Bifurcation of periodic orbits for the N -body problem, from a non geometrical family of solutions
aa r X i v : . [ m a t h . D S ] S e p BIFURCATION OF PERIODIC ORBITS FOR THE N -BODYPROBLEM, FROM A NON GEOMETRICAL FAMILY OFSOLUTIONS. OSCAR PERDOMO , ANDRÉS RIVERA , JOHANN SUÁREZ Abstract.
Given two positive real numbers M and m and an integer n > , itis well known that we can find a family of solutions of the ( n + 1) -body problemwhere the body with mass M stays put at the origin and the other n bodies, allwith the same mass m , move on the x - y plane following ellipses with eccentri-city e . It is expected that this geometrical family that depends on e , has somebifurcations that produces solutions where the body in the center moves on the z -axis instead of staying put in the origin. By doing analytic continuation ofa periodic numerical solution of the -body problem –the one displayed on thevideo http://youtu.be/2Wpv6vpOxXk –we surprisingly discovered that the originof this periodic solution is not part of the geometrical family of elliptical solutionsparametrized by the eccentricity e . It comes from a not so geometrical but easier todescribe family. Having notice this new family, the authors find an exact formulafor the bifurcation point in this new family and use it to show the existence ofnon planar periodic solution for any pair of masses M , m and any integer n . Asa particular example we find a solution where three bodies with mass movearound a body with mass that moves up and down. Introduction
The spatial isosceles solutions of the three body problem are solutions where two bo-dies of equal mass m have initial positions and velocities symmetric with respect tothe z -axis which passes through their center of mass at the origin, the third body ofmass M moves on the z Date : September 15, 2020.2010
Mathematics Subject Classification.
Key words and phrases. N -body problem, periodic orbits, bifurcations, analytic continuationmethod. periodic spatial isosceles solutions has been considered using variational methods,numerical methods or techniques of analytical continuation. In most of the casesit is assumed that the body on the z -axis has a small mass M or a mass M = 0 ,see [1, 3, 10, 15]. In the latter case we obtain the Sitnikov problem . For examplein [1] using the analytical continuation method of Poincarà c (cid:13) , the authors proveanalitically the existence of periodic and quasi-periodic spatial isosceles solutions,for
M > sufficiently small as a continuation of some well known periodic solutionsof the reduced circular Sitnikov problem (when the body in the z -axis has infini-tesimal mass M = 0 and the other bodies move in circular orbits of the two-bodyproblem). Using variational methods and for M > non necessarily small, theauthors in [18] prove numerically the existence of spatial isosceles solutions as min-imizers of the corresponding Lagragian action functional. A natural generalizationof the spatial isosceles solutions are those where n bodies move on the vertex of aregular polygon perpendicular to the z -axis and the ( n + 1) -th body moves alongthe z R [2 π/n ] = cos(2 π/n ) − sin(2 π/n ) 0cos(2 π/n ) sin(2 π/n ) 00 0 1 . A direct computation (also see [11]), shows that
Theorem 1.1.
The functions x ( t ) = (0 , , f ( t )) , y k ( t ) = R k [2( k − π/n ] y ( t ) , k = 1 , . . . , n, with y ( t ) = (cid:18) r ( t ) cos( θ ( t )) , r ( t ) sin( θ ( t )) , − Mmn f ( t ) (cid:19) , satisfying f (0) = 0 , ˙ f (0) = b, r (0) = r , ˙ r (0) = 0 , θ (0) = 0 , ˙ θ (0) = ar , provides a solution of the ( n + 1) -body problem with n ≥ and masses M ≥ forthe body moving along x ( t ) and mass m > for each body moving along the function y k ( t ) , if and only if FAMILY OF SOLUTIONS OF THE N-BODY PROBLEM 3 ¨ f = − ( M + mn ) fh , ¨ r = r a r − mλ n r − M rh , ˙ θ = r ar , (1.1) where h = r r + (cid:16) M + nmmn (cid:17) f and λ n = n − X k =1 − e i πkn | e i πkn − | = 14 n − X k =1 csc (cid:16) πkn (cid:17) . Related to the system (1.1) we mention some important results obtained in [11, 12,13, 14] by the first author of this paper. Firstly, a reduction of the system (1.1) to asingle second order differential equation, see [11]. Secondly, a rigorous mathematicalproof of the periodicity of a solution of the 3-body problem describe by (1.1), see[12]. This was achieved through a numerical method that keeps track of the round-offerror, developed in [14] and also by a lemma