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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

x ˙=(A+εQ(t,ε))x+εg(t,ε)+h(x,t,ε),

where A is elliptic and h is 𝒪(x 2 ). It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to ε, there exists a Cantorian set such that for all ε there exists a quasi-periodic solution such that it goes to zero when ε does. This quasi-periodic solution has the same set of basic frequencies as the perturbation. Moreover, the relative measure of the set [0,ε 0 ] in [0,ε 0 ] is exponentially small in ε 0 . The case g0, h0 (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)
34C27Almost and pseudo-almost periodic solutions of ODE
34D10Stability perturbations of ODE
34C99Qualitative theory of solutions of ODE
37C55Periodic and quasiperiodic flows and diffeomorphisms
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems