Unpredictable basin boundaries in restricted six-body problem with square configuration

doi:10.1016/j.newast.2020.101451

New Astronomy

Volume 82, January 2021, 101451

https://doi.org/10.1016/j.newast.2020.101451 Get rights and content

Highlights

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Restricted six-body problem with central-body square configuration is investigated.
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Cases of twelve and twenty libration points are explained in detail using contour plots and the distribution of potential funtion is also described.
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The multivariate form of Newton-Raphson method are used to explore the basin of attraction.
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Basin of attraction for possible values of the parameter mass ratio is explained and related data are presented in the form of tables and graphs.
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Pie-Charts are shown to explain the number of iterations needed for the convergence of initial conditions (one million in each case).
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Existence of unpredictable region throughout the basin of attraction and along the boundaries are examined using the method of basin entropy.
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Existence of Wada basin boundary is shown in case of twenty libration points.

Abstract

The present work deals with the recently introduced restricted six body-problem with square configuration. We have computed and verified the position of libration points with the help of Newton-Raphson method. It is observed that the total number of libration points are either twelve or twenty depends upon the mass parameter μ ∈ (0, 0.25). The multivariate form of Newton-Raphson scheme is used to discuss the basins of attraction. Different aspects of the basins of attraction are investigated and explained in detail. The complex combination of the different basins is found along the boundaries. The concept of basin entropy is used to unveil the nature of the boundaries. For μ ∈ (0.215, 0.238), the basins of attraction are unpredictable throughout. It is observed that for all values of the mass parameter μ, the basin boundaries are highly unpredictable. Moreover, we have investigated the presence of Wada basin boundary in the basin of attraction.

Introduction

In the field of Celestial Mechanics, the N-body problem has a very significant contribution. It has numerous applications in the field of galactic dynamics and motion of planetary objects. Many articles are available on the N-body problem for $N = 3, 4$ and 5 are by Michalodimitrakis (1981), Kalvouridis (1999), Celli (2007), Baltagiannis and Papadakis (2011), Shoaib and Faye (2011), Papadouris and Papadakis (2013), Arribas et al. (2016), Marchesin (2017), Alzahrani et al. (2017), Abouelmagd and Ansari (2013), Abouelmagd et al. (2013), Abouelmagd et al. (2015) and Abouelmagd et al. (2019). Recently, we have introduced a particular case of N-body problem known as restricted six-body problem with square configuration studied by Idrisi and Ullah (2020). Thus, a lot of work has to be done in this model. The restricted six-body problem is to study the motion of test particle under the gravitational fields of four primaries placed at the vertices of the square, while one primary is placed at the center of mass of the system. We have considered the mass ratio μ as the only parameter.

In general, for the N-body problem (N > 4), there is no specific method to determine the number of libration points. Therefore, we usually find it using numerical methods. Since, there are various numerical methods available to find out the libration points (or roots) of these dynamical systems. Among them, the N-R scheme is very well known and established method to determine the roots of nonlinear dynamical systems. In our case, we need the multivariate form of the N-R scheme. While going through recent articles on the applications of N-R method by Sprott and Xiong (2015) and Osorio-Vargas et al. (2020), we note that some initial conditions converge very quickly, some of them need more number of iterations, some of them even do not converge to any of the libration points. Thus, the study of the convergence of initial conditions is also a crucial aspect of the investigation. Also, it is essential to note that the initial conditions lying along the boundary need a higher number of iterations. Therefore, the detailed study of the BoA in R6BP is also one of the critical aspect.

The applications of the N-R scheme to the restricted problem of N bodies can be found in the work of Suraj, Sachan, Zotos, Mittal, Aggarwal, 2019, Suraj, Sachan, Zotos, Mittal, Aggarwal, 2019 and Zotos (2018). Based on that, we have investigated the BoA in R6BP using the multivariate form N-R scheme. In many cases, the BoA is found to be smooth except few. However, when the basins are not smooth, then we search for the degree of unpredictability in BoA and along boundaries of BoA. To measure this, we use the concept of basin entropy introduced recently by Daza et al. (2016). The configuration plane (x, y) can be divided into two parts; one is fractal region and the other is non-fractal region (based on log 2 graph shown in Fig. 7). We can decide about the region based on the values of S_b and S_bb, obtained using algorithm. When there is a coexistence of two or more attractors, there is a possibility for the occurrence of an important property called Wada. The concept and algorithm to show Wada basin boundary can be seen in the work of Aguirre et al. (2001), Aguirre and Sanjuán (2002), Aguirre et al. (2009), Seoane and Sanjuán (2013), Daza et al. (2015), Daza et al. (2017), Daza et al. (2018) and Bernal et al. (2018). We have also investigated the existence of Wada basin boundary in the R6BP.

Thus, in the present work, we have considered R6BP for investigation. To explore the BoA, we consider the permissible value of the parameter μ  ∈  (0, 0.25) and the N-R scheme. One of the crucial aspects is to explore the existence of an unpredictable region in BoA and along the boundaries. Further, the possibility of Wada basin boundary is also examined.

