Sitnikov Problem Research Articles

The photo-gravitational concentric Sitnikov problem

The current framework involves a configuration of two pairs of primary celestial bodies engaged in synchronized circular orbits around a central point of mass. Additionally, an infinitesimal mass is positioned along the z-axis, traversing through the system’s center of mass. This distinctive celestial scenario is designated as the Concentric Sitnikov Problem. Notably, it is postulated that the initial pair of primary bodies emit radiation, while the latter pair remains radiation-free. Within the scope of this study, we delve into an exploration of equilibrium points, periodic orbits, and the intriguing Newton–Raphson basins of convergence (N-R BoC) within the concentric Sitnikov model, all subject to the influence of radiation pressure. Remarkably, our investigation uncovers the presence of three equilibrium points, each exhibiting linear instability across the entire spectrum of mass parameter values μ∗∈(0,0.25). To visually comprehend the character of these celestial paths, we employ a graphical analysis technique known as the first return map. Varied values of the mass parameter μ∗ lead to the generation of diverse families of periodic orbits encircling both the primary celestial bodies and their equilibrium positions. Lastly, we embark on an exploration of the intricacies of the N-R BoC, intimately connected with the equilibrium points within this proposed celestial model.

The concentric Sitnikov problem: Circular case

A new configuration in the sense of concentric Sitnikov problem is introduced in this paper. This configuration consists of four primaries Pi(i=1,2,3,4) of masses mi, respectively. Here, all the primaries are placed on a straight line and moving in concentric circular orbits around their common center of mass under the restrictions of masses as m1=m2=m, m3=m4=ḿ and m>ḿ. The objective of the present manuscript is to study the existence of equilibrium points and their linear stability, first return map, periodic orbits, and N-R BoA in the proposed model. Under the restriction of mass parameter 0<μ́<1/4, it is found that there exist three equilibrium points and all the equilibrium points are linearly unstable for all values of mass parameter 0<μ́<1/4. Graphically, the nature of orbits have been examined with the help of first return map. The families of periodic orbits of infinitesimal mass m5 around the primaries and equilibrium points have been obtained for different values of mass parameter μ́. Finally, we explore the Newton–Raphson Basins of Attraction (N-R BoA), related to the equilibrium points in the concentric problem of Sitnikov.

Periodic oscillations in the restricted hip-hop $ 2N+1 $-body problem

This manuscript investigates a dynamical system in which $ 2N $ primary particles of equal masses move in space under Newton's law of gravitation forming the vertices of antiprisms while a particle of negligible mass moves along the common axis of symmetry of the antiprisms. This $ n $-body problem that we call the restricted hip-hop $ (2N + 1) $-body problem is an extension of the generalized Sitnikov problem studied in [17] for which the primaries remain in a plane. This work also relies on an early study [14] where certain families of periodic hip-hop solutions to a $ 2N $–body problem were constructed. We prove the existence of a continuous symmetric family of solutions of the restricted hip-hop $ (2N+1) $-body problem for each family of symmetric and periodic hip-hop solutions of the primaries studied in [14]. The main tools for proving our results are the implicit function theorem and a compactness argument. In addition, we present some numerical periodic solutions to the restricted $ 7 $-body problem.

On the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N + 1)-body problem

In this paper, we study the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N+1)-body problem. First, based on the relationship between the potential and the period as a function of the energy, we deduce the properties of the period of the solution of the corresponding autonomous equation (eccentricity e=0) in the prescribed energy range. Then, according to these properties and the stability criteria of symmetric periodic solutions of the time-periodic Newtonian equation, we analytically prove the linear stability/instability of the symmetric (m,p)-periodic solutions which emanated from nonconstant periodic solutions of the corresponding autonomous equation when the eccentricity is small, which indicates that the former stability criteria can be extended to the generalized Sitnikov problem with N≥3.

