On Quasi-Periodic Perturbations of Elliptic Equilibrium Points

Abstract

This work focuses on quasi-periodic time-dependent perturbations of ordinary differential equations near elliptic equilibrium points. This means studying \[ \dot x = (A + \varepsilon Q(t,\varepsilon ))x + \varepsilon g(t,\varepsilon ) + h(x,t,\varepsilon ), \] where A is elliptic and h is $\mathcal{O}(x^2 )$. It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to $\varepsilon $, there exists a Cantorian set $\mathcal{E}$ such that for all $\varepsilon \in \mathcal{E}$ there exists a quasi-periodic solution such that it goes to zero when $\varepsilon $ does. This quasi-periodic solution has the same set of basic frequencies as the perturbation. Moreover, the relative measure of the set $[0,\varepsilon _0 ]\backslash \mathcal{E}$ in $[0,\varepsilon _0 ]$ is exponentially small in $\varepsilon _0 $. The case $g \equiv 0$, $h \equiv 0$ (quasi-periodic Floquet theorem) is also considered. Finally, the Hamiltonian case is studied. In this situation, most of the invariant tori that are near the equilibrium point are not destroyed but only slightly deformed and “shaken” in a quasi-periodic way. This quasi-periodic “shaking” has the same basic frequencies as the perturbation.