3D billiards: visualization of regular structures and trapping of chaotic trajectories
Markus Firmbach, Steffen Lange, Roland Ketzmerick, Arnd Bäcker
33D billiards: visualization of regular structures and trapping of chaotic trajectories
Markus Firmbach,
1, 2
Steffen Lange, Roland Ketzmerick,
1, 2 and Arnd B¨acker
1, 2 Technische Universit¨at Dresden, Institut f¨ur Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187 Dresden, Germany (Dated: November 5, 2018)The dynamics in three-dimensional ( ) billiards leads, using a Poincar´e section, to a four–dimensional map which is challenging to visualize. By means of the recently introduced phase-space slices an intuitive representation of the organization of the mixed phase space with regularand chaotic dynamics is obtained. Of particular interest for applications are constraints to classicaltransport between different regions of phase space which manifest in the statistics of Poincar´erecurrence times. For a paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent withprevious results for maps we find that: (i) Trapping takes place close to regular structuresoutside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy.(iii) The dynamics of sticky orbits is governed by resonance channels which extend far into thechaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize thegeometry of stochastic layers in resonance channels explored by sticky orbits. I. INTRODUCTION
Billiard systems are Hamiltonian systems playing animportant role in many areas of physics. They are givenby the free motion of a point particle moving alongstraight lines inside some Euclidean domain with specu-lar reflections at the boundary. The dynamics is studiedin much detail [1–3] and ranges from integrable motion,e.g. for billiards in a circle, ellipse or rectangle, to fullychaotic dynamics, e.g. for the Sinai–billiard [4], the Buni-movich stadium billiard [5], or the cardioid billiard [6–9].Of particular interest is the generic situation with amixed phase space in which regular motion and chaoticmotion coexist [10]. This occurs for example when thebilliard is convex and the boundary is sufficiently smooth,e.g. a slight deformation of the circle such as the family oflima¸con billiards [6, 11]. For the class of mushroom bil-liards a sharply divided mixed phase space is rigorouslyproven [12]. Billiards also are important model systemsin quantum chaos [13, 14] and have applications in opticalmicrocavities for which the classical dynamics allows forunderstanding and tuning directed laser emission [15, 16].Three-dimensional ( ) billiards (see the upper rightinset in Fig. 1 for an illustration) have in particular beeninvestigated for establishing fully chaotic dynamics [17–27], and studying both classical and quantum proper-ties of integrable, mixed and fully chaotic systems, seee.g. [28–43]. Recent applications are in the context ofthree-dimensional optical micro-cavities [44–47]. Three-dimensional billiards are also of conceptual interest be-cause for systems with more than two degrees-of-freedomnew types of transport are possible, including the famousArnold diffusion [48–51].To understand the dynamics of billiards with a mixedphase space, for billiards the dynamics in the four–dimensional phase space is conveniently reduced to a area–preserving map using energy conservation anda Poincar´e section. This can be easily visualized andused for interactive computer explorations, see e.g. [52]. In contrast, for billiards the phase space is six–dimensional and, by energy conservation and a Poincar´esection, a symplectic map is obtained, which is dif-ficult to visualize. One method is to use the recentlyintroduced phase-space slice representation to visual-ize the regular structures of symplectic maps [53],e.g. of two coupled standard maps [54]. By this ap-proach it is possible to obtain a good overview of regularphase space structures and to demonstrate the general-ized island-around-island hierarchy [55] and the organi- − − − − tP ( t ) ∼ t − . FIG. 1. Poincar´e recurrence statistics P ( t ) in the bil-liard, defined in Eq. (3), for real flight time (red dashed line)and number of mappings (blue line). The dotted line indi-cates a power-law decay ∼ t − γ with γ = 1 .
2. Upper inset: paraboloid billiard shown with part of boundary removed forvisual reasons. Inside a sticky trajectory (blue line) is shownwhich starts and returns to region Λ (yellow ring). Lower in-set: trapped orbit (blue dots) and regular phase space struc-tures (gray) in a phase-space slice representation. a r X i v : . [ n li n . C D ] A ug zation in terms of families of elliptic tori [56].Another motivation comes from the important ques-tion on possible (partial) barriers limiting the trans-port between different regions in phase space in higher-dimensional systems. A sensitive measure for this isthe statistics of Poincar´e recurrence times P ( t ). Fullychaotic systems typically show a fast exponential decay,see e.g. [57–60], while for systems with a mixed phasespace the decay of P ( t ) is much slower, usually followinga power-law [61–78]. For recent results on the recurrencetime statistics in integrable systems see [79]. Closely re-lated to studying the Poincar´e recurrence statistics is thesurvival probability in open billiards, see e.g. [80–83].For two-dimensional systems the mechanism of thepower-law decay of the Poincar´e recurrence statistics P ( t ) is well understood: here regular tori are ab-solute barriers to the motion and thus separate differ-ent regions in phase space. Broken regular tori, so-calledcantori, form partial barriers allowing for a limited trans-port [65, 84–92]. Near a regular island formed by invari-ant Kolmogorov-Arnold-Moser (KAM) curves, there is awhole hierarchy associated with the boundary circle [93]and islands-around-islands [67]. These hierarchies of par-tial barriers are the origin of sticky chaotic trajectories inthe surrounding of a regular island and lead to a power-law behavior of the Poincar´e recurrence statistics P ( t ),see Refs. [66, 68, 69, 75, 78], and the reviews [91, 92].For higher-dimensional systems a power-law decay ofthe Poincar´e recurrence statistics is also commonly ob-served, see e.g. [94–102] and Fig. 1 for an illustra-tion. However, an understanding as in the case of two-dimensional systems is still lacking. The main reasonis that, for example for a map, the regular tori aretwo-dimensional and therefore cannot separate differentregions in the phase space. Thus broken regular tori alone cannot form a partial barrier limiting trans-port.In this paper we visualize the dynamics of billiardsusing the phase-space slice representation and basedon this investigate stickiness of chaotic orbits. Using the phase-space slice reveals how the regular region isorganized around families of elliptic tori and how un-coupled and coupled resonances govern the regular struc-tures in phase space. These can be related to trajectoriesin configuration space and the representation of the reg-ular region in frequency space. The Poincar´e recurrencestatistics shows an overall power-law decay. To investi-gate this decay, one representative long-trapped orbit isanalyzed in detail in the phase-space slice and in fre-quency space. We confirm the findings of Ref. [102] fora map also in the case of a billiard: (i) Trappingtakes place close to regular structures outside the Arnoldweb. