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I am given a system

$$ \dot{\theta_1} = C - \sin{\theta_1} + D\sin{(\theta_2-\theta_1)}, $$ $$ \dot{\theta_2} = C + \sin{\theta_2} + D\sin{(\theta_1-\theta_2)}, $$

$$ C,D \geq 0 $$

and asked to find the fixed points.

Right now I am rewriting it in terms of

$$\phi_1 = \theta_1 - \theta_2$$ and $$\phi_2 = \theta_1 + \theta_2$$ and using "brute force" to solve it. However, I am not very satisfied with the lengthy and messy calculation, so does anyone have any thoughts on how to solve this problem in some neat way?

basmatibiscuit
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  • Are you sure you stated it right? If $C$ is large and $D$ is not you are likely to have no fixed points at all. – demitau Apr 29 '15 at 06:51
  • Yes, it is correctly stated and the conditions for the existence of the different solutions are in fact a central part. – basmatibiscuit Apr 30 '15 at 10:05
  • Why they are a central part? Knowledge about existence of solutions (solvability of these algebraic equations with (co)sines ) does not automatically help you to find the solutions. – demitau Apr 30 '15 at 11:55
  • In fact if you just use brute force and the formula for a sum of sines you get a solution without to much hassle with your change of coordinates. – demitau Apr 30 '15 at 12:07
  • Maybe I was a bit unclear. They are central in the sense of existence of fixed points of the system, which certsinly is a central part in a course in dybamical systems. Also, I have found the solution and indeed used some trigonometric identities, but still, I was looking for a potentially more sophisticated method. – basmatibiscuit May 01 '15 at 18:36

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