Distance characteristics of ergodic trajectories in two-dimensional non-integrable systems: Reduced phase space
aa r X i v : . [ n li n . C D ] J a n Distance characteristics of ergodic trajectories in two-dimensionalnon-integrable systems: Reduced phase space
Jamal Sakhr
Department of Physics and Astronomy,University of Western Ontario, London, Ontario N6A 3K7 Canada (Dated: June 12, 2018)
Abstract
A fundamental process for any given chaotic flow is the deterministic point process (DPP)generated by any chaotic trajectory of the flow repeatedly crossing a canonical surface-of-section(herein referred to as a Σ-type DPP). This paper introduces the idea of using stochastic pointprocess models to describe and understand the spatial statistical features of Σ-type DPPs in two-degree-of-freedom (2D) Hamiltonian systems. In the specific context of 2D non-integrable systemspossessing ergodic components, it is proposed that, in an ergodic region, the pertinent modelfor describing the spatial statistical features of any typical Σ-type DPP is the two-dimensionalhomogeneous Poisson point process (herein denoted by P ). Of particular interest in this paperare the (Euclidean) k th-nearest-neighbor distance characteristics ( k = 1 , , , . . . ) of a given Σ-typeDPP. Employing the two-dimensional cardioid and semi-circular mushroom billiards as generictest cases, it is shown that typical sample Σ-type DPPs possess k th-nearest-neighbor distancecharacteristics consistent with model predictions for P . Deviations from this observed Poissonianbehavior are expected in strictly non-ergodic regions (i.e., the chaotic regions in generic Hamiltoniansystems), but in cases where the dynamics is strongly (but not fully) chaotic, such deviationsare posited to be negligible. The validity of the latter claim is demonstrated in the case of the2D H´enon-Heiles system at the critical energy. The results of the numerical experiments arecontextualized and their significances to both classical and quantum chaos are discussed. PACS numbers: 05.45.Ac, 02.50.Ey, 05.45.Pq, 05.45.Mt . INTRODUCTION Chaotic particle trajectories of classical conservative systems have an intricate and com-plex spatial structure. A key theoretical issue in the study of Hamiltonian systems (and innonlinear dynamics more generally) is the characterization of this spatial complexity. Ideasand techniques from a range of mathematical subjects, including topology [1, 2], differentialgeometry [3–5], and fractal geometry [6–10], have been profoundly useful in both identifyingand describing many fundamental spatial features associated with classical particle trajecto-ries. Computer simulations have also been instrumental in revealing the qualitative spatialstructure of both regular and chaotic trajectories in a variety of paradigmatic model sys-tems (see, for example, Ref. [11]). The spatial complexity frequently observed in computersimulations can, in part, be described using the language of fractal geometry, which has sofar been the predominate mathematical paradigm for investigating questions concerning thegeometrical structure of chaotic trajectories. There are however questions concerning thegeometrical structure of chaotic trajectories that fall outside the scope of both differentialand fractal geometry. One such question, of a statistical nature, is introduced below.The most prominent spatial attribute of the chaotic trajectories is the extreme spatialasymmetry that results from their seemingly unsystematic wandering through phase space.The seemingly haphazard evolution of these trajectories naturally elicits the use of stochasticmodeling for the purpose of characterizing their spatial structure. Indeed, the conventionalwisdom is that any evolving chaotic trajectory mimics some sort of random process (see,for example, Refs. [12–14]). This immediately raises a fundamental question: What type ofstochastic process(es) can aptly model the spatial characteristics of the chaotic trajectoriesin (for instance) two-degree-of-freedom (2D) Hamiltonian systems? Questions of this naturecommonly confront practitioners of “spatial analysis”. The idea of using stochastic processesto model the spatial structure of a spatially complex object (such as a chaotic particletrajectory) is actually one that is central to the mathematical fields of spatial and geometricalstatistics [15, 16]. The question being raised is thus (in essence) one that falls within thepurview of these disciplines. The words “spatial characteristics” (a verbalism frequentlyemployed in these fields) will likely be ambiguous to some readers. To be more explicit, itis useful to recast the above-posited question in more direct terms as the following two-partquestion: What is the spatial statistical (or geometrical-statistical ) structure of the chaotic2rajectories of a 2D (non-integrable) Hamiltonian system, and what stochastic geometricmodel(s) [17] (if any) can serve to elucidate this structure? To the author’s knowledge, thesequestions have not been addressed in the literature nor have (more generally speaking) theideas of spatial and geometrical statistics been explicitly applied to the study of Hamiltoniansystems.As mathematical objects classical particle trajectories can be subjected to a wide rangeof spatial statistical analyses and so the general study of their overall spatial statisticalstructure is a multifaceted problem. The intent here is to introduce what is perhaps themost elemental sub-problem: determining the spatial statistical properties of the points ona trajectory that intersect any representative surface-of-section. Rather than immediatelyproceeding to the specifics of the problem, it is conceptually useful to formulate the basicquestions involved using the language of point processes. In the nonlinear dynamics litera-ture, a (deterministic) process which reduces the dynamics to a point set using a series ofevent timings is often called a “point process” (see, for example, Ref. [18]). Unfortunately,the term “point process” more commonly refers to a certain type of stochastic process [17].Thus, following the authors of Ref. [19], the term “ deterministic point process” (DPP) willhere be used in reference to any deterministic process of the type alluded to earlier, whereasthe term “point process” shall here retain its traditional (stochastic) meaning. The succes-sive intersections of any (continuous) phase space trajectory with a surface-of-section aresuccessive point events that define a DPP. What are the spatial statistical properties ofsuch a DPP? Is there a (stochastic) point process that