On symplectic dynamics near a homoclinic orbit to $1$-elliptic fixed point

L. Lerman ; A. Markova

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 76 (2015) no. 2, pp. 309-342

Résumé

We study the orbit behavior of a four dimensional smooth symplectic diffeomorphism $f$ near a homoclinic orbit $\Gamma$ to an $1$-elliptic fixed point under some natural genericity assumptions. $1$-Elliptic fixed point has two real eigenvalues outside the unit circle and two on the unit circle. Thus there is a smooth $2$-dimensional center manifold $W^c$ where the restriction of the diffeomorphism has the elliptic fixed point supposed to be generic (no strong resonances and first Birkhoff coefficient is nonzero). Then the Moser theorem guarantees the existence of a positive measure set of KAM invariant curves. $W^c$ itself is a normally hyperbolic manifold in the whole phase space and due to Fenichel results every point on $W^c$ has $1$-dimensional stable and unstable smooth invariant curves smoothly foliating the related stable and unstable manifolds. In particular, each KAM invariant curve has stable and unstable smooth $2$-dimensional invariant manifolds being Lagrangian ones. Stable and unstable manifolds of $W^c$ are $2$-dimensional smooth manifolds which are assumed to be transverse along homoclinic orbit $\Gamma$. One of our theorems present conditions under which each KAM invariant curve on $W^c$ in a sufficiently small neighborhood of $\Gamma$ has four transverse homoclinic orbits. Another result ensures that under some Moser genericity assumption for the restriction of $f$ on $W^c$ saddle periodic orbits in resonance zone also have homoclinic orbits in the whole phase space though its transversality or tangency cannot be verified directly. All this implies the complicated dynamics of the diffeomorphism and can serve as a criterion of its nonintegrability. References: 42 entries.