How can this function be considered to have a saddle node bifurcation?

Question

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if:

• $f_{\mu_0}(0) = 0$
• $f'_{\mu_0}(0) = 1$
• $f''_{\mu_0}(0) \neq 0$
• $\frac{\delta f_\mu}{\delta \mu}|_{(x,\mu)=(0,\mu_0)} \neq 0$

I can see how the bifurcation points existence can be shown by proving the fixed points come into existence at a certain value of $\mu$. ie. when $\frac{1}{2} (\mu \pm \sqrt{\mu^2 -0.4})=0$ or when $\mu = \pm \frac{2}{\sqrt{10}}$. That is clear, what I am struggling with is maybe the understanding of the criteria.

I cannot see how bullet point number one is satisfied, surely $f_{\mu_0}(0) = 0.1$ ? I'm probably not understanding the definitions correctly, any help would be appreciated - thanks!

Answer 1

I think that you misunderstanding comes from the convention that in theory bifurcation state is usually placed at origin. It always can be done by simple shift of coordinates and it's usually done for the sake of having less ambiguous notation (as far as I understand that). So if you want to check whether it is a saddle-node bifurcation or not, you just have to shift your coordinate system such that steady state will appear at $0$ or reformulate conditions the following way.

If $\mu_0$ is a critical value of parameter and $x_{\mu_0}$ is the coordinate of appearing fixed point, then conditions should be rewritten as:

• $f_{\mu_0}(x_{\mu_0}) = x_{\mu_0}$
• $f'_{\mu_0}(x_{\mu_0}) = 1$
• $f''_{\mu_0}(x_{\mu_0}) \neq 0$
• $\frac{\delta f_\mu}{\delta \mu}|_{(x,\mu)=(x_{\mu_0},\mu_0)} \neq 0$