I am wondering if the following ODE belongs to a well-studied class and if anything is known about its solutions:

$\partial_t\theta = \sin(\theta)\cos(2\pi t) + \kappa$.

This equation roughly corresponds to an overdamped oscillator with applied constant torque. I am particularly interested in the values of $\kappa$ for which, for a given initial condition, the solution is pseudo-periodic in the sense that $\theta(1)-\theta(0)$ is a multiple of $2\pi$.

Some simple considerations that come to mind: if we change variables to $u=\tan(\theta/2)$, the equation looks Riccati

$\partial_t u = u\cos(2\pi t) + \kappa/2\cdot(1+u^2)$

which is equivalent to

$\partial^2_t y - \cos(2\pi t)\partial_t y +(\kappa/2)^2 y = 0$

where $\partial_t y/y = -\kappa u/2$. Now, if $\theta(0) = -\pi$, then the question of whether $\theta(t)$ is pseudo-periodic comes down to whether $y(1) = 0$. However, this second order linear equation is equally confusing to me, so any ideas, references, etc, are greatly appreciated!


By putting $y=w\exp(\int\cos(2\pi t)/2dt)$ you kill the first derivative term in the second order linear equation, and obtain the equation of the form $w^{\prime\prime}+Qw=0$, where $Q$ is a trigonometric polynomial. This is called Hill's equation and your problem is an eigenvalue problem for it. Such eigenvalue problems were studied a lot, and I recommend the book of Whittaker-Watson. There is also a book on Hill's equation, by Magnus and Winkler.

  • $\begingroup$ Thank you very much for the references! Do you think that the eigenvalues are analytically tractable in this case? $\endgroup$ – Ivan Jun 12 '14 at 22:18
  • $\begingroup$ I am not sure what this means "analyticaly tractable". There is no simple explicit formula, but there are estimates, approximations, qualitative properties, etc. $\endgroup$ – Alexandre Eremenko Jun 13 '14 at 8:06

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