A resolution of non-uniqueness puzzle of periodic orbits in the 2-dim anisotropic Kepler problem: bifurcation U S + U

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS A resolution of non-uniqueness puzzle of periodic orits in the 2-dim anisotropic Kepler prolem: ifurcation U S + U To cite this article: T. Shimada et al 28 J. Phys.: Conf. Ser Related content - Kepler Prolem in Lagrangian Formulation Discussed fromtopological Viewpoint Xu Gong-Ou and Xu Ming-Jie - Staility and instaility in the anisotropic Kepler prolem G Contopoulos and M Harsoula - A Method to Find Unstale Periodic Orits for the Diamagnetic Kepler Prolem Zuo-Bing Wu and Jin-Yan Zeng View the article online for updates and enhancements. This content was downloaded from IP address on 22//28 at 4:22

2 A resolution of non-uniqueness puzzle of periodic orits in the 2-dim anisotropic Kepler prolem: ifurcation U S + U T. Shimada, T. Nakajima, 2 K. Sumiya, 3 K. Kuo Department of physics, School of Science and Technology,Meiji University, Higashimita --, Tama, Kawasaki, Kanagawa , Japan 2 Faculty of Gloal Media Studies, Komazawa University, Komazawa -23-, Setagaya, Tokyo , Japan 3 Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan 529, Israel tshimada@gravity.mind.meiji.ac.jp Astract. Using inary coding of orit we introduce a finite level (N) surface over the initial value domain D of 2-dim AKP. It gives a tiling of D y ase rions. The scheme of the one-time map is studied and the properness of the tiling is proved. This analysis in turn resolves the long standing puzzle in AKP the non-uniqueness issue of a PO for a given code. We argue that the unique existence of a periodic orit (PO) for a given inary code generally holds (for inverse anisotropy parameter γ < 8/9) ut there is a remarkale exception in which a rion with a certain code escapes from shrinking at large N and emodies the Broucke-type stale PO (S). It comes along the ifurcation of an unstale PO (U): U(R) S(R) + U (NR) (R for retracing and NR for non-retracing). An analysis ased on orit topology clarifies the pattern of the ifurcation; we give a conjecture that it occurs among odd rank Y -symmetric POs.. Introduction Anisotropic Kepler Prolem(AKP), cultivated y Gutzwiller [, 2, 3, 4], is a vital testing ground of quantum chaos; we can control strength of chaos y a single tuning a parameter the mass anisotropy. (We use inverse of anisotropy, γ). Here we focus on 2D-AKP, where a inary coding of the orit is possile. It is proved y Devaney [5] mathematically (and showed in physical terms y Gutzwiller [2]) that there exist at least one periodic orit (PO) for a given rational code in the high anisotropy regin (γ < 8/9). Gutzwiller also conjectured from numerical results that the PO is unique. On the other hand, Devaney instigated Broucke to find a counterexample in the form of a stale PO [4]. Broucke reported two examples of stale PO [6], and more recently Contopoulos and Harsoula have reported rather many stale orits, which may defy the elief that AKP is an Anosov system [7]. In this report, we focus on the coarse-grained level N Devil s Staircase Surface (DSS), which defines a tiling of the properly compactified initial value domain on the Poincaré surface of section. (Let us call it Gutzwiller s rectangle D). We give a proof that the tiling is proper that is, the DSS is monotonic y an investigation of the generating mechanism of a level N + tiling from level N one y an investigation of the action of one-time map on D. It in turn shows that Content from this work may e used under the terms of the Creative Commons Attriution 3. licence. Any further distriution of this work must maintain attriution to the author(s) and the title of the work, journal citation and DOI. Pulished under licence y Ltd

