Effective Reducibility of Quasi-periodic Linear Equations Close To Constant Coefficients

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 quasi-periodic (and analytic) way. It is also assumed that the eigenvalues of $A$ and the basic fre...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 1997-01, Vol.28 (1), p.178-188
Main Authors: Jorba, Àngel, Ramírez-Ros, Rafael, Villanueva, Jordi
Format: Article
Language:English
Subjects:
Anàlisi global (Matemàtica)
Eigenvalues
Global analysis (Mathematics)
Linear equations
Values
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Summary: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 quasi-periodic (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 a reduction is also quasi-periodic with the same basic frequencies as $Q$. The results are illustrated and discussed in a practical example.
ISSN:0036-1410
1095-7154
DOI:10.1137/S0036141095280967