Annick Dhooge

MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs

MATCONT is a graphical MATLAB software package for the interactive numerical study of dynamical systems. It allows one to compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, and fold bifurcation points of limit cycles. All curves are computed by the same function that implements a prediction-correction continuation algorithm based on the Moore-Penrose matrix pseudo-inverse. The continuation of bifurcation points of equilibria and limit cycles is based on bordering methods and minimally extended systems. Hence no additional unknowns such as singular vectors and eigenvectors are used and no artificial sparsity in the systems is created. The sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods. The MATLAB environment makes the standard MATLAB Ordinary Differential Equations (ODE) Suite interactively available and provides computational and visualization tools; it also eliminates the compilation stage and so makes installation straightforward. Compared to other packages such as AUTO and CONTENT, adding a new type of curves is easy in the MATLAB environment. We illustrate this by a detailed description of the limit point curve type.

New features of the software MatCont for bifurcation analysis of dynamical systems

Bifurcation software is an essential tool in the study of dynamical systems. From the beginning (the first packages were written in the 1970s) 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. Poincaré 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.

Cl_matcont: a continuation toolbox in Matlab

{\rm CL\_MATCONT}

Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles

and MATCONT are MATLAB continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems, in particular ODEs. MATCONT is an interactive graphical package and

Numerical continuation of fold bifurcations of limit cycles in MATCONT

{\rm CL\_MATCONT}

MATCONT: a Matlab package for numerical bifurcation analysis of ODEs

is a commandline version. Both packages allow us to compute curves of equilibria, limit points, Hopf points, limit cycles, flip, fold, and torus bifurcation points of limit cycles. We discuss computational details of the continuation of limit cycles and flip, fold, and torus bifurcations of limit cycles in MATCONT and

BIFURCATION, BURSTING AND SPIKE GENERATION IN A NEURAL MODEL

{\rm CL\_MATCONT}

Numerical Continuation of Branch Points of Limit Cycles in MATCONT

using orthogonal collocation. Instead of the more commonly used fully extended systems we use minimally extended systems. We further describe the use of the MATLAB sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems. Finally, we compare the use of the minimally and the fully extended systems in the MATLAB environment.

Numerical continuation of branch points of equilibria and periodic orbits

CL_MATCONT is a Matlab continuation package for the numerical study of a range of parameterized nonlinear problems. In the case of ODEs it allows to compute curves of equilibria, limit point, Hopf points, limit cycles and period doubling bifurcation points of limit cycles. All curves are computed by the same function that implements a prediction-correction continuation algorithm based on the Moore - Penrose matrix pseudo-inverse. The continuation of bifurcation points of equilibria and limit cycles is based on bordering methods and minimally extended systems. Hence no additional unknowns such as singular vectors and eigenvectors are used and no artificial sparsity in the systems is created.The inherent sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods.CL_MATCONT furthermore allows to compute solution branches to underdetermined systems of nonlinear equations and parameterized boundary value problems.