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dc.contributor.authorJorba, Angel
dc.contributor.authorSimó Torres, Carlos
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
dc.description.abstractThis 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.
dc.rightsAttribution-NonCommercial-NoDerivs 2.5 Spain
dc.subject.lcshDifferential equations
dc.subject.lcshGlobal analysis (Mathematics)
dc.subject.otherquasiperiodic perturbations
dc.subject.otherelliptic points
dc.subject.otherquasiperiodic solutions
dc.subject.othersmall divisors
dc.subject.otherquasiperiodic Floquet theorem
dc.subject.otherKAM theory
dc.titleOn quasiperiodic perturbations of elliptic equilibrium points
dc.subject.lemacEquacions diferencials ordinàries
dc.subject.lemacVarietats (Matemàtica)
dc.contributor.groupUniversitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
dc.subject.amsClassificació AMS::34 Ordinary differential equations::34C Qualitative theory
dc.subject.amsClassificació AMS::58 Global analysis, analysis on manifolds
dc.rights.accessOpen Access

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