#45 Toolbox capabilities

[Mark]

Hi,
The po toolbox says it can detect saddle-node bifurcations. Is this saddle node bifurcations of equilibria, or saddle nodes of periodic orbits?
And, what types of dynamical system are the po and ep toolboxes able to analyse? I've seen ODEs, discretised PDEs, and hybrid systems in the examples. Are the provided toolboxes able to handle any other system types (differential algebraic equations, stochastic differential equations, etc.)?

Many thanks!

[Harry Dankowicz]

Hello. These are excellent questions.

A couple of things are worth noting:

1. The COCO platform is not restricted to any particular class of problems, as long as they (or a discretized version) can be formulated in the general structure expected by COCO.
2. The ep, coll, and po toolboxes collect particular classes of problems to provide ready-to-deploy constructors and monitor functions.
3. The book Recipes for Continuation describes how to build toolboxes for other problem classes.

To your specific question, ep provides constructors and monitor functions and other utilities for analyzing roots of nonlinear equations, especially when those roots are associated with equilibria of a continuous-time dynamical system of the form x'=F(x,p) for a smooth function F. This is described in the introduction to the ep tutorial (EP-Tutorial.pdf in the coco/help folder).

The po toolbox is designed to study periodic orbits of dynamical systems of the form x'=F(t,x,p) for a smooth function F, or periodic multisegment orbits defined by a hybrid dynamical system in terms of multiple smooth vector field. This is described in the introduction to the po tutorial (PO-Tutorial.pdf in the coco/help folder).

As explained in the po tutorial, when the po toolbox says that it can detect saddle-node bifurcations, it means cyclic fold bifurcations, i.e., saddle-node bifurcations of periodic orbits. If you want to detect saddle-node bifurcations of equilibria, use the ep toolbox constructors.

My research group has developed constructors for other problem types, and with other choices of discretization than that implemented in coll and po. This includes differential algebraic equations. COCO provides great flexibility, but it can only solve problems that we (mathematicians) have developed a solution technique for.

I hope that helps.

[Mark]

This is all very helpful, thanks!