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky or-bits is governed by resonance channels which extend farinto the chaotic sea. Clear signatures of partial barriersare found in frequency space and phase space. More-over, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.This paper is organized as follows: The first aim is toobtain a visualization of the mixed phase of a generic billiard. For this we briefly introduce in Sec. II A billiardsystems, the Poincar´e section, and as specific examplethe paraboloid billiard. In Sec. II B we review andillustrate phase-space slices and compare with tra-jectories in configuration space. The representation infrequency space is discussed in Sec. II C. The generalizedisland-around-island hierarchy is discussed in Sec. II Dand properties of resonance channels in Sec. II E. Under-standing the transport in a higher-dimensional system isthe second aim of this paper. For this the Poincar´e re-currence statistics is introduced and numerical results forthe paraboloid billiard are presented in Sec. III A. Theorigin of the algebraic decay of the Poincar´e recurrencestatistics are long-trapped orbits, which we analyze in de-tail in Sec. III B using different representations in phasespace and in frequency space. II. VISUALIZING THE DYNAMICS OF 3DBILLIARDSA. Billiard dynamics and Poincar´e section A billiard system is given as autonomous Hamilto-nian system H ( p , q ) = (cid:40) p , q ∈ Ω ∞ , q ∈ ∂ Ω , (1)which describes the dynamics of a freely moving pointparticle within a closed domain Ω ⊂ R with spec-ular reflections at the boundary ∂ Ω. The boundary ∂ Ω = ∪ ni =1 ∂ Ω i is assumed to consist of a finite numberof piece-wise smooth elements ∂ Ω i . Every point q ∈ ∂ Ωhas a unique inward pointing unit normal vector ˆ n ( q ),except for intersections of boundary elements.After the free propagation inside the domain the parti-cle collides at a specific point q ∈ ∂ Ω with the boundary.With respect to the normal vector ˆ n ( q ) the normal pro-jection of the momentum vector changes its sign, whilethe tangent component remains the same. Therefore thenew momentum p (cid:48) after a reflection is given by p (cid:48) = p − p · ˆ n ( q )) ˆ n ( q ) , (2)where p is the momentum before the reflection.The dynamics of a billiard takes place in a phasespace with coordinates ( p x , p y , p z , x, y, z ). As the Hamil-tonian (1) is time–independent, energy is conserved, i.e. H ( p , q ) is constant so that the dynamics takes place ona sub-manifold of constant energy. As the characterof the dynamics does not depend on the value of (cid:107) p (cid:107) ,we fix the energy shell by requiring (cid:107) p (cid:107) = 1. A furtherreduction is obtained by introducing a Poincar´e section.This leads to a discrete-time billiard map on a phasespace. A good parametrization of the section dependson the considered billiard. Note that for billiards thephase space is four-dimensional and the whole boundary ∂ Ω usually provides a good section. Here the section isconveniently parametrized in Birkhoff coordinates [103]by the arc-length along the boundary and the projectionof the (unit) momentum vector onto the unit tangentvector in the point of reflection. In these coordinates oneobtains a area-preserving map [103].As an explicit example we consider the paraboloidbilliard whose domain is defined by a downwards openedparaboloid ∂ Ω cut by the plane z = 0, leading to anellipsoid surface as boundary ∂ Ω , ∂ Ω = (cid:26) z = 1 − (cid:18)(cid:16) xa (cid:17) + (cid:16) yb (cid:17) (cid:19) , z ≥ (cid:27) ∂ Ω = (cid:26) z = 0 , (cid:18)(cid:16) xa (cid:17) + (cid:16) yb (cid:17) (cid:19) ≤ (cid:27) , (3)with parameters a = 1 .
04 and b = 1 .
12. These param-eters are chosen such that the billiard has no rotationalsymmetry, a (cid:54) = b , and that the central periodic orbit (go-ing along the line x = 0, y = 0) is stable as a, b >
1. Theshape of the system is illustrated in the upper inset inFig. 1 and in Fig. 4 where only one half of the paraboloid ∂ Ω is drawn and ∂ Ω is shown in green and yellow. Notethat for a = b the z component of the angular momentumis conserved. Numerically this billiard allows for a par-ticularly convenient implementation as reflection pointscan be computed by solving a quadratic equation [104].As Poincar´e section we choose the plane z = 0, so thatan initial condition is uniquely specified by its location( x, y ) within the ellipse ∂ Ω and the momentum com-ponents ( p x , p y ) since the third component follows frommomentum conservation (cid:107) p (cid:107) = 1. This reduces the billiard flow with phase space to a symplectic Poincar´emap ( p x , p y , x, y ) (cid:55)→ (cid:0) p (cid:48) x , p (cid:48) y , x (cid:48) , y (cid:48) (cid:1) on a phase space, M = { ( p x , p y , x, y ) | ( x, y ) ∈ ∂ Ω , ( p x , p y ) ∈ R with p x + p y ≤ } (4)with invariant measure d µ = | ∂ Ω | π d p x d p y d x d y . Notethat the trajectory can be reflected several times at thecurved boundary ∂ Ω before returning to ∂ Ω . Thereare two different time measures, namely the number t ofapplications of the Poincar´e map and the real flight time τ , which is the sum of the geometric lengths betweenconsecutive reflections at the billiard boundary ∂ Ω.Let us first discuss two special cases for the dynam-ics of the paraboloid billiard. Corresponding to themotion in the x - z and the y - z plane there are two em-bedded billiards with boundary given by a straightline and parabola with parameters a and b , respectively.The central periodic orbit has perpendicular reflectionsat the boundaries and geometric length 2. As a, b > x, p x ) and( y, p y ), respectively, is shown in Fig. 2.For the billiard map the stable periodic orbitcorresponds to an elliptic fixed point at the cen-ter u fp = ( q i , p i ) = (0 , r : s resonance chains, as implied by the Poincar´e-Birkhoff theorem [107, 108], leading to small embeddedsub-islands. Note that we choose the numbers r and s such that r is the number of sub-islands of a reso-nance. The phase space of the billiard map in ( x, p x ),see Fig. 2(a), shows a prominent 6 : 2 and a smaller 8 : 3resonance near the fixed point. Further outside a 10 : 3 − − xp x (a) − − yp y (b) FIG. 2. Phase space for the two billiards embedded inthe paraboloid billiard. Regular tori (red rings) are ar-ranged around an elliptic fixed point (black dot) in the center u fp = (0 , a = 1 .
04 two resonances 8 : 3 and 3 : 1 within the regu-lar region and one 10 : 3 resonance at the edge of the regularisland are shown. (b) For b = 1 .
