## Resonance tongues in the linear Sitnikov equation.(English)Zbl 1391.70032

Summary: 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 $$(0,(n/2)^2)$$, $$n=1,2,\dots$$ 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.

### MSC:

 70F10 $$n$$-body problems 70G60 Dynamical systems methods for problems in mechanics
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### References:

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