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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:

$\stackrel{˙}{x}=Ax+f\left(t,x,\epsilon \right),\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{2}$

where $A$ is a real $2×2$ constant matrix, and $f\left(t,0,\epsilon \right)=O\left(\epsilon \right)$ and ${\partial }_{x}f\left(t,0,\epsilon \right)=O\left(\epsilon \right)$ as $\epsilon \to 0$. With some nonresonant conditions of the frequencies with the eigenvalues of $A$ and without any nondegeneracy condition with respect to $\epsilon$, by an affine analytic quasiperiodic transformation we change the system to a suitable normal form at the zero equilibrium for sufficiently small perturbation parameter $\epsilon$.

MSC:
 34C20 Transformation and reduction of ODE and systems, normal forms