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MAT 331, Spring 2000
Due Wednesday, May 3
For the project # 2, unlike for the previous ones, you have a choice of topics. Please pick one from the following list and email me about your choice. Please do it no later than on April, 25. You will be expected to present a 15 minutes description of the project to the class at the very end of the semester. You are welcome to work in (small) groups.
In your presentation, you should include a number of relevant pictures and
graphs, preferably (but not necessarily) produced by Maple. These pictures
should be used to illustrate your exposition, not merely included without
comment or reason (remember: ``a picture is worth a thousand words, but a
thousand pictures are worthless''). Please pay attention to clarity of
exposition; do not merely hand in an annotated Maple worksheet. While
including relevant Maple commands is useful, your goal is to explain what
you are doing from a mathematical point of view, not to describe how
to use Maple to perform a certain task. You need not (although you can)
state or prove the relevant theorems, but you should explain how you are
using them to do your analysis.
Choice of topics for the project
- Gliders and Differential Equations
In class, we have been discussing a system of differential equations which
approximately models the flight of a glider. In this project, you are to
explore a similar system, which models such a glider with a propeller or
small engine attached. In this case, our model is
Here, as in class, v>0 is the speed of the glider through the air (not the horizontal speed), and
is the angle the nose makes with
the horizontal direction. Note that we have fixed the drag coefficient to
be 0.3, but have added an additional term k to account for acceleration
caused by the propeller.
You are to analyze and classify the solutions of this system for all
(note that k=0 was covered in class). This means that you should
find the various ranges of k where the behavior is qualitatively
Such analysis should include a discussion of the existence of fixed points
(equilibrium solutions) and their linearizations (i.e., a discussion of
eigenvalues and eigenvectors and how this relates to the solutions), as well
as a discussion of the long-term behavior of the solutions.
In addition to the description of the trajectories in the
-plane, you should also relate the solutions to properties
of glider flight. For example, discuss whether the glider eventually must
crash or can stay aloft indefinitely, whether the glider does loops, etc.
- Competing Species
Consider the competing species model, where x(t) and y(t) represent the sizes of two competing populations at time t.
Find all the critical points of the system. At each critical point, calculate the corresponding linear system and find the eigenvalues of the jacobian. Then, identify the type and stability of the critical points.
Use several initial data points in the first quadrant to draw the vector field and then phase portrait of the system. Identify the directions on the trajectories that correspond to time increasing.
- Suppose the initial state of the population is given by
x(0)=2.5, y(0)=2. Find the state of the population at
- Explain why practically there is no ``peaceful coexistence'', i.e., with the exception of an atypical set of initial conditions (the so called separatrix curves), one or the other population must die out. For which nonzero initial populations there is no change? Sketch the separatrices, i.e., the solution curves that approach the unstable equilibrium point where both populations are positive and form the boundary between the solution curves that approach each of the two stable points where only one population survives.
- Complex differential equation
Consider the differential equation for complex-valued function z=z(t):
is a real parameter.
- Show that the equation is rotationally invariant, i.e., if we let
then w satisfies the same differential equation.
- Plot the trajectories of the differential equation for parameter values
Determine the direction of the trajectories. What kind of symmetries can you see in the plots?
- Can you detect, by looking at the plots above, whether the parameter
passed through a bifurcation value? What exactly is the qualitative change in the behavior of the solutions?
- Estimate the actual bifurcation value of the parameter by plotting the trajectories of the system for
satisfies the differential equation
and that critical point of this differential equation corresponds to a solution of the initial one that stays fixed distance from the origin. By finding the critical points of the differential equation for ,
explain the results of (c) and (d).
- Fractal of preset dimension
In this project, you are asked to define a fractal of your choice
whose box-counting dimension is
Describe clearly how
you construct such a set and prove that its dimension is the
requested number. Also, plot the first few approximating
curves to the fractal.
- Julia set of a transcendental function
We remind that, by definition, Julia set of a map
is boundary of the set of the points that do not approach infinity after f is repeatedly applied.
Consider the family of maps
Ec(z)=cez, where c is a complex parameter.
- Write a procedure that computes and draws the Julia set of the member of the family, given the value of the parameter c.
- Compute the Julia set for the following values of the parameter:
c= .2; 1; -4+i; 1+2i.
- Let c be real for a moment. For which c values would you expect the corresponding member of the family to have an attracting fixed point? For which c values would you expect all real numbers to have orbits that escape? Now plot the Julia set for either situation. Do you see any changes in the Julia set structure? Find the exact value of c when the bifurcation occurs.
- Lorenz system
Consider the system
This system is known as Lorenz system and is related to a description of 2-D flow of fluid (and things like turbulence). Study the system as deeply as you see fit, but make sure to mention the following effects that the system is notorious for: sensitive dependence of the solutions on the initial conditions (so called butterfly effect; plot some convincing graphs!) and presence of so called strange attractor, which is neither a fixed point, nor a limit cycle (again, show convincing plots). You might want to use DEplot3d alongside with DEplot.
- The Game of Life.
http://www.math.com/students/wonders/life/life.html and learn the rules of the Game of Life. Implement the game with Maple. Show the behavior of a few interesting objects of your choice (but do include at least Gliders and Guns!).
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