functionality

The project Continuation Core and Toolboxes (COCO) is a development platform that provides a large amount of standard functionality required for investigating bifurcation problems and implementing toolboxes for new types of problems. The core was recently finished and the book Recipes for Continuation describes the underlying mathematics, the interface and toolbox development in great detail.

A new release of COCO is planned for November, 2015. This will include toolboxes for the following problems:

1. Continuation of equilibrium points (ep). The ep toolbox provides algorithms for continuation of equilibria of autonomous vector fields. The ep toolbox is fully embeddable. Usage is demonstrated in multiple demos and fully documented in a tutorial manual. The toolbox supports

   • detection of codimension-1 bifurcation points (saddle-node bifurcations, Hopf bifurcations, and neutral saddle points)
   • detection of codimension-2 bifurcation points (Bogdanov-Takens bifurcation along branch of Hopf bifurcations)
   • branch switching at branch points
   • continuation of saddle-node and Hopf bifurcation curves

2. Continuation of trajectory segments (coll). The coll toolbox provides algorithms for continuation of constrained collections of trajectory segments for autonomous and non-autonomous vector fields, including multisegment boundary value problems. The toolbox relies on segment-specific adaptive discretizations using piecewise-polynomial interpolants and orthogonal collocation. The coll toolbox is fully embeddable. Usage is demonstrated in multiple demos and fully documented in a tutorial manual. The toolbox supports

   • monitoring of solutions to the corresponding variational problems
   • continuation of solutions to the corresponding variational problems
   • monitoring of the corresponding trajectory segment discretization errors
   • branch switching at branch points

3. Continuation of periodic orbits (po). The po toolbox provides algorithms for continuation of single-segment periodic orbits in smooth autonomous or non-autonomous dynamical systems and of multi-segment periodic orbits in hybrid autonomous dynamical systems. The po toolbox is fully embeddable. Usage is demonstrated in multiple demos and fully documented in a tutorial manual. The toolbox supports

   • detection of codimension-1 bifurcation points (saddle-node bifurcations, period-doubling bifurcations, torus bifurcations, and neutral saddle points)
   • branch switching at branch points
   • branch switching at Hopf bifurcations of equilibria
   • branch switching at period-doubling bifurcations
   • continuation of saddle-node, period-doubling, and torus bifurcation curves

Available toolbox previews:

1. Continuation in experiments (CONTINEX). This toolbox implements a continuation method for noise contaminated zero problems. Examples are control based continuation in experiments and continuation for equation-free problems. The development of this toolbox was supported by the Danish research council (FTP) under the project number 0602-00753B.