The Hopf bifurcation with bounded noise

Abstract

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.

Conditioned Lyapunov exponents for random dynamical systems

“…1 We note that random dynamical systems with bounded noise suffer less from this problem, but such systems are generally not amenable to techniques from stochastic analysis and we will not consider these here. We refer to [2,13,14,19] for alternative approaches to bifurcations of random dynamical systems in the bounded noise context.…”

On the nonautonomous Hopf bifurcation problem

“…This is the reformulation of equation (1.4) referred to above. Let us remark that the family of equation (1.5) is formally similar to the random differential system ∂x ∂t = f (x, ε) + µξ t which was studied in [6]. Here ξ t represents a generic path of some noise process.…”

“…Here ξ t represents a generic path of some noise process. If the function g in (1.5) does not depend on x, then it defines a process by varying p ∈ P (for fixed values of ε and µ); this process differs from that of [6] in that, for the functions g we consider, it will be "less random". Note that we are mainly interested in the case where the origin is a fixed point for the perturbed system, while this assumption does not hold in [6].…”

“…If the function g in (1.5) does not depend on x, then it defines a process by varying p ∈ P (for fixed values of ε and µ); this process differs from that of [6] in that, for the functions g we consider, it will be "less random". Note that we are mainly interested in the case where the origin is a fixed point for the perturbed system, while this assumption does not hold in [6]. However we give a result in this setting too, Lemma 2.2, which allows to start a comparison of the two approaches: a detailed discussion of this topic may be the object of future investigations.…”

“…Even if the main focus of the paper will be on the case where the origin is a critical point for (1.5) for any µ, we consider briefly the case where g(t, 0, ε, µ) may be different from 0. This way we can compare the results obtained via our deterministic but non-autonomous approach, with those obtained via a stochastic analysis in [6].…”