A periodic bifurcation problem depending on a random variable

Abstract

We consider an abstract bifurcation equation $P(x)+\varepsilon Q(x,\varepsilon, \omega)=0$, where $P$ and $Q$ are operators, $\varepsilon$ is the bifurcation parameter, $\omega \in \Omega$, is the random variable and $(\Omega, \mathcal{F})$ is a measurable space. The aim of the paper is to provide conditions on $P$ and $Q$ to ensure the existence, for any $\omega \in \Omega$, 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 $\varepsilon = 0$ has a limit cycle. As a consequence we obtain the existence, for any $\omega \in \Omega$, of a branch of periodic solutions of the perturbed equation emanating from the limit cycle.