# MATCONT

**MATCONT** and **CL_MATCONT** are MATLAB numerical continuation packages for the interactive bifurcation analysis of dynamical systems. CL_MATCONT forms the computational core of MATCONT, but can also be used independently as a general-purpose non-interactive continuation toolbox in MATLAB. The software development started in 2000 and the first publications appeared in 2003.

## Overview

MATCONT provides means for continuing equilibria and limit cycles (periodic orbits) of systems of Ordinary Differential Equations (ODEs), and their bifurcations (including branch points). MATCONT also provides access to all standard ODE solvers supplied by MATLAB, as well as to two new stiff solvers, *ode78* and *ode87*.

MATCONT computes Poincare maps, as well as phase response curves for limit cycles and their derivatives as a byproduct of the continuation of limit cycles. These curves are fundamental for the study of the behavior of oscillators and their synchronization in networks.

For equilibria, the software supports the computation of critical normal form coefficients for all codimension 1 and 2 bifurcations. For limit cycles, it supports the computation of critical coefficients of periodic normal forms for codimension 1 bifurcations.

Finally, the continuation of homoclinic orbits (both to hyperbolic saddles and to saddle-nodes) is supported by MATCONT, together with detection of a large number of codimension 2 bifurcations along the homoclinic curves.

Most curves are computed with the same 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. Besides sophisticated numerical methods, MATCONT provides data storage and a modern graphical user interface (GUI).

Relationships between objects of codimension 0, 1 and 2 computed by MATCONT are presented in these figures, while the symbols and their meaning are summarized in the table below, where the standard terminology is used.

An arrow in the figures from an object of type *A* to an object of type *B* means that that object of type *B* can be detected (either automatically or by inspecting the output) during the computation of a curve of objects of type *A*. Each object of codimension 0 and 1 can be continued in one or two system parameters, respectively.

The symbol '*' stands for either *S* or *U*, depending on whether a stable or an unstable invariant manifold is involved.

In principle, the graphs presented in the figures are connected. Indeed, it is known that curves of codimension 1 homoclinic bifurcations emanate from the *BT*, *ZH*, and *HH* codimension 2 points. The current version of MATCONT fully supports, however, only one such connection: *BT* to *HHS*.

## References

- Dhooge A., Govaerts W. and Kuznetsov Yu. A. (2003) MatCont: A MATLAB package for numerical bifurcation analysis of ODEs.
*ACM TOMS***29**:141-164. - Govaerts W., Kuznetsov Yu.A., and Dhooge A. (2005) Numerical continuation of bifurcations of limit cycles in MATLAB.
*SIAM J. Sci. Comp.***27**:231-252. - Kuznetsov Yu. A., Govaerts W., Doedel E. J., and Dhooge A. (2005) Numerical periodic normalization for codim 1 bifurcations of limit cycles.
*SIAM J. Numer. Anal.***43**:1407-1435. - Govaerts W. and Sautois B. (2006) Computation of the phase response curve: A direct numerical approach.
*Neural Computation***18**:817-847. - Govaerts W. and Sautois B. (2006) Phase response curves, delays and synchronization in MATLAB.
*LNCS***3992**:391-398.

**Internal references**

- Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation.
*Scholarpedia*,**1**(10):1858. - James Murdock (2006) Normal forms.
*Scholarpedia*,**1**(10):1902. - Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit.
*Scholarpedia*,**1**(7):1358. - Carmen C. Canavier (2006) Phase response curve.
*Scholarpedia*,**1**(12):1332. - Yuri A. Kuznetsov (2006) Saddle-node bifurcation.
*Scholarpedia*,**1**(10):1859. - Philip Holmes and Eric T. Shea-Brown (2006) Stability.
*Scholarpedia*,**1**(10):1838.

## External links

- MATCONT is freely available at http://www.matcont.UGent.be. It requires MATLAB version 6.* or higher.

## See also

Bifurcation, Dynamical Systems, Equilibria, Periodic Orbit, Phase Response Curve, Stability