Introduction to AUTO 20001

AUTO 2000 ~ numerical bifurcation software

Presented by Patrick Longhini

1. 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

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.

)),(()( ptuftu

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

Main AUTO files

The equation-file xxx.cContains 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

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)).

Solution of Example abc

10 15 20 25 30 35 40 45 500

0.5

1u 1-

3 p

1 = 0

.15

10 15 20 25 30 35 40 45 500

2

4

6

u 1-3

p1 =

0.3

3

10 15 20 25 30 35 40 45 500

2

4

6

8

time

u 1-3

p1 =

0.3

9

15 16 17 18 19 20 21 22 23 24 250.97

0.975

0.98

0.985

0.99

0.995p1 = 0.15

p1 = 0.33

p1 = 0.39

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

Define system: abc.c

Parameters

Initial Conditions

Required Subroutines

Define constants: c.abc

Constants briefly described in: auto2000_quick_ref.pdf

Run 1: save as c.abc.1

MOST USED CONSTANTS

Command LineTo begin type

auto

2 2 2 0 10

3 80

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.

b.abc

d.abc

Using thecomand:

plot(‘abc’)

Looking at the solution for label 7

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 whichis contained in lor.dat

us(‘lor’) creates AUTO output file s.dat

Creating file: xxx.datUse fprintf to print the data to a file. Make sure that you only have one period.

time u1 u2 u3

lor.dat

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.

Makefile

Plotting using Plaut @p lor

Type: Commands

d3 (show stability)

then

BD0 (default settings)

Appendix B of manual contains all commands

Useful commands: (using command mode)

@fl lor – list Floquet multipliers in d.lor

@ev abc – list eigenvalues of the Jacobian d.abc

@lp xxx, @bp xxx, @hb xxx, lists limit points, branch points and hopf bifurcations, respectively.

using @fl lor

Graphical User Interface.Type

@auto

or its abbreviation @a

Appendix C of manual contains help for GUI

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