# Introduction to AUTO AUTO 2000 ~ numerical bifurcation software

## Presentation on theme: "Introduction to AUTO AUTO 2000 ~ numerical bifurcation software"— Presentation transcript:

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

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

Main AUTO files The equation-file xxx.c The constants-file c.xxx
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

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 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 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 u u 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, lists limit points, branch points and hopf bifurcations, respectively.

lor

Graphical User Interface.
Type @auto or its 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