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On quasi-periodic perturbations of elliptic equilibrium points. (English) Zbl 0863.34043

From the authors’ summary: This work focuses on quasi-periodic time-dependent perturbations of ordinary differential equations near elliptic equilibrium points. This means studying

$\stackrel{˙}{x}=\left(A+\epsilon Q\left(t,\epsilon \right)\right)x+\epsilon g\left(t,\epsilon \right)+h\left(x,t,\epsilon \right),$

where $A$ is elliptic and $h$ is $𝒪\left({x}^{2}\right)$. It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to $\epsilon$, there exists a Cantorian set $ℰ$ such that for all $\epsilon \in ℰ$ there exists a quasi-periodic solution such that it goes to zero when $\epsilon$ does. This quasi-periodic solution has the same set of basic frequencies as the perturbation. Moreover, the relative measure of the set $\left[0,{\epsilon }_{0}\right]\setminus ℰ$ in $\left[0,{\epsilon }_{0}\right]$ is exponentially small in ${\epsilon }_{0}$. 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 of ODE 34D10 Stability perturbations of ODE 34C99 Qualitative theory of solutions of ODE 37C55 Periodic and quasiperiodic flows and diffeomorphisms 37J40 Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
##### Keywords:
quasi periodic solutions; Floquet theory; KAM theory