Question

1. For each of the following systems, (i) determine all critical points, (ii) determine the corresponding linear system near each critical point, and (ii) determine the eigenvalues of each linear system and the corresponding conclusion that can be inferred about the nonlinear system. (a) dz/dt x- - zy, dy/dt 3y- xy-2y (b) dr/dt r2 + y, dy/dt=y-ay

Answer 1

Slow Count Sadurny 2x critical point a dx x - x² - xy = 0 & dy 3y - xy - 292 = 0 do, at points (0,0) & (-1,2) gives ax = d. elan (0,0) 4 (-12) are critical points Now let f= dx = x - x - ny 19 dy - sy-ay-aye So The Jacobian Fy is In the matrix "1-2xry - Gn 3-X-44 - Sli 162 :) ) at point (0,0) at poing (2) Ik System near critical point (o give V = 3V where uza 4 va y

that we known (A-XI) x=0 To find eign value det (A-XI) = 0 So. , we have to solve (1-x)(3-x) =0 Then, det (A-XI) = 11- 0 o 03-11 3) 3-1-3x + x2 = 0 = x² - x = 0 = => 0 4 1 x (x-1)=0 So, eignvalue d linear Hence the la critical nodal Seurce : system point are og 1. (0,0) is Improper - Now, linear system near 1 ui - 4 lv critical point lutu w (1,2) : gives 1-2 -24-4u 4 = utu v' = -2U-4V where u= -itx u =2 +y Also, det (A-XI) = 1- 0 (1-x) (-4-x) + 2 = 0 2 -4 7 L >> -A-+ + x + 2 20 X2-X + 2 = 0 Etsit T- Fü 2 de eignvalue of linear system are ithii , 1-57 Hence the critical point (-1,2) is spiral Source.

22 atx #4² 4 dj-y-eg o As Critical points, we dx = x + x² + y² =0 have & dy yaxy =o Then, At points (0,0) 4 (-1,0) gives da - dy zo ar (0.o) & (-1,0) critical points CM Now, let f. dx - x + x² + y2 & G = dy = Joxy C Jacobian at is ale The mahic : 1 +2x -y 24 1-2 Gx at point (0:0) lo 0- at point (-1,0) Then linear linear a system near critical point (0,0) : l gives I u = 4 V = V where uzx & vay Also, det (A-XI) 1-2 o To so >> (1->(1-x) - o intolaa =) = Sei eignvalues of Hence, the critical Source : f linear system point 10,0) are is 1 1 Nodalo dal or spiral

Now, lineal system near critical point (-1,0) u gives Then, To 2 I u = -4 V = av where u= -1 +2 v=y det (A-XI) = 1-1- olo - (-1-x) (2-x) = 0 To 2x - - 2 + x - 2x + x² = 0 -) 12- -2-0 => 1 - 2 &=-1 -1 Eign value Hence , the of lineal critical system are point (-1,0) 92 saddle is point.

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