On the global families of periodic solutions of a generalized Sitnikov problem

On the global families of periodic solutions of a generalized Sitnikov problem

On the Dumb-Bell Equilibria in the Generalized Sitnikov Problem

We also investigate the linear stability of the trivial equilibrium of a symmetrical dumb–bell in the elliptic Sitnikov problem. In the case of the dumb–bell length $l\geq 0.983819$, an instability of the trivial equilibria for eccentricity $e\in(0,1)$ is proved.

Translational-Rotational Motions of a Rod in the Circular Sitnikov Problem

Translational-Rotational Motions of a Rod in the Circular Sitnikov Problem

Odd periodic oscillations in Comb-drive finger actuators

In this work we consider the existence of solutions of a Dirichlet problem with a prescribed number of zeros for a large class of second order differential equations with singularities. Additionally, for the model of a well known kind of micro-electromechanical system, the transverse Comb-drive finger device, we provide sufficient conditions in terms of control parameters that lead to odd periodic oscillations with a certain number of zeros that ranges from zero to some positive integer. This work is motivated and follows the lines of R. Ortega (2016) where the existence of odd periodic oscillations was studied for second order differential equations without singularities, in particular, for the Sitnikov problem. The main results of this paper are obtained through a truncation process, the application of classical tools as the shooting method, and the Sturm comparison theory. A numerical approach based on a bisection method is employed to illustrate the main results with simulations by using literature reference parameters.

Periodic solutions of a generalized Sitnikov problem

In this paper, we study the problem of determining whether a global family of even periodic solutions of a generalized Sitnikov problem, which emerges from a circular generalized Sitnikov problem, continues for all values of eccentricity in [0, 1) or ends in the equilibrium.

A Proof for a Stability Conjecture on Symmetric Periodic Solutions of the Elliptic Sitnikov Problem

In this paper, we aim at proving a conjecture proposed in [X. Cen, X. Cheng, Z. Huang, and M. Zhang, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1271--1290] on the stability of symmetric periodic solu...

Subharmonic Oscillations in the Near-Circular Elliptic Sitnikov Problem

We have considered the case of a restricted three-body problem (material points), when the masses of the two main attracting bodies are equal. Their orbits have been assumed to be ellipses. The problem regarding the third body’s motion of negligible mass under the influence of the main body’s gravitational attraction (the Sitnikov problem) allows particular solutions for the following: the third body moves along a straight line passing through the center of mass of the main bodies and perpendicular to the plane of their orbits. We have assumed that the eccentricity of the main body orbits is small and we investigated the nonlinear problem of the existence of the third body’s periodic motion with a period multiple of the revolution period of the main bodies in their orbits. The problem of stability (in Lyapunov’s sense) of these periodic motions under both perturbations keeping the trajectory of the third body rectilinear and arbitrary spatial perturbations has also been solved.

Influence of the gravitational radius on asymptotic behavior of the relativistic Sitnikov problem.

The Sitnikov problem is a classical problem broadly studied in physics which can represent an illustrative example of chaotic scattering. The relativistic version of this problem can be attacked by using the post-Newtonian formalism. Previous work focused on the role of the gravitational radius λ on the phase space portrait. Here we add two relevant issues on the influence of the gravitational radius in the context of chaotic scattering phenomena. First, we uncover a metamorphosis of the KAM islands for which the escape regions change insofar as λ increases. Second, there are two inflection points in the unpredictability of the final state of the system when λ≃0.02 and λ≃0.028. We analyze these inflection points in a quantitative manner by using the basin entropy. This work can be useful for a better understanding of the Sitnikov problem in the context of relativistic chaotic scattering. In addition, the described techniques can be applied to similar real systems, such as binary stellar systems, among others.

