MATH2010 - Analysis of Ordinary Differential Equations Final Examination, Semester 1, 2019 1. (15 marks) Consider the following system of equations, 3x2 - y x+ y ? = f(+) (a) Determine the stability and nature of the critical points. (b) Compute the information you need to sketch the phase portrait close to the critical points. Sketch the phase portrait of the nonlinear system. Explicitly sketch and name trajectories which connect fixed points. a) fixed points : f(x, y+)= (0): 3x2-y = 0 => y=3x2 X + y = 0 => x+ 3x=0 => x+3 x2 =x(1+3x)=0=> x=0,-3 => either x=0 or 1+3x=0 = > fixed points : (0), (13) => y=x=> y=0,1/3 Approximation by linear system: Tocobian: Df= (6) 1 1 ad (0): Df(0,0) = 10, 1 1. det spirals improper Salad le trA -1) == A => tr A = 1 olet A - 0-(-1) = 1>0 deAA_ CtrA )2 4 = 1-4 = 300 => (3) is a spiral node since trA>> => unstable ad [1]3): Df(c1/3,1/3)= (-2 -\=B =>tv B =- 2+1=1 det B: - 2-1) =- 1 => saddle node (always unstable) <0 Page 2 of 8
MATH2010 - Analysis of Ordinary Differential Equations Final Examination, Semester 1, 2019 (Working Space Only) b) ad (8): either 10 or Determine the direction of the vector field $ close to (9): pick (10): f (1/10,0)= 100 1 10 => enti- clockwise -1/2 ad 1 13 1/2 a : Eigenva vest-vectors -2-X - 1 1 1=(+2)(1-1/+1= 1B- x 11/ == = 1 1-1 = >2+1-2+1=x2+1-1=0 => >12 === +1 -1-75 2 eigenvectors x (12) -2-11,2 -1 \11/2) => , > 0 , 12<0 -11/5-1 1-2- 2 - x(12)- -11/51 1 -37/5 1-12/ 1 1- 2 1 1 2 -1 1+(12) === > x(1,2)= - 3 4 / 5 1 +/51 2 2 1-3+2.2 1 2 +(1) ... unstable eigen direction + (2) ... stable 4- 1, 1 (1) = 1 1 1 and x (2)2 -2.6, -0.4 Page 3 of 8
MATH2010 - Analysis of Ordinary Differential Equations Final Examination, Semester 1, 2019 (Working Space Only) X heteroclinic orbit (2) 1/3 -1/3 Page 3 of 8