MTH 480 LAB 3

In this lab you will explore qualitative analysis of planar systems of ODEs moving towards nonlinear systems.

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, LAB2:
    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)
Random planar systems with MATLAB: DEMONSTRATION 14:05-14:15
  1. How to set up random matrices and plot individual points in the trace-determinant plane
    % Prepare for plotting in trace-determinant plane
    tau=linspace(-4,4);plot(tau,tau.^2/4);
    % decrease that randomness so you can reproduce your experiments !
    rand('seed',myOSUID);
    A = rand(2,2), A = randn(2,2),
    % now how do you get numbers with a given mean and standard deviation ?
    % compute trace and determinant of the matrix
    hold on;
    % plot the point in the trace-discriminant plane and watch where it lands
    scatter(t,d,'*');
  2. Now repeat the above $M=10$ times recording whether the points lie in the stable or unstable part of the trace-determinant plane. (You are doing Monte-Carlo simulations here.)

    What we are doing here is estimating probability that for the random matrix from a given distribution the origin is (for example), a saddle.


    You could use a loop as below to do this.


    for n=1:M %%% do your thing %%%% end

    Or do this by hand. REPORT TO your ROW MANAGER how many fell in each category.
  3. Now do it again, this time for a matrix as in
    1. Section 4.3 of your book
    2. Problem 3 from Chapter 4.
  4. If you know enough probability, can you calculate the probability that for the random matrix from uniform distribution on (-1,1) the origin is a saddle ?
Linear, linearized, and nonlinear systems with PPLANE: DEMONSTRATION 14:35-14:40
  1. Derive the linear first order ODE system equivalent to x''+kx=0
  2. Explore with PPLANE and confirm the expected behavior (harmonic oscillator).
  3. Now do the same for the damped system x''+bx'+kx=0.

    Fix k=1 and consider all possible values of the parameter b ranging over real numbers. What are the values of b for which bifurcations arise ?

  4. Now explore the system which is NOT a linear system with constant coefficients
    1. x''+x = 5 sin(a*t)

      What happens as you change a ? What does the term on the righ hand side mean physically ?

    2. Now consider (A) x''+x =0 and (B) x''+sin(x)=0. Record carefully the similarities and differences for these two systems. What are the equilibria for (A) and for (B) ?

      The system (A) can be seen as a linearized system (B) in which nonlinearity is replaced by the first term in Taylor expansion about an equilibrium. Which equilibrium ?
      What other possible approximations to sin(x) can you consider ?

    3. Come up with some other ways to modify x''+bx'+x=0 so the system is no more linear with constant coefficients.

To get my attention, please raise your hand.