The perturbed restricted three-body problem with angular velocity: Analysis of basins of convergence linked to the libration points
Md Sanam Suraj, Rajiv Aggarwal, Amit Mittal, Md Chand Asique
tto be inserted manuscript No. (will be inserted by the editor)
The perturbed restricted three-body problem with angularvelocity: Analysis of basins of convergence linked to thelibration points
Md Sanam Suraj · Rajiv Aggarwal · Amit Mittal · Md Chand Asique
Received: date / Accepted: date
Abstract
The analysis of the affect of angular veloc-ity on the geometry of the basins of convergence (BoC)linked to the equilibrium points in the restricted three-body problem is illustrated when the primaries are sourceof radiation. The bivariate scheme of the Newton-Raphson(N-R) iterative method has been used to discuss thetopology of the basins of convergence. The parametricevolution of the fractality of the convergence plane isalso presented where the degree of fractality is illus-trated by evaluating the basin entropy of the conver-gence plane..
Keywords
Restricted three-body problem · Radiationforces · Fractal basins of convergence · Newton-Raphsonmethod · The Basin Entropy
One of the most celebrated problem in the field of Ce-lestial Mechanics is the restricted three-body problem
Md Sanam SurajDepartment of Mathematics, Sri Aurobindo College, Univer-sity of Delhi, New Delhi-110017, Delhi, IndiaE-mail: [email protected]: [email protected] AggarwalDepartment of Mathematics, Deshbandhu College, Universityof Delhi, New Delhi-110019, Delhi, IndiaE-mail: rajiv [email protected] MittalDepartment of Mathematics, ARSD College, University ofDelhi, New Delhi-110021, Delhi, IndiaE-mail: [email protected] Chand AsiqueDeshbandhu College, University of Delhi, New Delhi-110019,Delhi, IndiaE-mail: [email protected] (R3BP). Many researchers and scientists are attractedtowards it due to its applications in various other fields(e.g. Abouelmagd and Abdullah (2019a), Abouelmagdet al. (2019b), Abouelmagd (2012), Abouelmagd et al.(2016), Alzahrani et al. (2017), Go´zdziewski and Ma-ciejewski (1987), Sano (2007), Pathak et al. (2015), Se-lim et al. (2019)). In addition, several modificationshave been proposed by various researchers to be morerealistic in the classical R3BP which make the appli-cations of this problem in wider sense. In this pro-posed problem we have considered two modificationsi.e., the radiation effects of both the primaries and thevariation in the angular velocity (Chermnykh problem,see Chermnykh (1987)). A generalization of the Euler’sproblem of two fixed gravitational centers and the re-stricted problem of three bodies where the third body,whose mass is negligible in comparison of the other bod-ies, orbits in the configuration plane of dumbbell whichrotates around their center of mass with a constant an-gular velocity ω , is always referred as Chermnykh prob-lem. Many authors have studied this problem due to itsimportant applications in the field of Chemistry (Pros-miti et al (1996)), Celestial mechanics and DynamicalAstronomy.One of the paramount issue in the dynamical sys-tem is to know the geometry of the ”basins of conver-gence” linked with the equilibrium points of the dynam-ical system. The domain of the BoC unveils the fact thathow the different initial conditions on the configurationplane are enticed by the particular equilibrium pointwhen an iterative method is applied to solve the sys-tem of equations. Undoubtedly, to solve the system ofsimultaneous equations with two or more variables, theN-R iterative scheme is contemplated as classical one.Previously, many authors have studied the BoC by ap-plying the N-R iterative method to reveal the numerous a r X i v : . [ n li n . C D ] M a y Md Sanam Suraj et al. intrinsic properties of different dynamical system (e.g.,the R3BP and the Hill’s problem with oblateness andradiation effects( Zotos (2016), Zotos (2017), Douskos(2010)), the restricted four-body problem (Baltagiannisand Papadakis (2011), Suraj et al. (2017b), Suraj etal. (2017a), Suraj et al. (2019d)), the restricted prob-lem of five bodies ( Zotos and Suraj (2018), Suraj etal. (2019), Suraj et al. (2019), Suraj et al. (2019), Surajet al. (2019)).Presently, we wish to analyze the effect of angularvelocity on the topology of the BoC when both of theprimaries are source of radiation. Moreover, the frac-tality of the BoC is also discussed as the function ofangular velocity.The present paper has following structure: alongwith the literature review regarding the R3BP presentedin Sec. 1, the description of the mathematical model ispresented in Sec. 2. The parametric evolution of the lo-cations of the equilibrium points is depicted in Sec. 3whereas the influences of the angular velocity on thegeometry of the BoC by using the bivariate sort of theN-R iterative method are illustrated in detail in Sec. 4.The degree of fractality of the BoC is depicted in Sec.5. The paper ends with Sec. 6 where the analysis of thestudy and the obtained results are discussed.