12 two resonances 3 : 1 and10 : 3 within the regular region and one 14 : 4 resonance at theedge of the island are shown. resonance is visible. Note that the 6 : 2 resonance chainconsists of two symmetry-related 3 : 1 resonance chainswhich only differ in the sign of the initial momentum p x .For the billiard system in ( y, p y ), see Fig. 2(b), there isa 3 : 1 resonance near the fixed point and further outsidea 10 : 3 and a 14 : 4 resonance. For both billiards, thecentral regular island is embedded in a chaotic sea withirregular motion (blue dots in Fig. 2). The central is-land is enclosed by a last invariant torus called boundarycircle [93].Note that for both billiards continuous families ofmarginally unstable periodic orbits exist in the chaoticpart of phase space at p i = 0 [109–112]. Such familiesare not of relevance for our study, as they are part of therecurrence region Λ for the Poincar´e recurrence statisticsdiscussed in Sec. III.Considering the dynamics of the billiard, beyondthe invariant planes, we have to investigate the full phase space. The central invariant object is the elliptic-elliptic fixed point u fp = ( p x , p y , x, y ) = (0 , , ,
0) result-ing from a direct product of the fixed points u fp of thetwo billiards. In the neighborhood of this elliptic-elliptic fixed point there is a high density of regular tori [113]. The regular tori form a whole “ regular re-gion ”, similar to the regular islands in the billiards.However, note that this regular region is not a connectedregion but just a collection of regular tori, permeated bychaotic trajectories on arbitrarily fine scales, see Sec. II Efor a more detailed discussion. B. 3D phase space slice
Since a direct visualization of the phase space ofthe Poincar´e section of the three-dimensional billiard isnot possible, we use a phase-space slice [53] whichis defined using a hyperplane in the phase space.Specifically we choose in the followingΓ ε = (cid:110) ( p x , p y , x, y ) (cid:12)(cid:12)(cid:12) | p y | ≤ ε (cid:111) (5)with ε = 10 − . Whenever a point ( p x , p y , x, y ) of an or-bit lies within Γ ε , the remaining coordinates ( p x , x, y ) aredisplayed in a plot. Objects of the phase space usu-ally appear in the phase-space slice with a dimensionreduced by one. Thus, a typical torus leads to a pair(or more) of lines. The (numerical) parameter ε de-fines the resolution of the resulting phase-space slice.For smaller ε longer trajectories have to be computedto obtain the same number of points in Γ ε . For furtherillustrations and discussions of the phase-space slicerepresentation see Refs. [53, 55, 56, 114].Fig. 3 shows a phase-space slice representation forthe paraboloid billiard. For this a few representativeselected initial conditions on regular tori are iterated un-til 5000 points fulfill the slice condition (5). Regular tori appear as two (or more) distinct rings, see the colored xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyp x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x ab cdeFIG. 3. phase-space slice of the billiard defined in Eq. (3)with a = 1 .
04 and b = 1 .
12 for | p y | ≤ ε = 10 − . Regular toriappear as and labeled tori a – d in Fig. 3. Note that the reflec-tion symmetry of the billiard at the x - z plane leadsto a symmetry in the phase-space slice with respectto x - p x plane. The reflection symmetry at the y - z planecorresponds to a symmetry in the phase-space slicewith respect to the y - p x plane.All regular tori are embedded in a chaotic sea (notshown), similar to the case in Fig. 2. Thus, the chaoticsea is a volume in phase space and appears as a volume in the phase-space slice.The phase-space slice representation resembles inlarge parts the phase space ( x, p x ) of the billiardshown in Fig. 2(a). This results from the chosen slicecondition (5), | p y | ≤ ε . Alternatively one could considerthe slice condition | p x | ≤ ε and display the remainingcoordinates ( x, y, p y ). This would resemble in large partsthe phase space ( y, p y ) of the billiard shown inFig. 2(b).To obtain a better intuition of the phase-space slicewe relate some regular tori of Fig. 3 to the correspond-ing trajectories in configuration space, see Fig. 4:a The pair of red rings correspond to a regular torus in the phase space and in configuration spaceto the trajectory shown in Fig. 4(a). This trajectory isclose to the x - z plane and may be considered as contin-uation of a trajectory of the billiard dynamics shownin Fig. 2(a) with additional dynamics in y -direction.b For the torus shown in Fig. 4(b) the trajectoryis close to the y - z plane and therefore similar to the xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz (a) a ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω ∂ Ω xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz (b) b xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz (c) c xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz (d) d xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz (e) eFIG. 4. paraboloid billiard with boundaries ∂ Ω (green) and ∂ Ω for which only the part with y < M fp x . b Trajectory (blue) close to M fp y . c Trajectory (orange) of an uncoupled resonance related to the billiard system with a = 1 .
04. d Trajectory (cyan) ofa coupled resonance. e Periodic trajectory (pink) of a double resonance with period 35. billiard dynamics shown in Fig. 2(a) with additional dy-namics in x -direction.c The six orange rings correspond to the trajectoryof Fig. 4(c), which is located in the 3 : 1 island chain ofFig. 2(a), again with additional dynamics in y -direction.Note that this trajectory has a symmetry related partner,which is obtained by inverting the initial momentum. Inthe phase-space slice this corresponds to the symme-try with respect to the reflection at the x - y plane.d A type of dynamics only occurring in billiardsis the cyan torus shown in Fig. 3 with trajectory dis-played in Fig. 4(d). Here the coupling between both de-grees of freedom can be nicely seen in the twisting enve-lope of the trajectory in configuration space. Note thatthis torus has a symmetry-related partner obtainedby inverting the initial momentum, i.e. in configurationspace one obtains the same type of trajectory passed inopposite sense. In the phase-space slice the symmetryrelated orbit is obtained by reflection at the x - p x plane.e Moreover, there are also trajectories of the typeshown in Fig. 4(e). This is a periodic orbit with period35 extending in both degrees of freedom and correspondsto a double resonance (see Sec. II C), which is not possiblein a billiard.The regular tori in the phase space of a mapare organized around families of elliptic tori [55, 56].Most prominently one has the so-called Lyapunov fami-lies [115–117] of elliptic tori which emanate from thecentral elliptic-elliptic fixed point u fp . For the billiardthese two families M fp x and M fp y corresponds to the reg- ular dynamics of the two embedded billiards shownin Fig. 2. These two families of elliptic tori form a“skeleton” around which the regular tori are orga-nized. For example, the orbit shown in Fig. 4(a) is aregular torus which is close to the Lyapunov family M fp x , while the orbit in Fig. 4(b) is close to the Lyapunovfamily M fp y .In the chosen phase-space slice (5) the family M fp x is completely contained in the x - p x plane of Fig. 3. Notethat only a few selected trajectories of Fig. 2(a) are dis-played. In contrast M fp y coincides with the y -axis whichis easily seen by applying the slice condition (5) to thephase space shown in Fig. 2(b). The closeness of the tori shown in Fig. 4(a) to M fp x and in Fig. 4(b) to M fp y is also clearly seen in the phase-space slice in Fig. 3.Note that in general the Lyapunov families not necessar-ily coincide with conjugate variables of the system, seee.g. [55]. C. Frequency space
The frequency space representation is an impor-tant complementary approach for understanding higher-dimensional dynamical systems. The basic idea is toassociate with every regular torus in the phasespace its two fundamental frequencies ( ν x , ν y ) ∈ [0 , and display them in a frequency space. Numericallythis is done using a Fourier-transform based frequencyanalysis [118–120]. The mapping from phase space to . . .
335 0 .
30 0 .
35 0 . ν x ν y M fp x M fp y M fp x M fp y : : : : : : : : : − : : : b c fp e Fig. 8(a) Fig. 11(a) d a
FIG. 5. Frequency space of the paraboloid billiard defined in Eq. (3) for a = 1 .
04 and b = 1 .
