On the perturbed photogravitational restricted five-body problem: the analysis of fractal basins of convergence
Md Sanam Suraj, Rajiv Aggarwal, Amit Mittal, Md Chand Asique, Prachi Sachan
OOn the perturbed photogravitational restrictedfive-body problem: the analysis of fractal basinsof convergence
Md Sanam Suraj, Rajiv Aggarwal, Amit Mittal, Md Chand Asique, Prachi SachanMay 22, 2019
Corresponding author: Md Sanam SurajDepartment of Mathematics, Sri Aurobindo College, University of Delhi, NewDelhi-110017, Delhi, IndiaEmail: [email protected], Email: [email protected]:In the framework of photogravitational version of the restricted five-body problem,the existence and stability of the in-plane equilibrium points, the possible regions formotion are explored and analysed numerically, under the combined effect of smallperturbations in the Coriolis and centrifugal forces. Moreover, the multivariate versionof the Newton-Raphson iterative scheme is applied in an attempt to unveil the topologyof the basins of convergence linked with the libration points as function of radiationparameters, and the parameters corresponding to Coriolis and centrifugal forces.Keywords: Five-body problem, Radiation forces, The Coriolis and centrifugal forces,Libration points and Zero-velocity curves, Newton-Raphson method.
In the recent decades, the few-body problem and particularly the circular restrictedfive-body problem is one of the most fascinated as well as the challenging problem inthe fields of dynamical astronomy and Celestial mechanics. The restricted problem offive bodies describes the dynamics of the test particle moving under the gravitationalinfluence of the four primaries. For describing the problem in very realistic manner,the dynamics of the test particles has been studied by adding several perturbing termsin the effective potential of the circular restricted five-body problem. The study of thefive-body problem with several modification is motivated by the various previous studyof the restricted three or four-body problem where the same modifications are proposedand significant results are obtained (e.g., [4], [20], [7], [17], [1], [3], [10], [2]).A particular restricted five-body problem is analyzed by [18] where he exploredthe dynamics of the infinitesimal body ( i.e., the fifth body) of negligible mass, in com-parison to remaining four bodies, and showed the existence and stability of libration1 a r X i v : . [ n li n . C D ] M a y oints. Further, [16] have extended the study of same problem by including the ef-fect of perturbation due to the radiation of the some or even all of the primaries. TheNewton-Raphson iterative scheme is applied to analyze the domain of the basins ofconvergence associated with the libration points in restricted five-body problem by theRef. [32]. This problem is one of the special case where N = N much bigger bodies. In which N − L i , i = , , L , are stable when the relation µ c = µ + ε √ is satisfied where µ = . ... , µ c is the critical mass ratio and thechange in the Coriolis force is controlled by ε . Moreover, in the paper, only the effectof small perturbation in the Coriolis force has been considered to analyze the stabilityof the libration points whereas the centrifugal force remains constant. In an extensionof this study, [8], [9] have discussed the stability of the libration points in the linear aswell in non-linear sense by taking the effect of small perturbations in the Coriolis aswell as in centrifugal forces. They further stated that the Coriolis force is not always astabilizing force. The collinear libration points are always unstable whereas the non-collinear libration points are stable for all mass ratios in the range of linear stabilityexcept the three mass ratios. Moreover, many authors have also extended their studyto unveil the effect of small perturbations in the Coriolis and centrifugal forces in theframe of the restricted four-body problem as well as in their photogravitational ver-sion, where they have discussed the existence and stability of libration points, regionsof possible motions, and the basins of convergence connected to the libration points ofthe system (e.g., [5], [14], [15], [21], [26] ).Recently, the existence as well as stability of the libration points are discussed byvarious authors in the context of the five-body problem, i.e., the axisymmetric five-bodyproblem, see Ref. [11], the basins of the convergence associated with the equilibriumpoints, which acts as attractors, in the axisymmetric five-body problem: the convexcase, see Ref. [22], in the concave case, see Ref. [23], the five-body problem whenthe mass of the test particle is variable, see Ref. [24] and the effect of Coriolis andcentrifugal force in the axisymmetric five-body problem, see Ref. [25].In the past few years, the Newton-Raphson basins of convergence have been studiedby many authors in various dynamical system, i.e., the restricted three-body problem(e.g., [29], [33], [34]), the restricted four-body problem (e.g., [26, 27, 28, 30]), theaxisymmetric restricted five-body problem (e.g., [22, 23]), restricted five-body problem(e.g., [24], [32]). The basins of convergence, linked with the libration