Question

3) For the given system restrict attention to the first quadrant, (x, y = 0). dx x(-4x – y +160) dt dy = y(-12 - y2 + 2500) dt Find and classify all equilibria.

Answer 1

(3) sol., Given system is age dt 4 (-11-y + 160) = y (-22 - y2 + 2500) (218,20) at have find and classify all equilibria, we put equility equilibrium Paints for da ahe dy HC-Ju-y tibo) = 0 n =0,0% (-en-y + 160) to y (-2² y2 +2500) = 0 y co as (-21²_y²+2500) = 0 and foi eaulibrium paines are aco and & Co,o) if_حن ig Palht 커 egh and y to then, @ from is co)2 - y2 + 2500 → Y2 - 2500 +50 but y zo y = 50. parht 18 co, 5o). a to and y co -4u-lo tibo then form is

40 9 1722 - 12800 40 +160 =0 Itu = 160 91-460 -"1 Hz to paintis (40,0). Mow a to and y to, then we solve the simultaneous equations. -40-y +160 sot luty = 160 and any2y² +2500 to Ja²ty2 = 2500 160-44 a2+ ( 160-44)2 = 2500 x² + (460)2 2Xaux160 + (4312 = 2500 22+ 16212 - 12800 = 2500- 25600 - - 23,100 17212_12804 + 23, 20o = 0 1722 5101 - 7701 +23,400 =0 175 (2-30) 770 (a-30) (a-30) (172 - 770) = 0 H- 30-o 17a-770 =0 I a - 30 g 172 = 770 g a = 770/17 Tf 1-30 y = 160-4x30=160-120 - y = 40 " Daint 18 (301 40), Now, if 2- 770/17 then y = 160 4x770 -160x17 3080 T7

2720-3080 -360 17 but you hat possible equilibrium pond paints are Coro), (050) (4000) and (30, 40). Now, discuss about stability of these pad. from given system, have fchey) = #c-14-y +160) = -2 E-10²-yut 1604 and g (h1y) = y 1-22_y²250) =-22y_4325007 find Jacobian matriy of for almost linearized the system, aut paint (hoito) I(Hoyo) fn (hoe yo) fy crop to) In (nor Tol Jy (hoto) no of the function f (ny) and g Chiy) UUU I (suy) - 8u - Y+160 -2-44 _d²_3y² +2500 2722

for equilibrium Paint C) (010), J (0,0) = 160 2500 The associated linear system 16 и du at and oldu at - 2500 u The characteristic eah of Ico, o) is I bord 250rd (160-2) (25000) Fo 1160 18:2500 Let d - 160 and t2 = 2500. Here, 2 > >o. i by theorem, stability of almost linear systems equilibrium put is unstable impoper hodlo for coi 50) J20150) = 1-501160 -3 (50)² + 2500 [ 1 1500 +2500

I co150) = Ito Asso aated lineas suffem is Itou du at and du at 5000 v The Chara chest ffic eah of 7 (0,50) Hod co o 5000 - (10-1) (-500-d) to az = -5000 A = 120 2 dz c - 5 ore I lo Here Her Ageyh lineas Aco by system are de zon Stallity of almost Ungfable saddle point at 10,50) (ill for (0o, o) -8800 + 160 J (40, o = 40 -80)²+2500 C 1-320+160 40 - 1600 +2500

eo дото) - Зор lineas suspem is Associested du df - - 16ou - oy. du Зор The characteristic egh is ( до , о ) - 1,04 о). с gy 160 — ( 3 - ) to — о - 1 () о х9 1 160 — 94 - 0 0 74-ба - 443 +00 - 0 1-40 4 — 14-19 60 - 0 о Я 1 14 38 ( 0-90-9xt-939xaw 2 x ) 1939 60 + V T1 23, 240 2 14:396ot 1068 329- 2 ahl, da to Теле

ng le hode. stable Impoper hode (to, o stable impopes til I for (30,40) J (30,40 ) = T- 8x30 - 10+160 -2x30x40 8-10123 (40) -30 2500 -200-40ti6o 30 - 2400 goo - 4800t 2500 -zgot 160 2000 - 5 joo +2500 a J ( 30, 40) = 120 30 -2000 3200 Also crated system is 12ou 3ou du at du do - 2400 u 3200 e

The characteristic egh is - 120-4 ö 13130140)| - 2 yeo 3200 - (-120d) (-3200 - 72000 = 0 120x3200 +120173200, tizzarzo 2 + 33201 + 384000-7200²0 ² + 33201 t 312000 t - 3320 I /32012_4x 31200 2x) - 3320 + 11022400 - 124800 d 2 -3320 & Vio 1897, 600 - 3320 + 3301.1513 got 12 (30100) 118 stable impropes hode 2