MATCONT

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Willy Govaerts et al. (2006), Scholarpedia, 1(9):1375. doi:10.4249/scholarpedia.1375 revision #123862 [link to/cite this article]
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Curator and Contributors

1.00 - Willy Govaerts

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.

Contents

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

External links

See also

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

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