3 there is a special case that a rion with certain code does not shrink at large N, and this case occurs along with a transition U S + U with decreasing mass anisotropy. The transition is found to occur among Y -symmetric orits, which comes from the fact that non-shrinking overlap of rions is produced when future (F) and past (P) rions ecome tangent each other. A topology consideration explains the detailed pattern of the ifurcation. 2. Devil s Staircase Surface and the One-time Map 2.. Gutzwiller s rectangle The Hamiltonian of two-dimensional AKP is given y H = u2 2µ + v2 2ν r, (r x 2 + y 2, µ > ν) () where u p x, v p y and the mass anisotropy is proportional to γ with γ ν/µ <. Because ÿ/ẍ = (µ/ν)y/x > y/x, the orit tends to cross the heavy x-axis more frequently. Thus, the Poincaré surface of section (PSS) is specified y the condition y =, and we can encode the orit y the sequence of a i = ±, the sign of x i at the i-th crossing of the orit with the x-axis. The future sequence is (a, a, ) and the past is (a, a, ). The constant energy condition determines the kinematically allowed region on the PSS (the physical region for short). As the potential is a homogeneous in coordinates, the system has a scaling property. Any value for the energy is equivalent and we take H = /2 y convention. The physical region of PSS is compactified into a rectangle { } D = (X, U) µπ X 2 and U B (2) 2 y an area preserving map; X = x ( + u 2 /µ ), U = µ arctan (u/ µ). (3) The collision manifold is the I-shaped ackone of D: I = { (X, U) X 2 and U = ±B } { (X, U) X = and U B }. (4) 2.2. Symolic coding of orits and Devil s staircase surface The future and past surfaces over D at level N are respectively descried y the height functions ζ F N(X, U ) N j= 2 j+ a j(x, U ), ζ P N(X, U ) N j= 2 j+ a j(x, U ). (5) As Fig. shows, the surface defined y (5) has rather simple structure. The level N surface y definition consists of 2 N+ distinct heights and the ase of it is divided into rions. Furthermore, the height of the step is monotonously increasing, if we traverse rions from left to right. We express that the rions are tiling D properly. The edge, separating two adjacent rions, can e worked out y Gutzwiller s technique of the collision parameter analysis. They are the unstale and stale manifolds of the collision for the future and past DSS respectively [2]. See related discussion for Fig. 3. 2

4 Orit PO PO step U rion (a) () (c) X Figure. (a), ():The first quadrant of future DSS. (c) PO and level N surface. Now, the code of a PO is cyclic; a F PO = (a, a,, a 2n ; a 2n = a ); n is the rank of the PO. 2 If the initial point of the PO is (X, U ) D, its height is naturally defined y The following holds: ζ F PO(X, U ) = 2 2n 2n j= 2 j+ a j(x, U ). (6) A step of the level N DSS whose height is given y the first N + its of the string (the primary part or its repetition) is the closest to the PO in height. See Fig. (c). To see this, consider the amount of misfit etween (5)and (6); δ ζ F PO ζ F N = j=n+ 2 j+ a j(x, U ) (7) We find δ /2 N+. On the other hand, the difference of height etween neighoring steps is N = /2 N. Therefore, the selected step is the closest 3. At the large N limit, the misfit δ vanishes and the PO asymptotically sits on the selected step One-time map We now discuss the one-time map F: F : D D (X, U ) (X, U ). (8) Because the AKP flow involves oth hyperolic and elliptic singularities, separation as well as low up inevitaly occur. We have to ack up the numerical calculation y the map restricted to the collision manifold I. D D F I : I I. (9) 2 The length of PO must e even (2n), ecause it must cross the x-axis even times to complete its period. 3 This statement holds irrespective of whether the tiling y rions is proper or not. 3

