I have this system of differential equations:
\begin{equation} \frac{dx}{dt}=1-(b+1)x+x^2 y\\ \frac{dy}{dt}=bx-x^2 y \end{equation}
I now that we will have a bifurcation when $b$ grows and passes $2$ and we know that the stationary point is $(1,b)$.
I know that we need to change the variables in our systems to our new variables $u = x - 1$ and $v = y - b$
I understand that it will be three conditions:
non-hyperbolicity condition: conjugate pair of imaginary eigenvalues) transversality condition: the eigenvalues cross the imaginary axis with non-zero speed genericity condition
Can you help med attack this problem?