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For nonlinear systems, I know the phase portrait at a fixed point is a spiral when the eigenvalues are complex conjugates with real parts, and centre when they have no real parts. But how should I determine if it's "left-handed" or "right-handed" spiral, or which way the centre is turning?

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  • $\begingroup$ You might find this relevant. $\endgroup$ – amd Dec 28 '17 at 22:53
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    $\begingroup$ Perhaps one should add that in general the statement "the phase portrait at a fixed point is a [...] center when they have no real parts" is false. $\endgroup$ – John B Dec 28 '17 at 22:57
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Find the direction of the velocity vector $d{\bf x}/dt$ at some point away from the origin in the linearized system. See e.g. these notes and these.

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