On the reducibility of a class of almost periodic Hamiltonian systems (Q2033772)

A linear system of the form \[ \dot{x} = A(t)x ,\qquad x\in{\mathbb{R}^{n}}, \] is reducible if there exists a quasi-periodic non-singular transformation \(x = \phi(t)y\), where \(\phi(t)\) and \(\phi(t)^{-1}\) are quasi-periodic and bounded, such that \(\dot{x} = A(t)x\) is transformed into \[\dot{y} = By,\qquad y\in{\mathbb{R}^{n}}, \] where \(B\) is a constant matrix. Here the authors consider the system \[ \dot{x} = (A + \epsilon Q(t,\epsilon))x , \qquad x\in{\mathbb{R}^{2}},\] where \(A\) is a constant matrix with distinct eigenvalues, and \(Q(t,\epsilon)\) is analytic and almost periodic with respect to \(t\) and analytic with respect to \(\epsilon\). Without any non-degeneracy condition, it is proved that the system is reducible for any choice of a sufficiently small parameter \(\epsilon\) by an almost periodic symplectic mapping.