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Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation. (English) Zbl 1218.34050
There is considered the differential equation $$ \dot{x}=x^{2n+1} + h(x,\omega t) + f(x,\omega t)$$ with $x\in \Bbb R$, $n\geq 0$ an integer, $h=O(x^{2n+2})$ is a higher order term and $f$ a small perturbation term. Let $\Delta_{r,s} = D(0,r)\times T_s$, where $D(0,r)=\{x\in C: |x|\leq r\}$ and $T_s$ is the strip domain defined by $T_s=\{\theta\in C^m/2\pi Z^m: |\Im m\theta_j|\leq s\;\;j=\overline{1,m}\}$. Let $h$ and $f$ be real analytic on $\Delta_{r,s}$ and analytic quasi-periodic in $t$ with vector frequency $\omega= (\omega_1,\dots,\omega_m)$. If $f=0$, the equation has 0 as degenerate equilibrium. It is proved that, when $f$ is sufficiently small, the differential equation can be reduced via an affine quasi-periodic transformation to a suitable normal form with 0 as equilibrium so it has a quasi-periodic solution near 0.

34C27Almost and pseudo-almost periodic solutions of ODE
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
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