Exit basins for the Sitnikov problem with variable mass

In the present paper, we perform a numerical study of the Sitnikov problem aiming to characterize the orbits of a variable mass particle (e.g., comet, rocket, asteroid or spacecraft) and determine the uncertainty in the prediction of the final state of the test particle. The classification of final states was done through the well-known exit basins, while the determination of the uncertainty was calculated using a new tool named Basin entropy. It is found that for small values of the initial mass of the test particle, the number of initial conditions leading to bounded orbits gets increased, thus reducing the uncertainty in the final states. The same behavior in uncertainty is observed for increasing values of the exponent in Jeans law for the variation of the mass. Our results allow us to conclude that: i) an accelerated fuel consumption in the initial stages of stabilization of a satellite can keep the object in an oscillatory state around the primaries and ii) if the mass of the satellite is less than one hundredth of the mass of each primary, it is possible to predict with a very high certainty the final state of the satellite, regardless of the accuracy in the initial conditions of the system.

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

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 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 (2,1)-periodic solutions of the elliptic Sitnikov problem are hyperbolic and therefore are Lyapunov unstable when the eccentricity is small, while the corresponding even (2,1)-periodic solutions are elliptic and linearized stable.

Periodic solution of the nonlinear Sitnikov restricted three-body problem

The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time.

On the Convergence Dynamics of the Sitnikov Problem with Non-spherical Primaries

We investigate, using numerical methods, the convergence dynamics of the Sitnikov problem with non-spherical primaries, by applying the Newton-Raphson (NR) iterative scheme. In particular, we examine how the oblateness parameter $A$ influences several aspects of the method, such as its speed and efficiency. Color-coded diagrams are used for revealing the convergence basins on the plane of complex numbers. Moreover, we compute the degree of fractality of the convergence basins on the complex space, as a relation of the oblateness, by using different computational tools, such the fractal dimension as well as the (boundary) basin entropy.

Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem

Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881–901) proved the existence of an eight-shaped orbit of the planar three-body problem with equal masses. Since then a number of solutions to the N-body problem have been discovered. On the other hand, symbolic dynamics is one of the most useful methods for understanding chaotic dynamics. The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and was studied by using symbolic dynamics (J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973). In this paper, we study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it. We also prove the existence of periodic orbits.

Stability and bifurcations of even periodic orbits in the Sitnikov problem

We study different families of even periodic solutions in the classical Sitnikov problem that emanate from the circular case as the eccentricity is increased. The families can be classified by the number N of full revolutions of the primaries and labelled by the number of zeroes p of the vertical coordinate of the massless body in half a period. We give a linear stability criterion of these branches depending on even N, based on the sign for the initial slope of the discriminant function for the associated Hill’s equation, in a computable interval of eccentricities. All families for $$N=2$$ are linearly stable for small and computable e. The results show a fundamental symmetry-driven difference between the even and odd N cases.

Revealing the Newton–Raphson basins of convergence in the circular pseudo-Newtonian Sitnikov problem

In this paper we numerically explore the convergence properties of the pseudo-Newtonian circular restricted problem of three and four primary bodies. The classical Newton–Raphson iterative scheme is used for revealing the basins of convergence and their respective fractal basin boundaries on the complex plane. A thorough and systematic analysis is conducted in an attempt to determine the influence of the transition parameter on the convergence properties of the system. Additionally, the roots (numerical attractors) of the system and the basin entropy of the convergence diagrams are monitored as a function of the transition parameter, thus allowing us to extract useful conclusions. The probability distributions, as well as the distributions of the required number of iterations are also correlated with the corresponding basins of convergence.

The Sitnikov problem for several primary bodies configurations

In this paper we address an $$n+1$$ -body gravitational problem governed by the Newton’s laws, where n primary bodies orbit on a plane $$\varPi $$ and an additional massless particle moves on the perpendicular line to $$\varPi $$ passing through the center of mass of the primary bodies. We find a condition for the described configuration to be possible. In the case when the primaries are in a rigid motion, we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period as the primary bodies. We show that this fact is related to the existence of a certain pyramidal central configuration.