12. In total 9 . × frequencies ( ν x , ν y ) for the tori are displayed (gray points). The rightmost tip ( ν fp x , ν fp y ) = (0 . , . u fp . Two Lyapunov families of elliptic tori M fp x (red) and M fp y (blue) emanate from thispoint. Some important resonance lines are shown as magenta dashed lines. The insets show magnifications of the frequencyspace. Colored points marked by a – e correspond to the examples shown in Fig. 3 and Fig. 4. frequency space allows for explaining features observedin the phase-space slice and identifying resonant mo-tion.For a given initial condition an orbit segment { (cid:0) p jx , p jy , x j , y j (cid:1) } with j = 0 , , ..., N seg − N seg = 4096 is used. From this orbit two complex sig-nals s jx = x j − i p jx and s jy = y j − i p jy are constructed,and for each signal its fundamental frequencies ν x and ν y are calculated. Note that usually the computed frequen-cies are only defined up to an unimodular transformation[53, 121, 122]. For the considered billiard system notransformations have to be applied to get a consistentassociation in frequency space.To decide whether the motion for a given initialcondition is regular or chaotic, another orbit segment { (cid:0) p jx , p jy , x j , y j (cid:1) } for j = N + N seg , ..., N + 2 N seg − N = 10 giving fundamental frequencies(˜ ν x , ˜ ν y ). As chaos indicator we use the frequency crite-rion δ = max ( | ν x − ˜ ν x | , | ν y − ˜ ν y | ) < δ reg . (6)This should be close to zero for a regular orbit, whilefor a chaotic orbit the frequencies of the first and second segment will be very different. While N = 0 was used in[53], using N = 10 leads to a more sensitive measure, inparticular excluding short-time transients. As threshold δ reg = 10 − has been determined based on a histogramof the δ -values, computed for many initial conditions to-gether with a visual inspection of selected orbits in the phase-space slice. This leads to a total of 9 . × regular tori with corresponding frequency pairs shown inFig. 5. Based on a visual check using the phase-spaceslice, initial points for the sampling of the frequency spaceare chosen within an ellipse ( x, y, z = 0) with half-axes r a √ r b √
2, and radius r = 0 . p x + p y ≤
1. Choosing initial condi-tions outside of this region leads to chaotic dynamics andthus the frequency criterion (6) is not fulfilled. From thefraction of accepted regular tori the size of the regularregion is estimated as 1 .
4% of the phase space.Note that even though the frequency criterion (6) is avery sensitive chaos-detector, it uses finite-time informa-tion and therefore some of the accepted regular toriare actually chaotic orbits. This is of course common toany tool for chaos detection, see Ref. [123] for a recentoverview.
1. Regular tori and Lyapunov families
The geometry of the frequency space is governed by afew organizing elements:First, the frequencies of the central fixed point u fp can be obtained by a linear stability analysis for eachof the two billiards which gives an analytic ex-pression for the frequency of u fp [124], evaluating to u fp = ( ν fp x , ν fp y ) ≈ (0 . , . tori M fp x and M fp y . For such tori only thelongitudinal frequency ν L , corresponding to ν x for M fp x and ν y for M fp y , can be determined directly. The otherfrequency ν N , called normal or librating frequency, canbe computed by contracting a surrounding torus [55]or using a Fourier expansion method [117, 125]. Goingaway from the fixed point along the families, i.e. eitheralong M fp x or M fp y , corresponds in Fig. 2 to move fromthe central fixed point u fp towards the boundary of theregular island. For the particular geometry of the paraboloid billiard the Lyapunov families M fp x and M fp y coincide with the dynamics of the billiards shown inFig. 2.These lower-dimensional dynamical objects providethe skeleton of the regular dynamics, both in frequencyspace and in phase space, around which the regular mo-tion on tori is organized. In the vicinity of the fixedpoint u fp the frequency pairs of regular tori have ahigh density and quite densely fill the region betweenthe Lyapunov families, also see upper inset in Fig. 5.With increasing distance from the fixed point, e.g. be-low ν x ≈ .
35, regular tori only persist close to thefamilies of tori, see the lower inset in Fig. 5. Anotherimportant observation are the numerous gaps, i.e. regionsnot covered by regular tori, which are arranged aroundstraight lines. The origin of these will be discussed in thefollowing section.
2. Resonance lines
The frequency space is covered by resonance lines , onwhich the frequencies fulfill the resonance condition m x ν x + m y ν y = n (7)for m x , m y , n ∈ Z without a common divisor and at least m x or m y different from zero. In the following a res-onance condition is denoted as m x : m y : n and the or-der of a resonance is given by | m x | + | m y | . The res-onance lines form a dense resonance web in frequencyspace. Some selected resonance lines are shown in Fig. 5.For a given frequency pair ( ν x , ν y ) the number of inde-pendent resonance conditions, the so-called rank, deter- mines the type of motion occurring in maps [126, 127](also see [55] for an illustration): • If the frequency pair fulfills no resonance condition,it is of rank-0 and the motion on the corresponding torus is quasi-periodic, filling it densely. Suchfrequencies for example correspond to KAM toriof sufficiently incommensurate frequencies. Exam-ples are the red ( a ) and blue ( b ) marked pointswhich correspond to the tori in phase-space sliceshown in Fig. 3 and the trajectories in Fig. 4(a) andFig. 4(b). • If only one resonance condition is fullfilled (rank-1case), the resonance is either (a) uncoupled, i.e. m x : 0 : n or 0 : m y : n , or (b) coupled, i.e. m x : m y : n with both m x and m y non-zero. The motion isquasi-periodic on a invariant set which eitherconsists of one component in the case of coupledresonances, or of m x (or m y ) dynamically con-nected components in the case of uncoupled res-onances. Note that in this rank-1 case one has (atleast) one pair of elliptic and hyperbolic tori[56].An example of an uncoupled resonance isthe orange marked frequency pair ( ν x , ν y ) =(0 . , . ν x , ν y ) = (0 . , . projections encod-ing the value of the projected coordinate by colorscale [128, 129], see e.g. Fig. 5 in [55] for a detailedillustration. • If two independent resonance conditions are ful-filled (rank-2 case) one has a double resonance. Thefrequency pair lies at the intersection of two res-onance lines and leads to (at least) four periodicorbits with different possibilities for their stability.As an example, we consider the frequency pair( ν x , ν y ) = ( n / m , n / m ) = ( / , / ) in Fig. 5which is the intersection of the 5 : 5 : 3 and5 : − m , m ) = 35. The corresponding elliptic-elliptic trajectory is shown in configuration spacein Fig. 4(e).Resonances also lead to gaps within the areas coveredby regular tori, see e.g. the white regions in Fig. 5.Of particular strong influence are resonances of low or-der as they typically lead to the largest gaps. Whenresonance lines intersect the families of tori, this alsoleads to gaps within these families. So strictly speak-ing, they form one-parameter Cantor families of tori[115, 130]. If either | m x | ≤ M fp x ) or | m y | ≤ M fp y ) this leads to gaps or strong bends in the corre-sponding families of tori [56]. For example, due to the3 : 0 : 1 resonance there is a large gap in M fp x and due tothe 0 : 10 : 3 resonance a smaller one in M fp y . Note that inthe case of the considered paraboloid billiard the fre-quencies of the families of tori cross near these gaps,see Fig. 5. D. Hierarchy
For systems with two degrees of freedom the phasespace shows a hierarchy of islands-around-islands on everfiner scale which are organized around elliptic periodic or-bits [67]. In higher-dimensional systems the organizationof phase space is based on higher-dimensional elliptic ob-jects. For example for a map, families of elliptic tori form the skeleton of surrounding regular tori, asdiscussed above. Thus the generalization of the island-around-island hierarchy can be fully described in terms ofthe families of elliptic tori [55]: There are two possibleorigins of such families which either ( α ) emanate from anelliptic-elliptic periodic point or ( β ) result from a familyof broken tori fulfilling a rank-1 resonance condition.For the first case one further distinguishes: ( α
