A periodic bifurcation problem depending on a random variable - Kamenskiĭ, Mikhail I.; Nistri, Paolo; de Fitte, Paul Raynaud (Topological Methods in Nonlinear Analysis, 2019)
We consider an abstract bifurcation equation $P(x)+ε Q(x,ε, ω)=0$, where $P$ and $Q$ are operators, $ε$ is the bifurcation parameter, $ω \in Ω$, is the random variable and $(Ω, \mathcal{F})$ is a measurable space. The aim of the paper is to provide conditions on $P$ and $Q$ to ensure the existence, for any $ω \in Ω$, of a branch of solutions originating from the zeros of the operator $P$. We show that the considered abstract bifurcation is the model of a random autonomous periodically perturbed differential equation having the property that the unperturbed equation corresponding to $ε = 0$ has a limit cycle. As a consequence we obtain the existence, for any $ω \in Ω$, of a branch of periodic solutions of the perturbed equation emanating from the limit cycle.