** Note: PDDE-CONT has been superceded by
Knut.**
The VFGEN command for PDDE-CONT is now unsupported, and it may
be removed in a future version.

PDDE-CONT is a software package developed by Róbert Szalai for continuation and bifurcation analysis of periodic solutions to delay differential equations.

A C++ system definition file for the PDDE-CONT package
is created by the command

$ vfgen pddecont vector_field_file.vf

The vector field must have at least one **delay** expression.

The file ShayerCampbell2000.vf defines the following system of equations:

x_{1}'(t) |
= | -κ x_{1}(t) + β tanh(x_{1}(t-τ_{s})) + a_{12} tanh(x_{2}(t-τ_{2})) |

x_{2}'(t) |
= | -κ x_{2}(t) + β tanh(x_{2}(t-τ_{s})) + a_{21} tanh(x_{1}(t-τ_{1})) |

This example was taken from the DDE-BIFTOOL manual. The reference given there for these equations is:

L. P. Shayer and S. A. Campbell. Stability, bifurcation and multistability in a system
of two coupled neurons with multiple time delays,
*SIAM J. Applied Mathematics*, **61**(2):673-700, 2000.

The command

$ vfgen pddecont ShayerCampbell2000.vf

creates the file
sys-ShayerCampbell2000.cpp.
This example uses these constants files:

The command$ pdde -c cfile-ShayerCampbell2000-eq -o sc-eq.mat

computes a family of equilbria in which a Hopf bifurcation occurs.
Then the next two commands compute the family of period orbits that
arises at the Hopf bifurcation. The family is computed in two stages;
during the second stage the number of discretization intervals is increased
from 50 to 320.
$ pdde -c cfile-ShayerCampbell2000-po-a -i sc_eq.mat -o sc-po-a.mat

$ pdde -c cfile-ShayerCampbell2000-po-a -i sc-po-a.mat -o sc-po-b.mat

The following plots show max($ pdde -c cfile-ShayerCampbell2000-po-a -i sc-po-a.mat -o sc-po-b.mat

The equation

The equation is not a delay equation; this example is simply a test of using PDDE-CONT to compute a family of solutions to a periodic vector field.

This equation can be written as the system

x ' | = | y |

y ' | = | -x - ε y - ε x^{3} +
ε a cos(ω t ) |

x '(θ) | = | Ty |

y '(θ) | = | T [-x - ε y - ε x^{3} +
ε a cos(2π θ )] |

x | = | 1.85 cos(θ - 0.34) |

y | = | 1.79 cos(θ + 1.2) |

We create a system definition file for PDDE-CONT with the command

$ vfgen pddecont wn2.vf

This creates the file
sys-wn2.cpp.
We edit this file and add the approximate solution in the function We use the constants files

to compute the family of periodic orbits with the commands$ pcompile sys-wn2-edit.cpp

$ pdde -c cfile-wn2a -o wn2a.mat

$ pdde -c cfile-wn2b -o wn2b.mat

$ pdde -c cfile-wn2c -i wn2b.mat -o wn2c.mat

The first pdde command computes the curve for $ pdde -c cfile-wn2a -o wn2a.mat

$ pdde -c cfile-wn2b -o wn2b.mat

$ pdde -c cfile-wn2c -i wn2b.mat -o wn2c.mat

Finally, we use the MATLAB script wnplot.m to create the following graph of the amplitude of the periodic orbit as a function of ω.

Copyright © 2005-2015 Warren Weckesser