Persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems (Q1359581)

The following nearly integrable Hamiltonian differential system \[ \dot x=H_y=\omega+P_y,\quad\dot y=-H_x=-P_x, \] \[ \dot u=H_v=\Omega v+P_v,\quad\dot v=-H_u=-\Omega u-P_u \] is considered. Here the Hamiltonian \[ H=N+P,\quad(x,y,u,v)\in T^n\times\mathbb{R}^n\times\mathbb{R}^m\times\mathbb{R}^m\quad(1\leq m,n<+\infty), \] \(T^n\) the usual \(n\)-torus, \(N=\sum_{1\leq j\leq n} \omega_jy_j+(1/2)\sum_{j=1}^m\Omega_j(\omega)(u_j^2+v_j^2)\) is a normal form, and \(P=P(x,y,u,v;\epsilon)\) is a small perturbation term, where \(\epsilon\) is a small perturbation parameter. The frequency vectors \(\omega=(\omega_1,\omega_2,\dots,\omega_n)\in O\subset\mathbb{R}^n\) are also regarded as parameters, where \(O\) is a bounded closed subset of \(\mathbb{R}^n\) with nonempty interior. The corresponding nonperturbed system admits a family of invariant tori \(\{T_\omega\mid\omega\in O\}\). Some of the invariant tori can be destroyed by an arbitrary small perturbation. The problem whether some invariant tory can persist a under small perturbation is considered. The persistence of elliptic invariant tori for a class of small perturbations of degenerate integrable Hamiltonian systems is proved.