Question

INSTRUCTIONS: SHOW ALL WORKING IN THE SPACE PROVIDED FOR EACH QUESTION; NO CREDIT GIVEN FOR UNSUBSTANTIATED ANSWERS 1. (a) Locate all critical points for the nonlinear system z'=x-y=f(x,y) =-x +y" = g(x, y). (b) Determine the 2 × 2 matrix 4 ) 34 of the linearization z' = Az at each critical point (2,j) for the system (1). culating the eigenvalues and associated eigenvectors of theses matrices de the nature of each critical point, and sketch the corresponding local traj in a neighborhood of each critical point.

Answer 1

x' = x - y y' = -x + y^2 C p are points for which x' = 0 & y' = 0 x - y = 0 - x + y^2 = 0 Rightarrow x = y - y + y^2 = 0 i e y (y - 1) = 0 y = 0 or 1 C P are (0, 0) & (1, 1) rightarrow [x = y] A = [f_x g_x f_y g_y] = [1 - 1 - 1 2 y] for C P (0, 0) A = [1 - 1 - 1 0] Characteristics polynomial of A is lambda^2 - (trace A) lambda + det A = 0 lambda^2 - lambda oplus - 1 = 0 by quadratic formula lambda = 1 plusminus squareroot 1 + 4/2 = 1 plusminus 5/2 lambda_1 = 1 + squareroot s/2 lambda_1 = 1 - squareroot 5/2 For Eigenvector w r t 3 Now (A - 3, 3) = [1 - 1 + squareroot 5/2 - 1 - 1 - 1 + squareroot 5/2] = [1 - squareroot 5/2 - 1 - 1 - 1 - squareroot 5/2] \r\n (- (1 + squareroot 5/2), 1) is eigenvector