5 C ISQS25 F I is worked out y Gutzwiller [2]. Polar coordinates (r, ψ) and (χ, ϑ) for oth the coordinates and momenta are introduced; u = µe χ cos ϑ, v = νe χ sin ϑ, x = r cos ψ, y = r sin ψ. () The kinetic energy is K = e 2χ /2 and under the canonical choice of energy H = /2 it follows r = 2/( + e 2χ ). Slowing down 4 the orit y dt = e 3χ dt and taking the limit χ, the equation of motion is reduced to an autonomous form dϑ dt = ( ) dψ µ cos ϑ sin ψ ν sin ϑ cos ψ, dt = 2 ( ν cos ϑ sin ψ µ sin ϑ cos ψ ). () This gives the map M : (ϑ, ψ ) (ϑ, ψ ), where (ϑ, ψ ) and (ϑ, ψ ) parameterize respectively the initial and final I in (9). This invokes a new critical angle ϑ c, and region 2 is divided into three su-regions 2A, 2B, 2C. X U X U ϑ v 2 X v B 2 X h B ϑ v π/2 2A X v B 2A X h X c B π/2 π ϑ c 2B X c B 2B X c B π ϑ c π ϑ h 2C X c X h B 2C X v B π ϑ h π 3 X h 2 B 3 X v 2 B Tale. Action of F I on five regions. X v = 2 cos 2 ϑ v, X h = 2 cos 2 ϑ h, X c = 2 cos 2 ϑ c. (B = π µ/2). Arrow represents the direction of increasing ϑ. At the separation (solid line) y H v (H h ), the signs of oth X and U flip, while at the su-criticality (dashed line) y ϑ (ϑ ) passing through π/2 (3π/2), only the U sign is flipped. The scheme of F I is shown in Tale and it is comined with the numerically calculated interior map in Fig (-+) (++) (-+) (++) B -- (--) (+-) (--) +- (+-) B Figure 2. The scheme of one-time map. Points that lows up (V +± C +±) (or focused in C +± v +±), are depicted y quarter-circles. Figure 3. Inset shows orit from collision. Dashed line is projection to (X, U) plane. 4 The collision occurs at χ =. This is equivalent to the low up technique used y Devaney [5] to remove the singularity due to the collision. See also McGehee [8]. 4

6 The oundary information is faithfully reflected in the interior result. Most important is the following; the separator curve C is doule sided, one side is mapped to v ++ and the other to v +. The reason why a curve is contracted into a point and a point is lown up to a curve can e understood as follows. In Fig. 3, we show the result of collision parameter analysis which follows [9]. Each orit emanating from the collision is laeled y a parameter (corresponding to the emission angle) and the position of its first arrival on the Poincaé surface forms a curve in the (X, U ) Gutzwiller s rectangle. This is nothing ut the core of the doule line C ++ and C Generation of Rion-tiling and a Proof of its Properness Let us prove the following: For any N, the rions are tiled properly. Firstly N =. It is just determined y the one-time map y itself (Fig. 2) and the heights of (future) rions y (5) distriute from left to right as ζ N= Rion : (a, a ) : : ( ) 2 : ( +) 3 : (+ ) 4 : (++) L : : R (2) Clearly the rions are properly tiled at N =. Now let us assume the level N tiling is proper and show that the properness of the level N + tiling follows from it. It is a crack of nutshell as shown in Fig. 4; place D in front of D, and project the level N tiling on D y F ack to D. Then, F N+ induces on D the level N + tiling. Remarkaly this way keeps the data from D upto D N intact. (By time translation symmetry, it is equivalent to the level N + on D ). Figure 4. Construction of the level N + tiling on D. In Fig. 5 the scheme 5 of F is laid over the level N tiling (y 2 N+ rions) of D. It shows succinctly how F creates the tiling of D. Most important is the contraction of the separator lines C ++ F v ++, C + F v +, (3) where the focus points v++ and v + are respectively on the right-top and left-down oundary of D. Thus, rions in the right-half (left-half) of D are transversely separated into two groups R I and R II (R III and R IV ), and they create new tiling of D. Here it should e noted that 5 The scheme of F is the reverse of F in Fig. 2, except that the superscript of all parts in the scheme are one step ack in time. 5

7 the order of R II and R III is left-right switched y the action of F. Note that the numer of rions are douled and each ecomes full-height extending from U = B to B. Thus, they constitute in general finer tiling after F. F - Figure 5. F : D D. Regions R K, K = I,, IV correspond to respectively (++), ( +), (+ ), ( ) in Fig. 2 Now, from a j (X, U ) = a j (X, U ), we find ζ N+ (X, U ) = 2 sign (X ) + 2 ζn (X, U ), (4) which calculates ζ N+ (X, U ). It is just half of previous heights except for an overall shift y the first term. Now, ecause the F is orientation preserving, this tells that the properness of tiling in each region is preserved via F. Taking into account of the first term, which is (+/2, /2, +/2, /2) for I, II, III, IV (due to the left-right switch mentioned aove), we find the height distriution of N + tiling is ζ N+ ( ) (, 2 2, ) (, ) ( 2 2, ) Region in D : F (R IV ) F (R II ) F (R III ) F (R I ). L : : R Apparently, the tiling is not-only within each region proper ut also gloally proper. (5) 4. Stale and Unstale Periodic Orits in AKP 4.. Use of Rions to Locate the Initial Point of a Periodic Orit Now, we are ready to resolve the long standing issue on the non-uniqueness puzzle of PO at a given code. In short, there are two cases; (A) Normally rions shrink roughly /2 N+, and the corresponding PO uniquely singled out at a cross junction of F and P asymptotic curves. See Fig. 6. 6

