1. (a) For the dynamical system

x˙ = (x−2)(x+1)x,

sketch the direction fields and the qualitative trajectories in the domain (−3,4)×(0,10).

(b) Consider the dynamical system

x˙ = x 2 +2x−c−2, x ∈ R,

where c ∈ R, constant, is a control parameter of the system.

(a) Determine the number and location of the equilibrium points of the above system for all values of the control parameter c.

(b) Using linear stability analysis determine the stability of the equilibrium points you have found in part (i). If linear stability analysis fails, use a graphical argument to determine stability.

(c) State the location and nature of any bifurcations of the system.

(d) Sketch the bifurcation diagram of the system.