I want to find all bifurcations for the system: \begin{align} x' =& -x+y\\ y' =& \frac{x^2}{1+x^2}-\lambda y \end{align} So far, I am using the idea that the intersection of the nullclines ($y'=0$ or $x'=0$) gives equilibrium points. So: \begin{align} x'=& 0 \implies x=y \\ y'=& 0 \implies y=\frac{x^2}{\lambda(1+x^2)} \end{align} Clearly $(0, 0)$ is an equilibrium point but there is also the possibility of two more equilibrium points when the parabola given by the above equation intersects the line $y=x$. To find the intersection I solved: \begin{align} x =& \frac{x^2}{\lambda(1+x^2)}\\ \lambda(1+x^2) =& x\\ \lambda x^2-x+\lambda =& 0 \\ x =& \frac{1\pm \sqrt{1-4\lambda^2}}{2\lambda} \end{align} I have these two intersection points. Now I need to vary $\lambda$ to find where the curve passing through the intersection points becomes tangent to $y=x$ and hence we would expect one equilibrium point instead of these two at this particular value of $\lambda$. Then continuing the variation of $\lambda$ in the same direction we would expect no equilibrium points from these two. Hence we originally had $2$ equilibrium points and then they coalesced and finally annihilated each other. This is a saddle-node bifucation.
How do I show this for my variation of $\lambda$? Are there any other bifurcations?
EDIT: Consider the discriminant of the equation:
\begin{align}
x = \frac{1\pm\sqrt{1-4\lambda^2}}{2\lambda}
\end{align}
\begin{align}
1-4\lambda^2 =& 0 \\
1 =& 4\lambda^2 \\
1 =& \pm 2\lambda \\
\pm\frac{1}{2} =& \lambda
\end{align}
So, I plotted the system with $\lambda = \frac{1}{2}$: sage: P = plot(x^2/ .5*(1+x^2), 0, 2) + plot(x, 0, 2)
We no longer have two bifurcations and instead have one. I was expecting the curves to be tangent. When does it happen that the curves are tangent? Actually I just realized that SAGE was not dividing the $1+x^2$ term, so I added the extra set of parens and everything works as expected!
Notice that x is real only when the discriminant 1-4λ^2 > 0. I.e. The curves of x' and y' do not intersect at points when 1-4λ^2 > 0 does not hold.