8 Figure 6. Longitudinal rion splitting process at γ =.2. Circle dot; PO with code (++ ), rank n = 2. (Id=3 in [3]). (ζ F, ζ P ) = (3/5, /5) y (6) and enclosed in the junction of F and P rions with step heights (5/8, /8) ((9/6, 3/6)) at N = 2 (N = 3) as calculated y (5). (B) Exceptional case: a certain rion escapes from shrinking. The est example is the so called Broucke s Island; let us start from a case study of PO Example:Broucke stale PO It is rank n = 3 with the code (+ + ). In Fig. 7, the evolution of a rion which encloses the initial point of (X, U ) of the Broucke s PO (call it Broucke s rion and give a mark B ) along with two neighoring rions is tracked y successively acting F on them. Remarkaly, only the Broucke s rion survives, while others (even the adjacent ones) diminish rapidly. S S B B B S B B B S B B B Figure 7. Evolution of future rions in the case of Broucke s PO3. Snap shots T N at N = (6, 7, 8; 2, 3, 4; 42, 43, 44). γ =.6. Circle-dots: Broucke s stale (S) and associated unstale (U ) PO. B : the rion enclosing them. Why can it survive under the chopping y the separator? It is in a very intriguing way; 7

9 Broucke s rion is protecting itself from shrinkage y changing at an early stage its tail to a line, so that the chopping ecomes immaterial. Now let us follow the decrease of the anisotropy y increasing γ and investigate how the Broucke stale PO3-6 comes out. As shown in Fig. 8, there is a threshold γ th where the unstale PO (U) changes into the stale one (S) following the advent of non-shrinking rions enclosing S. At the same time, a new unstale PO (U ) is orn. The overlap of non-shrinking F and P rions is a region where the orit from it repeats infinitely the codes, and it is natural that the stale PO is in the center of it. On the other hand, the unstale PO locates at the corner of the overlap. It must e inside the overlap, ecause the code must e repeated to e periodic, ut it must e at the edge, ecause a slight shift of the initial position must lead to an exponential low up of the shift for the PO to e unstale. Summing up, the stale PO emerges in a ifurcation process U S + U. B Future Past U(R) (R) U -B.25.5 B S(R) S -B Figure 8. Broucke PO3-6(N = 48) U S + U. γ th = Figure 9. Period. Broucke PO3-6: Lyapunov exponents and It is noteworthy that all of the POs (U, S, U ) are symmetric under the Y transformation: Y : x x, y y. The non-shrinking rions emerge just when the F and P rions ecome tangent each other at U =. U = implies p x =, therefore we consider the Y -symmetry is essential feature of the process. Also, we note that U and S are self-retracing (R) and U is self-non-retracing (NR); this ifurcation is actually U(R) S(R) + U (NR), (6) Finally, the ifurcation diagram is shown in Fig. 9. We oserve the followings. () Near the threshold, λ max γ γ th /2 gives a good description. (2) U for unstale ones also exhiits the typical threshold ehaviour U (γ γ th) /2. (3) The periods of the periodic orits are insensitive to the transition. The period of U as a function of γ elow γ th smoothly continues to that of S aove γ th. This itself may e natural since oth POs are of the same pattern (self-retracing), ut, curiously, the period the NR PO (U ) has also degenerate period in very good approximation. 8

