattice differential equations

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