MATCONT

• Dr. Willy Govaerts, Ghent University, Belgium.

• Prof. Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

• Dr. Bart Sautois, Ghent University, Belgium

Figure 1: Screenshot of MATCONT showing the main window, 2D-plot, starter and continuer windows in the case of continuation of an LP-curve with detection of CP, BT and BP's (see table below).

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).

Figure 2: The graph of adjacency for equilibrium and limit cycle bifurcations in MATCONT.

Figure 3: The graph of adjacency for homoclinic bifurcations in MATCONT; here * stands for S or U.

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.

Equilibrium- and cycle-related objects and their labels within the GUI Type of object Label Point P Orbit O Equilibrium EP Limit cycle LC Limit Point (fold; saddle-node) bifurcation LP Andronov-Hopf Bifurcation H Limit Point (fold; saddle-node) bifurcation of cycles LPC Neimark-Sacker (torus) bifurcation NS Period Doubling (flip) bifurcation PD Branch Point BP Cusp bifurcation CP Bogdanov-Takens bifurcation BT Zero-Hopf bifurcation ZH Double Hopf bifurcation HH Generalized Hopf (Bautin) bifurcation GH Branch Point of Cycles BPC Cusp bifurcation of Cycles CPC 1:1 Resonance R1 1:2 Resonance R2 1:3 Resonance R3 1:4 Resonance R4 Chenciner (generalized Neimark-Sacker) bifurcation CH Fold--Neimark-Sacker bifurcation LPNS Flip--Neimark-Sacker bifurcation PDNS Fold-flip Bifurcation LPPD Double Neimark-Sacker Bifurcation NSNS Generalized Period Doubling Bifurcation GPD

Objects related to homoclinics to equilibria and their labels within the GUI Type of object Label Limit cycle LC Homoclinic to Hyperbolic Saddle HHS Homoclinic to Saddle-Node HSN Neutral saddle NSS Neutral saddle-focus NSF Neutral Bi-Focus NFF Shilnikov-Hopf Bifurcation SH Double Real Stable leading eigenvalue DRS Double Real Unstable leading eigenvalue DRU Neutrally-Divergent saddle-focus (Stable) NDS Neutrally-Divergent saddle-focus (Unstable) NDU Three Leading eigenvalues (Stable) TLS Three Leading eigenvalues (Unstable) TLU Orbit-Flip with respect to the Stable manifold OFS Orbit-Flip with respect to the Unstable manifold OFU Inclination-Flip with respect to the Stable manifold IFS Inclination-Flip with respect to the Unstable manifold IFU Non-Central Homoclinic to saddle-node NCH

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

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Accepted on: 2006-08-21 07:42:14 GMT