2. Consider the nonlinear system of differential equations dy dr = ry - ry dt dt = y-e (a) Determine all critical points of the system. (b) For each critical point not on the y-axis: i. Determine the linearisation of the system with the critical point translated to (0,0) and discuss whether it can be used to approximate the behaviour of the non-linear system. ii. Find the general solution of the linearised system using eigenvalues and eigenvectors. iii. Sketch by hand the phase portrait of the linearised system. Include all working of any special cases corresponding to straight line orbits, behaviour of general orbits as t and t +-0, and slopes of orbits across the axes. iv. Identify the type and stability of the critical point of the linearised system (c) Using PPLANE, produce a global phase portrait of the non-linear system in a region that includes all of the critical points. The phase portrait should illustrate the behaviour of orbits in the vicinity of each critical point. 2. Consider the nonlinear system of differential equations dy dr = ry - ry dt dt = y-e (a) Determine all critical points of the system. (b) For each critical point not on the y-axis: i. Determine the linearisation of the system with the critical point translated to (0,0) and discuss whether it can be used to approximate the behaviour of the non-linear system. ii. Find the general solution of the linearised system using eigenvalues and eigenvectors. iii. Sketch by hand the phase portrait of the linearised system. Include all working of any special cases corresponding to straight line orbits, behaviour of general orbits as t and t +-0, and slopes of orbits across the axes. iv. Identify the type and stability of the critical point of the linearised system (c) Using PPLANE, produce a global phase portrait of the non-linear system in a region that includes all of the critical points. The phase portrait should illustrate the behaviour of orbits in the vicinity of each critical point