that can be viewed as a numerical versionof the implicit function theorem, see [12]. Finally, in [13] the author describes a newfamily of symmetric periodic solutions of (1.1) that contains a nontrivial bifurcationpoint. The diagram of periodic solutions in the space of initial conditions (similar tothe one in Figure 4.2) shows four branches emanating from this bifurcation solution.One is unbounded, another has as a limit point a solution where there is collision oftwo bodies, the third branch has as a limit point a solution where there is collisionof the three bodies and the fourth one ends on a family of trivial solutions.The main purpose of this paper is to provide an analysis of the properties of the familyof periodic solutions in the fourth branch mentioned above and to provide an explicitformula for its limit point. To this end, assuming nonzero angular momentum for(1.1) we will consider an appropriate system which symmetries can be used to obtainthe periodic solutions in the fourth branch; see Section 2. In Section 3, we proveTheorems 3.1 and 3.2 which are the statements of our main results. It is interestingto compare our results in this paper with those in [1]. Both papers analytically showthe existence of periodic solutions using analytic continuation techniques. The newsolutions in [1] emanate from solutions of the Sitnikov problem while those in thispaper emanates from what we can call “ part of circular solutions ” where the twomasses m and M , are arbitrary. We call them part of circular solutions because n of the ( n + 1) bodies move on a circle but they do not necessarily cover the wholecircle. Finally, numerical validation of the theoretical results obtained in Section 3are presented in Section 4. OSCAR PERDOMO, ANDRÉS RIVERA AND JOHANN SUÁREZ The reduced problem and symmetries
A simple inspection of the system (1.1), shows that r ˙ θ is the first integral of the an-gular momentum C = r a . Having said this, throughout this document we consideronly solutions of (1.1) with nonzero angular momentum, i.e., C = r a = 0 . Thisassumption allows us to consider only the initial value problem, ¨ f = − ( M + mn ) fh , f (0) = 0 , ˙ f (0) = b, ¨ r = r a r − mλ n r − M rh , r (0) = r , ˙ r (0) = 0 . (2.1)We can check that, if φ ( t ) = ( f ( t ) , r ( t )) is a solution of (2.1) and θ ( t ) = Z t r ar ( s ) ds, then φ ( t ) = ( f ( t ) , r ( t ) , θ ( t )) is a solution of (1.1). From now on, we denote by F ( a, b, t ) = f ( t ) , R ( a, b, t ) = r ( t ) , Θ( a, b, t ) = θ ( t ) , the solutions of the system (1.1) with initial conditions(2.2) f (0) = 0 , ˙ f (0) = b, r (0) = r , ˙ r (0) = 0 , θ (0) = 0 , ˙ θ (0) = ar . Remark . If for some a and b , f ( t ) and r ( t ) are T -periodic of the system (2.1) then ( f ( t ) , r ( t ) , θ ( t )) defines a periodic solution of the ( n + 1) -th body, if and only if θ ( T ) is equal to n π/n with n and n whole numbers. See [13]. In general, T -periodicsolutions of the systems (2.1) define reduced-periodic solutions of the ( n + 1) -bodyproblem. This is, solutions with the property that every T unites of time, thepositions and velocities of the ( n + 1) bodies only differ by an rigid motion in R . The existence of periodic solutions of (1.1) becomes simpler if we restrict our seachto periodic solutions with symmetries. In such a case, the following lemma, see [13]provides a useful result.
Lemma 2.2.
Let φ ( t ) = ( F ( a, b, t ) , R ( a, b, t )) be a solution of (2.1). ⊲ If for some < T we have F ( a, b, T ) = 0 and R t ( a, b, T ) = 0 , ( † ) then f ( t ) = F ( a, b, t ) and r ( t ) = R ( a, b, t ) are both T -periodic functions. FAMILY OF SOLUTIONS OF THE N-BODY PROBLEM 5 ⊲ If for some < T we have F t ( a, b, T ) = 0 and R t ( a, b, T ) = 0 , ( †† ) then f ( t ) = F ( a, b, t ) and r ( t ) = R ( a, b, t ) are both T -periodic functions. It is worth to mentioning that the solutions φ ( t ) = ( F ( a, b, t ) , R ( a, b, t )) that satisfies ( † ) are called odd solutions because f ( t ) = F ( a, b, t ) is an odd function with respectto t = 0 . On the other hand, if φ ( t ) satisfies ( †† ) they are called odd/even solutionsbecause f ( t ) = F ( a, b, t ) is an odd function with respect to t = 0 , but with respect to t = T , both functions f ( t ) = F ( a, b, t ) and r ( t ) = R ( a, b, t ) are even. Furthermore,we point out that every odd/even solution is also an odd solution.3. Main results
Periodic solutions for the reduced problem.
