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Zbl 1151.34030
Wang, Xiaocai; Xu, Junxiang
On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point. (English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 7, A, 2318-2329 (2008). ISSN 0362-546X

Summary: This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system: $$\dot x= Ax+f(t,x,\varepsilon),\quad x\in\Bbb R^2$$ 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 2000:: *34C20 Transformation of ODE and systems

Keywords: quasiperiodic solution; nonresonance condition; nondegeneracy condition; KAM iteration

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Zentralblatt MATH Berlin [Germany]