### Problem 1

Consider the coupled system:

dx/dt = x y - y
dy/dt = y^2 + 3 y - 1

1) Find the fixed points (the steady-state solutions).
2) Find the behavior of solutions that start near the fixed points.
3) Study the system numerically using fourth-order Runge-Kutta to do the time-stepping. First, give graphs demonstrating the behavior near the fixed points (one graph per fixed point). Second, give some graphs demonstrating more global behavior.
4) Prove that your fourth-order Runge-Kutta code is correct by showing me an error study. (I.e. compute to a fixed time using three different time-steps and show that the appropriate ratio is close to 16.)

### the Pendulum

Consider the coupled system:

dx/dt = y
dy/dt = - sin(x)

1) Find the fixed points and find the behavior of solutions that start near the fixed points.
2) Study the problem numerically using a fourth-order Taylor's approximation to do the time-stepping. First, give me local graphs for the behavior near each fixed point. Second, give me a global graph for the system.
3) Prove that your fourth-order Taylor code is correct by showing me an error study. (I.e. compute to a fixed time using three different time-steps and show that the appropriate ratio is close to 16.)
4) How is this system related to the single equation x_tt = - sin(x)?
5) Check that H(x,y) = y^2/2 + (1-cos x) is a constant of motion. This means that H(x(t),y(t)) = H(x(0),y(0)). Whatever value H initially has, it will always have.
6) This equation models a pendulum: a rigid rod which is pinned at one end and is free to spin. x corresponds to the angle from hanging straight down:
If it's hanging straight down then x = 0.
If it's pointing straight up then x = pi.
7) Now that you know what the equation models physically, what does H(x,y) correspond to? Explain the different types of behavior that your solution has in terms of what they mean the rod is doing.
8) Realistically, we expect there to be some friction in the pin that pins the rod and we expect the rod to settle down. Friction is included in the equation:
x_tt = - sin(x) - 1/10 x_t
Re-write this as a system, find the fixed points, and do the local analysis near the fixed points. Study the problem numerically and find how friction affects the behavior of the solutions. Is H still a constant of motion? If not, why not?