10 4.3. Symmetry-consideration Since all POs (U, S and U ) are Y -symmetric in the transition of the Broucke type, let us now contemplate on the Y -symmetric POs. By a simple topology consideration, we can explain that the pattern U(R) S(R) + U (NR) is natural. We prepare two concepts. Firstly, in order to distinguish R and NR orits, it is useful to extend the homotopy idea. We consider that a NR PO is as usual homotopic to S, ut that R PO, where the particle is going ack and forth etween two turning points, is homotopic to a squashed S. See Fig.. NR t.p. R t.p. squashed Figure. A NR PO (such as U in PO3-6) is homotopic to S, while a R PO (U and S) to a squashed S. Secondly, Let n the numer of perpendicular crossing of a Y -symmetric PO with the heavy x-axis. Then, n must e either 2 or, ecause an orit with odd n cannot e closed while n = 4, 6, can close ut in disconnected loops. Now we are ready to prove a remarkale fact: Any Y -symmetric periodic orit is suject to one of the following three classes; (a) R with n = 2, () NR with n =, (c) NR with n = 2. To see this, it is sufficient to consider how to realize an appropriate n for the topology of R PO and NR PO respectively. See Fig.. Figure. Left : three classes of Y -symmetric PO. Right upper diagram; Broucke PO3-6 transition U(R) S(R) + U (NR). Right lower; sample orits in class (c). 9

11 (i) For R PO (a squashed S ) to e Y -symmetric, a perpendicular crossing of the x-axis is needed. But just a single perpendicular crossing already saturates n = 2; the crossing is multiplicity 2 in itself. Other crossings are X-type junction each with multiplicity 4. This is class (a). (ii) On the other hand, for a NR PO, it can e Y -symmetric without perpendicular crossing; either n = () or n = 2 (c). In (), all the crossings are X-type junction with multiplicity 2. In (c), all the crossings are X-type except for two distinct perpendicular crossings each with multiplicity one. Having proved the classification, let us reconsider the Broucke transition U S + U in the light of it. The PO (U) at high anisotropy (γ < γ th ) is self-retracing and in class (a). Under the decrease of the anisotropy, it gradually changes its initial value (X, U ) = (X, ) to remain closed. When γ exceeds γ th, the PO starts transition. It can proceed in three routes:(a) (a), (), (c), call them type (I),(II),(III) respectively. Routes II and III involve a topology change from squashed S to S as shown in Fig. 2. That is, a small shift at the crossing implies gloal deformation of the PO. [I] (a) 2 2 [II] (a) () [III] (a) (c) Figure 2. Class-(a)-PO can change in three routes (I),(II) and (III). Note in route (I), PO remains in class (a) and only the orit profile changes U(R) S(R). Broucke-type transition occurs when the F and P rions ecome tangent each other at U =. As we saw in Fig. 8, PO (S) remains with U = and with small X enveloped y the overlap of F and P rions, while PO (U ) is suject to rapid increase of U (U (γ γ th) /2 ). This means U S proceed in route (I), while U U in route (II). And this is exactly what is oserved. See the Broucke PO3-6 transition in right diagram in Fig.. We can now tell from the classification why PO (S) is R and PO( U ) is NR in (6). For a PO in class (a), the total numer of crossings of the x-axis (i.e. the length) is 4n + 2, n is the numer of cross-junction with multiplicity 4 and 2 from a single perpendicular crossing with multiplicity 2. The rank n of a PO is half of its length. Thus, the rank of class (a) PO is 2n + and we conjecture that the Broucke-type transition, associated with the non-shrinking rion, will occur in odd rank Y -symmetric PO The case of PO-5 Now we apply the topology analysis to a PO at rank 5 (length 3) whose code is ( ) 2 ). We have chosen this PO as the longest one among stale POs recently discovered y Contopoulos and Harsoula [7] 7. We note that the analysis in [7] is ased on one-dimensional shooting (p x = assumed). Therefore, it cannot catch up the ifurcation U U + S, where U has non-vanishing U. The ifurcation diagram is shown in Fig Precisely, the tangency of the F and P rions at U = implies Y -symmetry of involved POs, and n = 2, ut the possiility of the pre-po (U in the Broucke case) eing in class (c) is not logically excluded. 7 We have read the (x, y = ) y eye from their Figs. 4 and 6, and refined it y 2-dimensional shooting for higher accuracy.

