MatCont: Matlab software for bifurcations of dynamical systems.
Annick Dhooge, Ghent University
Supervisor: Willy Govaerts
Co- supervisor: Yuri A. Kuznetsov
Phd. Abstract:
The study of differential equations requires good and powerful mathematical
software.Also, flexibility and extendibility of the package are important.
However, most of the existing software all have their own way of specifying the
system or are written in a relatively low-level programming language, so it is
hard to extend it.
In 2000, A. Riet started the implementation of a continuation toolbox in
Matlab. The aim of this toolbox was to provide an interactive environment for
the continuation and normal form analysis of dynamical systems. In 2002, the
toolbox was extended and improved by W. Mestrom. This toolbox was the base of
the toolbox we further developed and named Cl_MatCont.
We focussed on the software development, including GUI design, on the
development of new algorithms and on the optimization of known and new
numerical algorithms. Cl_MatCont and MatCont have many features and support
many functions that were not found in earlier packages. Several new algorithms
were developed and implemented, for example:
* for the detection of branch points of equilibria on curves of limit points
of equilibria. It is remarkable that no other software supports this detection,
though branch points generically appear on curves of limit points of
equilibria.
* for the detection of branch points of cycles on curves of limit points of
cycles. This is technically much more complicated and the remark of the
previous case applies here as well.
* for the continuation of branch points in three parameters. We implement this
for equilibria and limit cycles. Again, this is not provided in any other
software though it has many applications, in particular in systems with
symmetry or invariance.
* for the universal use of minimally extended systems. A detailed comparison
of minimally and fully extended systems is made to highlight the advantages of
the former systems.
* for the normal form coefficients for bifurcations of limit cycles. Again,
this is not provided in any other software. It is a very useful feature for the
applications. For example, a negative normal form coefficient for a torus
bifurcation in a chemical model means that the stability of the periodic orbit
is taken over by the invariant torus so that for the experimental observer it
does not disappear suddenly. On the other hand, a positive normal coefficient
means that for the observer the stability is lost in a sudden and nearly
unexpected way.
The results from this Phd. were already published or accepted for publication
in several specialized journals or Proceedings.
The software packages MatCont and Cl_MatCont were made public at
http://www.matcont.UGent.be from December 2003. 258 and 377 users downloaded
Cl_MatCont and MatCont, respectively. In July 2004, we released a new version.
Today, there are already 431 and 718 downloads of Cl_MatCont and MatCont,
respectively. MatCont has already been used for educational purposes.