Generalization of geometric Poincare theorem for small perturbations

    1998, Volume 3, Number 2, pp.  46-66

    Author(s): Kudryavtseva E. A.

    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 $B$.
    Citation: Kudryavtseva E. A., Generalization of geometric Poincare theorem for small perturbations, Regular and Chaotic Dynamics, 1998, Volume 3, Number 2, pp. 46-66

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