We have organised the present work as follows: The configuration and equations of motion of R6BP are explained in Section 2. The distribution of the potential function around the libration points is discussed in Section 3. In Section 4, we have mentioned the concept and algorithm used to find out BoA, basin entropy and boundary basin entropy. Results based on these concepts and the presence of wada basin boundaries are discussed in detail in Section 5. Concluding remarks based on numerical simulations and results are given in Section 6.

Section snippets

Configuration of the restricted six-body problem

The four particles P_i of equal masses m_i, $i = 1, 2, 3, 4,$ respectively, are placed at the vertices of square, revolving with angular velocity ω in circular orbit about their common center of mass O and the fifth particle P₀ of mass m₀ rests at the center. Let $O P_{i} = a$ be the distances of primaries from the center of mass, and an infinitesimal mass $\overset{´}{m}$ is moving under the gravitational field of m_i in the xy-plane. In such a system the motion of the infinitesimal is two dimensional. In the inertial frame

Lower limit of potential function Ω and jacobian constants

The libration points are the solution of the Eqns. $Ω_{x} = 0$ and $Ω_{y} = 0,$ i.e., $Ω_{x} = n^{2} x - {\frac{(1 - 4 μ) x}{r_{0}^{3}} + \frac{μ (x - 1)}{r_{1}^{3}} + \frac{μ x}{r_{2}^{3}} + \frac{μ (x + 1)}{r_{3}^{3}} + \frac{μ x}{r_{4}^{3}}} = 0 .$ $Ω_{y} = n^{2} y - {\frac{(1 - 4 μ) y}{r_{0}^{3}} + \frac{μ y}{r_{1}^{3}} + \frac{μ (y - 1)}{r_{2}^{3}} + \frac{μ y}{r_{3}^{3}} + \frac{μ (y + 1)}{r_{4}^{3}}} = 0 .$

On solving Eqs. (4a) and (4b), we get twelve libration points, out of which four libration points are collinear and eight libration points are non-collinear and all libration points lie on the concentric circles C₁, C₂ and C₃ centered at origin (Idrisi and Ullah, 2020). The libration points L₁, L₃, L₅ and L₇ are lying

N-R BoA

We can determine various aspects of dynamical system with the help of N-R BoA. Recently, few researchers have applied N-R method in various dynamical system including different perturbing terms in the effective potential. We have applied N-R iterative scheme (multivariate form) to study the BoA associated with the libration points. To reveal the domain of convergence of a specific libration point, we examine a set of initial conditions. To solve the systems of multivariate function $f (x) = 0,$ we

N-R BoA And existence of fractal

In this section, we have discussed the BoA for different values of the parameter μ. We have chosen μ in the interval (0, 0.25). Results regarding all the cases are presented in Table 3 and Figs. 4, 5 and 6(a-f). We have discussed each case to understand the impact of a parameter on BoA.

Concluding remarks

We have considered R6BP with a square configuration for the investigation. The evidence of either twelve or twenty libration points for the different values of parameter μ is crucial enough for further, in-depth study. It provides sufficient motivation for the investigation of N-R BoA. As the BoA appears to be smooth except boundaries, we have explored the unpredictable region in BoA along the boundaries. The multivariate form of the N-R method and the method of basin entropy is used to study

CRediT authorship contribution statement

Vinay Kumar: Conceptualization, Writing - review & editing. M. Javed Idrisi: Funding acquisition, Writing - original draft. M. Shahbaz Ullah: Formal analysis, Writing - review & editing.

Declaration of Competing Interest

None.

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Citation Excerpt :
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View full text

Unpredictable basin boundaries in restricted six-body problem with square configuration

Highlights

Abstract

Introduction

Section snippets

Configuration of the restricted six-body problem

Lower limit of potential function Ω and jacobian constants

N-R BoA

N-R BoA And existence of fractal

Concluding remarks

CRediT authorship contribution statement

Declaration of Competing Interest

Phys. D.

J. Differ. Equ.

New Astron.

Int. J. Non-Linear Mech.

Int. J. Non-Linear Mech.

The motion properties of the infinitesimal body in the framework of bi-circular sun perturbed earth-moon system

New Astron.

The effect of oblateness in the perturbed restricted three-body problem

Meccanica.

Periodic orbits for the perturbed planar circular restricted 3-body problem

Discret. Contin. Dyn. Syst.-Ser.B.

A first order automated lie transform

Int. J. Bifurc. Chaos.

Wada basins and chaotic invariant sets in the hénon-heiles system

Phys. Rev. E

On the fractal structures in nonlinear dynamics

Rev. Mod. Phys.

On the libration collinear points in the restricted three-body problem

Open Phys.

Out-of-plane equilibria in the symmetric collinear restricted four-body problem with radiation pressure

Astrophys. Space Sci.

Equilibrium points and their stability in the restricted four-body problem

Int. J. Bifurc. Chaos.

Uncertainty dimension and basin entropy in relativistic chaotic scattering

Phys. Rev. E