Question

Given the ordinary differential equation =1+re+ where the parameter r is a given real number. (a) Sketch all the qualitatively different vector fields that occur as r is varied. (b) Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. (c) Sketch the bifurcation diagram of fixed points r* versus r.

Nonlinear differential equations and Bifurcation theory.

Answer 1

is, Giren equation as ltrx tar where risa farameters for fired point foint azo gert YX+1=0 > - 11 Tora - - + TA 2. where - Yoo 4 Y ! >> 177 2. gf ry2 Then a Then a of 8=2 has mokvestised point. has only one tres fixed point fas no fixed point. has only one (tre) fixed point, Then a 'if -2 <r<2 it r=-2 Then a has the fores fixed point . if r2.2 Then on (by phase diagram à vo ayo är niso giyo x intable unstable. Semintable 1-2 T-2 n ge o -2<r< 0 0 < <2.

à = (2+1) ~ 귄 format 722 iyo 2 (-1,0) so semistable Stable unstable. ry2 1:2 (b) from are as / moves each other an are the besee we see that and the phase diagram we see that fixed points. created and destroyed parameter varied two fixed points towards collide and mutually annihilate. The's mechanism of saddle node. bifurcation : Here as & approches 2 from up, parabola moves up the two fixed points towards to each other. the fixed point coalesce into a half stable print at x = -5/2 and when off <2 then fixed point ranishes . Similar property occurs at x = /2 [040] saddle node bifurcation occurs at prat2. (C) Bifurcation when r2 more fixed point So, diagram curve Bifucation in ren artrating ☆ ram (*"+1) 2 KI Twhen xf ot ir when x → 0 rto

unstable. ry2 ९ unstable stable. r2-2 7=2 r= -2 Stable. This is a Sa dole node also bifurcation diagram. Here. bifurcation occurs at 5-12 know that bifurcation curve for a saddle no de bifureation Es tan gent to rata denote fixed we * Here att fixed point