2017/02/10 by Michel Benaı̈m, Benaïm, Michel, Édouard Strickler +1 · 2 citations
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Advanced Differential Equations and Dynamical Systems
paper · doi:10.48550/arxiv.1702.03089
Let E be a finite set, \Fi\i ∈ E a family of vector fields on ℝd leaving positively invariant a compact set M and having a common zero p ∈ M. We consider a piecewise deterministic Markov process (X,I) on M × E defined by Xt = FIt(Xt) where I is a jump process controlled by X: Pr(It+s = j | (Xu, Iu)u ≤ t) = ai j(Xt) s + o(s) for i ≠ j on \It = i \. We show that the behavior of (X,I) is mainly determined by the behavior of the linearized process (Y,J) where Yt = AJt Yt, Ai is the Jacobian matrix of Fi at p and J is the jump process with rates (aij(p)). We introduce two quantities Λ- and Λ+ respectively %called the \em minimal and \em maximal average growth rate. Λ- (respectively Λ+) is defined as the \em minimal (respectively \em maximal) \em growth rate of ‖Yt‖, where the minimum (respectively maximum) is taken over all the ergodic measures of the angular process (Θ, J) with Θt = (Yt)/(‖Yt‖). It is shown that Λ+ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of (Y,J) and that under general assumptions Λ- = Λ+. We then prove that, under certain irreducibility conditions, Xt → p exponentially fast when Λ+ < 0 and (X,I) converges in distribution at an exponential rate toward a (unique) invariant measure supported by M ∖ \p\ × E when Λ- > 0. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.