Introduction to AUTO 20001

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Introduction to AUTO 20001

• AUTO 2000 numerical bifurcation software
• Presented by Patrick Longhini

• E.J. Doedel, R.C. Paffenroth, A.R. Champneys,
  T.F. Farirgrieve, Yu. A Kuznetsov, B. Sandstede
  X.Wang (2001)
• AUTO 2000 Continuation and bifurcation
  software for ODEs (with HomCont) Technical
  Report, Caltech, Feb. 2001

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Getting Started

• Download
• http//sourceforge.net/projects/auto2000
• The AUTO manual gives details on how to install
  the program on your Linux/UNIX based machine.

AUTO can do bifurcation analysis on a system of
ordinary differential equations (ODEs) of the
form
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Main AUTO files

• The equation-file xxx.c
• Contains the C subroutines func, stpnt, bcnd,
  icnd,
• func defines the function f(u,p)
• stpnt defines the starting solution (u,p)
• The constants-file c.xxx
• Controls the dynamics of AUTO

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Example 1 abc reaction
This demo illustrates the computation of
stationary solutions, Hopf bifurcations and
periodic solutions in the A ? B ? C reaction
(Doedel Heinemann (1983)).
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Solution of Example abc
p1 0.15
p1 0.33
p1 0.39
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Basic Steps

• Step 1 step up abc.c and c.abc
• Step 2 first run -- stationary solutions
• Step 3 restart at hopf and compute periodic
  solutions

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Define system abc.c
Required Subroutines
Parameters Initial Conditions
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Define constants c.abc
Run 1 save as c.abc.1
Constants briefly described in
auto2000_quick_ref.pdf
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MOST USED CONSTANTS
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Command Line
To begin type auto
2 2 2 0 10
3 80
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save(abc)

• Using save(abc) you get 3 output files
• b.abc bifurcation branches
• s.abc solution
• d.abc dynamics (stability)

Relating labels to output file.
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b.abc
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d.abc
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Using the comand plot(abc)
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Looking at the solution for label 7
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Example 2 lor.c

• Starting an orbit from numerical data

with p1 280, p2 8/3 and p3 10.
Numerical data must have one complete periodic
oscillation which is contained in
lor.dat us(lor) creates AUTO output file s.dat
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Creating file xxx.dat
Use fprintf to print the data to a file. Make
sure that you only have one period.
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time u1 u2 u3
lor.dat
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Command line steps
Show how to write a Makefile to run command line
steps. This is better because you can running
all the steps with one command and keep track
of your work.
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Makefile
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Plotting using Plaut _at_p lor
Type Commands d3 (show stability) then
BD0 (default settings)
Appendix B of manual contains all commands
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Useful commands (using command mode)

• _at_fl lor list Floquet multipliers in d.lor
• _at_ev abc list eigenvalues of the Jacobian d.abc
• _at_lp xxx, _at_bp xxx, _at_hb xxx, lists limit points,
  branch points and hopf bifurcations, respectively.

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using _at_fl lor
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Graphical User Interface.
Type _at_auto or its abbreviation _at_a
Appendix C of manual contains help for GUI
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Other useful tools

• xppaut www.math.pitt.edu/bard/xpp/xpp.html
• Solves ODEs,DDEs,also AUTO built in
• winpp
• Windows version of xppaut but used LOCBIF instead
  of AUTO
• matcont allserv.rug.acbe/ajdhooge/research.html
• Continuation software in Matlab July 9th 2004
  (lastest version)
• DDE-BIFTOOL
• Matlab package for numerical bifurcation analysis
  of delay equations