Presentation on theme: "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."— Presentation transcript:
Introduction to AUTO 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 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 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 1 = 0.15 p 1 = 0.33 p 1 = 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 constants: c.abc Constants briefly described in: auto2000_quick_ref.pdf Run 1: save as c.abc.1
MOST USED CONSTANTS
Command Line To begin type auto
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.
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.
Plotting using lor Type: Commands d3 (show stability) then BD0 (default settings) Appendix B of manual contains all commands
Useful commands: (using command lor – list Floquet multipliers in abc – list eigenvalues of the Jacobian xxx, lists limit points, branch points and hopf bifurcations, respectively.
Graphical User Interface. or its Appendix C of manual contains help for GUI
Other useful tools xppaut 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 allserv.rug.acbe/~ajdhooge/research.html Continuation software in Matlab July 9 th 2004 (lastest version) DDE-BIFTOOL Matlab package for numerical bifurcation analysis of delay equations