Edit Question: Question: How can we tell from the phase plots if the system is oscillating?

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If the phase plot takes the form of a spiral or a circle, then the system is oscillating. Take a look at any of the spiral plots and trace one of the lines around an equilibrium point, looking at either x1 or x2. You'll ntoice that coordinate increasing as you spiral around, then decreasing as you come back towards the equilibrium point, increasing again as you move past it, then decreasing again as you come in on your spiral and so on.

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Edit Question: Question: How can we tell from the phase plots if the system is oscillating?

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