Transcript Introduction to AUTO20001 - Nonlinear Dynamical Systems

1
Introduction to AUTO 2000
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.
AUTO can do bifurcation analysis on a system of ordinary
differential equations (ODEs) of the form
u(t )  f (u(t ), p)
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
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
p = 0.15
1
0.5
0. 995
1-3
p1 = 0.15
1
0. 99
u
0. 985
0. 98
0. 975
15
20
15
20
25
30
35
40
45
50
15
20
25
30
time
35
40
45
50
0. 97
15
16
25
17
18
30
19
35
20
40
21
45
22
23
50
6
4
2
0
10
8
1
1-3
6
4
2
u
p1 = 0.39
p = 0.39
u
1-3
1
p = 0.33
p1 = 0.33
0
10
0
10
24
25
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
Required Subroutines
Parameters
Initial Conditions
Define constants: c.abc
Run 1: save as c.abc.1
Constants briefly described in: auto2000_quick_ref.pdf
MOST USED CONSTANTS
Command Line
To begin type
auto
2
2
3
80
2
0 10
save(‘abc’)
 Using save(‘abc’) you get 3 output
files
 b.abc
 s.abc
 d.abc
Relating labels to
output file.
bifurcation branches
solution
dynamics (stability)
b.abc
d.abc
Using the
comand:
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 which
is contained in lor.dat
us(‘lor’) creates AUTO output file s.dat
Creating file:
xxx.dat
Use 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