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.


70F10 \(n\)-body problems
70G60 Dynamical systems methods for problems in mechanics
Full Text: DOI


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