points of the2ynamical system, provide some of the most intrinsic properties of these systems. TheNewton-Raphson iterative scheme is applied to scan the set of initial conditions in anattempt to unveil the final states. The basins of convergence is collections of all thoseinitial conditions which converge to one of the particular attractor.Therefore, we believe that it would be interesting to analyze the effect of smallperturbations in the Coriolis and centrifugal forces on the existence as well as on thepositions of the libration points in the photogravitational version of the restricted prob-lem of five bodies. Moreover, the effect of the radiation parameter on the basins ofconvergence associated with the libration points are illustrated in a systematic manner.Therefore, we believe that the present study and obtained results are novel and this isexactly the contribution of our work. The paper is constructed in following pattern:the Sec. 2 provides the idea about of the mathematical model as well as the equationsof motion of the test particle. In the following section, the positions of the librationpoints as function of involved parameters and their stability are discussed, whereas,the regions of possible motion are explored in Sec. 4. In addition, a systematic studyof the topology of the domain of basins of convergence connected with the librationpoints are explored in Sec. 5. The paper ends with Sec. 6 where the discussion andconclusions regarding the main results are presented. The circular restricted five-body problem consists of three primaries where P i = , , P is present at the center. The fifth body,which act as an infinitesimal test particle has a significantly smaller mass in comparisonof the masses of the primaries, does not disturb the motion of the primaries.We choose the rotating co-ordinate system in which the origin coincides with thecenter of mass of the system of primaries. The positions of the center of the primariesare: ( x , y , z ) = ( , , ) , ( x , y , z ) = ( / √ , , ) , ( x , y , z ) = ( − x / , / , ) , ( x , y , z ) = ( x , − / , ) , while the dimensionless masses of the primaries are m = β m , m = m = m = m =
1. Moreover, the three primaries P , , with mass m are placed at the vertices of anequilateral triangle with unity side, whereas the primary P with mass β m , is presentat the center of the equilateral triangle (see Fig. 1). The line joining the centre ofmass of the primaries P and P are taken as the x − axis while the line passing throughorigin and perpendicular to x − axis is taken as y − axis and the z − axis is the line passingthrough the origin and perpendicular to the plane of motion of the primaries.We have further assumed that some or all the primary bodies are sources of radia-tion in context of the restricted five-body problem.3igure 1: The planar configuration of the circular restricted five-body problem. Thethree bodies with equal masses m are located at the vertices of an equilateral triangle,while the fourth primary, with mass β m , is located at the center of the equilateraltriangle. 4e, further, apply the transformation to scale the physical quantities, from the in-ertial to the synodic coordinates system. Therefore, in the rotating frame of reference,the motion of infinitesimal test particle is governed by the following equations:¨ x − y = U x , (1a)¨ y + x = U y , (1b)¨ z = U z . (1c)where U ( x , y , z ) = κ ∑ i = m i q i r i + (cid:16) x + y (cid:17) . In addition, we have introduce the small perturbations ϑ and υ in the Coriolis andcentrifugal forces respectively, using the parameters ε and ε (cid:48) , respectively by ϑ = + ε , | ε | (cid:28) υ = + ε (cid:48) , | ε (cid:48) | (cid:28)
1. Therefore, the Eqs. (1a-1c) become:¨ x − ϑ ˙ y = Ω x , (2a)¨ y + ϑ ˙ x = Ω y , (2b)¨ z = Ω z , (2c)where Ω ( x , y , z ) = κ ∑ i = m i q i r i + υ (cid:16) x + y (cid:17) , (3) κ = ( + β √ ) , and r i = (cid:113) ˜ x i + ˜ y i + ˜ z i , i = , , , , ˜ x i = ( x − x i ) , ˜ y i = ( y − y i ) , ˜ z i = ( z − z i ) , are the distances of the infinitesimal test particle from the respective primaries.The effect of radiation pressure of a radiating source on a particle is expressed bythe radiation factor q i = − b i where b i , always known as radiation coefficient (see[19]), describe the ratio of the radiation force F ri to the gravitational force F gi , i.e., b i = F ri F gi , thus there are four reduction factors q i , i = , , ..., ( β (cid:44) ) reduces to Ref. [16] when ϑ = υ =
1, further reduces to Ref. [18]when q i = , i = , , , β =
0. The Eqs. (2a-2c), i.e., the equations of motion of thetest particle correspond to an integral of motion, i.e., the Jacobi integral which readsas: J ( x , y , z , ˙ x , ˙ y , ˙ z ) = Ω ( x , y , z ) − ( ˙ x + ˙ y + ˙ z ) = C , (4)where ˙ x , ˙ y , and ˙ z represent the velocities, while C is linked to the conserved numericalvalue of Jacobian constant. 5igure 2: The positions of the collinear libration points L xi , i = , ,
5, on the x − axisas functions of µ ( q i = , i = , , Black lines correspond to the gravitational case, blue lines correspond to the case where the central primary radiate with q = . green lines correspond to the case where q = .
