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The answer by Nathaniel is clear and correct, yet let me add: it is obvious by inspection that this is motion in a potential, which can be derived from a Hamiltonian $$\frac{\dot r^2+ r^2\dot\theta^2}{2} + V(r,\theta)\;.$$ All you need to do now is to establish whether, around the equilibrium points, $V$ has a minimum or a maximum/saddle: in the first case ...

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I take the core of the question to be Is it possible to do linear stability analysis on 2nd order differential equations by finding eigen values of Jacobian matrix? The answer is yes, but first you have to convert your second-order equations into first-order ones. This is actually pretty easy to do: every time you see a second derivative, e.g. ...

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You need to linearize your equations around any stationary point. Then you can indeed treat it like by finding eigen values.

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The answer to your question is yes, the logistic map definitely has an attractor. To show this for general maps/dynamical systems, we can use a direct numerical study, delve into analytical arguments which might not be entirely complete, or a combination of both. An attractor is a set in the phase space of the dynamical system or map which attracts a ...

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In addition to the classic Weinberg paper cited above, there's this shorter version, and then follow ups by Peres 1989 on how it violates the 2nd law, by Gisin on how it allows superluminal communications, and by Polchinski on how it would allow for an 'Everett' phone. More recently, there's this mathematical argument against nonlinear QM by Kapustin.

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