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 Ω, is the random variable and (Ω, 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 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 Ω, of a branch of periodic solutions of the perturbed equation emanating from the limit cycle.