Knut

Capabilities

The software is designed to analyse explicitly time-dependent delay-differential equations with time dependent delays of the form \[ \boldsymbol{M}\,\dot{\boldsymbol{x}}(t) = \boldsymbol{f} (t, \boldsymbol{x}(t-\tau_0(t)), \boldsymbol{x}(t - \tau_1(t)), \dots , \boldsymbol{x}(t - \tau_m(t))) \] The mass matrix $\boldsymbol{M}$ can be singular, hence Knut can handle algebraic equations and neutral delay-differential equations. State dependent delays are not yet supported, see DDE-BIFTOOL for that.

The features include

Continuation of periodic orbits along a parameter
Calculation of Floquet multipliers of periodic orbits
Automatic detection of bifurcations
Continuation of torus, period doubling and fold bifurcations of periodic orbits along two parameters
Continuation of invariant tori with quasi-periodic orbits along two parameters
Continuation of steady states (as periodic orbits)

Significant differences from DDE-BIFTOOL

No need for any programming, equations entered using simple formulae
Stand-alone software
Graphical User Interface
Written in C++, hence faster
The only software that calculates quasi-periodic tori

The Graphical User Interface also includes a plotting tool that displays the calculation live.

Sample equation specification

The Mackey-Glass equation \[\dot{x} ( t ) = ax ( t ) + b \frac{x ( t - \tau )}{1 + x^{10} ( t - \tau )}\] is represented by the following code


vfname()=mackey_glass;

time()=t;

period(T)=2.0;

par(a)=-1.0;

par(b)=1.5;

par(tau)=2.0;

DX=delay(X,tau);

dot(X)=a*X + b*DX/(1+DX^10);

init(X)=((1.0-1.5)/(-1.0))^(1.0/10.0);

Bifurcation diagram of the Mackey-Glass equation

Bifurcation diagram

Figure 1. Fold bifurcations of the period-two orbits. Black curves are periodic solutions continued along $b=$constant lines. Red curves refer to fold bifurcations, and the blue line is the period doubling curve, where the period-two orbits arise.

Package downloads

Mac Application (El Capitan, Intel x86_64) Knut-10.dmg
Windows application (Intel 64 bit) Knut-11-w64.7z
Source code Knut-11.tar.bz2

Documentation

See the Users' manual

Compilation instructions

Dependencies are OpenBLAS, CMake and optionally Qt

To build the software use the following commands:


    git clone https://github.com/rs1909/knut.git

    cd knut/knut

    mkdir build

    cd build

    ../build-unix.sh

    make install

On Mac OSX replace the penultimate command with ../build-mac.sh
To make an installable package use make install and make dmg which will produce a disk image

On Fedora Linux with the MinGW cross-compiler installed the ../build-win-cross.sh will compile the software for Windows. To make a package use makensis wininstaller.nsi.

Please report any bugs and wishes on the issue tracker, I'll try to address them as my time allows it.