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Math Help - Nonlinear terms - Phase plane

  1. #1
    Newbie mukmar's Avatar
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    Nonlinear terms - Phase plane

    Problem: Consider the system \dot{x} =  -y  -  x^3, \dot{y} = x. Show that the origin is a spiral, although the linearization predicts a center.

    I tried converting to polar coordinates by doing x = r\cos{\theta}, y = r\sin{\theta}. But the equations I got were:
    \dot{r} = -r^3\cos^4{\theta} and \dot{\theta} = 1 + r^2\cos^3{\theta}\sin{\theta}, which aren't independent equations, so it didn't really help in analyzing the system.

    The two identities used were: x\dot{x} + y\dot{y} = r\dot{r} and \dot{\theta} = \frac{x\dot{y} - y\dot{x}}{r^2}

    Any hints or suggestions would be greatly appreciated!
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  2. #2
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    Well, this is what you have. First

    \dot{r} = -r^3 \cos^3 \theta \le 0 which means you'll either be on a decaying orbit or a fixed orbit depending on whether \dot{r} < 0 or \dot{r} = 0. However, if \dot{r} = 0 then \cos \theta = 0 so either \theta = \pi/2 \; \text{or}\; 3 \pi/2 so that \dot{\theta} = 1 which means that you'll pass through \theta = \pi/2 \;\text{or}\;  3 \pi/2 and then you're back to a decaying orbit thus giving a decaying spiral.
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  3. #3
    Newbie mukmar's Avatar
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    Thank you for explaining it so succinctly!
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