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Introduction This week we will use MAPLE to investigate some of the more subtle properties of 2-D continuous flows. Exercise 1. The Two-eyed Monster Open MAPLE in our classfile and enter the worksheet from Fig.1. How many rest points do you see? What kind are they? Change ini3 to x(0)=-1, y(0)=-0.4 and switch the time interval in the last line from 0..12 to 0..5. What do the relative lengths of the three trajectories tell you about the strength of the vector field near each initial condition? Try different initial conditions and see if you can locate any other fixed points. Exercise 2. A Sensitive Center The example of lions and gazelles in Lab 2 provided us with a fixed point called a center. The trajectories close to the fixed point were all closed, i.e. (cycles). Sometimes such fixed points change their character when the corresponding vector field is changed, even slightly. Consider the following set of differential equations:
x' = -y+ax(x^2+y^2)
When a is very small this system of differential equations can be thought of as a small perturbation of the system that corresponds to a=0:
x' = -y
Enter the worksheet from Fig.2 which plots the vector field when a=0. The origin (0,0) is clearly a center. See what happens when you change the values of a to 0.001 and -0.001. Exercise 3. The Simple Pendulum We will practice extracting useful information from the plot of a vector field. As an example we will use the differential equations that describe the motion of the simple pendulum, the most fundamental mechanical system in physics. These equations are :
x' = y
Here x represents the angular displacement of the arm of the pendulum from the vertical and y measures the angular velocity of the arm. Enter the worksheet from Fig.3 which plots the corresponding vector field. Print out a copy of this vector field and on the same piece of paper write down the answers to the following questions. (a) How many types of rest points do you see and which types are they? (b) Describe the state of the pendulum (the position of the arm) at each type of rest point. (c) Describe what the pendulum is doing along the trajectory starting from initial condition 1. (d) Describe what the pendulum is doing along the trajectory starting from initial condition 2. Exercise 4. The Pitchfork Bifurcation Consider a set of differential equations that depends on a parameter. For example
x' = sx-x^3
As we change the value of s the vector field will change in response. Sometimes these changes are small and the system will behave in much the same way. Other times the system behaves in a decidedly different way and we say that a bifurcation has occured. One type of bifurcation that can occur, and is easily recognized, is a change in the number of rest points. Enter the worksheet from Fig.4 which plots the vector field for s=-0.5. Choose any negative number for s and you will get the same general behavior : every point is attracted to the origin. See what happens when you let s=0 and s=0.1. Print the vector field for s=-0.5 and for s=0.1. On one of these plots describe the differences between the two vectorfields. |