Celest Mech Dyn Astr (2009) 103:49–78 DOI 10.1007/s10569-008-9165-2 ORIGINAL ARTICLE
Improved grid search method: an efficient tool for global computation of periodic orbits Application to Hill’s problem G. A. Tsirogiannis · E. A. Perdios · V. V. Markellos
Received: 1 March 2008 / Revised: 1 August 2008 / Accepted: 24 October 2008 / Published online: 5 December 2008 © Springer Science+Business Media B.V. 2008
Abstract We present an improved grid search method for the global computation of periodic orbits in model problems of Dynamics, and the classification of these orbits into families. The method concerns symmetric periodic orbits in problems of two degrees of freedom with a conserved quantity, and is applied here to problems of Celestial Mechanics. It consists of two main phases; a global sampling technique in a two-dimensional space of initial conditions and a data processing procedure for the classification (clustering) of the periodic orbits into families characterized by continuous evolution of the orbital parameters of member orbits. The method is tested by using it to recompute known results. It is then applied with advantage to the determination of the branch families of the family f of retrograde satellites in Hill’s Lunar problem, and to the determination of irregular families of periodic orbits in a perturbed Hill problem, a species of families which are difficult to find by continuation methods. Keywords
Families of periodic orbits · Grid search · Hill’s problem · Irregular families
PACS 45.50.Pk · 07.05.Kf · 91.10.Sp
1 Introduction The Grid Search Method (GSM) was first introduced in 1974 (Markellos et al. 1974) as a systematic way to compute complete networks of families of periodic orbits of non-integrable systems of two degrees of freedom, contained in a given region of the relevant space of initial G. A. Tsirogiannis · E. A. Perdios (B) · V. V. Markellos Department of Engineering Sciences, University of Patras, Patras, Greece e-mail:
[email protected] G. A. Tsirogiannis e-mail:
[email protected] V. V. Markellos e-mail:
[email protected]
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conditions. Since then it has been used for the numerical treatment of a number of dynamical systems, among which are, notably, the restricted three-body problem and Hill’s problem including their variants (see e.g. Markellos et al. 1974; Markellos 1974b, 1975; Hénon 1997, 2003; Kanavos et al. 2002; Russell 2005). The determination of periodic orbits requires the numerical integration of the equations of motion x˙ = f (x),
(1)
where we consider an autonomous dynamical system of two degrees of freedom, x = (x, y, x, ˙ y˙ ), possessing a conserved quantity (integral of motion): F(x) = c.
(2)
In this paper we consider periodic orbits symmetric with respect to the x-axis. The criterion employed to the determination of such an orbit is that it should have two perpendicular crossings of the x-axis, which is an axis of symmetry of the problem. This means that the x˙ component of the solution has the value x˙ = 0 at two distinct points in time, initially (t = 0) and at the half period (t = T /2). The existing integral of motion can be used to reduce the four-dimensional phase-space to a three-dimensional space of “isoenergetic” solutions corresponding to a particular value of the integral constant c. Thus, in the case of symmetric periodic orbits the initial conditions (x0 , 0, 0, y˙0 ) can be fully represented by a point in the (c, x) plane. The periodic orbits of the dynamical systems possessing such an integral are classified into monoparametric sets which are represented by continuous curves (called family characteristics) in the space of initial conditions. Probably the most important features of a periodic orbit are its stability parameters, which are important for the determination of its stability properties as well as for the detection of the branch families bifurcating from the stable segments of a given family (see Hénon 1965, 1973; Markellos 1976). The vast improvement of digital computers, in computation speed and storage capacity, in the last decades, makes the GSM a very attractive method for the computation of complete networks of families of periodic orbits of the problems arising in Dynamics and in particular in Celestial Mechanics and Dynamical Astronomy. These networks can be of critical importance in so far as, according to Poincaré, periodic orbits are the appropriate means to penetrate into the immense complexities of problems such as the celebrated restricted three-body problem and its variants. Three decades after the first appearance of the method, we now present the Improved Grid Search Method (IGSM). The major improvements are as follows: (i) the enhancement of the sampling technique so as to increase the number of computed periodic orbits in a given area of the (c, x0 ) plane (equivalently the space of initial conditions) without substantially increasing the computational cost, and (ii) the clustering (classification) of the periodic orbits found in the sampling phase into families by means of a data processing procedure without further numerical integrations. Addition of the clustering procedure constitutes an improvement which makes the method complete in the sense that it makes unnecessary the numerical continuation of the families with predictor-corrector techniques that would otherwise be required for the determination of their continuously evolving properties (such as period and stability). Thus, substantial economy of computing time, and user intervention, is gained in achieving the final task of a full description of the families and their properties, while maintaining the method’s advantage over continuation methods, of a full search in the space of initial conditions carried out in the sampling phase. Continuation is more versatile as it can be used for problems of more degrees of freedom and for non-symmetric orbits (see e.g. Doedel et al. 2007; Muñoz-Almaraz et al. 2007). Thus continuation can be employed to extend the results of the present method in these directions (at the relevant bifurcations).