1, the solid lines show the case when ε (cid:48) = ε (cid:48) = .
25. The positionsof the central primary body m is denoted by vertical yellow lines. For blue µ ∗∗ = . , µ ∗ = . , and for black µ ∗∗ = . , µ ∗ = . , and for green µ ∗∗ = . , µ ∗ = . µ = / ( + β ) and correspondingto the classical mass parameter of the restricted three-body problem, we have µ ∈ ( , ] for β ∈ [ , ∞ ) . We will try to illustrate how the effect of small perturbations in the Coriolis and cen-trifugal forces influence all the dynamical properties of the libration points. The neces-sary and sufficient conditions, which must satisfy for the existence of libration points,are: ˙ x = ˙ y = ˙ z = ¨ x = ¨ y = ¨ z = . The linked coordinates ( x , y , z ) of the equilibrium points can be evaluated by solvingthe system of following equations, numerically: Ω x ( x , y , z ) = , Ω y ( x , y , z ) = , Ω z ( x , y , z ) = , (5)where Ω x ( x , y , z ) = ∂ Ω ∂ x = − κ ∑ i = m i q i ˜ x i r i + υ x , (6a) Ω y ( x , y , z ) = ∂ Ω ∂ y = − κ ∑ i = m i q i ˜ y i r i + υ y , (6b) Ω z ( x , y , z ) = ∂ Ω ∂ z = − κ ∑ i = m i q i ˜ z i r i . (6c)In the same vein, the second order partial derivatives of the effective potential are asfollows: Ω xx = − κ ∑ i = m i q i (cid:32) r i − x i r i (cid:33) + υ , (7a) Ω yy = − κ ∑ i = m i q i (cid:32) r i − y i r i (cid:33) + υ , (7b) Ω zz = − κ ∑ i = m i q i (cid:32) r i − z i r i (cid:33) , (7c) Ω xy = κ ∑ i = m i q i x i ˜ y i r i = Ω yx , (7d) Ω xz = κ ∑ i = m i q i x i ˜ z i r i = Ω zx , (7e) Ω yz = κ ∑ i = m i q i y i ˜ z i r i = Ω zy . (7f)7igure 3: The parametric evolution of the positions of the libration points, Lxi , i = , , , and L xyi , i = , , ..., q i = , i = , , , µ = .
09; and (a) q ∈ ( , ] , ε (cid:48) = ε (cid:48) ∈ [ , ) , q = P i , i = , , , and the radiating primary P , respectively.More precisely, when we restrict this study only to the coplanar libration points, i.e.,the points which are located on the configuration ( x , y ) plane with z = q i = , i = , ,
3. Therefore, the intersections of the nonlinear equations Ω x ( x , y , ) = Ω y ( x , y , ) = µ is function of q , i.e., the radiationfactor when the central primary P is source of radiation. When q =
1, then µ ∗ = . q = .
5, then µ ∗ = . q = .
1, then µ ∗ = . q increases,the value of µ ∗ increases. Where the µ ∗ is the critical value where the number ofcollinear libration points changes. The problem admits three collinear libration pointsfor µ ∈ ( , µ ∗ ) on the x − axis in which L x lies out side the primary m whereas thelibration point L x lies between m and m and the libration point L x lies on negative x − axis. For µ ∈ [ µ ∗ , ) the problem admits two more collinear libration points L x , when ε (cid:48) =
0, i.e., the effect of centrifugal force is neglected and therefore there existsfifteen libration points in total.In Fig. 2, we have depicted how the positions of the collinear libration points L x , , change when the radiation parameter as well as the parameter ε (cid:48) which controlled thecentrifugal force change. It is observed that as the value of q increases, the position oflibration point L move far from the origin whereas the position of this libration pointmoves towards the origin when ε (cid:48) increases. When we have taken the effect of the8igure 4: The stability diagram for the collinear libration points for q i = , i = , , ,
3, and (a); the black and red colour dots show the criticalvalue of β i.e., ( L x , β ∗ ) = ( − . , . ) ; and ( L x , β ∗∗ ) =( − . , . ) ; for the classical case and the case when ε = ε (cid:48) = .
25 respectively in restricted five-body problem. The gray dashed lines show the po-sitions of collinear libration points in the classical case while the solid blue lines showthe positions of the libration points in the case when the effect of small perturbationsin the Coriolis and centrifugal forces have been considered. (b); the stability diagramfor µ = . q = . ε (cid:48) = .
