2017/07/19 by Dang H. Nguyen, Dang Nguyen, Nguyen H. Du +5
Economics, Econometrics and Finance · Mathematics · Medicine · #34C05 #60H10 #92D25 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and financial applications #math.DS #math.PR #msc:34C05 #msc:60H10 #msc:92D25
paper · pdf · doi:10.48550/arxiv.1707.06179
25 pages
arxiv created 2017/07/19 · openalex publication_date 2017/07/19 · arxiv updated 2017/07/20 · openalex created_date 2017/07/31 · openalex updated_date 2026/07/28
This work is devoted to examining qualitative properties of dynamic systems, in particular, limit cycles of stochastic differential equations with both rapid switching and small diffusion. The systems are featured by multi-scale formulation, highlighted by the presence of two small parameters ε and δ. Associated with the underlying systems, there are averaged or limit systems. Suppose that for each pair of the parameters, the solution of the corresponding equation has an invariant probability measure με,δ, and that the averaged equation has a limit cycle in which there is an averaged occupation measure μ0 for the averaged equation. Our main effort is to prove that με,δ converges weakly to μ0 as ε→ 0 and δ→ 0 under suitable conditions. Moreover, our results are applied to a stochastic predator-prey model together with numerical examples for demonstration.