On the reducibility for a class of quasi-periodic Hamiltonian systems with small perturbation parameter (Q638118)

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.