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

TL;DR

This paper investigates the complex orbit behavior near a homoclinic orbit to a 1-elliptic fixed point in a 4D symplectic system, revealing conditions for transverse homoclinic orbits and resonance zone dynamics.

Contribution

It provides new conditions for the existence of transverse homoclinic orbits near a 1-elliptic fixed point in symplectic dynamics, extending understanding of local and global orbit structures.

Findings

Existence of four transverse homoclinic orbits for each KAM invariant curve near the homoclinic orbit.

Conditions under which saddle periodic orbits in resonance zones have homoclinic orbits.

Demonstration of homoclinic orbits' prevalence under genericity assumptions.

Abstract

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 out of unit circle and two others 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). Moser's 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 forming two smooth foliations. In particular, each KAM invariant curve has stable and unstable smooth 2-dimensional invariant manifolds being Lagrangian. The related stable and unstable manifolds of $W^c$ are 3-dimensional smooth manifolds which are supposed to be transverse along homoclinic orbit $\Gamma$. One of our theorems presents 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 Birkhoff genericity assumption for the restriction of $f$ on $W^c$ saddle periodic orbits in resonance zones also have homoclinic orbits though its transversality or tangency cannot be verified directly.