25 and the black dots show the critical value of ε where ( L x , ε ∗∗ ) = ( − . , . ) , ( L x , ε ∗ ) = ( − . , . ) ; (c); q i = , i = , , , ; q = . ε = . µ = . A ∗ = . B ∗ = . C ∗ = . D ∗ = . E ∗ = . q = . q = . q = . ε = . ε (cid:48) = .
35, and µ = . q ∗ = . q ∗∗ = . q i = , i = , , , and ε = . ε (cid:48) = . µ = . q is denoted by q ∗∗∗ = . A = . B = . µ = . ε (cid:48) = . q = . q = . q = .
4, andthe radiation parameter q ∈ ( , ] . 9igure 5: The stability diagram for the libration points for q = . q = . q = . ε = . ε (cid:48) = . µ = . q ∈ [ . , ] the critical value of q for L x is q ∗ = . L xy , is q ∗∗ = . blue arrows show themovement of the libration points while the purple dots show the critical value of q .The green thick lines show the stable libration points. (colour figure online)10igure 6: The zero velocity curves for µ = . , q i = , i = , , ,
3, and C = . ε (cid:48) =
0; (b) ε (cid:48) = .
15; (c) ε (cid:48) = . ε (cid:48) = .
52; (e) ε (cid:48) = . ε (cid:48) = .
9, and with µ = .
89, (g) ε (cid:48) = .
1; (h) ε (cid:48) = .
8. The blue colour dots show thepositions of the three primary and red dot shows the radiating primary while the teal colour dots show the positions of the libration points.centrifugal force into the consideration, it can be easily observed that for µ ∗∗ < µ ∗ inall the considered three cases which means that the interval µ ∈ ( , µ ∗∗ ] there exist onlythree collinear libration points whereas for µ ∈ ( µ ∗∗ , ) , two more libration points occur.Therefore, the interval shrinks where the nine libration points exist and consequentlythe interval increases in which the fifteen libration points exist when the effect of thecentrifugal force is taken into consideration.It is observed that circular restricted five body problem has an axis of symmetry,i.e., x-axis. Indeed, if the x − axis is rotated successively through an angle of 2 π /
3, twoadditional lines of symmetry y = ± √ x − axis. Therefore, we have a totalof nine or fifteen libration points of the problem when q i = , i = , , q , and Fig. 3b, isillustrated for varying value of parameter ε (cid:48) , i.e., the effect of perturbation in the corio-lis force parameter. It is further observed that as the radiation parameter q increases allthe nine libration points L xi , i = , , L xyi , i = , , ..., ε (cid:48) increases.In the case when the radiation pressure due to other primaries are taken into con-sideration i.e., q i (cid:44) , i = , ,
3, there exist no collinear libration points when q (cid:44) q .It is observed that when q (cid:44) q there exist only five or seven non collinear libration11igure 7: The zero velocity curves for µ = . , q i = , i = , , , ε (cid:48) = C = . C = . C = . blue colour dotsshow the positions of the three primary and red dot shows the radiating primary whilethe teal colour dots show the positions of the libration points.12oints depending on the combination of the parameters ε (cid:48) , q i , i = , , , Ω are freefrom the parameter ε controlling the Coriolis force. Therefore, the positions of thecoplanar libration points remains unaffected by this perturbation parameter. However,this parameter effect significantly the stability of these libration points.The study of the stability of libration points in any dynamical system plays a crucialrole to unveil the various properties of that system. In an attempt to study the linearstability of the coplanar libration points ( x , y ) , a displacement is given to the positionsof infinitesimal mass as ( x + φ , y + ϕ ) where φ , and ϕ denote the perturbations alongthe ox and oy axes respectively.Therefore, expanding the equations of motion ( a and 2 b ) into first-order terms,with respect to φ and ϕ , we have¨ φ − ϑ ˙ ϕ = φ Ω xx + ϕ Ω xy , (8a)¨ ϕ + ϑ ˙ φ = φ Ω yx + ϕ Ω yy , (8b)where Ω xx , Ω yy , and Ω xy are the second order derivatives of Ω , with respect to x and y , evaluated at the libration points.Substituting the solutions of the variational equations, i.e., φ = ξ e λ t and ϕ = ξ e λ t in Eqs. (8a, 8b), we obtain: ( λ − Ω xx ) ξ − ( ϑ λ + Ω xy ) ξ = , (9a) ( λ − Ω yy ) ξ + ( ϑ λ − Ω yx ) ξ = . (9b)The Eqs. (9a, 9b) have nontrivial solution if the determinant of the coefficients ma-trix of the system vanishes. Therefore, the characteristic equation associated with thesystem of linear equations (9a, 9b) is quadratic in Ξ = λ and given as follows: ℘ Ξ + ℘ Ξ + ℘ = , (10)where ℘ = , (11a) ℘ = ϑ − Ω xx − Ω yy , (11b) ℘ = Ω xx Ω yy − Ω xy Ω yx . (11c)The associated libration points are said to be stable if all the four roots of the character-istic equation are pure imaginary. This happens only when the conditions, illustratedbelow, satisfy simultaneously: ℘ > , ℘ > , ℘ − ℘ ℘ > . (12)Indeed, if λ = Ξ then the characteristic equation has two real negative roots whichconsequently give four pure imaginary roots in λ .In Fig. 4, the stability diagrams of the collinear libration points are illustrated. InFig. 