Question

Please be as descriptive as you can

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 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 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. dy

Answer 1

Let ☺ (97-99) = ( 1 ) ) X = (2) then de our system is i = Aā. Eigen values of A: SO -a -b 0 → (ec-a)(1+1)+b=0 Y+ » x+(-a) x-a+b=0 + x = 2-1 I (-1-416-a) 2 so xao = (2-1) I Ja²+1-2a-ubtya - (0-1) = (a + 1)²-4b 2 2 90 2 = (2-1) + (0-1)²-4b 22 = (2-1)-Q-1) 2-4b > 2 2 as SO a>l we get 4 a SO a sink. Let D= (2-1) ²-4b. Case (i ► If (a-1) >0 then a 22 > 0 (a-1 > D for & (2-1) ² > 4b b< (2-1² 21, 2 >0 so Source ac1 then as, 22 <o for al Ee be (a-1) we get cannot have a>0 & X₂<0 or vice verse so we Case (u) D<o x =((2-1) + 8 D/2, 2 = ((a)-i Fo)/2 If a<I then we get a spiral sink ce axi & b>(2-1) for we get a spiral sink (2) If axt. & byla- get a spiral source (3) If a=t then get a center 3 we do not get sadde Loe Ee bo

2 = 220 then Caselui) D=0 2 = 22 = a-1 (1) Ifasi then Almost spiral sink then Almost spăral get (2) If all 1,= 42<o then get source

ob center spiral sink Source = spiral Source sink spiral Almost sink Almost spiral source.