***************************************************** A simulation software for the stochastic Brusselator Gabriele Bleckert bleckert@oppel2.mathematik.uni-muenchen.de http://oppel2.mathematik.uni-muenchen.de/~bleckert and Klaus Reiner Schenk-Hoppe' schenk@wiwi.uni-bielefeld.de http://www.wiwi.uni-bielefeld.de/~boehm/members/klaus November 4, 1998 ****************************************************** README.TXT (accompanies the software for the stochastic Brusselator) This software, which comes in two packed files called brus_unx.zip and brus_dos.zip, is a version of the software that has originally been developed for a numerical study of the stochastic Brusselator. For published visualizations, see the paper "The Stochastic Brusselator: Parametric Noise Destroys Hopf Bifurcation" Ludwig Arnold, Gabriele Bleckert, and Klaus Reiner Schenk-Hoppe' Report No. 425, Institut für Dynamische Systeme, Fachbereich Mathematik und Informatik, Universität Bremen, 1998. Forthcoming in Stochastic Dynamics (eds. V. M. Gundlach and H. Crauel), Springer-Verlag, Berlin, 1998. Consult this paper for further information on the models and the interpretation of the numerical results. The file README.TXT describes how to install the package, the program calls, and the visualization procedures. Installation under UN*X-Systems (Linux, Solaris, Unix, ...) 1) Unpack Unpack the file "brus_unx.zip" with the command unzip brus_unx which creates the directory "brusselator" with the files of the package "brus_unx.zip". Move to the directory "brusselator" by the command cd brusselator If Gnu's C-compiler is not available on your system edit the file "makefile.unx" and set "CC=cc". Additionally, if you want to process the numerical results with the program MATLAB, then you have to specify the TYPE_FLAG in the file "makefile.unx" as follows For PC: TYPE_FLAG=PC For Sun, Mac, HP, Apollo: TYPE_FLAG=SUN For VAX D-float: TYPE_FLAG=2000 (not tested) For VAX G-float: TYPE_FLAG=3000 (not tested) Compile the package by the command make -f makefile.unx which generates the executable program "bruspars" and auxiliary routines. If the above command returns error messages, try the command "gmake -f makefile.unx" instead. Installation under DOS/WINDOWS-systems 1) Unpack Unpack the file "brus_dos.zip" with the command pkunzip -d brus_sim or unzip brus_sim which creates the directory "brusselator" with the files of the package "brus_dos.zip". 2) Compile Move to the directory "brusselator" by the command cd brusselator Before compiling, please edit the file "makefile.dos" and set "CC" to your available compiler. We strongly recommend to use a 32-bit-compiler, otherwise you might have memory problems. Compile the package by the command make -fmakefile.dos which generates the executable program "bruspars" and auxiliary routines. Further steps are identical for DOS/WINDOWS and UN*X 3) Test You can test wether the installation is correct by executing the command bruspars traject.prs which calculates one trajectory of the Brusselator according to the parameters given in the file "traject.prs". This call generates the files "traject.dat", "traject.mat", and "traject.log". 4) Examples One example for every type of computation is provided with the package. Each has to be called by the command bruspars where is one of the files mentioned in the following. To compute a trajectory of the stochastic brusselator call "traject.prs". To compute a trajectory of the stochastic brusselator backward in time call "trajbwd.prs". To compute the stationary density of the stochastic brusselator call "histogrm.prs". To compute the Lyapunov exponents of the stochastic brusselator "lyapexp.prs". To compute the rotation number of the stochastic brusselator call "rotnumb.prs". To compute the backward explosion times of the stochastic brusselator call "bwdexpl.prs". To compute the backward flow of the stochastic brusselator "flowbwd.prs". To compute the pullback of the stochastic brusselator call "pullback.prs". To compute the lyapunov exponents of a scalar stochastic differential equation call "lyap_1d.prs". To compute the backward flow of the scalar stochastic differential equation $dx = (\beta x - x^3) dt + \sigma dW$ call "flow_1d.prs". To compute the backward explosion times of a scalar stochastic differential equation $dx = (\beta x - x^3) dt + \sigma dW$ call "expl_1d.prs". 5) Documentation In each example listed under 4) a short description is given, explaining the type of simulation and which parameters have to be specify. For your own computation you can copy the file to "myfile.prs", modify the parameters according to your needs, and finally call bruspars "myfile.prs" to carry out the simualtion. The numerical results are stored in two formats: *.dat *.mat In the *.dat file the numerical data are stored in plain ASCII format and in the *.mat file the numerical data are stored in MATLAB format. We provide script-files for visualizing the data with the visualization programs GNUPLOT and MATLAB, which are both available for Uni*x- and DOS/WINDOWS-systems. a) GNUPLOT The gnuplot-script-files have the postfix ".gp". You can invoke the interactive plotting program GNUPLOT by the command gnuplot and after the prompt the script-file is loaded, e.g. by load "traject.gp" which opens one (or more) windows and visualizes the numerical results. b) MATLAB The matlab-script-files have the postfix ".m". You can invoke the interactive plotting program MATLAB by the command matlab and after the prompt the script-file "traject.m" is loaded by traject which opens one (or more) windows and visualizes the numerical results. The messages, e.g. overflow messages, are logged in the file *.log. 7) Demo You can run a demo including every type of computation which has to be called as: Start the gnuplot program by the command gnuplot and load the script-file "demo.gp" by the command load "demo.gp" which opens several windows and visualizes the numerical results.