On the non-autonomous Hopf bifurcation problem: systems with rapidly varying coefficients.

Abstract

We consider a $2$-dimensional ordinary differential equation (ODE) depending on a parameter $\epsilon$. If the ODE is autonomous the supercritical Andronov–Hopf bifurcation theory gives sufficient conditions for the genesis of a repeller–attractor pair, made up by a critical point and a stable limit cycle respectively. We give assumptions that enable us to reproduce the analogous phenomenon in a non-autonomous context, assuming that the coefficients of the system are subject to fast oscillations, and have very weak recurrence properties, e.g. they are almost periodic (in fact we just need that the associated base flow is uniquely ergodic). In this context the critical point is replaced by a trajectory which is a copy of the base and the limit cycle by an integral manifold. The dynamics inside the attractor becomes much richer and, if one asks for stronger recurrence assumptions, e.g. the coefficients are quasi periodic, it can be (partially) analyzed by the methods of [M. Franca, R. Johnson, V. Muñoz-Villarragut, Discrete Contin. Dyn. Syst. Ser. S 9(2016), No. 4, 1119–1148]. The problem is in fact studied as a two parameters problem: we use $\epsilon$ to describe the size of the perturbation and $1/\mu$ to describe the speed of oscillations, but the results allows to set $\epsilon=\mu$.

Keywords

• andronov–hopf bifurcation
• fast varying coefficients
• integral manifolds
• non-autonomous bifurcation