Elena Kudryavtseva

We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.

Relatively recently in works [3], [4] the topological classification of smooth Hamiltonian systems with one degree of freedom was obtained. When we study the stability of obtained topological invariants, the following natural question arised: is the space of all Morse functions with fixed number of minima and maxima on a closed surface connected? The present paper discusses this question and gives an algorithm of reduction of any Morse function on a closed orientable surface to the so-called canonical form.

We consider dynamical system in phase space, which has closed submanifold filled by periodic orbits. The following problem is analysed. Let us consider small perturbations of the system. What we can say about the number of survived periodic orbits, about their number and about their location in the neighborhood of a given submanifold? We obtain the solution of this problem for the perturbations of general type in terms of averaged perturbnation. The main result of the paper is as follows. Theorem: Let us consider the Hamiltonian system with Hamiltonian function $H$ on symplectic manifold $(M^{2n},\omega^2)$. Let $\Lambda \subset H^{-1}(h)$ be the closed nondegenerate submanifold filled by periodic orbits of this system. Then for the arbitrary perturbed function $\tilde{H}$, which is $C^2$-close to the initial function $H$, the system with the Hamiltonian $\tilde{H}$ has no less than two periodic orbits on the isoenergy surface $\tilde{H}^{-1}(h)$. Moreover, if either the fibration of $\Lambda$ by closed orbits is trivial, or the base $B=\Lambda /S^1$ of this fibration is locally flat, then the number of such orbits is not less than the minimal number of the critical points of smooth function on the quotient manifold $B$.