Consider a dynamical system:
$\dot{x} = f(x,\lambda,\Lambda)$.
For a specific value of $\Lambda=\Lambda_1$, the system admits a fold bifurcation at $(\lambda_1,x_1)$. I want to find out two things, I want to find how the fold bifurcation evolves, i.e I want to:
- find $(\lambda(\Lambda),x(\Lambda))$ such that $(\lambda(\Lambda),x(\Lambda))$ is a fold bifurcation for the dynamical system for all $\Lambda < \Lambda_1$.
- I want find the limit of $(\lambda(\Lambda),x(\Lambda))$ as $\Lambda \to 0$?
I know the first of these problems can probably be solved by using continuation of the system $f$ and $\det{J_f}$, but I've not yet found anything which easily answers the second problem. Does anyone know of some numerical asymptotic method I could potentially use to find the limit as $\Lambda \to 0$?