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I have an assignment question based around a system of nonlinear differential equations,

$$ x' = f(x, y) \\ y ' = g(x, y) $$

The first part of the question is to locate and classify all the critical points (type and stability), which I have done.

The second part of the question then says that the system corresponds to a physical situation, involving a pendulum of a particular length. It defines the variables as meaning:

$x = \theta$ is the angle the [pendulum] wire makes with the verticle, and $y = \frac{-\theta'}{4}$ where $\theta'$ is the angular velocity at the end of the pendulum.

It asks me to explain the results of the first part in terms of that physical situation - specifically, "[my] answer should explain in terms of the pendulum, what the critical points are (without using the phrase 'critical points'), and something about their stability".

I am confident of being able to explain the stability based on the learning materials, but I'm not sure what a critical point exactly is in these terms - especially since I found some of them to be unstable. All of the information I can find, both in the learning materials and by googling, focuses on finding the critical points, which I have already done.

How can I understand what a critical point represents, and how that differs from some other point?

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It depends entirely on what your equations and variables represent, but a critical point always represents a state where the system is unchanging.

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  • $\begingroup$ I've added information about what the variables mean to the question. But having thought about it explicitly in that context, then it does seem quite obvious what it means for those to be constant functions. $\endgroup$ – lvc Jul 28 '15 at 15:39

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