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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.

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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:

1 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

2 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

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 p 1 = 0.15 p 1 = 0.33 p 1 = 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 Parameters Initial Conditions Required Subroutines

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

9 MOST USED CONSTANTS

10 Command Line To begin type auto

11

12 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.

13 b.abc

14 d.abc

15 Using the comand: plot(abc)

16 Looking at the solution for label 7

17 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

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

19 time u1 u2 u3 lor.dat

20 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.

21 Makefile

22

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

24 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.

25 lor

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

27 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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