In this section we prove theexistence of a one parametric a family of periodic solutions for the reduced ( n + 1) -body problem (2.1). Theorem 3.1.
For any n ≥ , let us define λ n = 14 n − X k =1 csc (cid:0) kπ/n (cid:1) . Assume that m, M and n satisfy that λ n = np + Mm ( p − for every positive integer p . Then,for any positive real number r there exist b = 0 near , T > near π q r nm + M ,and a > near q λ n m + Mr that provides an odd T -periodic solution of the reduced ( n + 1) -body problem with initial conditions described in Theorem 1.1.Proof. For fixed values of m, M, r , let F ( a, b, T ) , R ( a, b, T ) be the solutions of (2.1)-(2.2) evaluated at t = T . By Lemma 2.2 it follows that if for some a, b and T wehave that F ( a, b, T ) = 0 and R t ( a, b, T ) = 0 , then t −→ f ( t ) = F ( a, b, t ) and t −→ r ( t ) = R ( a, b, t ) are odd and even functionsrespectively and both functions are periodic with period T . An easy computationshows that F ( a, , T ) = 0 , ∀ T ∈ R . From here we can deduce the followinga) For all ( a, b, T ) we can express F as F ( a, b, T ) = b ˜ F ( a, b, T ) , (3.1) OSCAR PERDOMO, ANDRÉS RIVERA AND JOHANN SUÁREZ with ˜ F some smooth function. Therefore, an easy a direct computation showsthat ˜ F ( a, , T ) = F b ( a, , T ) . Moreover, for all ( a, T ) it follows ˜ F t ( a, , T ) = F bt ( a, , T ) , ˜ F a ( a, , T ) = F ba ( a, , T ) , and ˜ F b ( a, , T ) = F bb ( a, , T ) / . b) If a = a = r λ n m + Mr , we have that F ( a , , T ) = 0 and R ( a , , T ) = r solves (2.1). Therefore, F ( α ( T )) = R t ( α ( T )) = 0 , with α ( T ) = ( a , , T ) , T ∈ R . The path α ( T ) , more precisely, the pseudo periodic solutions of the ( n + 1) -body problem induced by the equations ( † ) , can be viewed geometrically asthe pseudo periodic solutions where n of the ( n + 1) -bodies move along partof a circle and the ( n + 1) -body stays put in the center. From this collectionwe will find a bifurcation point, a particular value for T , that will providethe nontrivial periodic solutions.The previous observations suggest to study the solutions of the system ˜ F ( a, b, T ) = 0 and R t ( a, b, T ) = 0 . (3.2)To this end we will use relation (3.1) to compute derivatives of ˜ F and R t . Recallthat F tt = − ( M + mn ) Fh , (3.3) R tt = a r R − mλ n R − M Rh , (3.4)where h = r R + (cid:16) M + nmmn (cid:17) F . Taking the partial derivative with respect to b on both sides of Equation (3.3) and evaluating at α ( t ) give us that F b ( α ( t )) satisfies ¨ u = − ( M + mn ) r u, with u (0) = 0 , ˙ u (0) = 1 . Therefore, F b ( α ( t )) = (cid:18) M + mnr (cid:19) − / sin (cid:18) M + mnr (cid:19) / t ! . (3.5) FAMILY OF SOLUTIONS OF THE N-BODY PROBLEM 7
The equation above shows that, if T = π s r mn + M , (3.6)then ˜ F ( α ( T )) = F b ( α ( T )) = 0 . In consequence, from a ) and b ) it follows that α ( T ) satisfies ˜ F ( α ( T )) = 0 , and R t ( α ( T )) = 0 , showing that the point ( a , , T ) solves (3.2). Finally, from Equation (3.5) we getthat ˜ F t ( α ( T )) = F bt ( α ( T )) = − . (3.7)Now, we take the derivative with respect to a on both sides of Equation (3.4) andevaluate at α ( t ) . Then R a ( α ( t )) satisfies ¨ v = 2 (cid:18) λ n m + Mr (cid:19) / − ( λ n m + M ) r v, with v (0) = 0 , ˙ v (0) = 