zbMATH — the first resource for mathematics

New features of the software MatCont for bifurcation analysis of dynamical systems. (English) Zbl 1158.34302
Summary: Bifurcation software is an essential tool in the study of dynamical systems. From the beginning (the first packages were written in the 1970’s) it was also used in the modelling process, in particular to determine the values of critical parameters. More recently, it is used in a systematic way in the design of dynamical models and to determine which parameters are relevant. MatCont and Cl_MatCont are freely available matlab numerical continuation packages for the interactive study of dynamical systems and bifurcations. MatCont is the GUI-version, Cl_MatCont is the command-line version. The work started in 2000 and the first publications appeared in 2003. Since that time many new functionalities were added. Some of these are fairly simple but were never before implemented in continuation codes, e.g. PoincarĂ© maps. Others were only available as toolboxes that can be used by experts, e.g. continuation of homoclinic orbits. Several others were never implemented at all, such as periodic normal forms for codimension 1 bifurcations of limit cycles, normal forms for codimension 2 bifurcations of equilibria, detection of codimension 2 bifurcations of limit cycles, automatic computation of phase response curves and their derivatives, continuation of branch points of equilibria and limit cycles. New numerical algorithms for these computations have been published or will appear elsewhere; here we restrict to their software implementation. We further discuss software issues that are in practice important for many users, e.g. how to define a new system starting from an existing one, how to import and export data, system descriptions, and computed results.

34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37M20 Computational methods for bifurcation problems in dynamical systems
65P30 Numerical bifurcation problems
PDF BibTeX Cite
Full Text: DOI
[1] Beuter A., Nonlinear Dynamics in Physiology and Medicine (2003) · Zbl 1050.92013
[2] Hoppensteadt F. C., Weakly Connected Neural Networks (1997)
[3] Diekmann O., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000) · Zbl 0997.92505
[4] Nowak M. A., Virus Dynamics: Mathematical Principles of Immunology and Virology (2000) · Zbl 1101.92028
[5] Kuznetsov Yu.A., Elements of Applied Bifurcation Theory, 3. ed. (2004)
[6] DOI: 10.1007/BF00275501 · Zbl 0628.92016
[7] DOI: 10.1098/rstb.1993.0121
[8] Bertram R., Bull. Math. Biol. 57 pp 413– (1995)
[9] DOI: 10.1186/1748-7188-1-11
[10] Doedel E. J., auto97-auto2000, Continuation and bifurcation software for ordinary differential equations (with HomCont). User’s guide (1997)
[11] Kuznetsov Yu.A., content: integrated environment for analysis of dynamical systems (1997)
[12] DOI: 10.1145/779359.779362 · Zbl 1070.65574
[13] Dhooge, A., Govaerts, W., Kuznetsov, Yu.A., Mestrom, W. and Riet, A. M. Cl_matcont: a continuation toolbox in Matlab. Proceedings of the 2003 ACM Symposium on Applied Computing. Melbourne, Florida, USA. pp.161–166.
[14] DOI: 10.1137/S0036142902400779 · Zbl 1066.65154
[15] DOI: 10.1137/030600746 · Zbl 1087.65118
[16] DOI: 10.1016/0022-0396(88)90063-0 · Zbl 0656.58028
[17] DOI: 10.1137/040611306 · Zbl 1103.34025
[18] DOI: 10.1142/S0218127405012491 · Zbl 1081.37054
[19] Friedman M., LNCS 3514 pp 263– (2005)
[20] Arnol’d V. I., Geometrical Methods in the Theory of Ordinary Differential Equations (1983)
[21] Broer H., Ergodic Theory Dyn. Sys. 4 pp 509– (1984)
[22] DOI: 10.1007/BF02234758 · Zbl 0185.41302
[23] DOI: 10.1016/0771-050X(81)90010-3 · Zbl 0449.65048
[24] DOI: 10.1063/1.459996
[25] Govaerts W., LNCS 3992 pp 391– (2006)
[26] DOI: 10.1162/neco.2006.18.4.817 · Zbl 1087.92001
[27] DOI: 10.1137/1.9780898718195 · Zbl 1003.68738
[28] DOI: 10.1016/S0006-3495(81)84782-0
[29] DOI: 10.1137/S0036142998335005 · Zbl 0931.34024
[30] Borisyuk R. M., Stationary Solutions of a System of Ordinary Differential Equations Depending Upon a Parameter (1981) · Zbl 0514.34001
[31] DOI: 10.1016/S0167-2789(03)00097-6 · Zbl 1024.37037
[32] DOI: 10.1142/S0218127494000587 · Zbl 0873.34030
[33] DOI: 10.1142/S0218127496000485 · Zbl 0877.65058
[34] DOI: 10.1093/imanum/14.3.381 · Zbl 0804.65086
[35] Beyn, W. J., Champneys, A. R., Doedel, E., Govaerts, W., Kuznetsov, Yu.A. and Sandstede, B. 2002.Numerical continuation and computation of normal forms inHandbook of Dynamical Systems: Vol 2, Edited by: Fiedler, B. 149–219. Amsterdam: Elsevier. · Zbl 1034.37048
[36] DOI: 10.1137/S1064827598344868 · Zbl 0967.34008
[37] DOI: 10.1016/0167-2789(95)90061-6 · Zbl 0889.34034
[38] DOI: 10.1016/S0006-3495(77)85598-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.