Homework 7

1) Chaotic System:
Reproduce T. Pang's figures 3.2 and 3.3 on page 66 and 67. Then do T. Pang, Problem 3.3. Pick Theta=0. (50 points)

2) Initial Value Problem:
Solve the following third order ordinary differential equation with the Runge-Kutta method: u_ttt + u_tt + u_t + u +1 = 0 . (Here "u_ttt" denotes the third derivative of u with respect to t, and so on.) The initial conditions are: u(0) = 0, u_t(0) = 1, u_tt(0) =2. You may modify the program 'pendulum.f' (or its analogue in C), given in the textbook. Integrate this differential equation from t=0.01 to t=1.0, using 100 time steps of size 0.01. Make a combined plot of u(t), u_t(t), and u_tt(t) in this region. Use Mathematica to obtain the exact solution for u(t) (write down the analytical expression), and compare it with your numerical result. Please hand in a printout of your program. (50 points)