0 , therefore R a ( α ( t )) = 2 (cid:18) λ n m + Mr (cid:19) − / − cos (cid:18) λ n m + Mr (cid:19) / t !! From this last expression we deduce R at ( α ( t )) = 2 sin (cid:18) λ n m + Mr (cid:19) / t ! . Applying the same ideas, the function R b ( α ( t )) can be obtained by solving ¨ w ( t ) = − ( λ n m + M ) r w, with w (0) = 0 , ˙ w (0) = 0 , therefore R b ( α ( t )) = R bt ( α ( t )) = 0 , for all t ∈ R . Further, from (3.4) we can deduce that R tt ( α ( t )) = 0 for all t ∈ R . These computations shows that the gradient of the function R t at α ( T ) is given by ∇ R t ( α ( T )) = π r λ n m + Mmn + M ! , , ! . (3.8) OSCAR PERDOMO, ANDRÉS RIVERA AND JOHANN SUÁREZ
Notice that our condition of λ n guarantees that ∇ R t ( α ( T )) does not vanish. Onthe other hand, using a ) and the equation (3.7) we obtain ∇ ˜ F ( α ( T )) = (cid:18) F ba ( α ( T )) , F bb ( α ( T )) , − (cid:19) . (3.9)Since η = ∇ ˜ F ( α ( T )) × ∇ R t ( α ( T )) = (cid:18) , R ta ( α ( T )) , R ta ( α ( T )) F bb ( α ( T )) (cid:19) , has its second entry different from zero, by the Implicit Function Theorem thereexists ǫ > and a pair of continuous functions such that T, a : ( − ǫ, ǫ ) → R b −→ T = T ( b ) , b −→ a = a ( b ) , with T (0) = T , a (0) = a , such that ˜ F ( γ ( b )) = 0 , and R t ( γ ( b )) = 0 , with γ : ( − ǫ, ǫ ) → R , given by(3.10) b → γ ( b ) = ( a ( b ) , b, T ( b )) , γ (0) = ( a , , T ) . Therefore, for each b ∈ ( − ǫ, ǫ ) it follows F ( γ ( b )) = b ˜ F ( γ ( b )) = 0 , and R t ( γ ( b )) = 0 . In consequence, using Lemma 2.2 we get that for any b , the functions f ( t ) = F ( a, b, t ) y r ( t ) = R ( a, b, t ) define a T ( b ) -periodic solution of the reduced problem (2.1). (cid:3) Theorem 3.2.
For any n ≥ , let us define λ n = 14 n − X k =1 csc (cid:0) kπ/n (cid:1) . Assume that m, M and n satisfy that λ n = nq + Mm ( q − for every positive even integer q . Then,for any positive real number r there exist b = 0 near , T ∗ > near π q r nm + M and a > near q λ n m + Mr that provides an ood/even T ∗ -periodic solution of the reduced ( n + 1) -body problem with initial conditions described in Theorem 1.1.Proof. We follows the same lines of the proof of Theorem 3.1, but now consideringconsider the system(3.11) F t ( a, b, T ) = 0 and R t ( a, b, T ) = 0 . FAMILY OF SOLUTIONS OF THE N-BODY PROBLEM 9
By (3.1) there exists a function W ( t, a, b ) such that F t ( a, b, t ) = bW ( a, b, t ) ,F bt ( a, b, t ) = W ( a, b, t ) + bW b ( a, b, t ) , for all ( a, b, t ) . In particular F bt ( a, , t ) = W ( a, , t ) , for all ( a, t ) . Now we search for points ( a, b, T ∗ ) such that W ( a, b, T ∗ ) = 0 and R t ( a, b, T ∗ ) = 0 . To this end, we need to study the zeroes of the function F bt ( a, b, T ∗ ) . From (3.5) wehave F bt ( α ( t )) = cos (cid:18) M + mnr (cid:19) / t ! , Therefore, F bt ( α ( T / and, W ( a , , T /
2) = 0 and R t ( a , , T /