In the present study, we have considered the dynamicalmodel same as in Ref.Perdios et al. (2015) which canbe reviewed as follows: the rotating, barycentric, anda dimensionless co-ordinate system with origin ”O” isconsidered as the centre of mass of the system. Thetwo primaries namely m and m rotate in circular or-bit around ”O” with angular velocity ω ≥ x − axis withco-ordinate ( x ,
0) = ( − µ,
0) and ( x ,
0) = (1 − µ, m = 1 − µ and m = µ where the mass parameter µ = m m + m ≤ .The restricted problem of three bodies reduces to theCopenhagen problem when µ = . We analyse the mo-tion of the third body m whose mass is negligible incomparison of the primaries. In addition, it is also con-sidered that both the primaries are source of radiation.Consequently, the motion of the infinitesimal mass isgoverned by two type of forces i.e., the gravitationalforces of the primaries and the repulsive force of thelight pressure. It is necessary to note that the radiationfactors can achieve the negative value as well whichmeans that these forces will give strength to the gravi-tational force. Fig. 1
The restricted three-body problem. (colour figure on-line).
In the dimensionless rectangular rotating co-ordinatesystem, the equations of motion of the third body, whichalso referred as test particle in the restricted three-bodyproblem with angular velocity, are (see Chermnykh (1987),Perdios and Ragos (2004) and Perdios et al. (2015)):¨ x − y = ( ω − Q ∗ ) x − M ∗ R ∗ , (1a)¨ y + 2 ˙ x = ( ω − Q ∗ ) y, (1b)where M ∗ = µ (1 − µ ) , (2a) Q ∗ = q (1 − µ ) r + q µr , (2b) R ∗ = q r − q r , (2c)while the time independent potential function Ω isgiven by: Ω = ω x + y ) + (cid:88) i =1 q i m i r i , (3a) r i = ˜ x i + ˜ y i , (3b)˜ x i = ( x − x i ) , (3c)˜ y i = ( y − y i ) , (3d)where r i represents the distances of the test particlefrom the primaries m i , respectively. The radiation pa-rameters q i ,(see Chernikov (1970)) due to the radiatingprimaries m i are defined as: q i = 1 − F p i F g i , where, F p i are the solar radiation pressure forces whereas F g i are the gravitational forces due to primaries m i , i =1 ,
2. The system admits the Jacobi integral i.e., C = 2 Ω − ( ˙ x + ˙ y ) . (4) itle Suppressed Due to Excessive Length 3 Fig. 2
The movement of the libration points for q = 0 . q = 0 .
25 and consequently ω ∈ (0 . , . L = Purple , L = Orange , L = Green , L = Olive , L = Cyan . (colour figure online).
