On the reducibility for a class of quasi-periodic Hamiltonian systems with small perturbation parameter. (English) Zbl 1222.37049

Li, Jia; Xu, Junxiang

Abstr. Appl. Anal. 2011, Article ID 354063, 17 p. (2011).

Summary: We consider the real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system \(\text{ẋ} = J \nabla_x H\), where \(H(x, t, \varepsilon) = (1/2)\beta(x^2_1 + x^2_2) + F(x, t, \varepsilon)\) with \(\beta \neq 0\), \(\partial_x F(0, t, \varepsilon) = O(\varepsilon)\) and \(\partial_{xx} F(0, t, \varepsilon) = O(\varepsilon)\) as \(\varepsilon \rightarrow 0\). Without any nondegeneracy condition with respect to \(\varepsilon\), we prove that for most of the sufficiently small \(\varepsilon\), by a quasi-periodic symplectic transformation, it can be reduced to a quasi-periodic Hamiltonian system with an equilibrium.

MSC:

37J40

Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion

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