LAB 3.1 Bifurcations in Linear Systems In Chapter 3, 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 = ax + by dt dy =-x-y, dt 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 b, 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 compu- tations 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.