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On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point. (English) Zbl 1151.34030
Summary: This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system:
\[ \dot x= Ax+f(t,x,\varepsilon),\quad x\in\mathbb R^2 \]
where \(A\) is a real \(2\times 2\) constant matrix, and \(f(t,0,\varepsilon)=O(\varepsilon)\) and \(\partial_xf(t,0,\varepsilon)=O(\varepsilon)\) as \(\varepsilon\to 0\). With some nonresonant conditions of the frequencies with the eigenvalues of \(A\) and without any nondegeneracy condition with respect to \(\varepsilon\), by an affine analytic quasiperiodic transformation we change the system to a suitable normal form at the zero equilibrium for sufficiently small perturbation parameter \(\varepsilon\).

MSC:
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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