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x
x
Exact solution
Interval satisfying Bolzano’s criterion
c
c
Fig. 1 Mesh of classical GSM (left)—exact computation of a symmetric periodic orbits in classical GSM (right)
In the following Sects. 2 and 3 we present in detail the above improvements in sufficiently technical terms. In Sects. 4 and 5 we apply the improved method to obtain new results in model problems of Celestial Mechanics, namely the classical and a perturbed variant of Hill’s lunar problem, and Sect. 6 concludes.
2 Improved grid search In this section we present the improvements of the GSM concerning the increase of the number of computed orbits as well as the speed-up of the computational procedure. 2.1 Multi-directional sampling—implicit grid search As mentioned in the introduction, GSM was designed for the computation of complete networks of families of periodic orbits. In this effort the number of computed orbits contained in a specific region of the (c, x0 ) plane is a critical parameter of the problem. Often the shape of a family characteristic is difficult to be traced correctly by techniques of the classical GSM due to the lack of sufficient member orbits. In the classical GSM, having as a goal low computational effort (in the 1970’s central processing units CPU were very slow doing floating point operations), we make sampling and correction only in one direction. Here, we extend the searching method to up to four directions in order to detect and compute more periodic orbits. To this end we build our algorithms in such a way that they re-use already computed data. For better description of the improvements, we give a short figure-based presentation of the sampling technique of the classical GSM. Our goal is to find symmetric periodic orbits of given multiplicities, and the kernel computational procedure is the numerical integration of the equations of motion. For this numerical integration we have chosen an 8th order Runge–Kutta method (but with computational complexity of a fifth order method) with dense output (see Dormand and Prince 1978; Hairer and Syvert 1987). This method provides the necessary accuracy with relatively low computational effort. In the classical GSM we draw a mesh as shown in Fig. 1 (left). For each point of the mesh we make a numerical integration of the equations of motion up to the given multiplicity of x-axis intersections by the orbit (see Fig. 2).
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y
y
y(t)
2nd cut X 3rd cut
X
x tNewton-Raphson Exact Solution of 2nd cut
2nd cut
1st cut
y(t) y(t-tstep) 0.366065 only one stable part exists. Concerning the vertical stability, for values of Q 1 < 0.25 the family is stable, but for values larger than this it has an unstable part. This pattern continues until Q 1 ∼ = 0.366, but for larger values the entire family is stable again (see Fig. 33). It appears that when the family terminates by shrinking to point size, the horizontal stability parameter is critical: ah = 1. In Fig. 34 we also show the evolution of family ir1 with respect to the multiplicity of its orbits. We note that we have followed the evolution of family ir1 toward its “focal point” (point of its birth—the “isola center”). It would be interesting to consider also its evolution as Q 1 tends to zero. But, this would be beyond the scope of the present case study, which has been to establish with concrete and detailed results the applicability of the present method, and in particular its clustering part, in the case of irregular families. However, we show in Fig. 35 a first phase of evolution as Q 1 is reduced below the value Q 1 = 0.23. It appears that the family splits up and doubles upon almost parallel segments. A possible scenario is that neighboring families break up, and join with each other to form new isolas. We intend to explore this evolution more systematically in a future work. 5.4 Horizontal critical orbits Suggestively we have chosen the value of the perturbing factor Q 1 = 0.3648 and computed all the critical orbits for family ir1 . Using the data clustering procedure described in Sect. 3 we have found the member orbits of this family. By making a search among them in order to find nearly critical orbits and using corrector algorithms, we found the critical orbits of Table 4. The positions of these orbits are shown in Fig. 36. In this figure we also show the
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evolution of the stability parameters αh (solid line), βh (dashed line) and γh (dotted line) of the family, where βh , γh are the bh , ch parameters transformed as in Subsect. 5.2. 5.5 The bifurcating family of asymmetric periodic orbits When IGSM has been applied we can use the information obtained to extend the results to types of orbits for which the method is not applicable, for example three-dimensional or nonsymmetric orbits. To such end, continuation methods can be applied. As an example, using as initial points to a suitable predictor-corrector algorithm (see Markellos and Halioulias 1977), the critical orbits ir1H C 4, ir1H C 5 (which mark bifurcations with asymmetric periodic orbits), we have computed the family a(ir1 ) of asymmetric periodic orbits (see Fig. 37). This family starts–ends at the horizontal critical orbits ir1H C 4, ir1H C 5. A mirror image of family a(ir1 ) also exists named a´(ir1 ). This is a family whose orbits are symmetric to the orbits of a(ir1 ) with respect to the x axis in the x − y plane, and its characteristic curve is symmetric to the characteristic curve of a (ir1 ) with respect to the x˙0 = 0 plane in the space of initial conditions (c, x0 , x˙0 ). Using the astability parameters for asymmetric orbits sh = (ah + dh )/2, sv = (av + dv )/2 (see Markellos and Taylor 1978), we find that a(ir1 ) consists of orbits which are stable both horizontally and vertically (Fig. 38). Some numerical data are given in Table 5.
