Overview
These tutorials are for our toolbox MatContM. The team consists of
- Hil G.E. Meijer, University of Twente, Department of Applied Mathematics
- Willy Govaerts, University of Ghent, Department of Applied Mathematics, Computer Science and Statistics
- Yuri A. Kuznetsov, Utrecht University, Mathematical Institute and University of Twente, Department
of Applied Mathematics
- Niels Neirynck, University of Ghent, Department of Applied Mathematics, Computer Science and Statistics
Our software is a Matlab-based toolbox to study bifurcations defined by iterated maps with parameters.
While MatContM is similar to MatCont, with the additional M indicating maps, it is a standalone toolbox,
with some functionality specific to maps. The toolbox is included (matcontm5p4.zip with docMatContMNov2017.pdf) and is also freely available from sourceforge; https://sourceforge.net/projects/matcont/files/matcontm/matcontm5p4/
and has been tested with Matlab versions 2016a up to 2017b.
It supports numerical continuation of local codim 1 bifurcations of fixed points and higher-period cycles, i.e.,
limit point, period-doubling and Neimark-Sacker. Also connecting orbits can be continued in one parameter,
as well as homoclinic and heteroclinic tangencies in two parameters. In addition, the toolbox computes
automatically critical normal form coefficients that can be used for non-degeneracy analysis and branchs witching.
In addition, (un)stable manifolds of saddles can be computed as well as Lyapunov exponents for
an orbit.
The intended audience for these tutorials consists of graduate or PhD students, and researchers. For students this material
could be part of computer sessions coming with a course on dynamical systems. After going through the
tutorials, researchers may explore bifurcations of their own model.
The toolbox can be used through the command line, e.g., with scripts, or a graphical user interface. On
Sourceforge there is a Documentation Folder, containing a Manual, as well as four Tutorials guiding a user
through all features via the GUI. These tutorials deal with the following:
- TutorialMIApr2018.pdf - Setting up a new model for simulations, and continuation of fixed points in one parameter.
- TutorialMIIApr2018.pdf - Continuation of codim 1 bifurcations of fixed points and cycles in 2 parameters;
- TutorialMIIIApr2018.pdf - Computing (un)stable manifolds and Continuation of Connecting orbits;
- TutorialMIVMar2019.pdf - Computing Lyapunov Exponents to characterize chaos and bifurcations of invariant curves or tori.
Acknowledgments
An early version of the toolbox was developed by Reza Khoshiar Ghaziani (Shahrekord, Iran/Ghent Belgium), Willy Govaerts (Ghent, Belgium), Yuri Kuznetsov (Utrecht, Netherlands) and Hil Meijer (Utrecht/Twente, Netherlands). A major contribution to the current version was made by Niels Neirynck (Ghent, Belgium) to both the algorithms and the user interface.
References
[1] A book by Kuznetsov and Meijer on "Numerical bifurcation theory of maps" including theory, numerics and examples, has been published by Cambridge University Press in March 2019, see http://www.cambridge.org/9781108499675.
[2] The design of the software and features are also specified in the PhD thesis of Niels Neirynck entitled “Advances in Numerical Bifurcation Software: MatCont”, which he successfully defended in January 2019, see https://biblio.ugent.be/publication/8615817.
| Keywords | Matlab |
| Model | |
| Software Type | |
| Language | |
| Platform | |
| Contact Person | |
| References to Papers | [1] A book by Kuznetsov and Meijer on "Numerical bifurcation theory of maps" including theory, numerics and examples, has been published by Cambridge University Press in March 2019, see http://www.cambridge.org/9781108499675.
[2] The design of the software and features are also specified in the PhD thesis of Niels Neirynck entitled “Advances in Numerical Bifurcation Software: MatCont”, which he successfully defended in January 2019, see https://biblio.ugent.be/publication/8615817. |