Question

3.1.2 please :)

EXERCISES FOR CHAPTER 3 3.1 Saddle-Node Bifurcation For each of the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. Finally, sketch the bifurcation diagram of fixed pointsversus 3.1.1 1+rx+ 3.1.2 ir-coshx 3.1.4 x=『竹x-x/(1+1) .1.sV (Unusual bifurcations) In discussing the nomal form of the saddle-node bi-

Answer 1

3.2 find the fined points, fa)o To osh T) an wsh(n) Now - osh x fined point wil be mo thun there Phare space hn 1-osh x = 1- omitting higher term order only fined poind in =O Phare space

2- oshx thum 2-(4) higher order terms fixed poimts are by Phare space the valu of sce that il chamges Y Sc Aigher) from 1 lower them neid finad points OY ereated dertroy ed ONe OY in the critical value Saddle no de bifurcation bifureation diagom Stable unstabla