MTH 480 LAB 1

In this lab you will explore basic computer tools helpful in understanding stability for scalar 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: (14:100-14:10)
  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:
    1. Locate a browser (Firefox or Chrome) and locate class website. Check if DFIELD by John Polking is working for you.

      These instructions are available at http://www.math.oregonstate.edu/~mpesz/teaching/480_W14/lab1.html

    2. Find MATLAB: as icon on desktop or in Start->Programs (Search for Matlab)
Exploring with DFIELD: Plot slope/direction fields and explore bifurcations (14:10-14:30)
  1. Get familiar with DFIELD using the following steps
    1. Click on the icon (and on the OK, Run etc. buttons that pop up)
    2. Get familiar with the different windows that appear
    3. In the window "DFIELD Direction Field Window" click at some point to see a trajectory (orbit).
    4. In the Window "DFIELD Equation Window" change parameters a,b and those of 'The Display Window'
  2. Use DFIELD to explore the equilibria for x'=ax+t, and x'=ax-t.
    Describe qualitatively what you see for different values of a.
  3. DEMONSTRATION. Explore equilibria and qualitative dependence on parameters for the ODE x'= x2 + b (type it as x^2+b)
    1. Find the equilibrium depending on b and classify it. Plot the phase line. (You can use DFIELD to guide you)
    2. Change b and do it again. And again ...
    3. Record your explorations on the (b,x) plot. Describe the nature of bifurcation (it is called a saddle node).
  4. Use DFIELD to explore the solutions to x'=ax^2+b

    In particular, consider the terminal velocity problem with a quadratic drag function, in which x denotes velocity, a=k/m, with k=0.2 denoting the drag coefficient, and b=-g=-10kg m/sec^2 denoting the gravity acceleration. If mass m=100kg, find the terminal velocity by hand (it should have the direction of gravity) and confirm using DFIELD. [You will have to search for the appropriate window].

  5. Now repeat 3). for the functions listed below and/or those from your homework
    1. x' = b - x^2 (saddle node bifurcation)
    2. x' = bx - x^2 (transcritical bifurcation)
    3. x' = bx - x^3, for b>0 (pitchfork bifurcation, supercritical)
    4. x' = bx + x^3, (pitchfork bifurcation, subcritical)
Exploring with MATLAB: Solve ODEs symbolically and plot trajectories.(14:30-14:50)
  1. Open MATLAB window and learn to navigate.
  2. Locate the Command Window, Help Window etc.
  3. In Command Window, type doc dsolve and read instructions. Type the examples from the help window in Command Window and learn how to solve ODEs symbolically with dsolve.
  4. Learn how to plot. Type doc plot .
  5. DEMONSTRATION Use dsolve to get a solution for the terminal velocity problem.

    Pick some initial condition, for example, x(0)=0 for the terminal velocity problem. Use dsolve to get the closed form solution, and plot the trajectory using plot

  6. Compare with trajectories obtained with DFIELD.
  7. Repeat for the problems in 5.

To get my attention, please raise your hand.