Dynamical systems under random perturbations with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles
Dynamical systems under random perturbations with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles
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具有快速切换和慢速扩散的随机扰动下的动力系统：双曲平衡和稳定极限环
DOI：
10.1016/j.jde.2021.05.032
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发表时间：
2021
影响因子：
2.4
通讯作者：
Yin, George
中科院分区：
文献类型：
--
作者：
Du, Nguyen H.;Hening, Alexandru;Nguyen, Dang H.;Yin, George
This work is devoted to the study of long-term qualitative behavior of randomly perturbed dynamical systems. The focus is on certain stochastic differential equations (SDE) with Markovian switching, when the switching is fast varying and the diffusion (white noise) is slowly changing. Consider the system d X ε, δ (t)= f (X ε, δ (t), α ε (t)) d t+ δ σ (X ε, δ (t), α ε (t)) d W (t), X ε, δ (0)= x, α ε (0)= i, where α ε (t) is a finite state Markov chain with irreducible generator Q=(q ι ℓ). The relative changing rates of the switching and the diffusion are highlighted by two small parameters ε and δ. Associated with the above stochastic differential equation, there is an averaged ordinary differential equation (ODE) d X‾(t)= f‾(X‾(t)) d t, X‾(0)= x, where f‾(⋅)=∑ ι= 1 m 0 f (⋅, ι) ν ι and (ν 1,…, ν m 0) is the unique stationary distribution of the Markov chain with generator Q. Suppose that for each pair (ε, δ), the process has an invariant probability measure μ ε, δ, and that the averaged ODE has a limit cycle in which there is an averaged occupation measure μ 0 for the averaged equation. It is proved in this paper that under weak conditions, if f‾ has finitely many stable or hyperbolic fixed points, then μ ε, δ converges weakly to μ 0 as ε→ 0 and δ→ 0. Our results generalize to the setting where the switching process α ε is state-dependent in that P {α ε (t+ Δ)= ℓ| α ε= ι, X ε, δ (s), α ε (s), s≤ t}= q ι ℓ (X ε, δ (t)) Δ+ o (Δ), ι≠ ℓ as long as the generator Q (⋅)=(q ι ℓ (⋅)) is locally bounded, locally Lipschitz, and irreducible for all x∈ R d. Finally, we provide two examples in two and three dimensions to showcase our results.
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