Before you can use AUTO, you must prepare your system for it. You must start your bifurcation analysis from either a fixed point of your model or a periodic orbit. AUTO seems to work best when you start from a steady state, but I have had success starting at periodic orbits. If you want to start at a steady state, find one and integrate so that the system is at rest. If you want to start at a periodic orbit, then find one and make sure that the total integration time is the ``period'' of your orbit. This is what the AUTO interface uses as an approximate starting period. Ther are several ways to do this; the best is to use the boundary value solver of XPP but just plain old integration often works fine. For the example file Morris-Lecar the parameter of interest is i and this has been set at a negative value so that the system has a stable rest state. The variables have been initialized to their rest states as well. Once you have prepared the problem as such, you are ready to run.
The first thing you should do is tell AUTO which parameters you might use in the bifurcation analysis. Up to 5 are allowed. Click on ``Parameter'' and a list of 5 parameters will appear. Type in the names of the parameters you want to use. The default is the first 5 or fewer parameters in your file. If you have fewer than 5 parameters, only the available ones will appear.
There are 7 choices
After clicking, a new window pops up with the following items:
Suppose you want to get plots at specific values of parameters or at fixed periods of a limit cycle. Then you can click on ``User'' which produces a menu 0-9 asking you how many points you want to keep. Click on 0 for none or some other number. A new window will appear with slots for 9 items. You can type in anything of the form:
Click on ``Run'' to run the bifurcation. Depending on the situation, a number of menus can come up. For inital exploration, there are two choices, starting at a new steady state or periodic. If you are running the example, click on the steady-state option and a nice diagram will show up and a bunch of points will move around in the stability circle. These indicate stability: for fixed points, they represent exponetials of the eigenvalues; for periodics, the Floquet multipliers. Thus those in the circle are stable and those out of the circle are unstable. Bifurcations occur on the circle. The outer ones are ``clipped'' so that they will always lie in the square, thus you can keep count of them.
The diagram,itself, has two different lines and two different circles. Stable fixed points are thick lines, stable periodics are solid circles, unstable fixed points are thin lines, and unstable periodics are open circles. Additionally, there are crosses occasionally dispersed with numbers associated with them. These represent ``special'' points that AUTO wants to keep. There are several of them:
You can use these special points to continue calculations with AUTO. The ``Grab'' item lets you peruse the diagram at a leisurely pace and to grab special points or regular points for importing into XPP or continuing a bifurcation calculation. Click on ``Grab'' and stuff appears in the info window and a cross appears on the diagram. Use the left and right arrow keys to cruise through the diagram. The right key goes forward and the left backward. At the bottom, information about the branch, the point number, the type of point, the AUTO label, the parameters, and the period are given. The points marked by crosses have lables and types associated with them. The type is one of the above. The label corresponds to the number on the diagram. If point is positive, it is an unstable solution and if it is negative it is stable. As you traverse the diagram, stability is shown in the circle.
You can traverse the diagram very quickly by tapping the (Tab) key which takes you the special points only. Type (Esc) to exit with no action or type (Enter) to grab the point. If it is a regular point (i.e., not special) then the parameters and the variables will be set to the values for that point within XPP. You can then integrate the equations or look at nullclines, etc. If you grab a special point, then you can use this as a restart point for more AUTO calculations, such as fixed period, two-parameter studies, and continuations. Then, you can run AUTO again. Bifurcation daigrams are cumulative unless you reset them in the ``File'' menu. That is, new stuff is continually appended to the old. The only limit is machine memory.
If you grab a special point and click on ``Run'' several possibilities arise depending on the point:
Before running, after a point is grabbed, be sure to set up the correct axes and ranges for the parameters.
Any calculation can be gracefully stopped by clicking on the ``Abort'' key. This produces a new end point from which you can continue. Note that if there are many branches, you may have to press ``Abort'' several times.
Clear clears the diagram and reDraw redraws it.
File allows you to do several things:
Table of Contents
Main Menu Items
ODE Files and Examples
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File Menu
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XPP Basics
Nonlinear ODEs
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Three-dimensions
and Beyond
Phase Equations
Chaos