The parametric evolution of the positions of librationpoints are presented in this section by using the sameprocedure given by Ref.Perdios et al. (2015). The collinearlibration points are those points which lie on x − axisand we can evaluate by system of equations (1a-1b), bysetting the velocity and acceleration components equalto zero and solving for x by taking y = 0, we get f ( x ) = ω x − (1 − µ ) q ( x + µ ) | x + µ | − µq ( x + µ − | x + µ − | = 0 , (5)by keeping the value of the parameters ω, µ and q i , i =1 ,
2, fixed. The presented problem reduces to the photo-gravitational version of the classical restricted problemwhen ω = 1. It is shown that the angular velocity ω has no affect on the existence of totality of number ofcollinear libration points (for detail see Ref.Perdios etal. (2015)) and these libration points are named as L i , i = 1 , , L < − µ < L < − µ < L , where − µ and 1 − µ are the positions of the primaries m and m respectively. As far as the non-collinear triangular libration pointsare concerned, their positions can be described as fol-lows: x = 12 (cid:110) (cid:16) q ω (cid:17) − (cid:16) q ω (cid:17) (cid:111) − µ, (6a) y = ± (cid:104)(cid:16) q ω (cid:17) − (cid:110) (cid:16) q ω (cid:17) − (cid:16) q ω (cid:17) (cid:111) (cid:105) , (6b)for detail see Ref. Perdios et al. (2015). In addition, it isunveiled that the planar non-collinear libration pointsi.e., y (cid:54) = 0, exist only when the following conditions aresatisfied simultaneously: q > , q > , and | q − q | < ω < ( q + q ) . (7)When the effect of the radiation pressure is neglectedthe positions of the non-collinear equilibrium points aredefined by the co-ordinates ( x, y ) (see Ref. Perdios etal. (2015), Perdios and Ragos (2004)) where x = 12 (1 − µ ) , y = ± (cid:114) ω − − , (8)and consequently, these points exist only when ω ∈ (0 , √ ω which depend on q and q (where q , q (cid:54) = 1), the tri-angular libration points exist only when ω ∈ ( ω , ω )where ω i = ω i ( q , q ) , i = 1 ,
2. However, the collinear li-bration points exist for ω ∈ (0 , ∞ ) and at ω = 2 √ q , q = 1) the non-collinear libration points coincidewith L . In Fig. 2, the movements of the position of li-bration points (as the value of parameter ω ∈ ( ω , ω ))are shown for constant values of the parameters q i ,and µ and different increasing values of ω . We canobserve that the libration point L move towards theprimary P whereas the libration points L , move to-wards the primary P as the value of ω increases. Itis also observed that the non-collinear libration pointsoriginate in the vicinity of the libration point L at ω ≈ . L at ω ≈ . ω are associated to the valueof q = 0 .
15 and q = 0 . We perform a numerical analysis of the influence of an-gular velocity, mass parameter, radiation parameterson the geometry of the BoC linked with the librationpoints of the dynamical system by using the bivari-ate version of the N-R iterative scheme. This iterative
Md Sanam Suraj et al.
Fig. 3
A characteristic example of the consecutive steps that are followed by the Newton-Raphson iterator and the corre-sponding crooked path-line that leads to an equilibrium point. (colour figure online). method can be applicable to the of system of bivariatefunction f ( x ) = 0, using the iterative method: x n +1 = x n − J − f ( x n ) . (9)Here, f ( x n ) denotes the system of equations, whereas J − is denoting the inverse Jacobian matrix.The iterative scheme for the x and y co-ordinatescan be decomposed as: x n +1 = x n − Ω x n Ω y n y n − Ω y n Ω x n y n Ω x n x n Ω y n y n − Ω x n y n Ω y n x n ,y n +1 = y n + Ω x n Ω y n x n − Ω y n Ω x n x n Ω x n x n Ω y n y n − Ω x n y n Ω y n x n , where the values of the x and y coordinates are repre-sented by x n and y n respectively at the n -th step.The philosophy which works in the background ofthe N-R iterative scheme is same as described in Zo-tos (2016). The collection of all those initial conditionswhich converge to the particular attractor (i.e., the sameroot of the equations) compose the so-called N-RBoC.Further, we apply color coded diagrams (CCDs), whereeach pixel is linked with a non-identical color, as per theconcluding state of the associated initial conditions, toclassify the nodes in the orbital plane. The color codesfor the domain of BoC linked to the respective librationpoints are same in each figure and the codes are sameas in Fig. 2. In Fig. 3, it is depicted that the successiveapproximation points move in a crooked path and alsofor different initial conditions but for same attractor thenumber of required iterations to converge are different.The numerical analysis with the Copenhagen casewhere the mass ratio µ = 0 . q = 0 . q = 0 .
25. We start our analysis with Fig. 4a, which is depicted for ω = 0 . . .
25% of the consideredinitial conditions) converges to the libration point L ,which has infinite extent as well. Whereas 17 .
