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Features

A discussion of features of continuation packages can be found in [16]. This toolbox is developed with the following targets in mind:

Upon the development of MatCont and CL_MatCont, there were multiple objectives:

A general comparison of the available features during computations for ODEs currently supported by the most widely used software packages auto97/2000 [6], content 1.5 [11] and MatCont/ CL_MatCont are indicated in Table .

Table 1: Supported functionalities for ODEs in AUTO (A), CONTENT (C) and MATCONT (M).

a c m
time-integration + +
Poincaré maps +
monitoring user functions along curves computed by continuation + + +
continuation of equilibria + + +
detection of branch points and codim 1 bifurcations (limit and Hopf points) of equilibria + + +
computation of normal forms for codim 1 bifurcations of equilibria + +
continuation of codim 1 bifurcations of equilibria + + +
detection of codim 2 equilibrium bifurcations (cusp, Bogdanov-Takens, fold-Hopf, generalized and double Hopf) + +
computation of normal forms for codim 2 bifurcations of equilibria +
continuation of codim 2 equilibrium bifurcations in three parameters +
continuation of limit cycles + + +
computation of phase response curves and their derivatives +
detection of branch points and codim 1 bifurcations (limit points, flip and Neimark-Sacker (torus)) of cycles + + +
continuation of codim 1 bifurcations of cycles + +
branch switching at equilibrium and cycle bifurcations + + +
continuation of branch points of equilibria and cycles +
computation of normal forms for codim 1 bifurcations of cycles +
detection of codim 2 bifurcations of cycles +
continuation of orbits homoclinic to equilibria + +

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, see [10].

 

Figure 1: The graph of adjacency for equilibrium and limit cycle bifurcations in MatCont

 

Figure 2: The graph of adjacency for homoclinic bifurcations in MatCont; here * stands for S or U.

Type of object Label
Point P
Orbit O
Equilibrium EP
Limit cycle LC
Limit Point (fold) bifurcation LP
Hopf bifurcation H
Limit Point 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 LPPD
Double Neimark-Sacker NSNS
Generalized Period Doubling GPD

Table 2: Equilibrium- and cycle-related objects 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 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

Table 3: Objects related to homoclinics to equilibria and their labels within the GUI

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 cycle, respectively. In general, an arrow from an object of type A to an object of type 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. For example, the arrows from EP to H, LP, and BP mean that we can detect H, LP and BP during the equilibrium continuation. Moreover, for each arrow traced in the reversed direction, i.e. from B to A, there is a possibility to start the computation of the solution of type A starting from a given object B. For example, starting from a BT point, one can initialize the continuation of both LP and H curves. Of course, each object of codim 0 and 1 can be continued in one or two system parameters, respectively.

The same interpretation applies to the arrows in Figure 2, where * stands for either S or U, depending on whether a stable or an unstable invariant manifold is involved.

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