NDLI: Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Kuznetsov, Yu. A. Meijer, H. G. E. Govaerts, W. Ghaziani, R. Khoshsiar
Copyright Year	2007
Abstract	We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a MATLAB toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e., limit point, period-doubling, and NeimarkSacker) and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations. We provide two examples illustrating the developed techniques: a generalized Hnon map and a juvenile/adult competition model from mathematical biology.
Starting Page	2644
Ending Page	2667
Page Count	24
File Format	PDF
ISSN	10648275
DOI	10.1137/060653858
e-ISSN	10957197
Issue Number	6
Volume Number	29
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2007-11-02
Access Restriction	Subscribed
Subject Keyword	normal forms Bifurcation problems Transformation and reduction of equations and systems, normal forms Normal forms cycles branch switching bifurcations of fixed points
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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