On the reducibility of a class of quasi-periodic Hamiltonian systems with small perturbation parameter near the equilibrium
The authors consider nonlinear quasi-periodic Hamiltonian systems of the form \[ \dot x= (A+\varepsilon Q(t))x+\varepsilon g(t)+ h(x,t),\tag{\(*\)} \] where \(x\in\mathbb{R}^{2n}\), \(A\) is a constant matrix with multiple eigenvalues, and \(h(x,t)\), \(Q(t)\) and \(g(t)\) are all analytic and quasi-periodic in \(t\) in a certain region \(D_\rho\). In the authors' terminology a function \(f\) is called quasi-periodic with frequencies \(\omega= (\omega_1,\omega_2,\) \(\dots, \omega_r)\) if \(f(t)= F(\omega_1t, \omega_2t,\dots, \omega_rt)\), where \(F(\theta_1,\theta_2,\dots, \theta_r)\) is \(2\pi\)-periodic in all arguments and \(\theta_i= \omega_i t\) for \(i=1,2,\dots, r\). If \(F\) is analytic on \(D_\rho= \{\theta:|\text{Im}(\theta_i)|< \rho,\) \(i= 1,2,\dots, r\}\), then \(f\) is called analytic, quasi-periodic on \(D_\rho\). The main result is the following: under the conditions noted below there is a Cantor set \(E\) with positive Lebesgue measure such that for each \(\varepsilon\in E\) there is a quasi-periodic symplectic transformation \(\chi= \psi(t,\varepsilon) y+ \varphi(t,\varepsilon)\), where \(\psi\) and \(\varphi\) are quasi-periodic with frequencies \(\omega\), that transforms the original system \((*)\) into \[ \dot y= B(\varepsilon)y+ h_\infty(y,t, \varepsilon), \] where \(B\) is a real constant matrix and \(h_\infty(y,t,\varepsilon)= O(y^2)\) as \(y\to 0\). The necessary conditions include several technical limitations summarized briefly as: \(A\) is a diagonalizable \(2n\times 2n\) matrix, the system \((*)\) is non-resonant and nondegenerate, and \(D_{xx} h(x,t,\varepsilon)\) is bounded on a unit ball around the origin. The proof uses an extended KAM iteration.
scientific article; zbMATH DE number 6855269