6 Concluding remarks We have presented an improved grid search method for the global computation of families of periodic orbits in model problems of Dynamics. The method is improved, by comparison to its older version, in two respects: firstly, it is improved in efficiency-speed of its basic procedure of global sampling in the space of initial conditions. Secondly, it is improved by complementing it with a second phase which is a data processing procedure for the classification (clustering) into families of the orbits found in the first phase. This second phase makes the method complete in the sense that it makes unnecessary the numerical continuation of the families with predictor-corrector continuation techniques, that would otherwise be required for the determination of their continuously evolving properties (such as period and stability). Thus, substantial economy of computing time and user intervention is gained in achieving the final task of a full description of the families and their properties, while maintaining the advantage over continuation methods, of a full search in the space of initial conditions carried out in the first phase (global sampling). However, by contrast to continuation techniques, the present method is limited to problems in which the initial conditions of the sought orbits are represented by points in a plane (for example symmetric orbits in four-dimensional phase space with a conserved quantity). The method was tested-verified by using it to recapture some known families of the classical Hill problem, and was applied successfully to obtain some new results in Hill’s problem. In particular we computed the entire tree of branches (up to multiplicity 15) of the basic family f of retrograde satellites in the classical Hill problem, and verified the mechanism of their bifurcation from f based on the relevant self-resonance conditions. The method is of particular usefulness in the computation of irregular families of periodic orbits, due to the full search of its first phase (which is well suited for the discovery of irregular families), and we used it to compute several such families in the case of a perturbed Hill problem, and to explore the evolution of a typical such family toward its birth point, under variation of the size of the perturbation.
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We note that the information content (stability, bifurcations) obtained for the families is the same when using the present method as when using continuation methods. Continuation is, however, more versatile as it can be used for problems of more than two degrees of freedom and for non-symmetric orbits. It can therefore be employed to extend accordingly the results obtained by the Improved Grid Search Method. As an example of such possibilities, we have used numerical continuation starting from information obtained by the present method in our case study of the perturbed Hill problem, to compute a bifurcating family of asymmetric periodic orbits. Acknowledgements comments.
