pde2path - a Matlab package for continuation and bifurcation in 2D elliptic systems

$\def\R{\mathbb R}\def\lam{\lambda}\def\Om{\Omega}\def\pa{\partial}$ $\def\huga#1{\begin{gather} #1 \end{gather}}$ $\newcommand{\spr}[1]{\left\langle #1 \right\rangle}$ $\def\om{\omega}\newcommand{\reff}[1]{(\ref{#1})}$ $\def\er{{\rm e}}\def\ri{{\rm i}}\def\C{\mathbb C}$ $\newcommand{\bce}{\begin{center}}\newcommand{\ece}{\end{center}}$

pde2path - a Matlab package for continuation and bifurcation
in 2D elliptic systems

Webpages of the old version pde2path 1.0, outdated! We strongly recommend to switch to pde2path 2.*

Overview. pde2path uses (pseudo) arclength continuation to compute branches of stationary solutions and their bifurcations for PDE systems of the $\huga{\label{gform} G(u,\lam):=-\nabla\cdot(c\otimes\nabla u)+a u-b\otimes\nabla u-f=0, }$ where $u=u(x)\in\R^N$, $x\in\Omega\subset\R^2$ some bounded domain, $\lam\in\R$ is a parameter, $c\in\R^{N\times N\times 2\times 2}$, $b\in\R^{N\times N\times 2}$, $a\in\R^{N\times N}$ and $f\in\R^N$ can depend on $x,u,\nabla u$, and, of course, parameters. The software is based on the Matlab pdetoolbox. The standard assumption is that $c,a,f,b$ depend on $u,\nabla u,\ldots$ locally, e.g., $f(x,u)=f(x,u(x))$; however, the dependence of $c,a,f,b$ on arguments can in fact be quite general, for instance involving global coupling.
The current version supports "generalized Neumann" boundary conditions (BC) of the form \begin{align}\label{e:gnbc} {\bf n}\cdot (c \otimes\nabla u) + q u = g, \end{align} where ${\bf n}$ is the outer normal and again $q\in \R^{N\times N}$ and $g\in \R^N$ may depend on $x$, $u$, $\nabla u$ and parameters. These BC include zero flux BC for $q,g=0$, while large prefactors in $q$, $g$ can be used to generate "stiff spring" approximations of Dirichlet BC that we found to work reasonably well.

There are a number of predefined functions to specify domains $\Om$ and boundary conditions, or these can be exported from Matlab's pdetoolbox GUI, thus making it easy to deal with (almost) arbitrary geometry and boundary conditions.

The software can also be used to time-integrate parabolic problems of the form \begin{equation}\label{pprob} \pa_t u=-G(u,\lam), \end{equation} with $G$ as in \eqref{gform}. This is mainly intended to easily find initial conditions for continuation. Finally any number of eigenvalues of the Jacobian $G_u(u,\lam)$ can be computed, thus allowing stability inspection for stationary solutions of \eqref{pprob}.

Go to equation to see the equation and BCs in more detail, jump directly to Demos or Download, or see Basics and First steps for a little mathematical background and a short overview on the usage of pde2path.