Chaotic motion of the -vortex problem on a sphere: II. Saddle centers in three-degree-of-freedom Hamiltonians

This paper deals with complicated behavior in the N = 8 n vortex problem on a sphere, which is reduced to three-degree-of-freedom Hamiltonian systems. In the reduced Hamiltonians, the polygonal ring configuration of the point vortices becomes a saddle-center equilibrium which has two hyperbolic and four center directions in some parameter regions. Near the saddle center, there exists a normally hyperbolic, locally invariant manifold including a Cantor set of whiskered tori. For N = 8 we numerically compute the stable and unstable manifolds of the locally invariant manifold with the assistance of the center manifold technique, and show that they intersect transversely and complicated dynamics may occur. Direct numerical simulations are also given to demonstrate our numerical analysis.