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Introduction to AUTO 2000 1

Introduction to AUTO 2000 1. 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)

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Introduction to AUTO 2000 1

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  1. 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

  2. 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

  3. 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

  4. 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)).

  5. Solution of Example abc p1 = 0.15 p1 = 0.33 p1 = 0.39

  6. 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

  7. Define system: abc.c Required Subroutines Parameters Initial Conditions

  8. Define constants: c.abc Run 1: save as c.abc.1 Constants briefly described in: auto2000_quick_ref.pdf

  9. MOST USED CONSTANTS

  10. Command Line To begin type auto 2 2 2 0 10 3 80

  11. 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.

  12. b.abc

  13. d.abc

  14. Using the comand: plot(‘abc’)

  15. Looking at the solution for label 7

  16. 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

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

  18. time u1 u2 u3 lor.dat

  19. 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.

  20. Makefile

  21. Plotting using Plaut @p lor Type: Commands d3 (show stability) then BD0 (default settings) Appendix B of manual contains all commands

  22. 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.

  23. using @fl lor

  24. Graphical User Interface. Type @auto or its abbreviation @a Appendix C of manual contains help for GUI

  25. 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

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