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Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations $$ \sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n, $$ has finite number of solution on $\{\Theta\in{\mathbb T}^n; \theta_1+\ldots+\theta_n =0 \}$. This equation represents equilibrium point of a dynamical system for $n$ neurons modeled as a network of oscillators.

Any idea or reference suggestion is appreciated.

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    $\begingroup$ As it is written, this question does not seem to concern differential equations, so the tag 'differential-equations' appears to be misplaced. $\endgroup$ Dec 15, 2013 at 20:16
  • $\begingroup$ @RicardoAndrade: In theory of ODE, as you have $\dot{\rm x} = {\rm f}({\rm x})$, for calculating the critical points of system, you need to solve $f({\rm x})=0$. This s the origin of Q in theory of ODE. $\endgroup$ Dec 15, 2013 at 20:21
  • $\begingroup$ Just to clarify, what is $\mathbb{T}$? Is it $[0,1)$ or is it $\{ \exp{\imath t} \,|\, t \in \mathbb{R}\}$? $\endgroup$ Dec 16, 2013 at 2:29
  • $\begingroup$ @Vit: I'm assuming $\mathbb{T}=\mathbb{R}/(2\pi)$. So $\mathbb{T}^n$ is an $n$-dimensional torus. $\endgroup$ Dec 17, 2013 at 7:43
  • $\begingroup$ @VítTuček By ${\mathbb T}$ I mean ${\mathbb R}/(2\pi {\mathbb Z})$. The case $n=2$ is obvious and checked. $\endgroup$ Dec 21, 2013 at 13:35

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Here's an idea. Not a particularly good one, but it's too long for a comment.

Using summation formuli you can rewrite the term in your sum as a polynomial in $\cos(\theta_i)$ and $\sin(\theta_i)$ for $i=1,\ldots,n$. Then you can rationally parametrize your circles by $\cos(\theta_i) = \frac{1-t_i^2}{1+t_i^2}$, $\sin(\theta_i) =\frac{2t_i}{1+t_i^2}$ and obtain a sum of rational functions thus reducing the problem to a system of polynomial equations. (The last condition $\sum_i \theta_i = 0$ just reduces the number of your variables but you have to pay the price of much uglier expressions coming from $\sin$ containing say $\theta_n = -\sum_{i\neq n} \theta_i$.)

For fixed $n$ you may (hopefully) deduce finiteness of the solution set via Groebner basis.

Or maybe there's even better coordinate system for the cut of your torus (which is, if I am not mistaken, topologically just the torus of dimension one less $\simeq \mathbb{T}^{n-1}$) and the functions in your equations have some "geometrical meaning".

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