Question

For the following equations (i) Find the equilibrium points for any le(-00,00). (ii) Sketch the phase line diagram for the indicated. (iii) Find the bifurcation point and sketch the bifurcation diagram. (iv) State if the bifurcation is a saddle-node, transcritical, pitchfork, or none of these. (a) x' = x2 (1+ 24); 1=1, 1=-1.

Answer 1

Guiven problem is, x's ot x +41), disa parameter I w row for fixed points. . x=0. a. n( +49) 20 or, X220 oy 1 +24 20. **80 Xand for a=0 we get only one solution so. for d20 g only one fixed - © which is d) x*=0. xao. point of system for 170 we get , - 420 4 x4 and siree x9=1 has no real solution. so, x°20 is only solution of o. Therefore for 170 I only one fixed point of system 0, . for i 20 we get 1. x20 4 29=-) Tove d=-1, where uyo (FIR). Then atau » x= & ula = a (-)" . , for real solution so for aco. for three fixed points are, A nn-0, † (-1) 14

Phase Diagram: x'. x6. »st xa Semistable 2-0 ge'X (+224) » a semistable. x= x 1 +24) → unstable V unstan stable semintable ko Bifurcation Diagram : x=0 X12 - unstable. Stable stable - Jyo

from diagram 1. at dao . we Hence see that bifurcation bifurcation occurs point is dezo. iv) = A A+ x4). - since Here fixed point occurs & ME (-2,0). Hence. it is not saddle node bifurcation : Toko xc - Then - å= n(+XA). ar xs-+24) which not of the form o Hence ® is not symmetric system so does not give pitch fork bi furcation. Since fixed point occurs vaflosa). Jotencs gt es a not tram critical bifurcation, because a transcritical bifurcation is one in which a fixed point exists for all values of parameter and is never destroyed. Hower inuch a fixed point interchanges eto stability with another fixed print as the parameter in varied. In other words, both before and after bifurcation , there is one o unstable and me stable fixed frint. However, their stability is exchanged when they collide. So, anstable fixed point becames stable and vice versa. Abo normal form of tram critical bifurcation is - x'=rx-13, where risa parameter. Hence ① hegnat saddle -node, trane ritical and prichfork bifurcation