Quantitative Stability of Certain Families of Periodic Solutions in the Sitnikov Problem

Galán, J.; Núñez, Daniel; Rivera, Andrés

doi:10.1137/17m1113990

Cited by 9 publications

(3 citation statements)

References 20 publications

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“…A rigorous mathematical study on the existence and continuation of nonconstant periodic solutions of can be found in the studies . In recent years, the stability problem has been also studied …”

Section: Examples and Conclusionmentioning

confidence: 99%

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

Zhang

Cen

Cheng

2018

Math Methods in App Sciences

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This paper is motivated by the stability problem of nonconstant periodic solutions of time-periodic Lagrangian equations, like the swing and the elliptic Sitnikov problem. As a beginning step, we will study the linearized stability and instability of nonconstant periodic solutions that are bifurcated from those of autonomous Lagrangian equations. Applying the theory for Hill equations, we will establish a criterion for linearized stability. The criterion shows that the linearized stability depends on the temporal frequencies of the perturbed systems in a delicate way. KEYWORDSHill equation, Lagrangian equation, linearized stability, linearized instability, nonconstant periodic solution, Sitnikov problem where the variable length l(t) > 0 is 2 -periodic. Roughly speaking, if the linearized equation of (1.1) along x(t) = 0 x + l(t)x = 0 Math Meth Appl Sci. 2018;41 4853-4866.wileyonlinelibrary.com/journal/mma

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Section: Examples and Conclusionmentioning

confidence: 99%

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

Zhang

Cen

Cheng

2018

Math Methods in App Sciences

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“…In all cited papers, the existence of periodic spatial isosceles solutions has been considered using variational methods, numerical methods or techniques of analytical continuation. In most of the cases it is assumed that the body on the z-axis has a small mass M or a mass M = 0, see [1,3,10,15]. In the latter case we obtain the Sitnikov problem.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of periodic orbits for the $N$-body problem, from a non geometrical family of solutions

Perdómo¹,

Rivera²,

Suárez³

2020

Preprint

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Given two positive real numbers M and m and an integer n > 2, it is well known that we can find a family of solutions of the (n + 1)-body problem where the body with mass M stays put at the origin and the other n bodies, all with the same mass m, move on the x-y plane following ellipses with eccentricity e. It is expected that this geometrical family that depends on e, has some bifurcations that produces solutions where the body in the center moves on the z-axis instead of staying put in the origin. By doing analytic continuation of a periodic numerical solution of the 4-body problem -the one displayed on the video http://youtu.be/2Wpv6vpOxXk -we surprisingly discovered that the origin of this periodic solution is not part of the geometrical family of elliptical solutions parametrized by the eccentricity e. It comes from a not so geometrical but easier to describe family. Having notice this new family, the authors find an exact formula for the bifurcation point in this new family and use it to show the existence of non planar periodic solution for any pair of masses M , m and any integer n. As a particular example we find a solution where three bodies with mass 3 move around a body with mass 7 that moves up and down.

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“…See [11, §6] for details. As for nonconstant, even (in time) periodic solutions of (S e ) which are emanated from the corresponding solutions of (S 0 ), the stability and linearized stability are studied in very recent papers [5,6,14,21]. Most of these are based on the theory for Hill's equations.…”

Section: Introductionmentioning

confidence: 99%

On the Stability of Symmetric Periodic Orbits of the Elliptic Sitnikov Problem

Cen¹,

Cheng²,

Huang³

et al. 2019

Preprint

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Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which are emanated from nonconstant periodic solutions of autonomous equations. By using the theory of Hill's equations, we will first deduce in this paper a criterion for the linearized stability and instability of periodic solutions which are odd in time. Such a criterion is complementary to that for periodic solutions which are even in time, obtained recently by the present authors. Applying these criteria to the elliptic Sitnikov problem, we will prove in an analytical way that the odd (2p, p)-periodic solutions of the elliptic Sitnikov problem are hyperbolic and therefore are Lyapunov unstable when the eccentricity is small, while the corresponding even (2p, p)periodic solutions are elliptic and linearized stable. These are the first analytical results on the stability of nonconstant periodic orbits of the elliptic Sitnikov problem.

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Quantitative Stability of Certain Families of Periodic Solutions in the Sitnikov Problem

Cited by 9 publications

References 20 publications

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

Bifurcation of periodic orbits for the $N$-body problem, from a non geometrical family of solutions

On the Stability of Symmetric Periodic Orbits of the Elliptic Sitnikov Problem

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