12% ofconsidered initial conditions converge to the L , and4 .
66% of initial conditions converge to the librationpoint L . The majority of the area of the finite do-main of the BoC is occupied by those initial conditionswhich either converge to one of the non-collinear libra-tion points. In Fig. 4b, when the ω = 0 . L and L resembles to shape of exotic bugs with many legs andantennas which are separated by the chaotic strip com-posed of various type of initial conditions whose finalstates are not same. However, the entire xy -plane is cov-ered by the well formed BoC. The extent of the domainof BoC linked to the equilibrium point L is infinite onthe other hand for all other equilibrium points theseextents are finite. The domain of the BoC linked to thelibration points L and L looks like butterfly wingswhose wings boundaries are segregated by chaotic mix-ture of various types of initial conditions whose finalstate are different. It is observed that 6 .
87% of consid-ered initial conditions are converging to the equilibriumpoint L , 9 .
6% of initial conditions are converging to L whereas 11 .
74% of initial conditions are converging toeach of equilibrium points L and L and remaining itle Suppressed Due to Excessive Length 5 Fig. 4
The (BoC) linked with the libration points on ( x, y )-plane for µ = 0 . , q = 0 . , q = 0 .
25, and then the permissiblerange is 0 . < ω < . top left: for ω = 0 . . , (b) top right: for ω = 0 . middle left: for ω = 0 .
25, (d) middle right: for ω = 0 . bottom left: for ω = 0 .
95, (f) bottom right: for ω = 1 . are converging to L which has infinite extent. Further,when the value of ω increases, the domain of BoC as-sociated to the equilibrium points shrinks significantlyexcept the domain of BoC linked to the libration point L which consequently increases. Moreover, with theincrease in value of angular velocity, the domain of theBoC linked to L and L becomes more regular but de-creases. Further, 3 .
09% of considered initial conditionsconverge to the libration point L , 3 .
79% of initial con-ditions converge to libration point L whereas 11 . L and L and 69 .
1% of initial conditions converge to L when ω = 0 . L , look like a very small bugs without legs andantenna (see Fig. 4f ) when ω = 1 . .
16% of the initial conditions converge to the li-bration point L , 0 .
22% of initial conditions converge to L whereas 18 .
55% of initial conditions converge to eachof libration points L and L which is slightly higherthan the previous cases and remaining are convergingto L with infinite extent. It can be seen that each ofthe initial conditions converge to one of the attractorssooner or later.In Fig.5, the domain of BoC is depicted for thosevalues of ω for which there exist only collinear librationpoints, i.e., when 0 < ω < . ω > . ω = 0 .
02 (see Fig.5a) it is observed that the ex-tent of BoC corresponding to each of libration pointslooks infinite. We believe that this happens since thevalue of ω is very close to zero. It is seen that 53 . L whereas 14 .
1% and 32 .
28% of the investigated initialconditions converge to L and L respectively. It is no-ticed that in Fig. 5(b, c) the domain of BoC linked tolibration point L has infinite extent and for remainingequilibrium points the domain of BoC are finite. How-ever, in this case when three libration points exist, it isseen that for ω = 1 . .
6% and 2 .
23% ofinitial conditions converge to collinear libration points L and L respectively and rest of initial conditionsconverge to L which has infinite extent. Further, when ω = 3 . . . L and L , which unveil the fact thatthe domain of the BoC linked to these libration pointsreduces as value of ω increases. Indeed, it is very remarkable to compare the Fig.5aand Fig.4a, where the number of libration points changesfrom three to five, respectively. It can be noticed thatwhen value of ω is just above zero, the domain of theBoC linked to the libration point L (see Fig.5a) lookslike antennas of the exotic bugs shaped region, consti-tutes the domain of the BoCs linked to the librationpoints L , when the value of ω increases slightly fromthe critical value (see Fig.5a). This happens, since thenon-collinear libration points just originate in the vicin-ity of the L at the critical value of ω . Further, whenwe compare Fig.4f and Fig.5b, it can be noticed thatthe domain of the BoC linked to non-collinear librationpoints L , shrinks to the BoC linked to the collinear li-bration point L when the value of ω crosses the criticalvalue. The main reason for this is the libration points L , annihilate in the vicinity of L at the critical valueof the ω .In Fig. 6, the BoC are depicted for three differentvalues of the mass parameter µ in the presence of fixedangular velocity ω = 0 . µ = 0 .
