Consider the nonlinear system: yty (1) (A) Find the fixed points, write the linearized system around each and classify the fixed points of the linearized systems. In case of hyperbolic fixed points, find the matrix P that transforms the linearized systems in a canonical one. (B) Find a Hamiltonian for (1) and use it to classify the non-hyperbolic fixed points. (C) Using (A) and (B), sketch a phase portrait (hand-drawned) of the nonlinear system (1). It should include the fixed points and some trajectories nearby them, with the direction of the flow. In case of hyperbolic fixed points, use the linearized system from (A) to draw more precisely the phase portrait near those fixed points. (D) Describe which curve, or level set, corresponds to a homoclinic orbit. Can you find an initial condition Xo 0 such that the solution X(t) = (x(t), y(t)) to (1), with X(0) = Xo, reaches the origin in finite time? Explain