I'm looking at the system of ODEs \begin{align*}x' & = a - x - \frac{4xy}{1 + x^2} \\ y' & = bx(1 - \frac{y}{1 + x^2})\end{align*} and I've been asked to find values of $a$ and $b$ for which bifurcations occur. However, so far I've mainly learned to look for bifurcations where the number of equilibrium points changes, and for $b\neq 0$ there is only one equilibrium point, even if the phase portrait of the system changes qualitatively in some places.
Are there other sorts of bifurcations? And are there ways of quantitatively identifying what values of the parameters yield them?
Generally, for a 2D system of ODEs, we have fold, BT, and Hopf bifurcations at the equilibrium points. For Hopf bifurcation, we have to seek a pair of purely imaginary roots for the characteristic equation at one equilibrium point. Now it is easy to get the unique equilibrium point $E(\frac{a}{5},\frac{a^2+25}{25})$. (We suppose $a>0,b>0$.) At $E$, the Jacobian matrix is $$ A=\left( \begin{array}{cc} \frac{3 a^2-125}{a^2+25} & -\frac{20 a}{a^2+25} \\ \frac{2 a^2 b}{a^2+25} & -\frac{5 a b}{a^2+25} \\ \end{array} \right) $$ whose characteristic polynomial is $$ P(x)=x^2+\frac{-3a^2+5ab+125}{a^2+25}x+\frac{5ab}{a^2+25}. $$ So it is easy to check that $P(x)$ has a pair of purely imaginary roots $\pm w i$ iff $$ b=\frac{3a^2-125}{5a} $$ for $a>5$ where $$ w=\frac{\sqrt{15(a^2-25)}}{a^2+25}. $$ Thus for $a>5$, the system exhibits a Hopf bifurcation when $b$ passes through $b=\frac{3a^2-125}{5a}$ and hence a periodic solution bifurcating from $E$.