Bifurcacions de Hopf i de Neimark-Sacker

Abstract

[en] Dynamical systems that depend on one or more parameters can display different types of bifurcations under small variations of these. Particularly, we will deal with those which, under this perturbation, an equilibrium point (fixed point, for discrete systems) of the phase space changes its stability, that is, switching from stable to unstable, or vice versa, and an isolated periodic orbit (closed invariant curve, respectively) of small amplitude emerges around it. This description corresponds to the Hopf bifurcation, for continuous systems, and the Neimark-Sacker bifurcation, for discrete ones. Although it generically occurs, there is no guarantee that a periodic orbit (or closed invariant curve) will branch from that equilibrium (or fixed point). For both cases the needed existence and genericity assumptions will be studied in detail, starting with planar systems and concluding with the generalization for the n-dimensional case. To achieve these results we will use normal form and center manifold theories, which will also allow us to analyze the dynamics of the bifurcation itself.