Introduction to AUTO 2000 1 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
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Introduction to AUTO 2000 1 AUTO 2000 ~ numerical bifurcation software Presented by Patrick Longhini 1.E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F.
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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.