Pattern Formation in Periodic Gradient Systems

We consider the problem of stable stationary configurations in general gradient systems with periodic boundary conditions in two or three spatial dimensions. Such a problem arises in point vortices, classical mechanics of particles, and optimization problems in order to find the minimal state of som...

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Bibliographic Details
Published inTheoretical and Applied Mechanics Japan Vol. 57; pp. 227 - 232
Main Author UMEKI, Makoto
Format Journal Article
LanguageEnglish
Published Tokyo National Committee for IUTAM 2009
Japan Science and Technology Agency
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Summary:We consider the problem of stable stationary configurations in general gradient systems with periodic boundary conditions in two or three spatial dimensions. Such a problem arises in point vortices, classical mechanics of particles, and optimization problems in order to find the minimal state of some functions such as an energy or variables due to distances associated with every pair of two points. First, we show how triangular patterns appear in two-dimensional gradient systems. Then, the system extended into three dimensions is proposed and investigated numerically.Finally, it is shown that triangular patterns appear in the gradient system for the point vortices with the same strength.
ISSN:1348-0693
1349-4244
DOI:10.11345/nctam.57.227