Math 308 Computer Projects

• \frac{dy}{dx} = x y
• \frac{dy}{dx} = y^2
• x \frac{dy}{dx} + \sin(x) y = x^2

1. Graph the direction field in the neighborhood of the origin
2. Graph approximate integral curves for 10 different initial conditions (you can use dfield8.m)
3. Solve the differential equation, using Euler's method (e.g. with euler.m), for three initial conditions and two values of the stepsize, $h$
4. If possible compute the error, either using the exact solution (if available) or a very accurate numerical solution (from matlab)

Do your own work.

You will need the files lorenz_demo.m, g.m, and pplane8.m from the Matlab m-files page.

Phase Plane analysis. We will investigate the following systems of equations - specifically, the behavior of the systems as the various parameters are changed.

I. Two dimensional Analysis. In the first case, we can use phase plane analysis (nodes, saddle points, periodic orbits, etc).

2x2 Lotka-Volterra (LV) system:

\frac{dx}{dt} = x(\alpha-\beta y)
\frac{dy}{dt} = -y(\epsilon-\delta x)

1. Assuming the parameters (\alpha,\beta,\epsilon,\delta) are all positive, find the critical point of the LV system in terms of the parameters.
2. Identify the type of critical point (using eigenvalues). Illustrate each type of critical point with a phase plane plot, and component plots, including solution curves.
3. Extinction occurs if one or more of the species vanishes as t \rightarrow \infty. Can this happen for solutions of the LV equations? If so, give an example (initial conditions, phase plane plots and component pltos).

II. Three dimensional Analysis. In the second example, with three dimensions, things are much more complex (and interesting!). We can see evidence of a phenomenon called chaos

3x3 Lorenz Dynamical System:

\sigma=10, \rho=28, \beta=8/3
\frac{dx}{dt} = \sigma (y-x)
\frac{dy}{dt} = x (\rho - z) -y
\frac{dz}{dt} = x y -\beta z

Change the parameters \sigma, \rho and \beta from the values given above, and examine the behavior (especially near the critical point). You may change one or more values at a time. Find as many different sets of parameters that give rise to fundamentally different behavior. Describe the behavior in the phase plane, as well as in the component plane (x(t) vs t, y(t) vs t, z(t) vs t).

IF YOU HAVE QUESTIONS, PLEASE EMAIL ME, OR ASK ABOUT IT IN CLASS...

The equations of motion for the double pendulum are described here.

The details of the derivation can be found here.

The goal of the third project is to recreate the motion contained in the above document. Use Matlab to numerically integrate the system, with the given parameters and initial values, and compare the component plots to those given in the reference.

Investigate several other cases and determine if the motion appears to be chaotic.

Check out the literature on the web relating the double pendulum and chaos, and summarize your findings.

m1=2, m2=1, L1=1, L2=2, g=32.0, \theta_1(0)=1.57,

the two component plots for \theta_1 and \theta_2 are shown below:

\theta_2(0)=3.14, for 10 seconds.

m1=2, m2=1, L1=1, L2=2, g=32.0, \theta_1(0)=1.57, \theta_2(0)=3.14, for 100 seconds.

the two component plots for \theta_1 and \theta_2 are shown below:

The total energy (E=T+V) vs time is plotted below:

min(E)=63.9221, max(E)=63.9331,
(max(E)-min(E))/(max(E))=1.7215e-04 - this is less than 0.02% variation!!!!