4a, the stability diagram is presented for the case when none of the primary isradiating. The pink colour region shows the stability region for the classical case of13estricted five-body problem. It can be noticed that the only most negative librationpoint on the x − axis i.e., L x is stable for β ∗ = . β ∗ is the criticalvalue of β such that the libration point L x is stable for β ≥ β ∗ . The green colourregion shows the stability region for the case when the effect of small perturbationsin the Coriolis and centrifugal forces have been taken into account. It is observedthat the stable regions increase in comparison of the classical case (see [18]) and alsothe β ∗∗ = . < β ∗ i.e., the critical value of β also decreases significantlyin comparison of the β ∗ . However, for this set of values only the libration point L x isstable. In Fig. 4b, we have illustrated the stability region when only the central primaryis source of radiation. It is observed that as the effect due to small perturbation in theCoriolis force increases the libration points L x , become stable. The critical value of ε are denoted by ε ∗ and ε ∗∗ such that the libration point L x is stable for ε ≥ ε ∗∗ while L x is stable for ε ≥ ε ∗ . Therefore, it can be concluded that the libration point L x isstable for slightly higher value of ε . In Fig. 4c, the stability regions for the collinearlibration points are discussed for varying value of the parameter ε (cid:48) and fixed value ofthe other parameters. It is observed that the collinear libration point L x is stable forthe two different intervals.In Fig. 4d, the stability region is depicted for varying value of the radiation param-eter q where the other primaries are also source of radiation. It is observed that thelibration point L x is stable in two intervals, i.e., q ∈ ( , q ∗∗ ] and q ∈ [ q ∗ , ] .In Fig. 4e, the stability region is illustrated only when the central primary is radi-ating. It is observed that the collinear libration point L x is stable for the two differentintervals, i.e., q ∈ [ q ∗∗∗ , B ] and q ∈ [ A , ] .The movement of the positions of the libration points has been illustrated in Fig. 4f,where the radiation parameter of each primary is different, i.e., q = . q = . q = .
4, and the radiation parameter q ∈ ( , ] . It can be observed that there existonly non-collinear libration points and the number of libration points varies from fiveto seven as the value of the radiation parameter q due to the primary P increases. Thestability analysis shows that the libration points L xy , , , are stable in some interval ofradiation parameter q . The stable libration points are shown in green colour.For q i = , i = , , q = . ε (cid:48) = .
25; and µ = . L xi , i = , , ..., L xyi , i = , , ...
10. When we have analyzed the effect of the parameter ε on the stability of the these libration points, it is unveiled that the libration points L x and L xy , are stable for ε ∈ [ ε ∗∗ = . , ] while L x and L xy , are stable for ε ∈ [ ε ∗ = . , ] .For q i = , i = , , q = . ε = . µ = . ε (cid:48) , it is observed that in the interval ε (cid:48) ∈ ( , D ∗ ] and ( C ∗ , ] there exist nine librationpoints whereas for ε (cid:48) ∈ ( D ∗ , C ∗ ] there exist fifteen libration points. The numericalinvestigations for the stability of these libration points unveil that the libration points L x , L xy , are stable in ε (cid:48) ∈ ( , A ∗ ] , L x , L xy , are stable in ε (cid:48) ∈ ( B ∗ , C ∗ ] and L x , L xy , are stable in ε (cid:48) ∈ ( D ∗ , E ∗ ] .In figure 5, the parametric evolution of the movement of position of libration pointsare illustrated when the radiation parameters for the primaries m , are same and forvarying value of radiation parameter q ∈ ( . , ] . It can be observed that the positionsas well as the movement of the non-collinear libration points are symmetrical about the14igure 8: A characteristic example of consecutive steps that are followed by theNewton-Raphson iterative scheme and the associated crooked path-line that leads toa libration point L x . The starting point I ( x , y ) = ( − . , . ) is depicted by red dot while the libration point is pin pointed by green dot, to which the method con-verges. The Newton-Raphson method converges after 16 iterations to L x with accuracyof six digits. x − axis. Also, all the libration points move far from the central primary as the valueof the parameter q increases. The thick green lines show the stable region for thelibration points L x and L xy , . The non-collinear libration points L xy , are stable for q ∈ ( . , ] with the same set of values for the other parameters while the co-linear libration point L x is stable for q ∈ ( q ∗ , ) . The zero velocity surface (ZVS) is a three-dimensional surface defined by the relation2 Ω ( x , y , z ) = C . This ZVS provided the freedom to illustrate the energetically possibleorbits of the fifth body whereas the Hill’s regions are defined as the projections of thesesurfaces on the configuration ( x , y ) plane. Moreover, the zero velocity curves (ZVCs)are the locus where the kinetic energy vanishes, describe the boundaries of the Hill’sregions. Thus, we revealed that how the topology of the ZVSs, and of course the linkedZVCs change as the function of the parameters q , ε (cid:48) and Jacobian constant C .In Fig.6, we have illustrated the zero velocity curves for varying value of the param-eter ε (cid:48) and for fixed value of mass parameter µ where none of the primary is radiating.We have fixed the value of the mass parameter µ in the interval for which there existfifteen libration points and also the value of the Jacobian constant C is taken constant. Itis observed that when ε (cid:48) = ( x , y ) planefor the increasing value of the radiation parameter q , when nine libration points existfor µ = . ε (cid:48) = q i = , i = , ,
3; (a) q = .
