### Abstract

We give a detailed analytical description of the global dynamics of N points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group Dl of order 2l. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as McGehee transformation. After performing this change of coordinates that regularizes the total collision, we study the rest-points of the ow, the invariant manifolds and, with the help of a computer algebra system, we derive interesting information about the global dynamics for l = 2. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.

Original language | English (US) |
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Pages (from-to) | 3011-3042 |

Number of pages | 32 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 33 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2013 |

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### Keywords

- Dihedral N-vortex filaments
- Global dynamics
- Logarithmic potential
- McGehee coordinates
- N-body problems

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*33*(7), 3011-3042. https://doi.org/10.3934/dcds.2013.33.3011