HW assigned 12/4, due 12/11
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?