20; (b) q = .
25; (c) q = .
5; (d) q = .
85; (e) q = .
95; (f) q = .
99. The crimson colour dots show the positionsof the four primaries while the teal colour dots show the positions of the nine librationpoints. For the colour code denoting the attractors see the text.16unicate from one primary to either of the primary also the test particle is restricted tomove from inner regions to outer region and vice-versa, i.e., there is no communica-tion channels. In Fig. 6b, when ε (cid:48) = .
15, the channel at the libration points L x , L xy and L xy open and consequently the test particle can orbit to communicate between theprimaries P i , i = , , , ε (cid:48) to 0.153, the three morechannels around the libration points L x , , appear so that the infinitesimal mass cancommunicate from interior region to the exterior region and vice-versa. In the follow-ing panel, it is observed that as the parameter ε (cid:48) increases to 0.52, the forbidden regionshrinks significantly and constitutes three branches, each containing L x , , , L xy , , ,and L xy , , , respectively. Moreover, for ε (cid:48) = .
59, three limiting situations at the li-bration points L x , L xy and L xy occur while these forbidden regions further shrinkand constitute six branches each containing one libration point for ε (cid:48) = . ε (cid:48) . In Fig. 6g and h, the regions of possible motions are illustrated for the specific valueof the mass parameter µ for which nine libration points exist and observed that as thevalue of ε (cid:48) increases, the regions of possible motion also increase but even for highervalue of the ε (cid:48) = .
8, the forbidden regions do not disappear completely.It can be concluded that as the value of the parameter ε (cid:48) , which occur due to effectof small perturbation in the centrifugal force, increases the forbidden region decreasessignificantly and consequently the test particle can move freely on the entire configu-ration ( x , y ) plane except the six small islands shaped forbidden region.In Fig. 7, the zero velocity curves are drawn for the increasing value of the Jacobianconstant. It is observed that as the value of the Jacobian constant increases, the regionsof possible motion also decrease. In this section, we describe how the topology of the basins of convergence linked withthe coplanar libration points (i.e., the libation points lie on ( x , y ) plane) are affectedby the effect of small perturbations in the Coriolis and centrifugal forces and by theradiation parameters q i , i = , , ...,
3. We have adopted the same philosophy and pro-cedure discussed in Ref. [32], and [22]. The basin of convergence associated withthe libration points are collections of the set of points which converge to specific at-tractor after successive iterations. We use the multivariate version of Newton-Raphsoniterative method, which is defined by the map x n + = x n − J − f ( x n ) , (13)where x = ( x , y ) , f ( x n ) define the system of equations 6a-6b, and J − is the Jacobianmatrix.In this section, we will illustrate how the radiation parameter q i , i = , , , ε (cid:48) which occurs due to effect of small perturbation in the centrifugalforce, affect the topology of the Newton-Raphson basins of convergence linked withthe libration points (which act as attractors) in the circular photogravitational restrictedfive-body problem. We have used the color-coded diagrams to classify the nodes on17igure 10: The Newton-Raphson basins of attraction on the configuration ( x , y ) planefor the increasing value of the parameter ε (cid:48) , when nine libration points exist for µ = .
4; (a) q = , ε (cid:48) = .
2; (b) q = , ε (cid:48) = .
6; (c) q = . , ε (cid:48) = .
