REU Summer 2002: Dynamical Systems

• Katy Bold's web page
• Judith Hubbard's web page
• Saby Guharay's web page
• This postscript file describes our proposed summer projects.
• bifdiagnotes.ps Terse notes on some of the defining equations for the bifurcation diagram of H.
• hsidea.ps   hsidea.pdf: Horseshoes and Canards: Two Possibilities to Investigate
• VDP_slide.tar.gz Professor Guckenheimer's slides (contains lots of postscript files)
  You can access the individual files here.
• slides_example.tar.gz An example of using the "seminar" class in Latex to create transparencies. When unzipped and extracted, this will create a directory called "slides_example". In that directory, the main file to look at (and edit) it "oldtalk.tex". Run the commands
  > latex oldtalk
  > xdvi oldtalk &
  to see the output.
• reu_hs.ps Postscript version of a very rough and incomplete paper on our study of canards and horseshoes in the force van der Pol system. A tar file that of the directory that contains the Latex file and the postscript figures is: reu_hs0809.tar.gz

• MATLAB Web Page   (The Mathworks Web Page)
• An Introduction to MATLAB   (A PDF file)

• Hypertext Help with LaTeX
• Getting Started with LaTeX
• AMS-LaTeX (You can get the AMS-LaTeX User's Guide here.)
• TeX_cookbook.pdf This PDF document contains lots of tables that let you quickly look up the answer to "How do I do <...>?"

• AUTO2000 SourceForge Web Page
• auto.notes.020729   Some brief notes on installing AUTO.
• fcon.c   Bug fix for user data conversions. Put this file in auto/2000/src.
• autofvdp.tar.gz   When you unzip and extract this, you will have a directory called "autofvdp" that contains an example of computing periodic orbits for the forced van der Pol system. The file "script1" contains a set of auto commands that should work in the auto command line interface.
• autofvdpjac.tar.gz   These files are a slight variation of the previous example. Here I've included the derivatives in the C equations files, and in the constants files I've set IJAC=1. In theory, this should improve the performance of AUTO, but I haven't done a careful comparison. When you extract the files, a directory called "autofvdpjac" is created. This directory contains the C files, s.dat, a couple constants files, and the file script2. You can run the example with the command
     auto < script2
  After this, you can run AUTO again and enter the command pl('fvdpjac') to see the result. (Or you can take a look at the contents of script2, and enter the commands individually.)