The authors wish to thank the editor and two anonymous referees for their constructive
Appendix
Algorithm 1: Implicit horizontal–vertical–diagonal GSM Input: cmin , cmax , xmax , xmax % region of study %, VER_POINTS % number of vertical mesh points % HOR_POINTS % number of horizontal mesh points %, MAX_CUTS % maximum multiplicity % Output: Symmetric periodic orbits contained in (c, x) input region of study 1 1 ver _step = VER_POINTS , hor _step = HOR_POINTS ,i =1 1. For x = xmin : ver _step : xmax % for each line % 1a. i=i+1 % find second line % 2. For c = cmin : hor _step : cmax % for one line % 2a. Make classical GSM by checking points of the same line 2b. If i > 1 % we apply implicit search % 2ai. Fill the Data Matri x array as in Fig. 4 2aii. Apply Bolzano’s criterion as in Fig. 4 for multi plicit y = 1 : MAX_CUTS 2c. Move the contents of the first row of Data Matri x to the second row % prepare for next implicit search % 2d. Eliminate multiple found orbits % e.g. multiplicity 2 is re-found as 4, 6, . . . %
Algorithm 2: Phase 1 of data clustering Input: Set of computed periodic orbits from sampling phase of GSM, k % number of first orbits %, MIN_DIST % radius of the sphere % Output: k + 1 first member orbits { p0 , p1 , . . . pk } of a segment of the family pcur = p0 , dmin = ∞ 1. While i ≤ k 2. For each periodic orbit of input data 3. scur =current orbit 4. If ||(c pcur , x pcur ) − (cscur , x scur )||2 < MIN_DIST then 5. Compute the distance d = ||(c pcur , x pcur , y˙ pcur ) − (cscur , x scur , y˙ scur )||2 6. If d < dmin then 7. pi+1 = scur , pcur = scur , dmin = d 8. Remove pcur from input data 9. i = i + 1, dmin = ∞
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Algorithm 3: Phase 2 of data clustering Input: Set of computed periodic orbits from sampling phase of GSM, k % first orbits computed from algorithm 2 %, MIN_DIST % radius of the circle % Output: new member pnew orbit of the family Err ormin = ∞ x T /2 y˙0 T /2 x0 c 1. Construct approximation functions G appr ox , G appr ox , G appr ox , G appr ox , G appr ox % use splines (see de Boor 2001) % 2. For each orbit of the input data 3. scur =current orbit 4. If ||(c pcur , x pcur ) − (cscur , xscur )||2 < M I N _D I ST then 5. Compute functional values x T /2 y˙0 x0 c G appr ox (F(scur )), G appr ox (F(scur )), G appr ox (F(scur )), G appr ox (F(scur )), T /2 G appr ox (F(scur )) 6. Compute error: x T /2 scur scur scur x0 c Err or = |cscur − G appr − G appr ox | + |x T /2 − G appr ox | + | y˙0 ox | + |x 0 y˙
T /2
0 scur − G −G appr appr ox | ox | + |T /2 % for simplicity we omit F(scur ) % 7. If Err or < Err ormin then 8. Err ormin = Err or , pnew = scur
Algorithm 4: Grouping into families Input: Set of computed periodic orbits from sampling phase of GSM, Output: Family having member orbits p0 , p1 , . . . , pm % First phase % 1. Choose initial orbit p0 % see 3.1 % 2. Find first k orbits p0 , p1 , . . . , pm % use algorithm 2 % % Second phase % 3. Include a new member orbit % use algorithm 3 % 4. If the error is greater than estimated value, then 4a. Go to 5th step % one sub-segment completed, see Fig. 7% 4b. Else, go to 3r d step % see Fig. 11 % 5. If the two possible subparts have been completed, then % see lemma page 10 of [15] % 5a. Terminate 5b. Else, go to 2nd step % compute the second possible sub-segment, see Fig. 7 %
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Table 6 Data for critical orbits of the branch families Family name: f b2
f b3 f b6
f b7
f b10
f b11
f b13
f b14
f b16
f b18 f b24
f b25
T /2
x0
y˙0
x(T /2)
2.616748
−0.153655
3.342560
2.619800
−0.765869
