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. Poincare´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.
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.
Abstract. We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ODEs, namely the fold, the perioddoubling (or flip) bifurcation, and the torus (or Neimark-Sacker) bifurcation. In the fold and flip cases we append one scalar equation to the standard periodic BVP that defines the periodic solution; in the torus case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of linear BVPs, whose sparsity structure (after discretization) is identical to that of the linearization of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software auto and colsys.
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