Resonance tongues in the linear Sitnikov equation

Abstract

In this paper, we deal with a Hill’s equation, depending on two parameters $$e\in [0,1)$$ and $$\varLambda >0$$ , 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 $$(e,\varLambda )$$ -plane presents unusual resonance tongues emerging from points $$(0,(n/2)^2),\ n=1,2,\ldots $$ 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 $$e\in [0,1)$$ . Indeed, an interesting behavior of the tongues is found: almost all of them concentrate in a small $$\varLambda $$ -interval [1, 9 / 8] as $$e\rightarrow 1^-$$ . 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. We also study the Lyapunov stability of the equilibrium in the center of mass of a curved Sitnikov problem.