2) = 0 . Once again, with the aim of applying the Implicit Function Theorem we consider thegradiente vector ∇ W ( a , , T / and ∇ R t ( a , , T / . A direct computation showsthat ∇ W ( α ( T / ×∇ R t ( α ( T / (cid:18) , − πT R ta ( α ( T / , R ta ( α ( T / F bbt ( α ( T / (cid:19) . Notice that the second entry of above vector is given by R ta ( α ( T / π r λ n m + Mmn + M ! , which by hypothesis is different from zero. Then, there exist δ > and two conti-nuous functions T ∗ , a ∗ : ( − δ, δ ) → R such that b → T ∗ = T ∗ ( b ) , b → a ∗ = a ∗ ( b ) , with T ∗ (0) = T / , a ∗ (0) = a and W ( γ ( b )) = 0 and R t ( γ ( b )) = 0 , with γ : ( − δ, δ ) → R , (3.12) b → γ ( b ) = ( a ∗ ( b ) , b, T ∗ ( b )) , γ (0) = ( a , , T / . Then for b ∈ ( − δ, δ ) it follows F t ( γ ( b )) = bW ( γ ( b )) = 0 , and R t ( γ ( b )) = 0 , Once again, by Lemma 2.2 we have that for any b , the functions f ( t ) = F ( a, b, t ) y r ( t ) = R ( a, b, t ) define an odd/even T ∗ ( b ) -periodic solution of the reduced problem(2.1). (cid:3) Periodic solutions for the ( n + 1) -body problem. We will only considerthe case when the solutions of the ( n + 1) -body problem comes from solutions ofEquation ( † ) . The case when the solutions of the ( n + 1) -body problem come fromsolutions of equation ( †† ) is similar.In order to show that we can find a non trivial periodic solution with b > we just need to check that the function Θ( a, b, t ) is not constant along the curve γ ( b ) = ( a ( b ) , b, T ( b )) defined in Equation (3.10). This is true because every triple ( a ( b ) , b, T ( b )) solves the equation F = 0 and R t = 0 and by continuity, if b → Θ( a ( b ) , b, T ( b ))) is not constant, then we can find a b such that γ ( b ) satisfies that Θ( γ ( b )) = n π/n with n and n integers. See Remark 2.1.One way to study the behavior of the function Θ along the curve γ ( b ) is byreparametrizing the curve γ ( b ) as β ( τ ) = γ ( b ( τ )) where β is an integral curveof the vector field X defined as X = ∇ ˜ F × ∇ R t = (cid:16) ˜ F b R tt − ˜ F t R tb , ˜ F t R ta − ˜ F a R tt , ˜ F a R tb − ˜ F b R ta (cid:17) . As we have done before, we can explicitly compute as many partial derivatives ofthe functions ˜ F and R t at the bifurcation point p = β (0) = γ (0) . Therefore wecan compute as many derivatives as needed for the curve β at τ = 0 and since wecan also compute as many partial derivatives of the function Θ at p then using thechain rule we can compute the first and second derivative of the function ξ ( τ ) = Θ( β ( τ )) , at τ = 0 . A direct computation shows that ξ ′ (0) = 0 and a long direct computation,see [16], shows that ξ ′′ (0) = (cid:0) A ( n, m, M, r ) + B ( n, m, M, r ) R at ( α ( T )) (cid:1) R at ( α ( T )) , FAMILY OF SOLUTIONS OF THE N-BODY PROBLEM 11 with A = A ( n, m, M, r ) and B = B ( n, m, M, r ) given by A = 3 r / π (cid:20) λ n m + M ) (cid:16) λ n m + 24 M − mn + 16 M q λ n m + Mr (cid:17) − M ( M + mn ) (cid:16) q λ n m + Mr (cid:17)(cid:21) m n ( − λ n m + 4 mn + 3 M )( λ n m + M ) / B = 9 M r / ( M + mn ) / (cid:16) q λ n m + Mr (cid:17) m n ( λ n m + M )( λ n m − M − mn ) . Which implies that for most of the choices of n , M and m the second derivative of ξ ( τ ) = Θ( β ( τ )) at τ = 0 is different from zero.4. Numerical solutions
In this section we present the analytic continuation of a periodic solution that is closeto the solution displayed in the youtube video http://youtu.be/2Wpv6vpOxXk. Itcan be checked that m = 92 , M = 242 , n = 3 , r = 11 and q = ( a , b , T ) = (1 . , . , . , provides a periodic solution. Figure 4.1 shows the trajectory of one of the threebodies with mass 92. Figure 4.1.