5, the domain of BoC is symmet-rical about both the axes and also the BoC have finiteextent connected to all the equilibrium points except L which has infinite extent. Further, when the value ofthe mass parameter decreases, the domain of the BoCconnected to those libration points which have finiteextent, expands. The domain of the BoC connected tothe equilibrium points L , appear as exotic bugs hav-ing various legs and antennas whereas the domain ofBoC linked to non-collinear libration points appear asmultiple butterfly wings when the value of mass pa-rameter decreases these wings become larger. Further,only 9 .
92% and 6 .
21% of initial conditions converge tothe collinear libration points L and L respectively,whereas 11 .
78% of those initial conditions converge to L , and remaining 60 .
27% of initial conditions con-verge to L when µ = 0 .
25. In Fig. 6c, when µ = 0 . L de-creases. In addition, the domain of BoC associated to L , which looks like exotic bugs, increases significantly.If we compare Fig.3f in Zotos (2016) with the Fig.6a which is illustrated for the same value of µ = 0 . ω = 0 .
5, we can observe that the well formeddomain of the BoC associated to the libration pointshaving finite extent increases significantly. Moreover,the domain of BoC linked to L , which was regularbecomes more chaotic when ω (cid:54) = 1. Moreover, when ω (cid:54) = 1(in particular < µ decreases from 0 . ω = 0 . itle Suppressed Due to Excessive Length 7 Fig. 5
The (BoC) linked with the libration points on ( x, y )-plane for µ = 0 . , q = 0 . , q = 0 .
25, and (a) top left: for ω = 0 .
02, (b) top right: for ω = 1 .
5, (c) bottom: for ω = 3 .
5. The dots show the positions of libration points. (colour figureonline). and ω = 1. Furthermore, the Fig. 6 is done with a morerefined initial conditions grid with respect to Fig.3f inZotos (2016), and this allows better visualization of thefractal structure.In Fig. 7(a,b), the BoC are presented for two differ-ent values of radiation parameter q and fixed value of ω = 0 .
85. The topology of the domain of BoC associ- ated with the equilibrium points significantly changeswith the change in the radiation parameter. In both thecases the domain of BoC linked to libration point L has infinite extent. It is noticed that when the value of q increases from 0 .
004 to 0 .
01, the number of initialconditions which converge to L has infinite extent, in-creases from 43 .
29% to 52 .
56% and consequently, the
Md Sanam Suraj et al.
Fig. 6
The (BoC) linked with the libration points on ( x, y )-plane for ω = 0 . , q = q = 1, and (a) top left: for µ = 0 .
5, (b) topright: for µ = 0 .
25, (c) bottom: for µ = 0 .
05. The dots show the positions of libration points. (colour figure online).itle Suppressed Due to Excessive Length 9 area related to finite extent decreases. Moreover, theinitial conditions which compose the BoC of the finiteextent linked to the non-collinear libration points de-crease from 23 .
85% to 19 .
83% and the initial conditionswhich compose the BoC linked to the libration point L also decrease whereas for L , it increase. In Fig. 7(c,d), the BoC are presented for two different values ofradiation parameter q and fixed value of ω = 2. Wecan observe that in this case there exist only three li-bration points, and the domain of the BoC linked tolibration points L , increase and consequently, the do-main of BoC associated to L which has infinite extentdecreases with the increase in value of q .It is observed that the values of ω depend on thevalue of q i , i = 1 ,
2, when q = 0 . , q = 1, the valueof ω ∈ ( ω = 0 . , ω = 1 . ω = 0 .
85 is close to the critical value of ω and for 7a, where q = 0 .
004 and for Fig. 7b, where q = 0 .
01. Therefore, a comparison with Fig.4a andFig. 7a, which are both illustrated for the very closevalue of ω to its critical value, the topology of the do-main of BoC in Fig.4a is very noisy whereas in Fig.7ait looks much regular. It is also shown in Fig.8b, thatas the value of the ω is small, the value of the basinentropy increases. However, in both the cases the ex-tent of the BoC linked to the central collinear libra-tion point is infinite and for the remaining librationpoints it is finite. For Fig. 7c, where q = 0 .
