On quasi-periodic perturbations of elliptic equilibrium points.

*(English)*Zbl 0863.34043From the authors’ summary: 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 \([0,\varepsilon_0] \setminus{\mathcal E}\) 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.

Reviewer: P.Smith (Keele)

##### MSC:

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34D10 | Perturbations of ordinary differential equations |

34C99 | Qualitative theory for ordinary differential equations |

37C55 | Periodic and quasi-periodic flows and diffeomorphisms |

37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |