**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) **