1.011952
2.542543
−0.760729
2.619800 5.873414 5.421432
−0.699134
1.764284
0.276246
1.214351
7
Ah
4.867530
0.769234
−1.172118
−0.035203
3.001290
7
Bv
4.802133
0.765509
−1.260337
−0.052937
2.782203
7
Cv
4.802133
−0.765509
1.260337
0.052937
2.782203
7
Cv
c
N
−0.706802
1.914252
4
−0.013363
3.347032
4
Av
1.136199
−0.029989
3.074234
4
Dv
−0.013363
12.096245
−0.765869
3.347032
4
Dv
−0.787032
0.770984
0.787032
3.805034
5
Cv
Dh
4.867530
−0.769234
1.172118
0.035203
3.001290
7
Bv
6.220226
−0.735937
0.353126
0.735937
4.217736
7
Bv
6.097293
−0.730876
0.318209
0.730876
4.237725
7
Cv
4.348323
−0.670126
0.119738
0.670126
4.317383
7
Cv
4.251183
−0.665779
0.120244
0.665779
4.319326
7
Bv
5.331367
−0.666753
1.442281
0.111323
2.253115
9
Ah
5.431699
−0.714073
0.902047
0.005898
3.516846
9
Bv
9
Cv
5.344974
−0.711623
0.958919
0.010432
3.410173
26.391354
−1.416116
3.137715
−1.796614
−2.416787
10
Bv
26.368854
−1.413645
3.132821
−1.789885
−2.404614
10
Cv
26.368854
−1.789885
3.623958
−1.413645
−2.404614
10
Cv
26.391354
−1.796614
3.635032
−1.416116
−2.416787
10
Bv
8.351516
−0.692245
1.833160
−0.333164
0.966285
10
Ah
8.898435
−0.389282
2.199365
−0.674001
0.755080
10
Bv
8.932278
−0.392523
2.194015
−0.672836
0.743762
10
Cv
7.057895
−0.763382
1.305707
−0.063762
2.663305
10
Dv
7.113505
−0.766755
1.239755
−0.048389
2.835142
10
Av
29.680599
−1.480010
3.254403
1.480010
−2.668508
11
Bv
29.577107
−1.469762
3.232256
1.469762
−2.606108
11
Cv
29.576728
−1.824642
3.700023
1.824642
−2.606105
11
Bv
29.680128
−1.861175
3.759671
1.861175
−2.668617
11
Cv
8.501660
−0.673327
1.782589
0.673327
1.152810
11
Cv
8.659853
−0.664017
1.804234
0.664017
1.079466
11
Bv
8.663214
−0.308772
2.384332
0.308772
1.078253
11
Cv
8.497327
−0.290548
2.445681
0.290548
1.155440
11
Bv
5.884249
−0.713438
0.661179
0.000335
3.893151
11
Bv
5.826603
−0.712034
0.687646
0.000825
3.856975
11
Cv Dh
11.359751
−0.368132
2.237987
0.686806
0.830809
13
10.938086
−0.695552
1.833427
0.333366
0.965337
13
Av
11.007904
−0.694057
1.839958
0.339365
0.941307
13
Dv
11.007904
−0.339365
2.301644
0.694057
0.941307
13
Av
10.938086
−0.333366
2.316781
0.695552
0.965337
13
Dv
8.306982
−0.180877
3.063101
0.180877
1.772779
13
Bv
8.327511
−0.183793
3.037614
0.183793
1.756076
13
Cv
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Table 6 continued Family name:
T /2
x0
y˙0
x(T /2)
c
N
8.823696
−0.719838
0.733758
0.719838
3.794502
13
f b28
8.231207
−0.708677
0.867157
−0.003813
3.576869
14
Av
f b29
39.916913
−1.466325
3.198066
1.661834
−2.413347
15
Dv
39.894780
−1.466542
3.197076
1.655170
−2.405311
15
Av
f b32
10.827424
−0.232936
2.704383
0.696133
1.435146
15
Dh
10.410414
−0.189526
2.997023
0.718046
1.678264
15
Dv
10.760327
−0.226421
2.741794
0.699761
1.469455
15
Av
10.760327
−0.699761
1.690462
0.226421
1.469455
15
Dv
10.410414
−0.718046
1.629062
0.189526
1.678264
15
Av
2
0.5
0
0
−2
−0.5
−2
0
0.5
0.5
0
0
Bv
2
0
−2 −0.5
2
−0.5
0
0.5
−0.5
−0.5
0
0.5
−0.5
0
−2
0.5
0
2
0.5 0.5
0.5
0.5 0.2 0
0
0
0
0
−0.2 −0.5
−0.5
−0.5 −0.5 −0.5
0
0.5
−0.5
0
0.5
−0.5
0
0.5
2
−0.2
0.5
0
0.2
−0.5
3
1
0
0
0
0.5
0.2 0
0
0
−0.2
−2
−0.5
−2
0
2
−0.2
0
0.2
−1
−3
−0.5
0
−3
0.5
0
3
−1
0
1
4
0.5 0.5
0.5
0.2
0
0
0
−0.5
−0.5
−0.5
0
−0.5 −0.6
0.5
0
0
−0.2 −4
−0.4
−0.2
0
0.2
0.4
0.6
−0.5
0
0.5
0.5
3
−0.2
0
−4
0.2
0
4
0.5
0.5
0.2
0
0
0
0
−0.5
−0.5
0
−0.2
−3
−0.5 −3
0
3
−0.2
0
0.2
−0.5
1 0.2
0
0
−0.2
−1
−0.2
0
0
0.5
0
1
0
3
0
0
0
−3
−0.5
0
0.5
0
0
−0.5 −0.5 0
0.5
−0.5
0
0.5
Fig. 39 Sample orbits for families f b1 – f b32 (from left to right, top to bottom)
123
0
0.5
−0.2 −3
0.5
−0.5
−0.5
0.5
0.5
0
0.5
0.2
−0.5 −1
0.2
−0.5
3
−0.2
0
0.2
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