01, we get ω ∈ ( ω = 0 . , ω = 1 . q = 0 .
2, we get ω ∈ ( ω = 0 . , ω = 1 . ω for which five librationpoints exist, consequently in 7(c,d) the value of ω is setout of the range so that only three equilibrium pointsexist. Moreover, 7d is illustrated for very close valueof ω to its critical value. We compare Fig. 5(b,c) with7(c,d) and observe that the domain of BoC linked to L , increases in both cases when ω increases as wellas q increases. Further, in all the cases the topology ofthe BoC are symmetrical about the x − axis.If we compare Fig.9a of Ref. Zotos (2016) (when ω = 1) with Fig. 7b where the value of ω = 0 .
85, wecan notice that the basins boundaries are more chaoticin comparison of the previous case moreover, finite re-gions of the domain of the BoC also increases signifi-cantly, however, not much changes are noticed in thetopology of the BoC. Also the similar behaviour hasbeen observed for the domain of BoC linked to the li-bration points L , L , i.e., it increases with the increasein the value of q . On the contrary, when the value of ω (cid:54) = 1, the domain of BoC linked to L , decreases withthe increase in the value of q . In the analysis of color coded diagrams (CCDs), it isobserved that the basin of convergence is highly fractalin the locality of the basins boundaries which unveil thefact that it is quite impossible to judge the final stateof the initial conditions falling inside these fractal re-gions. The term ”fractal” is simply used in the text tounveil the particular area which shows the fractal-likegeometry, without evaluating the fractal dimension (seeAguirre et al. (2001), Aguirre et al. (2009)). Recently,a new tool to measure the uncertainty of the basins hasbeen presented in paper Daza et al. (2016), is namedas the basin entropy and refers to the geometry of thebasins and consequently explore the concept of unpre-dictability and fractality in the context of BoC.The philosophy that works in the background of themethod is to split the phase space into N small cellsin which every cell contains at least one of the totalnumber of final states N A . In addition, the probabilityto evaluate the state j in the k − th cell is denoted by p j,k . Using the Gibbs entropy formulae, the entropy for j − th cell is S j = N A (cid:88) k =1 p j,k log (cid:16) p j,k (cid:17) . (10)The average entropy for the total number of cells N iscalled as basin entropy, i.e., S b S b = 1 N N (cid:88) j =1 S j = 1 N N (cid:88) j =1 N A (cid:88) k =1 p j,k log (cid:16) p j,k (cid:17) . (11)It is necessary to mention that the result for the basinentropy is highly influenced by the total number of cells N , so that a precise value of S b can be obtained forlarger value of N . In an attempt to overcome this prob-lem, we use Monte Carlo procedure to select randomlysmall cells in the phase space, and we observe that for N > × cells, the final value of the basin entropyremain unchanged.In Fig. 8a, we have illustrated the parametric evo-lution of the basin entropy for various values of the an-gular velocity ω , with ω ∈ (0 , .
5) when values of radia-tion parameters are fixed i.e., q = 0 . , q = 0 .
25. Thegray dashed line shows the value of ω ≈ . S b is maximum. We believe thatthe value of S b is maximum as the value of ω is veryclose to the critical value. It is further observed thatthe unpredictability linked to the N-RBoC for the re-stricted three-body problem is higher when the value ofthe ω ∈ (0 . , . ω increases and at ω = 0 .
65 the value
Fig. 7
The BoC linked with the libration points on ( x, y )-plane. When ω = 0 . , q = 1 , µ = 0 .
5, and (a) top left: for q = 0 . top right: for q = 0 .
01. When ω = 2 , q = 1,(c) bottom left: for q = 0 .
01 (d) bottom right: for q = 0 .
