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On quasiperiodic perturbations of elliptic equilibrium points
Jorba, Angel; Simó Torres, Carlos
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
This work focusses on quasiperiodic 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 quasiperiodic solution such that it goes to zero when $\varepsilon$ does. This quasiperiodic 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 $g\equiv 0$, $h\equiv 0$ (quasiperiodic 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 quasiperiodic way. This quasiperiodic ``shaking" has the same basic frequencies as the perturbation.
Differential equations
Global analysis (Mathematics)
quasiperiodic perturbations
elliptic points
quasiperiodic solutions
small divisors
quasiperiodic Floquet theorem
KAM theory
Equacions diferencials ordinàries
Varietats (Matemàtica)
Classificació AMS::34 Ordinary differential equations::34C Qualitative theory
Classificació AMS::58 Global analysis, analysis on manifolds
Attribution-NonCommercial-NoDerivs 2.5 Spain

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