# Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

## TL;DR

This paper investigates the relationship between heteroclinic orbits and nonintegrability in two-degree-of-freedom Hamiltonian systems with saddle-centers, establishing conditions for transverse intersections of invariant manifolds and illustrating with a quartic potential example.

## Contribution

It provides new criteria linking heteroclinic orbit transversality to nonintegrability, extending understanding of Hamiltonian dynamics near saddle-centers.

## Key findings

- Transverse heteroclinic intersections imply nonintegrability under certain conditions.
- Quadratic tangencies or no intersections occur if eigenvalue conditions are not met.
- Numerical results support the theoretical criteria for specific Hamiltonian systems.

## Abstract

We consider a class of two-degree-of-freedom Hamiltonian systems with saddle-centers connected by heteroclinic orbits and discuss some relationships between the existence of transverse heteroclinic orbits and nonintegrability. By the Lyapunov center theorem there is a family of periodic orbits near each of the saddle-centers, and the Hessian matrices of the Hamiltonian at the two saddle-centers are assumed to have the same number of positive eigenvalues. We show that if the associated Jacobian matrices have the same pair of purely imaginary eigenvalues, then the stable and unstable manifolds of the periodic orbits intersect transversely on the same Hamiltonian energy surface when sufficient conditions obtained in previous work for real-meromorphic nonintegrability of the Hamiltonian systems hold; if not, then these manifolds intersect transversely on the same energy surface, have quadratic tangencies or do not intersect whether the sufficient conditions hold or not. Our theory is illustrated for a system with quartic single-well potential and some numerical results are given to support the theoretical results.