Mathematical Modeling
Spring 2012

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MATH 485/585 - Spring 2012 - Assignments

D2L Course Page

• Due by Friday January 27th 11:59PM Online (D2L) quiz on Plagiarism and related issues which will be posted soon.
  More than half of the grade for this course is for teamwork on a modeling project, based on research published in the scientific literature. The quiz provides a review of acceptable practices regarding authorship and references to the literature. It will also help you avoid plagiarism. This quiz will be graded and the number of attempts is unlimited. It is recommended that you familiarize yourselves with the guidelines described in Miguel Roig's book, entitled Avoiding plagiarism, self-plagiarism, and other questionable writing practices: A guide to ethical writing.

• Due Tuesday, January 24th: Problems # 1 and 2, page 7. Submit your write ups in my D2L drop box called HW1). I will provide strict comments and suggestions for improvement, but will grade this first assignment leniently

• Homework 2 will be due Thursday 2/16:

  Part I - LaOla

  Download LaOla.m and LaOla.fig and save them in a directory that you will use from within Matlab.  You should then be able to start Matlab and change the directory to the one where tou saved the two files you downloaded.  To start the program, type "LaOla" at he Matlab prompt.

  Explore the behavior of the model and try to answer the questions on pages 12 (at the bottom) and 13.  In particular, please write up (in complete sentenses) answers to the following:

  (a) What do you observe when you change R and R_m? 

  (b) What is the speed of the Wave?

  (c) Change the Activity threshold A.  Why does the wave die when A is too large?  What other parameter can you modify to prevent that from happening?

  (d) What can you change to make the wave propagate to the left (i.e. counter clockwise)?

  Part II - Dimensional Analysis

  Do problem 15 on page 36 and 23 on page 38.

  Part III - Phase plane and Solving ODE's with matlab

  Use the code samples in DampedPendulumSystem.m and M485Hw2.m as a starting point to duplicate the phase plane plots on pages 26 and 31 in the course notes.  Please note, if you want to add more curves to an existing graph, you can issue the command "hold on", and then use more plot commands to add additional curves.  When you want the next plot command to create a fresh plot, you can issue the command "hold off".  You should not try to include the little vector fields in the plots that you produce, just the curves showing the phase plane trajectories.  

  You should submit a listing of the commands that you used to generate the plots, as well as the plots that you produced.  You can save the plots from the file menu in the plot window using "save as" and chosing a jpeg or bmp filetype.  That image file can then be included in your document.  (You can also just print out the plots and hand in hard copy). 

• Homework 3 (Due date TBA)

  Part I:

  You can use Matlab to produce a bifurcation diagram for the logistic map similar to the figure that is on page 62 of the notes (reproduced quite poorly).  The trick is to vary the value of the bifurcation parameter a and for each value of the parameter, iterate the map x_n+1=f(x_n)=ax_n(1-x_n) a few hundred times so that the initial transient behavior passes, and then plot the next bunch of points that are produced from iterating the map.  In this case bunch should be something like 64 or 128 or 256.  These points are plotted together onto a single figure where the horizontal axis is the value of a, and the vertical axis is the value of x.  The following Matlab code will do the job:

  figure(1) % This brings the plot window to the front
  hold off  % clear earlier plots
  newplot
  hold on % now the plots following will be superimposed
  for a=0:.002:4 % loop over a values
  x=0.1;            % an arbitrary initial x
  for j=1:500 x=a*x*(1-x); end % iterate past the initial transient
  xx(1)=x;   % Now start storing the values to plot
  for j=2:256 xx(j)=a*xx(j-1)*(1-xx(j-1)); end  % iterate the map storing the values
  plot(a*ones(256,1),xx,'.','MarkerSize',1);   % plot the points, the first coords are all a, use a small marker.
  end % end of the loop on a values

  You can zoom in on a portion of the plot using

  axis([2.7,3.7,.3,1]) % the argument is a vector (xmin,xmax, ymin,ymax)

  Play with the picture and the code to produce plots similar to those in the course notes. 

  Once you are comfortable with that, you should print out your plots (you can save them in various formats from the file menu in the plot window (use save as...).  Now you can produce bifurcation plots for a different iterated map.  The new map is called the Henon map.  It is a map from R² to R².  It is given via (x_n+1,y_n+1)=(y_n+1-ax_n²,bx_n).

  Look at the page on wikipedia on the Henon map.  There are two figures that you should reproduce.  One is labeled Hénon attractor for a = 1.4 and b = 0.3, and the other is labeled as "Orbit diagram for the Hénon map with b=0.3."  The latter plot is a bifurcation diagram holding b fixed and varying a while plotting just the x value.  It captures the very complex behavior of this map.  The former is a plot of points (x,y) for an interesting pair of values of the parameters (one where there is chaotic behavior).

  Use the Matlab code above as a starting point for producing these plots.

  Your submission should include the figures for the logistic map, and those for the Henon map as well as the Matlab commands that you used to generate the plots.  If you put the Matlab into a file, you can run it from within the Matlab editor using the run command on the Debug menu. 

  Part II:

  Do problems #4, 6, 11, 13  on pages 69-72 and #4 on page 84.