2. The dots show thepositions of libration points.(colour figure online). of S b = 0 . ω , and again the value S b increasesalmost monotonically till ω = 1 . S b decreases monotonically when ω ∈ (1 . , . q inthe both cases i.e., when ω = 1 and when ω (cid:54) = 1. Weobserve that the value of the basin entropy remains al-ways higher when the angular velocity ω (cid:54) = 1. However,the similar tendency in the value of basin entropy hasbeen noticed for the increasing value of the radiation parameter q in both the cases. It is necessary to men-tion the fact that to illustrate this diagram we haveused the numerical results for various additional val-ues of the angular velocity ω which are not necessarilypresented in the Figs. 4,5, 6 and 7.The main observations can be summarized as fol-lows: – When ω ≈ . itle Suppressed Due to Excessive Length 11 Fig. 8
The evolution of the basin entropy S b , of the configuration ( x, y ) space with µ = 0 .
5: (a) left: as a function of theperturbation parameter ω . The vertical, dashed, green lines referred as the value of ω where the tendency of the parametricevolution of the basin entropy changes as these are the critical value of ω . (b) right: as a function of the perturbation parameter q when q = 1. The blue line shows the basin entropy when the value of ω = 0 .
85 and gray line shows the basins entropywhen ω = 1. (colour figure online). tion points remains three and in this case the valueof basin entropy S b decreases monotonically. – When ω = 3 . S b ≈ . ω shows the same tendency i.e., the smooth-ness in the basins increases. – When 0 . < ω < . ω it can be ob-served that the value of the basin entropy changesabruptly. Consequently, for this range of ω the un-predictability linked to the N-RBoC for the R3BPin the presence of angular velocity ω is higher. – When the value of radiation parameter q increasesthe value of the basin entropy decreases monoton-ically when q ∈ (0 , .
4) and increases monotoni-cally when q ∈ (0 . ,
1) and ω = 1. Whereas basinentropy decreases monotonically when q ∈ (0 , . q ∈ (0 . ,
1) and ω = 0 .
85. It is necessary to note that, although thecurves are different, their behaviour are same.
In the present paper, we numerically explored the BoCby applying the bivariate version of iterative scheme in the photo-gravitational version of restricted problem ofthree bodies when the angular velocity is not equal tounity. The main outcomes of the present study can besummarized as follows:* There exist either five or three libration points forthe system. For fixed values of the q i , µ and varyingvalues of ω , it can be seen that the libration point L moves towards the primary P whereas the li-bration points L , move toward the primary P asthe value of ω increases. It is observed that the non-collinear libration points originate in the vicinity ofthe libration point L at ω ≈ | q − q | and thesepoints annihilate in the neighbourhood of the libra-tion point L when ω ≈ ( q + q ) .* The attracting domains, linked to the equilibriumpoint L , extend to infinity, in all studied cases (ex-cept for Fig. 5 a ), while the domain of BoC asso-ciated to other libration points are finite. The BoCdiagrams, on the configuration ( x, y ) plane are sym-metrical in all the studied cases, with respect to thehorizontal x -axis.* The numerical investigations suggest that the multi-variate version of Newton-Raphson iterative schemeconverges very fast for those initial conditions whichlie in the vicinity of the libration point and convergevery slow for those initial conditions which are lying in the vicinity of the basin boundaries. However, allthe initial conditions converge to one of the attrac-tors sooner or later.* The numerical investigations unveil that for the in-terval of ω where only three libration points ex-ist, the lowest value of S b is attained near ω =3 .
5, whereas the highest value of basin entropy wasachieved near ω ≈ . ω where the number of libration pointschanges. Moreover, for those intervals of ω in whichfive libration points exist, the maximum value ofthe basins entropy S b is achieved for the value of ω ≈ . ω when there exit five libration points. This reveals theunpredictability, regarding the attracting regions, inthe photo-gravitational restricted three-body prob-lem with angular velocity.In addition, we have used the latest version 12 of Mathematica (cid:114) for all the graphical illustrations in this paper. In future,it is worth studying problem by using different iterativeschemes to analyze the similarity as well as differenceon the associated basins of attraction. Compliance with Ethical Standards - Funding: The authors state that they have not re-ceived any research grant.- Conflict of interest: The authors declare that theyhave no conflict of interest.
Acknowledgments
The authors would like to express their warmest thanks to theanonymous referee for the careful reading of the manuscriptand for all the apt suggestions and comments which allowedus to improve both the quality and the clarity of the paper.
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