Trajectory of one of the bodies with mass 92 for the so-lution of the 4 body problem with initial conditions q = ( a , b , T ) =(1 . , . , . When we extend this solution, similar to the results in [13], the collection of points ( a, b, T ) that satisfy the equations F ( a, b, T ) = 0 and R t ( a, b, T ) = 0 with b = 0 have the shape of a fork, where two of the edges goes to a points with a = 0 correspondingto periodic solutions with collisions, the other edge seems to be unbounded and thefourth edge goes to a point where b = 0 . The value of a for this limit point with b = 0 is a = q λ m + Mr = q
22 + √ ≈ . which was somehow expected becausethis is the a corresponding to the solution where the mass in the center stays putand the other masses move along a circle. The value of T for this limit point isnear 5.03224. In other words the limit point with b = 0 that we found by analyticcontinuation of the periodic solution displayed in the video is q = ( a , b , T ) = (5 . , , . . Figure 4.2 shows the points q and q as part of the family of reduced periodicsolutions of the 4-body problem. This paper comes from an effort to understand thereason of the value of T in the point above. Initially we were expecting a value for T equal to q √ π ≈ . which is half of the period of the circular solutionwhere the mass in the center stays still. The result of this explanation is Theorem3.1, it explain how actually this limit point is a bifurcation of the solutions comingfrom the line L = n ( a , , t ) = ( s
22 + 9211 √ , , t ) : t ∈ R o , which even though they do not provide geometric periodic solutions, they satisfy theequation F ( a, b, t ) = 0 and R t ( a, b, t ) = 0 . Our main theorem predicts the value ofbifurcation for T at q π ≈ . which completely explains our numericallyvalue of T in the point q . Figure 4.2 shows the solutions with b = 0 that form afork shape curve and the solution with b = 0 , the line L defined above. It is worthpointing out that this line L is the image of the curve α that we used in the proof ofthe main results of this paper.Just to give an application to our Theorem 3.1, we decided to consider periodicsolutions of the 4-body problem produced when m = 3 , M = 7 and r = 11 .According to Theorem 3.1, when we consider the system(4.1) (cid:26) F ( a, b, T ) = 0 ,R t ( a, b, T ) = 0 , it follows that, along the line of solutions of this system n ( a , b, T ) : a = r (cid:16) √ (cid:17) , b = 0 o , FAMILY OF SOLUTIONS OF THE N-BODY PROBLEM 13
Figure 4.2.
Two bifurcation points for the spatial 4-body problem.For these solutions the masses of three of the bodies is 92 and themass of the fourth body is 242. One of the bifurcations is explainedin the paper [13], and the other bifurcation, the one that takes placeon the line L , is explained in this paper.there is a bifurcation at p = ( a , , T ) = (cid:16)r (cid:16) √ (cid:17) , , √ π (cid:17) ≈ (0 . , , . . In order to find a non trivial solution near p , we took b = 0 . and we did a searchby doing small changes of a = 0 . and T = 28 . to solve the system (4.1).We found that the point p = ( a , b , T ) = (0 . , . , . , Satisfies that | F ( a , b , T ) | and | R t ( a , b , T ) | are smaller than − . Using thepoint p we did an analytic continuation to obtain solutions of the system (4.1) with b = 0 . Figure 4.3 shows about 12,000 solutions found numerically, along with thesolutions along the line b = 0 .We can compute the exact value for Θ( p ) , it is a T = π √ √ ≈ . . We alsohave that Θ( p ) ≈ . , this time the computation has to be done numerically. Figure 4.3.
Solutions of the system (4.1) near the bifurcation point p . The point p allowed us to start the analytic continuation bymoving along the integral curve of the vector field ∇ F × ∇ R t . Thisfigure shows about 12,000 points with b = 0 . The points p , p and p are points where Θ equals to π , π , and π respectively. ( / )( )( / )
35 40 T Θ Figure 4.4.
