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Given differential equation

$\frac{dy}{dt} = f(y, \alpha),$

we solve $f(y, \alpha) = 0$ to find equilibrium solutions.

By definition, $\overline y$ is a bifurcation point and $\overline \alpha$ is a bifurcation value if $f(\overline y, \overline \alpha) = 0 \text{ and } f_y(\overline y, \overline \alpha) = 0$.

Then my professor said that bifurcations can only happen at such points (but might not). I have difficulty trying to come up with an example where these conditions are satisfied, but bifurcation doesn't occur. Can you give me an example of such $f(y, \alpha)$? Providing some conceptual explanation would help as well.

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  • $\begingroup$ What you describe is kind of a working definition of bifurcation point... :) It is completely fine for a first course though. Actually, there is no universally accepted notion of bifurcation. Anyways, I am adding an answer with an example. $\endgroup$
    – John B
    Feb 20, 2016 at 21:22

1 Answer 1

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Take $f(y,\alpha)=\alpha y^2$, and the values $\overline y=0$ and $\overline \alpha=1$. There is no bifurcation in this case.

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  • $\begingroup$ Makes perfect sense. Thank you. $\endgroup$
    – Lidd88
    Feb 20, 2016 at 21:49

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