I am looking at a 1-D dynamical system in a generic form $\dot{y}=g(y,\rho),$ where $\rho$ is a parameter.
All I know about it is that this function has a fixed point at $y=0$ and when $\rho=0$. If I change $\rho$ slightly, how can I algebraically determine the impact on the fixed point?
Here is what I am trying:
Do a Taylor Series expansion with respect to the fixed point y* and $\rho=\epsilon$, some small value. Then
$\dot{y} = g(y*,\epsilon) + (y-y*)\frac{\partial g}{\partial y} \Bigr\rvert_{(y = y*,\rho=\epsilon)} + (\rho-\epsilon)\frac{\partial g}{\partial \rho} \Bigr\rvert_{(y = y*,\rho=\epsilon)} + \frac{1}{2}(y-y*)^{2}\frac{\partial^{2} g}{\partial y^{2}} \Bigr\rvert_{(y = y*,\rho=\epsilon)} $
Then I guess I would set $\dot{y}=0$, recognize the first RHS term is zero. But I'm confused as to how to proceed, and this form doesn't look very helpful.
I'm also wondering if the fixed point solution is unique only if g is continuously differentiable or if there are other conditions I need to consider.
Thank you.