1) the fixedpoint is either the central elliptic-elliptic fixed point u fp or it corresponds to an elliptic-elliptic periodic point re-sulting from a broken torus which fulfills a rank-2resonance. ( α
2) The families of tori emerge from anelliptic-elliptic periodic point resulting from a broken el-liptic torus when its longitudinal frequency ν L = nm fulfills an rank-1 resonance. This corresponds to an inter-section of a resonance line with a one-parameter familyof elliptic tori.As this hierarchy of elliptic tori is reflected in thesurrounding tori, Figs. 3–5 provide an illustration ofthe hierarchy: • ( α M fp x and M fp y , respectively, which emanate fromthe central elliptic-elliptic fixed point u fp . An ex-ample of a double resonance is the elliptic-ellipticperiodic point shown in e . From this periodic or-bit also two Lyapunov families of elliptic toriemerge. • ( α torus of M fp x with longitudinal frequency ν L = fulfillsthe 3 : 0 : 1 resonance (rank-1), which gives rise toa period-3 periodic orbit with attached Lyapunovfamilies. • ( β ): An example of a two-parameter family of tori fulfilling a rank-1 resonance condition is the 2 : 1 : 1 resonance, for which d shows one surround-ing torus.Analyzing the dynamics of this hierarchy in frequencyspace requires an adjusted frequency analysis as the fre-quencies collapse to either ( α ) a point or ( β ) a resonanceline [55]. E. Resonance channels and Arnold diffusion
Points on a resonance line correspond in phase space ei-ther to elliptic tori or the surrounding tori. Thusthe one-parameter family of elliptic tori forms the“skeleton” of the so-called resonance channel . The regu-lar part of the resonance channel consists of the elliptic tori and their surrounding tori. The chaotic partof the resonance channel consists of the correspondinghyperbolic tori and the chaotic motion in the stochas-tic layer , which is associated with the homoclinic tangleof the stable and unstable manifolds of the hyperbolic tori. For a detailed discussion of the geometry ofresonance channels in phase space and the relation tobifurcations of families of elliptic tori see [56].In these stochastic layers chaotic transport along theresonance channels is possible, which is commonly re-ferred to as Arnold diffusion [48–51]. As resonance linescover the whole frequency space densely, all stochastic re-gions of phase space are connected. Their network withinthe region of regular tori is referred to as
Arnold web .Perturbing an integrable system, Nekhoroshev theoryshows that in the near-integrable regime the speed ofArnold diffusion is exponentially small [131, 132], whichmakes its numerical detection very difficult. This regimeis called
Nekhoroshev regime . For stronger perturbationsregular tori become sparse and neighboring stochasticlayers begin to overlap with much faster transport, inparticular across channels. This regime is called
Chirikovregime [133, 134].The considered paraboloid billard does not qualifyas near-integrable. Still the dynamics within a given res-onance channel shows both the behavior of the Nekhoro-shev and the Chirikov regime depending on the locationalong the channel, see e.g. lower inset of Fig. 5: Nearthe intersection of the resonance line with M fp x or M fp y ,the stochastic layer is embedded in surrounding regu-lar tori and the chaotic dynamics along the channelshould be governed by the slow Arnold diffusion, i.e. thispart of the resonance channel belongs to the Nekhoro-shev regime. Further along the channel the neighboringregular tori become more sparse and one gets into theChirikov regime in which the stochastic layers of neigh-boring resonances overlap. Thus transport across reso-nance channels becomes possible and is more likely fur-ther along the channel. In addition other crossing res-onance channels may also be explored by a trajectorystarted within a stochastic layer.With this general background in mind it is also possibleto represent chaotic trajectories in frequency space andinterpret the results: Of course, for chaotic trajectories ina stochastic layer no frequencies exist in the infinite timelimit, however it is possible to associate “finite-time” fre-quencies. For example chaotic trajectories approachinga regular torus also acquire similar frequencies. And fortrajectories in a stochastic layer their finite-time frequen-cies will cover a small region in the surrounding of thecorresponding resonance line in frequency space. Thiswill be illustrated and discussed in detail in Sec. III B. III. STICKINESS AND POWER-LAWTRAPPINGA. Poincar´e recurrence statistics
In systems with a mixed phase space the transportbetween different regions can be strongly slowed downby so-called stickiness of chaotic trajectories takingplace in the surrounding of regular regions. A con-venient approach to characterize stickiness is based onthe
Poincar´e recurrence theorem . It states that for ameasure-preserving map with invariant probability mea-sure µ almost all orbits started in a region Λ of phasespace will return to that region at some later time [135].Based on the recurrence times t rec ( x ) of orbits with ini-tial conditions x ∈ Λ, one obtains the recurrence timedistribution ρ ( t ) = µ ( x ∈ Λ | t rec ( x ) = t ) µ (Λ) . (8)The average recurrence time follows from Kac’s lemma[136–138] as (cid:104) t rec (cid:105) := 1 µ (Λ) (cid:90) Λ t rec ( x ) d µ = µ ( M acc ) µ (Λ) , (9)where M acc is the accessible region for orbits starting inregion Λ.Instead of considering the distribution of recurrencetimes, it is numerically more convenient to use the Poincar´e recurrence statistics , which is the complemen-tary cumulative Poincar´e recurrence time distribution, P ( t ) = ∞ (cid:88) k = t ρ ( k ) , (10)i.e. the distribution of the recurrence times larger than t . Initially one has P (0) = 1 and by definition P ( t )is monotonically decreasing. Numerically, the Poincar´erecurrence statistics is determined by P ( t ) = N ( t ) N (0) , where N (0) is the number of trajectories initially startedin Λ and N ( t ) is the number of trajectories which havenot yet returned to Λ until time t . The nature of the decay of P ( t ) depends on the dy-namical properties of the systems. Fully chaotic systemsshow an exponential decay [57–60] whereas generic sys-tems with a mixed phase space typically exhibit a power-law decay [61–66, 68–78, 87, 139, 140]. Note that consid-ering the Poincar´e recurrence statistics with respect to Λcan also be seen as an escape experiment from an openbilliard so that the decay of P ( t ) agrees with the decay ofthe survival probability with trajectories injected in theopening Λ [80].To numerically study the Poincar´e recurrence statisticsthe region Λ in phase space should fulfill two prerequi-sites in order to obtain good statistics: First Λ should beplaced in the chaotic sea far away from the regular re-gion to ensure that trajectories are started outside of theexpected sticky region. Second, the volume of Λ shouldbe chosen sufficiently large to ensure that non-trappedorbits return quickly enough to avoid unnecessary com-putations. For the Poincar´e map of the billiardwe choseΛ = (cid:26) ( p x , p y , x, y ) : 12 (cid:18)(cid:16) xa (cid:17) + (cid:16) yb (cid:17) (cid:19) > r , ( x, y ) ∈ ∂ Ω , and p x + p y ≤ (cid:27) . (11)A point ( p x , p y , x, y ) ∈ Λ defines the initial condition( p x , p y , p z , x, y, z ) with z = 0 and p z > || p || = 1.In configuration space the region Λ corresponds to anelliptical ring in the z = 0 plane, defining the openingin ∂ Ω , marked in yellow in the inset of Fig. 1. Thisallows for starting trajectories into the billiard under alldifferent angles. After visual inspection of the regularstructures with the help of the phase-space slice wechoose r = 0 . phasespace of the Poincar´e map.For the determination of the Poincar´e recurrencestatistics N (0) = 10 random initial conditions are cho-sen uniformly in Λ. For each of them the real flight timeand the number of iterations of the Poincar´e map aredetermined until the trajectory returns to Λ.In Fig. 1 the Poincar´e recurrence statistics for realflight times and the number of mappings is shown. Ini-tially one has approximately an exponential decay forsmall times up to t (cid:46) P ( t ) exhibits an overall power-law decay P ( t ) ∼ t − γ with exponent γ ≈ .
