Many texts illustrate the theoretical explanation of bifurcations in one-dimensional systems with simple graphs. This extends, in a simple and natural way, to two-dimensional systems.

The reason is not hard to see. For a one-dimensional system, a periodic orbit is determined by fixed points of a composition of a map, and may be seen on a graph as intersections of the line y=x with the graph of that composition f_n. Alternatively, it is determined by the zeros of the function x-f_n. Zeros of functions of one variable are usually discrete points.

For a two-dimensional system, the map has two components (functions f and g of variables x and y) and a fixed point is the simultaneous zero of the functions x-f_n and y-g_n. Zeros of functions of two variables are usually curves in the plane; simultaneous zeros are intersections of two curves and usually discrete points.

This observation is at the heart of the comptational algorithms used to locate fixed points numerically in "Chaos for Java". The zero curves shown in the lower picture are found by computing the signs of these functions and then searching for the sign changes. Subsequently, the points are refined using Newton's method.

When passing through a bifurcation, the curves change their structure. The remarkable thing is that these changes are the same in two dimensions as in one, providing a useful visual aid. This is why the curves may be displayed, as well as their intersections.

The simple theory behind this is the subject of a paper which may be downloaded, as a preprint in pdf form, from this link. You may need Adobe Reader to read it.

These pages are hosted by Mathematical Sciences Institute, ANU, and maintained by Brian Davies .

e-mail to Brian.Davies@anu.edu.au
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