AUTO is a software package for numerical continuation. The AUTO web page at Concordia University is http://indy.cs.concordia.ca/auto/.
The newest version of AUTO is AUTO07p. The previous version was AUTO2000. The examples presented here were run with AUTO07p.
A C file that can be used by AUTO is created by the command
The name of the C file will be [name]_avf.c, where [name] is the name of the vector field given in the vector field file. Only the AUTO function called "func" will be complete. A "stpnt" function will created to serve as a template. The DefaultInitialConditions of the variables will be used for the values in the "stpnt" function; the file must be edited by hand to define an actual known solution. The remaining functions will be stubs that do nothing but return 0.
Any user functions defined in the vector field file are ignored in AUTO command.
lang |
By default, C code is generated by the VFGEN auto command.
If the option lang=fortran is given, FORTRAN code is generated,
for AUTO07p.
The name of the file containing the code is [name]_avf.f.
Default: lang=c |
We consider this variation of the van der Pol equation:
εx' | = | y+x-x^{3}/3 |
y' | = | a-x |
The file created by
The AUTO script vanderpola.auto first does continuation in a of the equilibrium that is (0,0) when a = 0. (We don't really need to use AUTO for this, since the formula for the equilibrium is simply x = a, y = -a+a^{3}/3.) There is a Hopf bifurcation when a = 1; the script computes the family of periodic orbits that arises from the Hopf point.
Here are plots (in the (x, y) plane) of the some of the periodic orbits computed by AUTO.
The newest version of AUTO is AUTO07p. This version of AUTO allows the equations file to be written in either C or FORTRAN. VFGEN can generate code for either case.
The file created by
In this example we first use VFGEN to convert the Mackey-Glass delay equation into a system of ordinary differential equations, and then use AUTO to find some of the bifurcations of the system. (See the DDE23 command or the DDE_SOLVER command for examples in which the Mackey-Glass equation is solved with a differential delay equation solver.)
We use the version of the Mackey-Glass equations defined in this file: mg.vf. The DefaultInitialCondition and DefaultHistory of the state variable x are both set to 1, because in the AUTO command, the DefaultInitialCondition is used for the default values of the STPNT function, and in the Mackey-Glass equation, x=1 is an equilibrium solution.
First, we use VFGEN to approximate the delay equation as a system of ordinary differential equations:
Next, we use VFGEN to create an AUTO C file for this system:
We also need an AUTO constants file: c.mg_2ode_avf. (This was created by hand.)
We use an AUTO script, mg.auto, to compute several bifurcation curves, in which the bifurcation parameter is the delay τ. The script is executed with the auto command:
AUTO finds a Hopf bifurcation near τ=4.703, and a sequence of period doubling bifurcations at τ=13.05, τ=15.78, and τ=16.31. Presumably, this is the beginning of a period doubling cascade to chaos.
Here is a plot of the bifurcation diagram, created by the Python script mg_bif_plot.py.