1. Give an example of a nonlinear system that has an equilibrium point for which the linearized system has a zero eigenvalue, but the nonlinear system has only one equilibrium point.
2. For the system
x' = y - 1
y' = y - x2,
determine the types of each equilibrium point.
3. For the system
x' = sin(x)
y' = cos(y),
find all equilibrium points and use linearization together with the nullclines to sketch the phase plane.