Note

This software is a new and renamed version of PDDE-CONT. This is also a developement version and is NOT considered final.

Knut can continue periodic solutions and periodic solution bifurcations of autonomous and periodically forced delay-differential equations that are in the general form

dx(t)/dt = f(t, x(t-τ₀), x(t-τ₁), ..., x(t-τ_r), λ).

To start the continuation, the software needs an initial periodic solution x₀(t) at some parameter value λ₀. This starting periodic solution has to be specified either in the system definition (C++ source file) or it can be loaded from an input file. The periodic solutions can bifurcate in several ways. If one of the three common codimension-one bifurcations (fold, period doubling, Neimark-Sacker) is found along the branch of periodic solutions, the point can be used as a starting point for continuing the branch of bifurcation points in a two parameter space.

Equilibria of autonomous systems can be also handled, as periodic solutions of periodic systems by keeping the period constant. Although it is very inefficient, this capability might be useful, when looking for periodic orbits arising at Hopf bifurcations points. Hopf bifurcations of fixed points are detected as Neimark-Sacker bifurcations, and the periodic solutions emanating from such points can be continued by switching to the periodic solution branch.

See the User's Manual.

Installation from the source code

The software can be built using the cross-platform build facility CMake. The build also requires a C and C++ compiler, preferably GCC, and the ATLAS linear algebra library. The (optional) graphical user interface requires the Qt GUI toolkit.

The software is available free of charge. Copying of the software is governed by the GNU General Public License (GPL) version 2. In addition to the GPL, the author of any scientific or non-scientific publication who used the software is required to refer to the User's Manual in the publication where the software or its modified version was used [1].

UMFPACK and AMD can be distributed under the GNU General Public License (GPL)

[1] R. Szalai, Knut: A continuation and bifurcation software for delay-differential equations, 2009.

[2] R. Szalai, PDDE-CONT: A continuation and bifurcation software for delay-differential equations, 2005.

[3] R. Szalai, G. Stepan, and S. J. Hogan. Continuation of bifurcations in periodic delay-differential equations using characteristic matrices. SIAM Journal on Scientific Computing, 28(4):1301-1317, 2006.