MTH 480 LAB 2

In this lab you will explore computer tools helpful in qualitative analysis of systems of ODEs.

Turning in your work is not mandatory but can be used for extra credit. If you choose to do so, make sure all the steps are labelled properly and described well. A bunch of graphs stapled together does not constitute a solution worthy of extra credit.

Instructions follow below. Please note approximate timing of the steps and demonstrations.
Set-up phase: (14:00-14:05)
  1. Log in to the computer using your ONID account. The computers can be started in Windows or in MAC mode.

    I will be presenting in Windows so if you prefer to use MAC, try to follow what I am doing but I won't be able to assist you.

  2. Locate the basic tools from LAB1:
    1. Locate a browser (Firefox or Chrome) and locate class website. Check if PPLANE by John Polking is working for you. If not (Java !) then copy the MATLAB files (for example to Desktop).
    2. Find MATLAB: as icon on desktop or in Start->Programs (Search for Matlab)
Transformations with MATLAB: DEMONSTRATION 14:05-14:20
  1. Learn the syntax of matrices and vectors in MATLAB by doing (or watch my demonstration)
    A = [3 1; 2 5]
    A (:,1)
    A(1,:)
    u = [2 1]
    u = [2,1]
    u = [pi,1]'
  2. What does A do to a vector ?
    theta = linspace(0,2*pi)
    u=[cos(theta);sin(theta)];
    size(u)
    plot(u(1,:),u(2,:)); axis equal;
    u=A*u;
    plot(u(1,:),u(2,:)); axis equal;
  3. Now do it again ...
  4. Explore matrices from your (past and current) HW or class examples

    Sometimes it may be necessary to start with an ellipse rather than a circle.

Exploring with PPLANE: Plot slope/direction fields and explore bifurcations (14:20-14:35)
  1. Explore with PPLANE the familiar system X'=AX with A=[2 1; 1 2] (in MATLAB format)
  2. Repeat for matrices A=[a 0; 0 1];.

    Consider all possible values of the parameter a ranging over real numbers. What are the values of a for which bifurcations arise ?

  3. Offline - you can use this tool in solving your next HW.
Eigenvalues with MATLAB and exploring phase portrait with PPLANE for a defective matrix: DEMONSTRATION 14:30-14:35

A = [2 1; 1 2]
eig(A)
[v,e]=eig(A)

  1. Study eigenvalues and eigenvectors for A = [0 1; -1 0]
  2. ... and for A = [1 0; 0 0] ...
  3. Now, the defective matrix A = [1 1; 0 1]
    What are the eigenvalues and eigenvectors ?
  4. Perturb A a bit and study A1 = [1 1; 0.01 1]. Compare the spectrum to that of A.
  5. Repeat for A2 = [1 1; 0.1 1]
  6. Now open PPLANE and study direction fields and trajectories for the systems with A2, A1, A. You can see how A arises as a limit of
    Aeps=[ 1 1; eps 1]
    when eps goes to zero.
    What is the qualitative behavior of the direction fields in that limit ?
    This type of an equilibrium is called a "degenerate node" .

To get my attention, please raise your hand.