Locating Fold Bifurcation Points Using Subspace Shooting

Hidetaka ITO  Akira KUMAMOTO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E81-A   No.9   pp.1791-1797
Publication Date: 1998/09/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)
Category: Chaos, Bifurcation and Fractal
Keyword: 
fold bifurcation,  periodic orbits,  shooting method,  subspace iteration,  center manifold,  

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Summary: 
A numerical method is proposed for efficiently locating fold bifurcation points of periodic orbits of high-dimensional differential-equation systems. This method is an extension of the subspace shooting method (or the Newton-Picard shooting method) that locates periodic orbits by combining the conventional shooting method and the brute-force method. Fold bifurcation points are located by combining a variant of the subspace shooting method with a fixed parameter value and the secant method for searching the parameter value of the bifurcation point. The target in the subspace-shooting part is an (not necessarily periodic) orbit represented by a Poincare mapping point which is close to the center manifold and satisfies the eigenvalue condition for the bifurcation. The secant-search part finds the parameter value where this orbit becomes periodic. Avoiding the need for differentiating the Poincare map with respect to the bifurcation parameter and exploiting several properties of the center manifold, the proposed method is both robust and easy to implement.