Seminario de Jóvenes Investigadores en Matemáticas

Event Details

This event finished on 19 December 2017.

Título: Resonance tongues in the linear Sitnikov equation
Impartida por: Mauricio Misquero Castro (Universidad de Granada - FPU)
Resumen: It is studied a Hill's equation, depending on two parameters $e\in [0,1)$ and $\Lambda>0$, that has applications to some problems in Celestial Mechanics of the Sitnikov-type. Due to the nonlinearity of the eccentricity parameter $e$ and the coexistence problem, the stability diagram in the $(e,\Lambda)$-plane presents unusual resonance tongues emerging from points $(0,(\frac{n}{2})^2),\ n=1,2,...$. The tongues bounded by curves of eigenvalues corresponding to $2\pi$-periodic solutions collapse into a single curve of coexistence (for which there exist two independent $2\pi$-periodic eigenfunctions), whereas the remaining tongues have no pockets and are very thin. Unlike most of the literature related to resonance tongues and Sitnikov-type problems, the study of the tongues is made from a global point of view in the whole range of $e\in[0,1)$. We apply the stability diagram of our equation to determine the regions for which the equilibrium of a Sitnikov $(N+1)$-body problem is stable in the sense of Lyapunov and the regions having symmetric periodic solutions with a given number of zeros.