# Chaotic motion of the N-vortex problem on a sphere I: Saddle-centers in two-degree-of-freedom

Preprint Series # 858
SAKAJO, Takashi and YAGASAKI, Kazuyuki Chaotic motion of the N-vortex problem on a sphere I: Saddle-centers in two-degree-of-freedom. (2007);

 PDF - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader3175Kb

## Abstract

We study the motion of $N$ point vortices with $N\in\Nset$ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with $N$ degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the \textit{$N$-ring}, in which they are equally spaced along a line of latitude of the sphere when it is unstable. When the number of the point vortices is $N=5n$ or $6n$ with $n \in \Nset$, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable $N$-ring. Utilizing a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we prove numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable $N$-ring is shown to be chaotic.

Item Type: Preprint Point vortex; Flows on sphere; Hamiltonian sytems; Chaos; Melnikov Method 76-xx FLUID MECHANICS70-xx MECHANICS OF PARTICLES AND SYSTEMS37-xx DYNAMICAL SYSTEMS AND ERGODIC THEORY 1711