Math 3226

Laboratory 4A
Bifurcations in Linear Systems

Consider the following system of first-order linear differential equations:

where a and b are real parameters.

1 For each value of a and b, classify the system's equilibrium points (as sinks, spirals, etc.). Draw a picture in the ``ab"-plane, and indicate the regions corresponding to the various types (for instance: shade all (a,b) values for which the origin is a sink red, the values for which the origin is a spiral sink orange, and so forth). Be sure to include all the computations necessary to draw the picture.

As the values of a and b are changed and the point (a,b) moves from one region to another, the ``equilibrium type" changes. Such a change is called a bifurcation. A typical bifurcation occurs when a harmonic oscillator changes from being underdamped to being overdamped.

2 Which of the bifurcations in your picture affect the long term behavior of the solutions?

3 What is happening at the boundary between regions?

This laboratory is based on Laboratory 3.1 in ``Differential Equations" by Blanchard, Devaney and Hall.