question as attached

Consider the Hamiltonian system: '=_y+ys, '=xx3 {1) {A} Write the linearized system around each of the three xed points: {(1, U), {i}, :|:l} and classify the fixed points of the linearized systems. {E} Find a Hamiltonian for [1} and use it to classify the nonhyperbolic fixed points {C} Using {A} and {B}. sketch a phase portrait of the nonlinear system {1}. It should include the three xed points and some traiectories nearby them. with the direction of the flow. {D} Describe which curve. or level set, corresponds to a homocliniciheteroclinic orbit {whichever is appropriate]. Can you nd an initial condition in. i [i such that the solution X) = (x(r}. y(t}} to ('11. with X0?) = X0, reaches the origin in nite lime? Explain