Question

Consider the system of coupled ODES: x' = x - y, y = x + xy - 6y (+) (a) Find the critical points (C+, Y*) € R2 of this system. [3 marks] Hint: One critical point is (0,0) and there are two more critical points. (b) For each critical point, find the approximate linear ODE system that is valid in a small neighbourhood of it. [6 marks] (c) Find the eigenvalues of each of the linear systems found in part (b). Also, for the real eigenvalues only, construct the corresponding eigenvectors. [6 marks] (d) The Figure below shows 5 actual trajectories on the phase plane (x,y) of the solutions to system (t). Copy this Figure by hand, to your answer book. Finish your plot by marking all 3 critical points and adding arrowheads on each of the 5 trajectories.

Answer 1

x=y ineqh. Critical e so l ý x'=x-y = f(x,y) y' = x +xy-6y = g(,y) (9) critifical points de la mo, y'=0 x-y=o, x²+xy-by=0 - y² + y²_by=0 = y(y²74-6)=0 e =) y=0, y:+3y-24-6=0, y = -3,9=2 points (0,0),(2,2), (-3,-3) (b) = 1,2-1, = 32'ty, 32 10,99 lreorie f = f(0,0) + tel. (8-9) + |(-0) = 0 + 1 x - y = x-y 10,0) Lineariz g= g(0,0) + de lo alon) + Dg (x-o) + 39 (4-0) 0+ 0 lx-o) +46)(y-o). 1 Ly= -by Linearize x'=x-y y' = -6y (C) 1-1 e eigenvalue - eigen verö = 1 = 1,36 -6- z! ro -1717 7 Figen 20) T59/6) SM Az-* :] []

-1 See Vam e A= -6 T=-5 see with eigen values and eigen vectors. saddle. 72 – 4A = 49 Eigenvalues: = -6 y = 1 Eigenvectors: v = (-4) v2 = (7) ✓ Eigenvectors - Linearice for (2,2). Lf = Lr- f(2,2) tot 1 1 2 4 2 ) Jy |(2, 2) f12.2l+Of | (x-2) + at (y-2) = 0 +1.(x-2) – 1 (y-2) = x-y La = 9(2,2)+ ag l (x-2) + dag l (4-2) - dinin onl(2, 2) . Jy (22) = 0 + 1462-2) + (-4) (4-2) =142-4y -20 As we get tocus at equilibrium point. X ' = 14 A = 10 T=-3 ² – 4A = -31 Complex Eigenvalues : fi 31 -3 -1 731 – 31 2 3 2 1 No real eigenvectors Eigenvectors for (-3, -3) (f = 0 + 1 (2+3) - 1 (y +3) -xy Lg = 0 +24(4+5) –9(4+3) = 24x-9y +45 corresponding eigen value and eigen vectors. X = 24 A=15 t= -8 7² – 4A=4 Eigenvalues: A1 = -5 A₂ = -3 Eigenvectors: Eigenvectors

(d) we plot the phase plane then corresponding soluhan phase plane with Solection anve on red anve. A 10,0) B(2,2) c(-3,-3). 1 A1 1 1 1 1 1 5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0. 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
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