A Computational Method of a Tangent Bifurcation Set by Using the Bisection Method

Yuji KATSUTA  George MIYAKE  

Publication
A - Abstracts of IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences (Japanese Edition)   Vol.J101-A   No.7   pp.170-177
Publication Date: 2018/07/01
Online ISSN: 1881-0195
DOI: 
Type of Manuscript: PAPER
Category: 
Keyword: 
nonlinear,  tangent bifurcation,  bifurcation set,  bisection method,  

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Summary: 
Many dynamic systems are described by nonlinear ordinary differential equations. The order of differential equations becomes higher, a numerical analysis for the equations involves a great deal of difficulty. Variations in parameters in the differential equation achieve a change in the property of solutions to the differential equations, which is called a bifurcation phenomenon. Values of the parameters at a bifurcation phenomenon are called bifurcation parameters. The Newton method in calculations of a bifurcation set requires differential coefficients of second order, which makes its dimension of differential coefficients extremely large in size, consequently it is difficult to deal with the coefficients. A computational algorithm using the bisection method for a bifurcation set is applied to a tangent bifurcation of an equilibrium point. The algorithm provides a way for the calculations of a bifurcation set without the differential coefficients of second order, and the difficulty as described above can be averted.