2. The only exceptionof the straight power-law is a small step for t ∈ [10 , ]which could be a manifestation of some more restrictivepartial barriers. Note that one could also consider othergeometries of the opening which only affects the initialexponential decay but not the exponent of the power-lawdecay [141]. The Poincar´e recurrence statistics of thenumber of mappings t and real flight time τ are shiftedby approximately a factor of ∼ .
94 which is close to thegeometric length τ = 2 of the stable periodic orbit in thecenter of the billiard.0An interesting application of Kac’s lemma is to esti-mate the size µ ( M reg ) of the regular region M reg , as itwas done in [138] for the H´enon map. Using the av-erage recurrence time (cid:104) t rec (cid:105) one gets from (9) µ ( M reg ) = 1 − µ ( M acc ) = 1 − (cid:104) t rec (cid:105) µ (Λ) . (12)With (cid:104) t rec (cid:105) = 2 .
696 and µ (Λ) = 0 .
36 we obtain µ ( M reg ) = 0 .
029 which gives an estimate of 2 .
9% for thesize of the regular region in the phase space. Thisis approximately twice as large as the regular fractiondetermined using the frequency criterion, see Sec. II C.Note that using Eq. (12) is expected to provide an upperbound to the size of the regular region, as orbits startedin Λ explore the chaotic region and thus approach theregular region from the outside. Moreover, there mightbe chaotic regions which are not accessed at all on theconsidered time-scales, while initial points in such regionscould already be detected as chaotic by the frequency cri-terion (6). Moreover, the threshold δ reg for the frequencycriterion has been chosen quite small and relaxing this to δ reg = 10 − gives comparable results for the size of theregular region.The overall slow decay in the Poincar´e recurrencestatistics is due to orbits with large recurrence times t rec .Therefore we want to analyze such long-trapped orbitswithin the phase space and frequency space introducedin Sec. II B and Sec. II C. B. Long-trapped orbits
To obtain a better understanding of the origin of theobserved power-law decay of the Poincar´e recurrencestatistics we consider one representative example of along-trapped chaotic orbit in the following. Analyzingthis long-trapped orbit both in the phase-space sliceand in frequency space allows to draw the following con-clusions: Trapping takes place (i) at the ”surface” of theregular region (outside the Arnold web) and is (ii) notdue to a generalized island-around-island hierarchy, asdiscussed in Sec. II D. We find that the dynamics of long-trapped orbits is (iii) governed by numerous resonancechannels which extend far into the chaotic sea. The re-sults suggest to decompose the dynamics in the sticky re-gion into (iii.a) transport across resonance channels and(iii.b) transport along resonance channels. For the trans-port across resonance we find clear signatures of partialbarriers. All these points support the results obtained inRef. [102] for the case of the coupled standard map.In particular we obtain a very clear example of the geom-etry of the trapped orbit in the phase-space slice andits signature in frequency space. Note that we only showone representative orbit here, while the essential featuresare observed for all of the about 100 orbits we analyzed.To arrive at these conclusions we make use of two time-resolved representations of the long-trapped orbit. In the phase-space slice points of the long-trapped orbit arecolored according to time (from blue at t = 0, starting xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyp x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x p x t rec t i m e FIG. 6. Trapped orbit with recurrence time t rec (cid:39) . × in the in Λ, to orange at t = t rec , returning to Λ, see color-bar in Fig. 6). It is also possible to analyze trappedorbits in frequency space [102, 118, 119, 142, 143]. Al-though for chaotic orbits no fundamental frequencies ex-ist, a numerical assignment of frequencies is still possiblebecause for short time intervals the dynamics of nearbyregular tori is resembled. For this a sticky chaotic or-bit is divided into segments of length N seg = 4096. Foreach segment the frequency analysis, see Sec. II C, is per-formed. This leads to a sequence of consecutive frequen-cies ( ν x ( t i ) , ν y ( t i )) with t i = iN seg , i ∈ N , which can bedisplayed either in frequency space with time encoded incolor or as frequency-time signals, see Fig. 8. Exemplar-ily, we consider one long-trapped chaotic orbit with largerecurrence time t rec (cid:39) . × in the phase-spaceslice, see Fig. 6 and Fig. 7, and in frequency space, seeFig. 8; another example is discussed in App. B, Fig. 11.Note that these trapped orbits are shown in a phase-space slice with slice parameter ε = 10 − , see Eq. (5), toobtain a higher density of points. (i) Trapping is at the surface of the regular region The long-trapped orbit is shown in the time-encoded phase-space slice in Fig. 6. It is close to the x - p x planeand thus close to M fp x . The coloring of the orbit ac-cording to time shows several bands with different colors,1 p x xy .
55 0 . .
50 0 . . . − chaotic sea t rec time FIG. 7. Magnification (rotated) of the trapped orbit of Fig. 6 with recurrence time t rec (cid:39) . × in the phase-space slicewith time [0 , t rec ) encoded in color. Regular tori of some important resonance channels 22 : 0 : 7, 14 : 2 : 5, 24 : − which means that the long-trapped orbit covers differentregions of phase space for specific time intervals. Further-more it is close to regular phase-space structures (gray).More precisely, the orbit is located close to the “surface”of the regular region, which is composed of the regular tori shown as grey rings in the phase-space slice.This is even better seen in a magnified (and rotated) side-ways view of the box indicated in the upper left part inFig. 6. This magnification is shown in Fig. 7, where thesurface of the regular region is indicated by the regular tori (black lines) at the left side. Going towards theregular region corresponds to decreasing x and p x × additional fre-quency pairs of regular tori are shown (grey dots).The long-trapped orbit spreads approximately parallel to the Lyapunov family M fp x , staying above the asso-ciated regular tori. Fig. 8(d) shows a magnification ofthe region indicated in Fig. 8(a), where the ordinate ˜ ν y is the distance to the lower side of the parallelogram,˜ ν y = ν y + k · ν x + ν s with k = − .