• odetools
• Integro and Continuo Guides

• vdp.m
• vdpscript1.m
• fvdp.m
• fvdpscript1.m

• fvdp_mplots.m
  This is a self-contained MATLAB function (with no arguments--edit the file to change parameters). It makes the 3D plot (among other things).
• sncontinue.tar
  MATLAB files for the continuation of saddle-node points of H. These do not use the odetools library. Save this file, and run "tar xvf sncontinue.tar". This will creat the directory "sncontinue" that contains the MATLAB files.
• zfuncs.tar   (Last update: August 1)
  MATLAB functions that implement the defining equations for homoclinic points in the half return map H. When the files are extracted, a directory called "zfuncs" will be created that contains the functions.
  August 1 update: I've included "getMaxStop.m", a slightly modified version of Chantal's function to find the theta coordinate from where the maximal canards jump; also, "mctan.m", a function of (a,omega) that is zero when the maximal canard lands at a point where the flow is tangent to x=-2.
• fvdp_singcan.m
  Creates the "singular canards" plot singcan.eps
• bdplot_0801.tar.gz  (Last update: August 1)
  In the August 1 version, I've update the bifurcation diagram to include two of the "maximal canard fixed point" curves that Chantal has computed (plotted in red), and the "saddle-saddle connection" that Ricardo computed (plotted in magenta). I've also changed the phase plane program so that it plots the x-nullcline in gray.
  Before downloading this, rename your old bdplot directory (if you have one) to something like bdplot.old (or just delete it).
  I've put all the stuff from fvdp2002 into the bdplot directory. When you extract this, the directory bdplot will contain the main functions:
         vdp_bd_plot2.m
         fvdp_hplot.m
         fvdp_h2plot.m
         fvdp_phport.m
         bdexplorer.m
  and lots of auxiliary MATLAB files and data files. The most interesting NEW thing here is the GUI interface to the bifurcation diagram, called bdexplorer. After downloading and extracting the files, run MATLAB in the bdplot directory, and after setting the odetools path (and executing add_vf(integro,'fvdp_slow_vf') and add_vf(integro,'fvdp_slow_sc_vf'), if necessary), enter the command
  >> bdexplorer
  This will plot the bifurcation diagram and bring up a GUI interface that you can use to "explore" the diagram. See me for further instructions, or if it doesn't work.
• fvdp2002_0717.tar   (Note: the contents of this tar file are a subset of the bdplot file above.)
  Save this in a directory. If you have a subdirectory named fvdp2002 in this directory, rename it using the shell command
     mv fvdp2002 fvdp2002.old
  Then execute "tar xvf fvdp2002_0717.tar". This will create a new fvdp2002 directory. The main MATLAB functions in this directory are fvdp_hplot and fvdp_h2plot. They use integro, so you must set the odetools path, and you will have to execute the commands
  >> add_vf(integro,'fvdp_slow_vf')
  >> add_vf(integro,'fvdp_slow_sc_vf')
  in MATLAB before using them. (You only have to do this once.)
• fvdp_findpo.m, findpo_script.m, fvdp_vf.c
  The MATLAB function fvdp_findpo attempts to find a stable periodic orbit in the full forced van der Pol system. This function uses the integro/mex interface; the C file that defines the vector field is fvdp_vf.c. You will have to enter the command
  >> add_vf(integro,'fvdp_vf')
  in MATLAB before using fvdp_findpo. The script findpo_script uses this function to find an orbit and save it in file that can be converted to AUTO format. See me for details.
• sigmamapphiext.m, getphiextstart.m, fvdp_phiextvf.c
  The MATLAB function "sigmamapphiext" computes the map (and its Jacobian) from x=x0 to x=-x0 for the forced van der Pol equation. (We are only using using x0=-1.) It uses integro; the C file that implements the vector field is fvdp_phiextvf.c. You will have to set up ODETOOLS and then execute the command add_vf(integro,'fvdp_phiextvf') before using this function.
  The file "getphiextstart.m" is a script that calls sigmamapphiext and saves the data to a file that can be read by AUTO.
  (The "phi" in the file names refers to the fact that the ODEs have been changed to make the period 2*PI rather than 1; that is, phi=2*PI*theta. This makes the data compatible with AUTO. I have versions of these files that use theta instead of phi if anyone needs them. The "ext" in the file names refers to these being extended to include the variational equations.)

• po_e0.0001a4.0o1.55p1.dat
  A stable period 1 orbit, with parameters epsilon=0.0001, a=4.0, omega=1.55.
• po_e0.0001a2.5o1.55p3.dat
  A stable period 3 orbit, with parameters epsilon=0.0001, a=2.5, omega=1.55.
• asymm.dat (about 1.3Meg!)
  asymm2.dat (a reduced version of the same trajectory)
  This file contains the data for a stable asymmetric periodic orbit. The file asymm.parameters contains some more comments and the parameters.

• extractb This perl script creates files that can be used by the MATLAB script bd2 (see below).
• extracts This perl script extracts a single solution from the AUTO solutions file. If you use this to create a file called, say, "sol99.dat", then in MATLAB you can use the following commands to read and plot this data:
  >> v = load('sol99.dat'); % Load the data (t,x,y,theta) into v.
  >> plot(v(:,4),v(:,2));   % Plot x vs. theta.
  >> plot(v(:,2),v(:,3));   % Plot y vs. x.
  
• bd2.m This MATLAB script reads the files created by extractb and plots the bifurcation diagram.