12 Amazingly the first ifurcation U U +S is followed y a second one S S +S. (It occurs in near integrale limit γ > 8/9). The location of POs inside the relevant rions are depicted in Fig. 4. The orit profiles are given in Fig. 5; remarkaly, the multiplicity characteristics are just in accord with our topology analysis. [II] [I] [I] [III] Figure 3. Figure 4. Figure 5. Fine structure of orit profile. From the numer of crossings we determine its topology. The region marked y an asterisk are resolved in the douly magnified diagram. 5. Conclusion The key of this work is an introduction of tiling y rions of the initial value domain D. This is a switch from a dual picture (Gutzwiller s collision parameter analysis). By the use of tiling

13 defined at finite N, it has ecome tractale to track the generation of higher level DSS. This in particular flags the advent of non-shrinking rions. Our resolution of the non-uniqueness puzzle of POs in AKP is quite simple. When the rions shrink at large N, a unique PO is singled out for a given code; if the rions escape from shrinking, then a stale PO is associated with an unstale PO in the ifurcation process U(R) S(R) + U (NR). A topology argument supports the pattern of this ifurcation. A conjecture is drawn that such a code-preserving ifurcation occurs among Y -symmetric rank odd POs. Where to proceed? First, we need a etter analytic understanding of the condition for the advent of the non-shrinking rions. To this end, a local ifurcation analysis using the normal form will e helpful. We envisage another interesting use of it. We have successfully traced the quantum-classical correspondence in AKP in terms of inverse chaology []. However, if one wishes to apply the Gutzwiller trace formula directly to the ifurcation process, one instantly fails ecause either the Lyapunov exponent (for U) or the rotation index (S) vanishes at the ifurcation threshold. We are planning to apply the inverse chaology to this challenging ifurcation process y stepping forward to the next order in ħ and y including the satellite contriution via the normal form analysis ([], [2] and [3]). Acknowledgement We thank W. Ochs at the Max-Planck Institute, München for unfailing encouragement, and P. H. Damgaard, N. E. J. Bjerrum-Bohr and P. Orland at the Niels Bohr Instute for warm hospitality. We also thank people working on dynamical systems, especially M. Saito, M. Shiayama, and K. Tanigawa for their interest in this work and various suggestions. Finally, we would like to thank, with deep respect, M. C. Gutzwiller for his kind correspondence in 28 (just once). This work is partially supported y Research Project Grant (B) y Institute of Science and Technology, Meiji University. References [] Gutzwiller M C 97 Periodic orits and classical quantization conditions J. Math. Phys [2] Gutzwiller M C 977 Bernoulli sequences and trajectories in the anisotropic Kepler prolem J. Math. Phys [3] Gutzwiller M C 98 Periodic orits in the anisotropic Kepler prolem Classical Mechanics and Dynamical Systems ed R L Devaney and Z H Nitecki (New York: Marcel Dekker) pp 69-9 [4] Gutzwiller M C 99 Chaos in Classical and Quantum Mechanics (New York: Springer) [5] Devaney R L 978 Collision orits in the anisotropic Kepler prolem Inventiones. Math [6] Broucke R 985 Dynamical Astronomy ed V G Szeehely and B Balazs (Austin, TX: Univiversity of Texas Press) pp 9-2 [7] Contopoulos G and Harsoula M 25 Staility and instaility in the anisotropic Kepler prolem J. Phys. A: Math. Gen [8] McGehee R M 974 Triple collision in the collinear three-ody prolem Inventiones. Math [9] Gutzwiller M C 988 From classical to quantum mechanics with hard chaos J. Phys. Chem [] Kuo K and Shimada T 24 Periodic orit theory revisited in the anisotropic Kepler prolem Prog. Theor. Exp. Phys. 23A6 [] Ozorio de Almeida A M and Hannay J H 987 Resonant periodic orits and the semiclassical energy spectrum J. Phys. A: Math. Gen [2] Main J 999 Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra Phys. Rep [3] Schomerus H and Haake F 977 Semiclassical spectra from periodic-orit clusters in a mixed phase space Phys. Rev. Lett