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Zbl 0863.34043
Jorba, Àngel; Simó, Carles
On quasi-periodic perturbations of elliptic equilibrium points. (English)
[J] SIAM J. Math. Anal. 27, No.6, 1704-1737 (1996). ISSN 0036-1410; ISSN 1095-7154/e

From 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 ${\cal O}(x^2)$. It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to $\varepsilon$, there exists a Cantorian set ${\cal E}$ such that for all $\varepsilon\in{\cal E}$ 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 $[0,\varepsilon_0] \setminus{\cal E}$ 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.
[P.Smith (Keele)]

MSC 2000:: *34C27 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, etc.
37J99 Finite-dimensional Hamiltonian etc. systems

Keywords: quasi periodic solutions; Floquet theory; KAM theory

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Zentralblatt MATH Berlin [Germany]