Dynamical systems under random perturbations with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles

Abstract

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.