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The only occurrence I see complex numbers used in dynamical systems is to analyse the eigenvalue $\lambda$ of the linearised approximation to determine the characteristics of equilibrium points. However in classical mechanics when plugging $x=Ae^{i(\omega t+\phi)}$ into the equation of the oscillator $m\ddot{x}-b\dot{x}-kx=0$, we basically solve it with complex analysis, and $v=\dot{x}=i\omega x$ is a mapping $\mathbb{C}\to\mathbb{C}$ with $\pi/2$ rotation and a stretch of $\omega$ times, hence the $(v,x)$ plane is basically a complex plane. The Fourier transform also involves circles in the complex plane.

Is there a relation between the two? Can we use the Riemann sphere to analyze dynamical systems?

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It’s very convenient to represent the phase of all sorts of oscillations as a complex angle:

  • The complex angle of the Fourier transform is the phase of the respective frequency component, i.e., $\phi$ in your example.

  • All sorts of phase statistics involves complex arithmetics. For example,

    $$R = \left| \frac{1}{n} \sum_{j=1}^{n} e^{i\phi_j} \right|$$

    is a popular measure of how much the phases $\phi_1, …, \phi_n$ align, which in turn is indicative of synchrony (keywords: mean phase coherence, Kuramoto order parameter). There are several other characteristics based on similar concepts.

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  • $\begingroup$ Do you know how these two relates more fundamentally, rather than just a convenient tool? See my new question: How would behaviors of Riemann spheres represent characteristics of physical dynamical systems? Thanks. $\endgroup$ – Ooker Sep 29 '17 at 11:14
  • $\begingroup$ I never encountered the Riemann sphere in dynamical systems. Also, complex numbers are nothing but a convenient tool (as are numbers in general). $\endgroup$ – Wrzlprmft Sep 29 '17 at 11:25
  • $\begingroup$ Isn't the phase portrait deeply connects with topology? $\endgroup$ – Ooker Sep 29 '17 at 12:26
  • $\begingroup$ Depends of what kind of phase you are talking about. The phase space (a.k.a. state space) is deeply connected with topology, if you so wish, but it is only equivalent to the complex plane in rather boring cases. In particular the complex phase representation only replies to one kind of topology, namely circles. $\endgroup$ – Wrzlprmft Sep 29 '17 at 15:58

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