This is the graph of Θ against T along the solution ofthe system F ( a, b, T ) = 0 , R t ( a, b, T ) = 0 . We have highlighted threepoints that represents periodic solutions. FAMILY OF SOLUTIONS OF THE N-BODY PROBLEM 15
Since the function Θ along the solutions of the system (4.1) changes, by continuitywe have that for some solution p = ( a, b, T ) of the system Θ( p ) = uv π with u and v whole numbers. Therefore we get that the functions x ( t ) = (0 , , F ( a, b, t )) y ( t ) = (cid:18) R ( a, b, t ) cos(Θ( a, b, t )) , R ( t ) sin(Θ( a, b, t )) , − F ( a, b, t ) (cid:19) y ( t ) = (cid:18) R ( a, b, t ) cos(Θ( a, b, t ) + 2 π , R ( t ) sin(Θ( a, b, t ) + 2 π , − F ( a, b, t ) (cid:19) y ( t ) = (cid:18) R ( a, b, t ) cos(Θ( a, b, t ) + 4 π , R ( t ) sin(Θ( a, b, t ) + 4 π , − F ( a, b, t ) (cid:19) , will eventually close and therefore the solution of the -body problem given by thesefunctions will be, not only pseudo periodic, but periodic. We have noticed thatalong the branch of solutions of the system (4.1) with b = 0 (see Figure 4.3), Θ is a function of T , see Figure 4.4. After noticing that the values of Θ increase upto numbers bigger that π , we searched for the solutions that satisfy θ ( T ) = 3 π/ , θ ( T ) = 4 π/ and θ ( T ) = π . We found that the points p = ( a, b, T ) = (0 . , . , . ,p = ( a, b, T ) = (0 . , . , . ,p = ( a, b, T ) = (0 . , . , . , satisfy that Θ( p ) = π , Θ( p ) = π , and Θ( p ) = π . Figure 4.5 shows the image ofthe function y ( t ) for these three solutions p , p and p . Figure 4.5.
This figure shows the graph of the function y ( t ) for theperiodic solutions provided by the points p , p and p respectively. For the periodic solution given by the solution p , Figure 4.6 shows the image of thefunctions x ( t ) , y ( t ) , y ( t ) and y ( t ) . Figure 4.6.
For this periodic solution m = 3 , M = 7 , r = 11 and n = 3 . This solutions was found by first finding the bifurcation pointgiven by Theorem 3.1 and then finding a solution with b = 0 that isnearby, and doing analytic continuation of this latter point. For thissolution θ ( T ) = π . References [1] Corbera, M. & LLibre, J. (2000)
Families of periodic orbits for the spatial isosceles -bodyproblem . Siam J. Math. Anal. . No 5, 1311-1346.[2] Devaney, R. Triple collision in the planar isosceles three-body problem . Inventions Math (1980) pp. 249-267[3] Galán, J., Núñez D. & Rivera, A. (2018) Quantitative stability of certain periodic solutions inthe Sitnikov problem.
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A family of periodic solutions of the planar three-body problem, and their stability .Celestial Mechanics (1976) pp. 267-285[5] Martinez, R. & Simo, C. [1987] Qualitative study of the planar isosceles three-body problem .Celestial Mechanics , (1987), pp 179-251.[6] McGehee, R. [1974] Triple collision in the collinear three-body problem . Inventions Math. ,(1974), pp 191-227.[7] Meyer, K. & Wang, Q. [1995] The global phase structure of the three-dimensional isoscelesthree-body problem with zero energy, in Hamiltonian Dynamical Systems (Cincinnati, OH,1992) . IMA Vol. Math. Appl. , Springer-Verlag, New York, 1995, pp 265-282.[8] Meyer, K. & Schmidt, D [1993] Libration of central configurations and braided saturn rings .Celestial Mechanics and Dynamical Astronomy
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Qual. Theory Dyn. Syst. (2017). https://doi.org/10.1007/s12346-017-0244-1.[14] Perdomo (2019)
The round Taylor Method
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Periodic solutions in the generalized Sitnikov ( N + 1) -body problem. Siam J.Applied Dynamical Systems. No 3, 1515-1540.[16] Suárez, Johann. (2020)
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New phenomenons in the spatial isosceles three-body problem . Internet. J.Bifur. Chaos Appl. Sci. Engrg. 25, (2015), , 15p Department of Mathematics. Central Connecticut State University. New Britain.CT 06050
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