Math 321.001: Lab 2

Two Parameter Families of Differential Equations


This lab is due Monday, November 8, 2016 in class.   The concepts in this lab are related to sections 1.6 and 1.7.  You may use any technology that you have available: a spreadsheet, Mathematica, Matlab, etc. You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.

Group Work Rules: This lab should be completed by a group of 3 to 5 students.  Turn in one copy of the completed assignment with the names of all group members.  After the graded assignment is returned, copies should be made so that each group member has a copy to keep.

In this lab you will investigate three families of first order differential equations that each depend on two parameters, a and r. The goal in each case is to give a sketch of the "bifurcation plane" (or "parameter plane"). The bifurcation plane is a picture in the a,r-plane of the regions (i.e., the values of a and r) for which there are different types of phase lines (specifically, different numbers of equilibria). The curves that separate these regions are the parameters where bifurcations occur (the "bifurcation curve"). For the first two families use analytic methods: first find all of the equilibrium points and determine their types (these equilibria will, of course, depend on a and r). Then sketch the bifurcation curve(s) and the regions in the a,r-plane where you find different numbers of equilibrium points, and, in each different region, draw a representative picture of the phase line for any a and r value in this region. Finally, describe in a sentence or two the bifurcations that occur as you move from each region to an adjacent region. For a sample of this type of analysis, look at the example below for the family y' = r + ay.


Family #1. y' = r + ay - y2     Hint: You can use the Phase Lines tool to help out on this one.


Family #2. y' = r + ay2


For the final family, using analysis as above becomes very complicated.  [Extra credit if you can work that out!!]  As an alternative, just use the Phase Lines tool to view the graphs, phase lines, and bifurcation diagrams for the members of the family. Then use these pictures to give an approximate sketch of the bifurcation plane with the corresponding phase lines included.


Family #3. y' = r + ay - y3


Example: Bifurcation diagram for y' = r + ay


The equilibrium points are where y' = 0, and so are given by r + ay = 0.    If  a = 0 that requires r = 0 as well, and then every y is an equilibrium point.  Otherwise,
y = -r/a
is the only equilibrium point.  We can classify this point as follows.  First, suppose a < 0.  Then the graph of  y' = r + ay versus y is a line with negative slope, so y' decreases as y increases, as in the left-hand graph below.  In particular, since y' is zero at ye = -r/a, it must be positive for y less than ye and negative for y greater than ye.  Therefore the equilibrium point is a sink.  This is shown on the phase line diagram on the right-hand graph below.

When a < 0

y' versus y
Phase line 


On the other hand, if a > 0,  the y' versus y graph is a line with positive slope, thus y' increases with y.  This time we conclude that y' is negative for y less than  ye and positive for y greater than  ye, as illustrated below.  In this case the equilibrium point is a source. 

When a > 0

y' versus y
Phase line 


To summarize:

So a bifurcation occurs as a passes through 0. There are three possible ways this can occur:

The bifurcation plane picture is: