Question

Problem 5. Consider the system = I-? – ry, = 3y - xy - 2y? Please answer the following questions. (a) Determine all critical points of the system of equations. (b) Find the corresponding linear system near each critical point. (c) Discuss the stability of each critical point for the nonlinear system.

Answer 1

The gw en ytem in, f (yy) Say) dt dy dt C Scy) 37-xy-2y Cy) The citical poinb are & 3 y-y-2o y(3--2Y)0 1-x-) 3-2-2Y 0 OY OY, 1--y= 0 Now f thun, 3--2Y o 3 3-2Y 0 y

thun 2-Y o y= 2 .^ K= 1 - 2 =-I critical points are, (0,0), (0,), (0),(-,2) The Jacobion of t em in of 1-2-Y J = -Y 3--2Y Jaobiamin J(o,0) = for 0,0) t wence the coeponding linar ystem for (o) », () O The eigen valuus of tu o-eftiient matn'n 1- A quen by, 3-A a-CA-3)o in, J (o at (0, te Jacobjan O The oTTAnonding imenr skemf 0) in ()1G) 0

The eigen valus the co-tieient matnx are quem by 0-A O >A=o, A= --4 for () the Ja cobiom in, J(,o)= Tue corenpording Limear tonin -I 2 qwen by, Tu igen values Cre 0 2- A+) (A-2) = o - = 2 for (-,2) Hhe jacobiom in -2 OTT ApOding ine ar sytem , the [:IG) -2 o qiven by, The igin voalues ore 2 D-A IT-8 2.

for (0,0) Hle eigin valus ore boti poitive ie (1,) 0,6) in unstable node Henee 0, A ope tor 0, the eigun valus are / 2 eigin valuu i o, we cannod classity tlu stability of , 2 e of opposite are for (v6) eign valus (40) in saddle point И gт. Henee eigen r plex oite values are tor (-1/2) ttu unstable spiral (-,2) in poritive real Hence part