Question:
Differential Equations : Bifurcations in Linear Systems
We have studied techniques for solving linear systems. Given the coeffi-cient matrix for the system, we can use these techniques to classify the system, describe the qualitative behavior of solutions, and give a formula for the general solution. In this lab we consider a two-parameter family of linear systems. The goal is to better under- stand how different linear systems are related to each other, or in other words, what bifurcations occur in parameterized families of linear systems. Consider the linear system
dx/dt = ax + by
dy/dt = -x -y.
where a and b are parameters that can take on any real value. In your report, address the following items:
1. For each value of a and //, classify the linear system as source, sink, center, spiral sink, and so forth. Draw a picture of the ab-plane and indicate the values of a and b for which the system is of each type (that is, shade the values of a and b for which the system is a sink red, for which it is a source blue, and so forth). Be sure to describe all of the computations involved in creating this picture.
2. As the values of a and b are changed so that the point (a, b) moves from one region to another, the type of the linear system changes, that is, a bifurcation occurs. Which of these bifurcations is important for the long-term behavior of solutions? Which of
these bifurcations corresponds to a dramatic change in the phase plane or the x(t)- and y(t)-graphs?
Your report: Address the items above in the form of a short essay. Include any computations necessary to produce the picture in Part 1. You may include phase planes and/or graphs of solutions to illustrate your essay, but your answer should be complete and understandable without the pictures.