無限次元力学系における大域分岐検証(<小特集>力学系の位相計算理論)

概要

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A rigorous numerical method to verify bifurcation branches in infinite dimensional dynamical systems is studied in this paper. Key tools of this method are the Conley index and interval arithmetic. At first, we briefly explain the Conley index theory and consider a numerical technique to rigorously verify the existence of equilibrium point in R^m. The essential points are to construct a box in R^m which is expected to contain an equilibrium point and to check the vector field on the boundary. If the Conley index determined by the vector field is the same as that for a hyperbolic equilibrium point, then the Conley index theory assures the existence of equilibrium point in the box. Next, this method is extended to infinite dimensional problems to prove the existence of stationary solutions for evolution equations. The difficulty caused by infinite dimensional problems is that it is impossible to directly check all the vector fields by using computers. This problem is overcome in such a way that we impose the power decay property on higer modes. Then the infinite dimensional problem can be essentially reduced to finite dimensional one.
日本応用数理学会の論文
2005-06-24