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Effective reducibility of quasiperiodic linear equations close to constant coefficients
Jorba, Angel; Ramírez Ros, Rafael; Villanueva Castelltort, Jordi
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
Let us consider the differential equation $$ \dot{x}=(A+\varepsilon Q(t,\varepsilon))x, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ where $A$ is an elliptic constant matrix and $Q$ depends on time in a quasiperiodic (and analytic) way. It is also assumed that the eigenvalues of $A$ and the basic frequencies of $Q$ satisfy a diophantine condition. Then it is proved that this system can be reduced to $$ \dot{y}=(A^{*}(\varepsilon)+\varepsilon R^{*}(t,\varepsilon))y, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ where $R^{*}$ is exponentially small in $\varepsilon$, and the linear change of variables that performs such reduction is also quasiperiodic with the same basic frequencies than $Q$. The results are illustrated and discussed in a practical example.
Differential equations
Global analysis (Mathematics)
quasiperiodic Floquet theorem
quasiperiodic perturbations
reducibility of linear equations
Equacions diferencials ordinàries
Varietats (Matemàtica)
Classificació AMS::34 Ordinary differential equations::34A General theory
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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