2023-03-21T08:23:08Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/696672022-11-17T02:08:08Zhdl_2115_45007hdl_2115_116Chaotic motion of the N-vortex problem on a sphere I: Saddle-centers in two-degree-of-freedomSAKAJO, TakashiYAGASAKI, KazuyukiHamiltonian systempoint vortexflow on spherechaosMelnikov method410We study the motion of N point vortices with Nāā 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 N-ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5n or 6n with nāā, 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. Using 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 show 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.Department of Mathematics, Hokkaido UniversityDepartmental Bulletin Paperapplication/pdfhttp://hdl.handle.net/2115/69667info:doi/10.14943/84008https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/69667/1/pre858.pdfHokkaido University Preprint Series in Mathematics8581342007-06-06engpublisher