Chapter 2.5, Problem 10P

Bifurcation Points. For an equation of the form

$dy/dt=f(a,y)$ (i)

where $a$ is a real parameter, the critical points (equilibrium solutions) usually depend on the value of $a$. As $a$ steadily increases or decreases, it often happens that at a certain value of $a$, called a bifurcation point, critical points come together, or separate, and equilibrium solutions may either be lost or gained. Bifurcation points are great interest in many applications because near them the nature of the solution of the underlying differential equation is undergoing an abrupt change. For example, in fluid mechanics a smooth (laminar) flow may break up and become turbulent. Or an axially loaded column may suddenly buckle and exhibit a large lateral displacement. Or, as the amount of one of the chemicals in a certain mixture is increased, spiral wave patterns of varying color may suddenly emerge in an originally quiescent fluid. Problems $10$ through $12$ describe three types of bifurcations that can occur in simple equations of the form (i).

Consider the equation

$dy/dt=a-y^2$ (ii)

Find all of the critical points for Eq. (ii). Observe that there are no critical points if $a<0$, one critical point if $a=0$, and two critical points if $a=0$, and two critical points if $a>0$.

Draw the phase line in each case and determine whethereach critical point is asymptotically stable, semistable, or unstable.

In each case, sketch several solutions of Eq. (ii) in the $ty$-plane.

If we plot the location of the critical points as a function of $a$ in the $ay$-plane, we obtain Figure $\mathrm{2.5.11}$. This is called the bifurcation diagram for Eq. (ii). The bifurcation at $a=0$ is called a saddle-node bifurcation. This is more natural in the context of second order systems, which are discussed in Chapter $7$.