Please use the stream plots in Wolfram Alpha. No need to do the phase portraits by hand.

Non-Linear Dynamics and Chaos Here are some simple Bifurcations. These are situations where the phase portraits are dependent on an outside parameter ,u which influences the stable points. 1) Saddle Node {x' = [J x2 3' = '3' i) Find the stable points for p 0 ii} Sketch some trajectories in the plane for each of u = -4, 0, 9. iii} Use Jacobians to classify stable points as source, sink, or saddle. 2) Pitchfork {x' = p X x2 y = '3' i) Find the stable points for p 0 ii) Sketch some trajectories in the plane for each of LL = -2, O, 2. iii) Use Jacobians to classify stable points as source, sink, or saddle. 3) Hopf Note: This one is in polar coordinates. {r' = [J r r3 6' = r2 i) Find the stable points for u <: u="0,"> 0 ii) Sketch some trajectories in the plane for each of u = -8, U, 1. iii) Use Jacobians to classify stable points as source, sink, or saddle