8. The crimson colour dots show the positions of the four primaries while the teal colour dots show thepositions of the nine libration points. For the colour code denoting the attractors seethe text.the configuration ( x , y ) plane. The different colour is assigned to each pixel accordingto the final state of the associated initial condition.In Fig. 8, the crooked path line is illustrated by the successive approximationpoints. The crooked path leads to a desired position of the equilibrium point whenthe iterative scheme converges for the particular initial condition. The collections ofall initial conditions, which lead to same specific libration point, compose a basins ofconvergence or attracting domain.The colour code for various attractors are as follows: L x → red, L x → cyan, L x → teal, L x → gray, L x → crimson, L xy → amber, L xy → black, L xy → pink, L xy → indigo, L xy → olive, L xy → yellow, L xy → lime green, L xy → purple, L xy → lightblue, L xy → orange, and the non-converging points are shown in white colour.In Fig. 9, we have depicted the basins of convergence associated with the co-planar libration points by using the multivariate version of the Newton-Raphson itera-tive scheme for six increasing values of the radiation parameter q i.e., when the centralprimary is radiating and also q i = , i = , , q , themass parameter µ is fixed in the interval for which there is only nine libration points.It is observed that in all the panels the extent of the basins of convergence associatedwith each of the libration point is infinite however the configuration ( x , y ) plane is fullycovered by well-formed domain of the basins of convergence. Additionally, the vicin-ity of the basins boundaries are composed of the mixture of the initial conditions whichhave fractal like geometry. It is observed that the slight change in the radiation param-eter q leads to the significant change in the structure of the basins of convergence. InFig. 9a, the basins of convergence is illustrated for parameter q = .
2, i.e., the centralprimary is source of radiation and the area enclosed by schematic yellow circle will benamed as interior region-I, and area enclosed by schematic green circle will be named18s interior region-II, from now. It is observed that, outside the interior region-I, threewings shaped area is very noisy. The domain of the basins of convergence associatedwith the libration points L x , and L xy , look like three exotic bugs, with many legsand antenna whereas the domain of the basins of convergence L x and L xy , look likebutterfly wings, which lie inside the region-I, are regular but their boundaries are sep-arated by thin strips which are composed of initial conditions. The region-II is alsonoisy except the three leaves shaped domain of the basins of convergence associatedwith the libration points L x and L xy , . Moreover, as the value of radiation parameter q increases, the three wings shaped regions, which were noisy in previous panel andlooks like chaotic sea of initial conditions, now started to convert into some small regu-lar islands. In Fig. 9c-f, we can observe that a large area of these wings shaped regionsconverted to well shaped basins of convergence as the value of radiation parameter q increases. Additionally, the three leaves shaped domain of the basins of convergenceinside the region-II increase and most of the region inside it, is now covered by wellformed basins of convergence.In Fig. 10, the basins of convergence are illustrated as the function of parameter ε (cid:48) when nine libration points exist. It can be observed that the entire configurationplane is covered by well formed basins of attraction, however the domain of each ofthe basins of convergence has infinite extent. As we increase ε (cid:48) , the interior region-I,shrinks whereas the number of leaves in the wings shaped region increases. Moreover,we observe the vicinity of the basins boundaries are composed of the mixture of initialconditions. This fact unveils that the basins boundaries are highly chaotic and hence,the initial conditions, inside these areas are extremely sensitive to its final state. In-deed, even a very slight change in the value of initial conditions will lead to a entirelyunexpected attractor. The domain of the basins of convergence associated with the li-bration points L x , L xy , look like exotic bugs with many legs and antenna, shrink asthe parameter ε (cid:48) increases.In Fig. 11, the basins of convergence are depicted for varying values of the radiationparameter q , when fifteen libration points exist. The configuration plane is coveredby well-formed basins of attraction whereas the basins boundaries are composed ofthe mixtures of the initial conditions. When we compare the Figs. 11a and b, it canbe observed that the three heart shaped regions, which occur at the joining point ofwing shaped area, separate into two parts and consequently hinges which connect theinterior region-I and the wings shaped region shrinks. Moreover, the area inside theinterior region-II becomes more regular in comparison of panel-b. Furthermore, theincrease in the parameter q = .
95 the basins boundaries which connect the leaves ofthe wings are now regular and the chaotic basin boundaries disappear.The basins of the convergence associated with the libration points, for the casewhen all the primaries are radiating and the values of these radiation parameters aredifferent, are illustrated in Fig. 12. As we have observed that in this case the sym-metry of the libration points about the x − axis is destroyed and only five non-collinearlibration points exist, the basins of convergence also look strange. We observe thata majority of the ( x , y ) plane is covered by the highly fractal mixture of initial con- The term fractal simply unveils the fact that the specific area has a fractal-like geometry, without con-ducting extra calculations, as in [6]. ( x , y ) planefor the increasing value of the radiation parameter q , when fifteen libration pointsexist for µ = . ε = . q i = , i = , ,
3; (a) q = .
