Chaotic motion of the \(N\)-vortex problem on a sphere. I. Saddle-centers in two-degree-of-freedom Hamiltonians
The motion of \(N (N\in N)\) point vortices on a sphere in the presence of fixed pole vortices is shown in this work. Special attention is paid to the evolution of their configuration, in which they are equally spaced along a line of latitude of the sphere. When \(N = 5n\) and \(6n\) with \(n\in N\), the system is reduced to a two degree of freedom Hamiltonian with saddle-center equilibria, one of which corresponds to the unstable \(N -ring\). To compute stable and unstable manifolds , it is shown numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. The basic results are presented in: 2 tables, 18 figures, 2 appendixes, 5 propositions, 2 lemmas and 2 theorems.
scientific article; zbMATH DE number 5552199
scientific article