The persistence of elliptic lower dimensional tori with prescribed frequency for Hamiltonian systems

Abstract

In this paper we consider the persistence of lower dimensional tori of a class of analytic perturbed hamiltonian system, $$H=\langle \omega(\xi), I \rangle +\frac12 \Omega_0(u^2+v^2)+P(\theta,I,z,\bar{z};\xi)$$ and prove that if frequencies $(\omega_0,\Omega_0)$ satisfy some non-resonant conditions and the Brouwer degree of the frequency mapping $\omega(\xi)$ at $\omega_0$ is nonzero, then there exists an invariant lower dimensional invariant torus, whose frequencies are the small dilation of $\omega_0$.

Keywords

• hamiltonian system
• kam iteration
• invariant tori
• non-degeneracy condition