1 Introduction
The study of differential equations requires good and powerful mathematical software. Also, a
flexible and extendib l e package is important. However, most of the existing software all have
their own manner of specifying the system or are written in a relatively low-level programming
language, so i t is hard to extend it.
A powerful and widely used environment for scientific computing is matlab[ 25 ] . The aim
of MatCont and CL
MatCont is to provide a continuation toolbox which is compatible
with the standard matlab ODE repres e ntation of differential equations. The user can easily
use h i s/he r models wi t hout re wri t i ng them to a specific package. The matlab programming
language makes the use and extensions of the toolbox very easy.
This document i s struc tur e d as follows. In section 2 the underlying mathematics of con-
tinuation will be treated. Section 3 intro du ce s how singularities will be represented. The
toolbox specification is explained in section 4 with a simple example in section 5. A more
complex applicat i on of the toolbox, continuation of equilibria, is des cr i bed in section 7. The
continuation of a solution to a boundary value problem in a free parameter with the 1-D
Brusselator as e x ampl e is described in Section 8. Section 9 describes the continuation of limit
cycles and the computation of the phase response curve. Section 10 descri bes the continua-
tion of codim 1 bifurc at i ons, at present limit points of equilibria, Hopf bifurcation points of
equilibria, period doubling bifurcation points of limit cycles, fold bifurcation points of limit
cycles and Neimark-Sacker bifurcation points of limit cycles. Section 11 describes the con-
tinuation of codim 2 bifurcations, at pr es ent branch points of eq u i l i br i a and branch points of
limit cycles. Section 12 deals with the continuati on of homoclini c orbits.
1.1 Features
Upon the development of MatCont and CL
MatCont, there were multiple objectives:
• Cover as many bifurcations in ODEs as possi bl e (e.g. now all bifurcations wi t h two
control parameters are covered).
• Allow easier data exchange b e tween programs and with matlab’s standard ODE solvers.
• Implement robust and efficient numerical methods for all computations, using minimally
extended systems where possible.
• Represent results in a form suitable for standard control, identification, and visualiza-
tion.
• Allow for easy ext end i bi l i ty.
A general comparison of the available features during computations for ODEs currently
supported by the most widely used software packages auto97/2000 [9], content 1.5 [24]
and MatCont/CL
MatCont are indicated in Table 1.
Relationships between objects of codimension 0, 1 and 2 computed by MatCont and
CL
MatCont are presented in Figures 1 and 2, while the symbols and their meaning are
summarized in Tables 2 and 3, where the standard terminology is used, s ee [23].
An arrow in Figure 1 from O to EP or LC means that by starting time integration from
a given point we can converge to a stable equilibrium or a stable limit cycl e , respectively. In
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