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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

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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

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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

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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)).

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Solution of Example abc p 1 = 0.15 p 1 = 0.33 p 1 = 0.39

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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

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Define system: abc.c Parameters Initial Conditions Required Subroutines

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

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MOST USED CONSTANTS

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Command Line To begin type auto

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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.

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b.abc

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d.abc

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Using the comand: plot(abc)

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Looking at the solution for label 7

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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

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

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time u1 u2 u3 lor.dat

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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.

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Makefile

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

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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.

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lor

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Graphical User Interface. or its Appendix C of manual contains help for GUI

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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

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