99 and ν s = 0 . tori M fp x , can be considered as inner partof the regular region. Thus decreasing ˜ ν y moves towardsthe surface of the regular region. Moreover, increasing ν x moves towards the central elliptic-elliptic fixed point.As the sticky orbit stays well outside of any regions withmany tori, it is effectively trapped at the surface ofthe regular region.In particular this means that it does not enter theArnold web of resonance lines which are embedded withinregular tori. (ii) Trapping is not due to a hierarchy Trapping is also not due to the generalized island-around-island hierarchy, summarized in Sec. II D. Thiscan be concluded from the phase-space slice represen-tation. Trapping deep in a hierarchy would imply suc-cessive scaling on finer and finer phase-space structuresas known from maps [68]. However, the long-trappedorbit spreads over the surface of the regular region andno signatures of a hierarchy are visible. This is alsosupported by the frequency-time signals ν x ( t ) and ν y ( t )2 . . . . . . .
314 0 .
315 0 .
316 0 .
317 0 .
318 0 . ν x ν y M fp x : − : : : : : : : : − : (a) (d) (d) × × . × t (b)05 × × . × .
314 0 .
315 0 .
316 0 .
317 0 .
318 0 . ν x t (c) 0 . . ν x ˜ ν y : − : M fp x . . . t t rec time FIG. 8. Frequency space representation of the trapped orbit, see Fig. 7, with t rec = 1 . × and time encoded in color. (a)Magnification of Fig. 5 with regular tori (grey dots) and selected resonance lines, (b, c) frequency-time signals ν y ( t ) and ν x ( t ),respectively, (d) magnification of frequency-space, see box in (a), in local coordinates. shown in Fig. 8(b, c). For trapping in the generalizedhierarchy the frequencies either collapse on a frequencypair ( α ) or on a resonance line ( α , β ) [55]. Neitherof these is observed for the considered example. Notethat for the second example of a long-trapped orbit dis-cussed in App. B, the frequency-time signals shown inFig. 11(b, c) collapse on the 3 : 0 : 1 resonance for somelonger time interval. Still the trapping is not dominatedby a hierarchy. (iii) Resonance channels The frequency-time signals shown in Fig. 8(b, c) donot collapse on a specific frequency or resonance linebut mainly fluctuate within specific frequency rangesover longer time intervals. These frequency ranges areconfined around certain resonance lines, as shown inFig. 8(a), for which 15 : 1 : 15, 24 : − − tori are displayed as stacks of3black rings in the phase-space slice representation inFig. 7. For the long-trapped orbit the bands of similarlycolored points in the phase-space slice are arrangedaround these regular parts of the resonance channels.This suggests that the long-trapped orbit is confined tothe stochastic layer of the resonance channels.Both representations and in particular the transforma-tion to local coordinates ( ν x , ˜ ν y ) as shown in Fig. 8(d),together with the animation of the long-trapped orbit inFig. 7, suggest a decomposition of the chaotic dynam-ics in transport (iii.a) across and (iii.b) along resonancechannels:(iii.a) In the phase-space slice of Fig. 7 the dis-tinct colored bands indicate that the sticky orbit staysfor extended time intervals within the stochastic layerof a given resonance channel, e.g. 14 : 2 : 5 or 22 : 0 : 7, andthen quickly jumps to a different resonance channel. Thishappens mainly within the x - p x plane, i.e. for approxi-mately constant y . These transitions across different res-onance channels are also clearly seen in frequency spacein Fig. 8(b, c).(iii.b) Transport along resonance channels is best seenfor the 24 : − y -direction. In frequency spacethis corresponds to the magnification shown in Fig. 8(d).We now discuss both types of transport in more detail. (iii.a) Across resonance channels In Fig. 8(a) and in particular in the frequency-timesignals ν x ( t ) and ν y ( t ) the importance of four resonancelines namely 15 : 1 : 5, 24 : − t ≈ , mainly inan interval around the 15 : 1 : 5 resonance and then up to t ≈ . × around the 22 : 0 : 7 resonance, followed by alonger time window up to t ≈ × around the 14 : 2 : 5resonance. Subsequent frequency intervals are aroundthe 24 : − ν x and ν y inFig. 8(b, c). For short time intervals (cyan and green) thesmall stochastic layer around the 26 : − tori as indicatedin the inset of Fig. 8(a). Even though this region isthreaded by resonance lines on arbitrarily fine scales, theeffective transport along these lines is expected to be veryslow. This is also suggested by the geometry in the phase-space slice, see Fig. 7, where this collection of reg-ular tori constitutes an effective surface.It is important to emphasize that each stochastic layer actually consists of a whole collection of reso-nances. For example, the stochastic layer around the22 : 0 : 7 resonance corresponds to the whole interval with0 . (cid:46) ν x (cid:46) . ν x ( t )is largest in its surrounding.The sudden transitions between different frequency in-tervals are manifestations of partial barriers. For com-parison this is illustrated in Appendix A for the bil-liard shown in Fig. 2(a). There, a sticky orbit approachesthe boundary circle in the so-called level hierarchy. Insuch a two-dimensional case partial barriers are well es-tablished as cantorus barriers (broken KAM curves) orbroken separatrices formed by stable and unstable man-ifolds [92]. However, these partial barriers do not gener-alize to systems with more than two degrees of freedom.Thus by using the frequency analysis it is possible to de-tect partial barriers without constructing them explicitly,in particular even if their dynamical origin is not known.These results show that long-trapped orbits explore thechaotic part of resonance channels and jump (iii.a) acrossresonances, i.e. trapping takes place in the Chirikovregime of overlapping resonances. We find both in fre-quency space and in phase space clear signatures of somekind of partial transport barriers. At present their dy-namical origin is not known. (iii.b) Along resonance channels Besides the transport across resonance channels alsotransport along resonance channels is present. This isbest visible for the considered long-trapped orbit aroundthe 24 : − tori [56]. The extent is smallest near the Lya-punov family M fp x and widens for increasing ˜ ν y . As dis-cussed in Sec. II E this corresponds to going from theNekhoroshev regime, where the channel is surrounded bymany regular tori and transport is governed by very slowArnold diffusion, towards the Chirikov regime of overlap-ping resonances for which the regular tori are sparse ornot present. Thus the distance to M fp x along the reso-nance channel takes the role of the perturbation strengthin the setting of perturbed integrable dynamics. Whileindividual tori in a map cannot confine chaotic mo-tion, a two-parameter family of them (with small gapsdue to higher-order resonances) effectively confines thechaotic motion around the resonance within the triangu-lar region in frequency space.During the time interval [7 . × , . × ] the orbit islocated in the stochastic layer around the 24 : − ν x ∈ [0 . , . ν y one can see that it initially decreases, i.e. the sticky or-4bit moves along the channel towards the Lyapunov family M fp x , see Fig. 8(d). This approach is followed by a longertime-interval with fluctuations around some constant ˜ ν y before the orbit moves along the channel away from theLyapunov family M fp x . The involved time-scales showthat the motion along the resonance channel is typicallymuch slower than the motion within the stochastic layer,i.e. see spreading in ν x -direction or animation of Fig. 7.The numerical results indicate that the transition ratesfor going across resonance channels depend on the posi-tion along a resonance channel, see Fig. 8(d). In theconsidered time interval the adapted frequency ˜ ν y ofthe long-trapped orbit first decreases, interpreted as ap-proaching the Nekhoroshev regime in which transitionsacross resonance channels become unlikely. Subsequentlythe orbit moves along the channel away from M fp x intothe Chirikov regime allowing for transitions across res-onance channels. Note that in Ref. [102] it is sug-gested that transport along channels can be modeled bya stochastic process with effective drift which gives onepossible mechanism of power-law trapping.Since the Arnold web of connected resonance channelsis not explored on the considered time scales, Arnold dif-fusion is not the origin of the long-trapped orbits. IV. SUMMARY AND OUTLOOK
