Chapter 5.1, Problem 18E

If a nonlinear system depends on a parameter, then the equilibrium points can change as the parameter varies. In other words, as the parameter changes, a bifurcation can occur. Consider the one-parameter system family of systems

$\begin{array}{l}\frac{dx}{dt}=x^2-a\hfill \\ \frac{dy}{dt}=-y\left(x^2+1\right)\hfill \end{array}$

where $a$ is the parameter.

(a) Show that the system has no equilibrium points if $a<0$

(b) Show that the system has two equilibrium points if $a>0.$

(c) Show that the system has exactly one equilibrium point if $a=0$

(d) Find the linearization of the equilibrium point for $a=0$ and compute the eigenvalues of this linear system.

Remark: The system changes from having no equilibrium points to having two equilibrium points as the parameter $a$ is increased through $a=0.$ We say that the system has a bifurcation at $a=0,$ and that $a=0$ is a bifurcation value of the parameter.