EFFECTIVE REDUCIBILITY OF QUASIPERIODIC LINEAR EQUATIONS CLOSE
TO CONSTANT COEFFICIENTS
Angel Jorba, Rafael Ramirez-Ros and Jordi Villanueva
Dept. de Matem\`atica Aplicada I, ETSEIB,
Universitat Polit\`ecnica de Catalunya, Diagonal 647,
08028 Barcelona, Spain
E-mails: jorba@ma1.upc.es, rafael@tere.upc.es, jordi@tere.upc.es
Abstract
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.
Keywords: quasiperiodic Floquet theorem, quasiperiodic perturbations,
reducibility of linear equations.
AMS Subject Classifications: 34A30, 34C20, 34C27, 34C50, 58F30