09; (b) q = . q = .
55; (d) q = .
95. The crimson colour dots show the positions of the fourprimaries while the teal colour dots show the positions of the fifteen libration points.For the colour code denoting the attractors see the text.20igure 12: The Newton-Raphson basins of attraction on the configuration ( x , y ) planewhen five libration points exist for µ = . ε (cid:48) = .
25; (a) q = . q = . q = . q = .
15; (b) q = . q = . q = . q = .
15; (c) q = . q = . q = . q = .
15. The crimson colour dots show the positions of thefour primaries while the teal colour dots show the positions of the five libration points.For the colour code denoting the attractors is as follows: L xy ( blue ); L xy ( green ); L xy ( red ); L xy ( black ); L xy ( yellow ); the non-converging points (white).ditions. The only regular island shown in green and red colour corresponds to thoselibration points which exist near the origin, i.e., L xy , (see Fig. 4). However, the basinsof convergence associated with the libration points L xy , , look like exotic bugs with-out legs and antennas. The Fig. 12b and c look like mirror images about the x − axiswhen the green and red colors are interchanged. This happens only because the valueof the radiation parameter q and q are interchanged. In the present manuscript, we have successfully illustrated the influence of the effectof small perturbations in the Coriolis and centrifugal forces on the positions of thelibration points, their stability and the regions of possible motion in context of pho-togravitational version of restricted five-body problem. We have further, investigatednumerically the basins of convergence, linked to the libration points, in the problem.We have used the multivariate version of the Newton-Raphson iterative scheme to un-veil the associated basins of attraction on the configuration ( x , y ) plane. The role ofthe attracting domains is very crucial as it can explain how each initial point of the ( x , y ) plane is attracted by the attractors of the dynamical system, which are, indeed, alibration point of the system. We have successfully managed to detect how the geom-etry of the Newton-Raphson basins of attraction changes as a function of the radiationparameters and the ε (cid:48) .In the problem, we have taken either the central or all of the primaries as source ofradiations. The main conclusions can be summarized as follows: • The number as well as the positions of the libration points are highly sensitivewith the change in the value of the parameter ε (cid:48) . The value of µ ∗ , ( i.e., the value21f mass parameter is critical value since it delimits the point where the numberof the equilibrium points changes) decreases as ε (cid:48) increases. • If only the central primary radiates, i.e., q i = , i = , , , and other perturbationsare neglected, all the libration points move away along a straight line from thecentral primary when q ∈ ( , ] whereas, all the libration points move towardsit along a straight line if the effect in the centrifugal force increases where noneof the primary is radiating. • The non collinear libration points L xyi , i = , , , q increases where the other primaries are alsothe source of radiation such that q (cid:44) q = q . • The stability analysis of the libration points have unveiled that in most of thecases only the collinear libration point L x is stable. Moreover, for various com-bination of parameters, the libration points L x , , and L xy , , , are also stablesometime or another. It is interesting to note that only those non-collinear libra-tion points are stable, which lie in the first or fourth quadrant. • If we compare our result from the classical five-body problem (see [18]), it isobserved that the critical value of β decreases significantly when the effect ofthe Coriolis and centrifugal forces is taken into consideration. • If we increase the parameter ε (cid:48) , the possible regions of motion increase whereasthey decrease when the Jacobian constant C increases. Further, these regions areunaffected by the change in value of ε , i.e., parameter occur due to presence ofthe Coriolis force as the effective potential is independent from this parameter. • The basins of convergence associated with all the libration points have infiniteextent. The the interior region-I, interior region-II, and the wing shaped regionbecome more regular as the radiation parameter due to the central primary in-creases. Moreover, the entire basins of convergence shrinks in whole as theparameter ε (cid:48) increases while the basins of convergence remains unaffected withthe change in the value of ε which control the Coriolis force. • The majority of the area of basins of convergence look like chaotic sea, com-posed of different type of initial conditions when the value of the radiation pa-rameters q i , i = , , .., of Mathematica (cid:114) is used for the graphical illustration of thepaper. It is interesting to use other types of iterative schemes in the present problemto discuss the similarities and differences in the context of the basins of convergencelinked with the libration points of the dynamical system. We believe that the above-mentioned results and ideas will be useful in the active field of attracting domains oflibration points in various dynamical system.22 ompliance with Ethical Standards - Funding: The authors state that they have not received any research grants. - Conflict of interest: The authors declare that they have no conflict of interest.
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