Answer Happy

1. Consider the non-linear first order system: dx x - Y dt dy sin(x + y) dt (a) Determine all of the critical points of the non-linear system. (b) For each of the critical points in part (a): i. Determine the linearised system. ii. Discuss whether the linearised system can be used to approximate the behaviour of the non-linear system. (c) For the linearised system(s) with real eigenvalues: (i) Determine the general solution of the linearised system using eigenvalues and eigenvec- tors. (ii) Determine the type and stability of the critical point. (d) Using PPLANE, produce a global phase portrait of the non-linear system in the region −10 ≤ x ≤ 10, −10 ≤ y ≤ 10. Show at least 4 orbits in the vicinity of each critical point. (e) Using your global phase portrait or otherwise, determine the equation of the path(s) in terms of x and y, where the orbits of the nonlinear system approach a critical point as t → ∞. |