Question

Consider the nonlinear system
?x′ = ln(y^2 − x) and y'=x-y-1
(a)Find all the critical points

(b)Find the corresponding linearized system near the critical points.

(c) Classify the (i) type (node, saddle point, · · · ), and (ii) stability of the critical points for the corresponding linearized system.

(d) What conclusion can you obtain for the type and stability of the critical points for the original nonlinear system?

Answer 1

ebolution Given that , x'- lalya-x) = f(x,y) y'= n-y-1 glug) (a) f(x, y) = 0 = y2-1=1 => x=42-1 gun,y)= 0 = x-y-1 =) 4²-4-2=0 => y = 2,-1 y=2) 14=4-1 - 3 points =(3,2), (0-1) y=-| ) X=0 - L. critical points (01-1) and (3,2)

= ( b ) fy y?re -1 fore y²-a Ja = 1 gy four (0,-). Coorousbonding linear System u's fa 10,-1). U + fy (0, -1). V V = 9,10,-1). U + 8y (0,-1). V u=-u -2v W'= u-v four 13,2). u= all of 4v U-v V'e 2 (C) tou 10,-) [ TA-AI)=05) (-1-252 +9=0 1 + 2 +9242=0 22 +92 +3=0 5) 9 + 14 - 12 -4 VB 2 a = -1 +iva - 20 » Critical point is stable Spiral.

fou 13.21, A: E - 1A-2 I)= 0 = (-1-752- 4= 0 22 +29-3=0. = (2+3) (2-1) = 0 n=1 1-3 Obbosite sign. Saddle boint always unstable -> (d) Original .non. linear system and Conxoted linear system have some property.