In this paper we visualize the mixed phase space ofa billiard and analyze restricted classical transport,manifested by a slow decay of the Poincar´e recurrencestatistics, due to long-trapped orbits. To understand thedynamics of a billiard its phase space is reducedby energy conservation and a Poincar´e section to a symplectic map. This map with mixed phase spaceis visualized using a phase-space slice which reducesthe dimension of orbits and invariant objects such thatthey can be displayed in a plot. This provides a goodoverview of the regular region and its organization. Acomplementary representation is the frequency spacein which both regular tori and sticky trajectories can berepresented and related to resonances. Moreover, the fre-quency computation provides a chaos indicator to distin-guish between regular and chaotic dynamics. The orbitsin the phase-space slice and in frequency space canbe related to trajectories in configuration space of the billiard which provides an instructive representationof objects in higher-dimensional systems.The second focus of the paper is to study transportproperties. A slow power-law decay of the Poincar´e re-currence statistics indicates the presence of sticky orbits.This is of particular interest as the mechanism of stick-iness for higher-dimensional systems is still not under-stood, in contrast to trapping in systems with two de-grees of freedom. By analyzing long-trapped orbits in the phase-space slice and in frequency space we find thattrapping takes place (i) at the ”surface” of the regularregion (outside the Arnold web) and is (ii) not due to a generalized island-around-island hierarchy. We find thatthe dynamics of long-trapped orbits is (iii) governed bynumerous resonance channels which extend far into thechaotic sea. The sticky orbits stay for long times within astochastic layer of a resonance channel, with fast transi-tions to other channels. These are clear signatures, thatbetween the stochastic layers there are some restrictivepartial barriers, whose dynamical origin is not yet clear.For the billiards the results in the phase-spaceslice, see Fig. 7 in particular, suggest the existence of aneffective (local) boundary surface formed by regular tori which is approached by the sticky chaotic orbits viaa sequence of coupled and uncoupled resonances.An important task for the future is to identify andcompute the relevant partial barriers. Based on this itshould be possible to define the different states and ulti-mately explain the origin of power-law trapping in higher-dimensional systems. Another interesting application ofthe visualization of the phase space of a billiard are optical microcavities where understanding the mixedphase space may guide how to tune their emission pat-terns. ACKNOWLEDGMENTS
We are grateful for discussions with Swetamber Das,Felix Fritzsch, Franziska Onken, Martin Langer, MartinRichter, and Tom Schilling. Furthermore, we acknowl-edge support by the Deutsche Forschungsgemeinschaftunder grant KE 537/6–1.All visualizations were created using Mayavi [144].
Appendix A: Signatures of partial barriers in 2Dbilliards In billiards, and more generally in autonomousHamiltonian systems with two degrees-of-freedom, theorigin for power-law trapping are partial transport bar-riers [91, 92]. We now illustrate how signatures of thesepartial barriers can be detected in the frequency-timeplots for a billiard. This allows for comparing with thecorresponding time-frequency plots of the billiard. Asan example we consider the billiard shown in Fig. 2(a)and determine the Poincar´e recurrence statistics as inSec. III A. The result in Fig. 9 shows an overall power-law with exponent γ ≈ .
5. The slower decay around t ≈ is presumably caused by some more restrictivepartial barriers.Fig. 10 shows a long-trapped orbit with t rec (cid:39) . × and time encoded by color (blueto orange) in a magnification of phase space and infrequency-time representation ν ( t ), computed as insection Sec. III B from the complex signal x − i p x forsegments of length N seg ν BC which can be approximated bythe convergents of its continued fraction expansion [93].For the boundary circle with frequency ν BC ≈ . , , , , . Notethat only every second approximant is smaller than ν BC ,giving the sequence of principal resonances , , and . Each of these fractions corresponds to a resonancechain with elliptic periodic orbits, surrounded by regularmotion, and hyperbolic periodic orbits with associatedchaotic layer. These chaotic layers correspond to thestates in a Markov model description of power-lawtrapping and separated from each other by partialbarriers. These partial barriers are cantori, broken KAMtori, with irrational frequency ν c which themselves canbe approximated by periodic orbits corresponding to theconvergents of the continued fraction expansion of ν c .The transition rates between the states correspondingto the stochastic layers become smaller and smallerwhen approaching the boundary circle. A long-trappedorbit is expected to approach the boundary circle viathis so-called level hierarchy of such states. Note thatthe stochastic component of each of these states usu-ally contains several other (non-principal) resonances.Moreover, trapping also takes place in the neighborhood − − − − tP ( t ) ∼ t − . FIG. 9. Poincar´e recurrence statistics P ( t ) for the bil-liard with a = 1 .
04 for 10 trajectories started in region Λfor the number of mappings (blue line) and real flight time(red dashed line). The dotted line indicates a power-law de-cay ∼ t − γ with γ = 1 .
5. Upper inset: opened billiardwith parabola as boundary and a short sticky trajectory with t rec = 131. Lower inset: Poincar´e section ( x, p x ) with regulartori (red curves), opening Λ (yellow rectangle), and corre-sponding trapped orbit (blue dots). . . . .
650 0 . . . xp x BC (a)0 . . . . . . . × × . × tν ν BC (b) 0 t rec time FIG. 10. (a) Trapped orbit of the billiard with a = 1 . t rec (cid:39) . × in a magnification ofphase space with time [0 , t rec ) encoded in color. Additionallyshown is the boundary circle (full, black curve) and impor-tant surrounding resonances are labeled as fraction n/m ν ( t ) ofthe sticky orbit. Frequencies of important resonances areshown as magenta dashed lines and the frequency ν BC of theboundary circle as solid horizontal line. of the resonance islands and their island-around-islandhierarchy [67, 68, 74], which leads to time-intervals withconstant frequency.The different stochastic layers correspond to the re-gions with different colors in Fig. 10(a). Signatures of thepartial transport barriers can also be clearly seen in thefrequency-time plot. Here the signal ν ( t ) randomly fluc-tuates within some interval around a principal resonant6frequency. Passing through a partial barrier, a differentfrequency interval around another dominant resonance isaccessed. For the example shown in Fig. 10(b) the fre-quencies of the sticky orbit are initially confined in aninterval around ν = nm = and then a sudden transi-tion to the stochastic layer around ν = occurs. This isone of the convergents of ν BC and is closer to the bound-ary circle, compare with Fig. 10(a). The stochastic layeraround the next convergent ν = is only accessed verybriefly. Finally the level-hierarchy is left via the stochas-tic layer around ν = and by passing through ν = and ν = (not shown). Note that in this example notonly stochastic regions associated with the convergents ofthe boundary circle, but also several other non-principalresonances and partial barriers appear to be of relevancefor the long-time stickiness of the trapped orbit. Appendix B: Further example of a trapped orbit inthe 3D billiard
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