Chapter 7.3, Problem 12P

Bifurcation points. Consider the system

$\begin{array}{ccc}x\text{'}=F(x,y,\alpha ),& y\text{'}=G(x,y,\alpha ),& \text{(i)}\end{array}$

Where $\alpha$ is a parameter. The equations

$\begin{array}{ccc}F(x,y,\alpha )=0& G(x,y,\alpha )=0& \text{(ii)}\end{array}$

determine the x and y nullclines, respectively. Any point where an $x$ nullcline and a $y$ nullcline intersect is a critical point. As $\alpha$ varies and the configuration of the nullclines changes, it may well happen that, at a certain value of $\alpha$, two critical points coalesce into one. For further variations in $\alpha$, the critical point may disappear altogether, or two critical points may reappear, often with stability characteristics different from before they coalesced. Of course, the process may occur in the reverse order. For a certain value of $\alpha$, two formerly nonintersecting nullclines may come together, creating a critical point, which, for further changes in $\alpha$, may split into two. A value of $\alpha$ at which critical points coalesce, and possibly are lost or gained, is a bifurcation point. Since a phase portrait of a system is very dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behaviour of the system’s solutions. Problems 11 through 17 illustrate some possibilities that involve bifurcations.

In each of Problem 11 through 14:

a) Sketch the nullcelines and describe how the critical points move as $\alpha$ increases.

b) Find the critical points.

c) Let $\alpha =2$ classify each critical points by investigating the corresponding approximate linear system. Draw a phase portrait in a rectangle containing the critical points.

d) Find the bifurcation point ${\alpha }_0$ at which the critical points coincide. Locate this critical point and find the eigenvalues of the approximate linear system. Draw a phase portrait.

e) For $\alpha >{\alpha }_0,$ there are no critical points. Choose such a value of $\alpha$ and draw a phase portrait.

$\begin{array}{cc}x\text{'}=\frac{3}{2}\alpha -y& y\text{'}=-4x+y+x^2\end{array}$