Applications of nonlinear systems

Take the damped nonlinear pendulum equation $\theta '' + \mu \theta' + (\nicefrac{g}{L}) \sin \theta = 0$ for some $\mu > 0$ (that is, there is friction). Suppose the friction is large, in particular $\mu^2 > 4 (\nicefrac{g}{L})$.

1. Find and classify the critical points.
2. Explain what your findings mean, and if it agrees with what you expect in reality.

Suppose we have the predator-prey system where the foxes are also killed at a constant rate $h$ ($h$ foxes killed per unit time): $x' = (a-by)x,$ $y' = (cx-d)y - h$.

1. Find the critical points and the Jacobian matrices of the system.
2. Put in the constants $a=0.4$, $b=0.01$, $c=0.003$, $d=0.3$, $h=10$. Analyze the critical points. What do you think it says about the forest?

Suppose the foxes never die. That is, we have the system $x' = (a-by)x,$ $y' = cxy$. Find the critical points and notice they are not isolated. What will happen to the population in the forest if it starts at some positive numbers. Hint: Think of the constant of motion.