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ID 114910
Author
Tsumoto, Kunichika Osaka University
Keywords
nonlinear dynamical system
bifurcation
boundary value problem
variational equation
Newton's method
Content Type
Journal Article
Description
In this paper, we explain how to compute bifurcation parameter values of periodic solutions for non-autonomous nonlinear differential equations. Although various approaches and tools are available for solving this problem nowadays, we have devised a very simple method composed only of basic computational algorithms appearing in textbooks for beginner's, i.e., Newton's method and the Runge-Kutta method. We formulate the bifurcation problem as a boundary value problem and use Newton's method as a solver consistently. All derivatives required in each iteration are obtained by solving variational equations about the state and the parameter. Thanks to the quadratic convergence ability of Newton's method, accurate results can be quickly and effectively obtained without using any sophisticated mathematical library or software. If a discontinuous periodic force is applied to the system, we can use the same strategy to solve the bifurcation problem. The key point of this method is deriving a differentiable composite map from the various information about the problem such as the location of sections, the periodicity, the Poincaré mapping, etc.
Journal Title
Nonlinear Theory and Its Applications, IEICE
ISSN
21854106
Publisher
The Institute of Electronics, Information and Communication Engineers
Volume
3
Issue
4
Start Page
458
End Page
476
Published Date
2012-10-01
Rights
© IEICE 2012
EDB ID
DOI (Published Version)
URL ( Publisher's Version )
FullText File
language
eng
TextVersion
Publisher
departments
Center for Administration of Information Technology
Medical Sciences
Science and Technology