Numerical Continuation of Bifurcations of Limit Cycles in MATLAB

${\rm CL\_MATCONT}$ 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 ${\rm CL\_MATCONT}$ 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 ${\rm CL\_MATCONT}$ 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.