DDE-Biftool demos

• neuron: Step-by-step walk-through of basic DDE-Biftool usage for a neuron model with parametric delays. This demo illustrates
  • computation and continuation of steady states,
  • computation of their stability,
  • branching off to periodic orbits,
  • continuation of periodic orbits,
  • computation of their stability,
  • continuation of Hopf bifurcations in two parameters,
  • computation of normal form coefficients for Hopf bifurcations and Hopf-Hopf interactions,
  • computation of connecting orbits
  An additional script demo1_simple.m demonstrates, how one can use convenience functions from the folder ddebiftool_utilities to perform the above tasks.
  The script demo1_POfold.m demonstrates the continuation of folds of periodic orbits.


• sd_demo: step-by-step walk-through of basic DDE-Biftool usage for a system with state-dependent delays. This demo illustrates
  • computation and continuation of steady states,
  • computation of their stability,
  • branching off to periodic orbits,
  • continuation of periodic orbits,
  • computation of their stability,
  • continuation of Hopf bifurcations in two parameters,
  • continuation of Hopf bifurcations in two parameters.
• hom_demo: Step-by-step walk-through of constructing and continuing connecting orbits between equilibria using DDE-Biftool.
• nmfm_demo: A demo focussing on the features of the ddebiftool_extra_nmfm extension. Along two branches of Hopf bifurcations, normal form coefficients (determining if the bifurcation is sub- or supercritical) are computed. Also, a number of codimension-two points along the Hopf branch are detected and their normal forms are computed, too.
• minimal_demo: This example (a Duffing oscillator with delayed feedback) demonstrates how to use the convenience functions from the folder ddebiftool_utilities and the extension in the folder ddebiftool_extra_psol to continue torus bifurcations and folds of periodic orbits in systems with parametric delays. It also demonstrates the option to vectorize the right-hand side to speed up computations, and computes normal forms along Hopf branches.
• nested: This example shows that DDE-Biftool can treat state-dependent delays with arbitrary levels of nesting.
• rotsym_demo: A demonstration of the DDE-Biftool extension in ddebiftool_extra_rotsym for rotationally symmetric systems using the Lang-Kobayashi equations (a model of a semiconductor lasers with delayed optical feedback). The demo illustrates
  • definition of the rotational symmetry with a skew-symmetric matrix and how to incorporate this into the system definition,
  • computation and continuation of relative equilibria (rotating waves),
  • computation of their stability,
  • branching off to relative periodic orbits (modulated waves),
  • continuation of relative periodic orbits,
  • computation of their stability,
  • continuation of Hopf and fold bifurcations of relative equilibria in two parameters,
  • continuation of period doublings and fold bifurcations of relative periodic orbits in two parameters,
• Mackey-Glass: This demo continues period doubling bifurcations in the well-known Mackey-Glass equation, a scalar DDE with parametric delay. It shows how one can branch off at a period doubling to track the period-doubled orbit. In this demo one can switch on or off vectorization to compare the speed of computations. Also included: normal form computation for Hopf bifurcation.
• Humphries et al: This demo continues folds and torus bifrucations in a scalar DDE with two state-dependent delays. It also demonstrates the option to vectorize right-hand side, delays, and their derivatives to speed up computations.
• phase oscillator (contributed by Azamat Yeldesbay): This demo shows that rotations (such as in phase oscillators or in a rotating forced pendulum) can be treated as periodic orbits.
• Holling-Tanner model - Demo for Takens-Bogdanov point normal form (contributed by Maikel Bosschaert). The demo shows the computation of test functions for the Takens-Bogdanov point on a model discussed by Liu, X., Liu, Y., and Wang, J. (2013): Bogdanov-Takens bifurcation of a delayed ratio-dependent Holling-Tanner predator prey system (in Abstract and Applied Analysis).
• System with cusp and Takens-Bogdanov bifurcations (contributed by Maikel Bosschaert) from Giannakopoulos, F. and Zapp, A. (2001). Bifurcations in a planar system of differential delay equations modeling neural activity. Physica D: Nonlinear Phenomena, 159(3):215-232.