can aptly model these properties?The spatial statistical properties of this particular type of DPP (henceforth referred to as aΣ-type DPP) will of course vary depending on the nature of the generating trajectories. (Inmixed systems, for example, island chains will have different spatial statistical propertiesthan trajectories which visit the chaotic sea.) In the present paper, the preceding questionsshall be addressed in the specific context of 2D non-integrable Hamiltonian systems that areeither fully ergodic or possess any number of ergodic components.The contents of the paper, in brief, are as follows. In Sec. II, it is proposed that, for anyergodic component of a 2D non-integrable system, the pertinent model for describing thespatial statistical structure of any typical Σ-type DPP is the two-dimensional homogeneousPoisson point process (henceforth denoted by P ). In particular, it is argued that the (Eu-clidean) k th-nearest-neighbor distance characteristics ( k = 1 , , , . . . ) of any such Σ-type3PP should be consistent with those theoretically predicted for P . A concise review ofpertinent details concerning P and the associated k th-nearest-neighbor distance distribu-tions for P is given in Sec. III. The ideas and arguments put forward in Sec. II are thennumerically tested using three exemplary model systems: the cardioid billiard (Sec. IV), thesemi-circular mushroom billiard (Sec. V), and the 2D H´enon-Heiles system (Sec. VI). Theresults of the numerical experiments as well as their significance to classical and quantumchaos are discussed in Sec. VII, and concluding remarks are given in Sec. VIII. II. INTERPOINT DISTANCE CHARACTERISTICS OF Σ -TYPE DPPS IN 2DNON-INTEGRABLE SYSTEMS: THE ERGODIC CASE In numerical Poincar´e surface-of-section (SOS) computations, the usual procedure is tocompute numerical trajectories (i.e., ‘pseudotrajectories’) for a large number of initial con-ditions and then observe the resulting point pattern that ensues from the intersection ofthese numerically-computed trajectories with a chosen SOS. In 2D conservative systems,the visual signature of fully developed chaos is an apparently random scatter of points (onthe SOS) generated from one initial condition. What are the spatial statistical propertiesof such a point pattern? For illustrative purposes, consider for a moment the point patternon the SOS shown in the top panel of Fig. 1, which was generated by iterating the Poincar´emap of the 2D cardioid billiard (see Sec. IV for background details). Interestingly, this pointpattern (obtained from classical deterministic laws and equations) is visually indistinguish-able from a realization of a Poisson point process in the SOS (see lower panel of Fig. 1).Are the spatial statistical properties of the point pattern obtained from classical mechanics(which is actually a discrete pseudotrajectory of the Poincar´e map) indistinguishable fromthose of any suitably-defined realization of a two-dimensional Poisson point process? Inother words: Are the spatial statistical properties of this Σ-type DPP consistent with thosetheoretically predicted for the two-dimensional homogeneous Poisson point process ( P )?Admittedly, there is a certain degree of vagueness in the preceding line of questioningsince “spatial statistical properties” is a rather broad term and can refer to any of a vastnumber of commonly encountered statistical quantities, relationships, and methodologiesbased on distance, area, orientation, and/or other geometric descriptors of a point process.For the present, suppose “spatial statistical properties” refers to any spatial statistical quan-4 s p −4 −3 −2 −1 0 1 2 3 4−1−0.500.51 X Y FIG. 1: (Top) A typical pseudotrajectory of the Poincar´e map of the cardioid billiard. (In thisinstance, the map was iterated 8000 times.) The phase space of the map is the Poincar´e surface-of-section (SOS) coordinatized using the canonical Birkhoff coordinates ( s, p ). For the cardioid,the SOS Σ = { ( s, p ) : s ∈ [ − , , p ∈ [ − , } . (Bottom) A realization of a two-dimensionalhomogeneous Poisson point process of intensity ρ = 500 generated from a binomial point processwith N = 8000 points in the rectangle [ − , × [ − , tity or relationship (pertinent to point processes) that involves only the distances betweenspecified points of a point process. The distributions of the k th-nearest-neighbor distances( k = 1 , , , . . . ), for example, are among the most rudimentary “distance characteristics”of a point process and are often used to characterize spatial point patterns that arise fromtheoretical models and physical data [16, 20, 21]. Although no canonical metric exists formeasuring distances between arbitrary points in classical phase space, the familiar Euclideanmetric can be used without restriction. (Other metrics could be potentially useful depend-ing on the specific system(s) of interest but non-Euclidean metrics will not be given furtherconsideration here.) One approach then to addressing the above-posited questions is tostudy the k th-nearest-neighbor distance distributions ( k th-NNDDs) of the points on a givenpseudotrajectory of the flow that intersect a chosen SOS, or equivalently [22], to study the5istributions of the k th-nearest-neighbor distances between the points of a given ‘discretetime’ pseudotrajectory of the 2D Poincar´e map.For a fully chaotic 2D system, almost any pseudotrajectory of the Poincar´e map willexplore the full phase space of the map ergodically, i.e., almost any pseudotrajectory of thePoincar´e map will densely and uniformly cover the entire SOS. The k th-nearest-neighbordistance distributions of such a pseudotrajectory should (due to denseness and uniformity)be well modeled by the corresponding distributions theoretically predicted for P . The sameshould be true of any chaotic pseudotrajectory that ergodically explores any positive-measuresubset of the total available phase space. (In other words, individual chaotic trajectoriesneed not densely cover the entire phase space.) Successive point intersections of any suchpseudotrajectory with a canonical SOS will (after sufficient time) uniformly cover some sub-set W of the SOS. The ensuing point set should be indistinguishable (insofar as its distancecharacteristics are concerned) from any suitably defined realization of a Poisson point pro-cess in W (of appropriate intensity). Even in the case of generic mixed systems where noergodic components exist (i.e., no positive-measure regions of phase space are completelydevoid of islands), there generally exist conditions under which the chaotic pseudotrajec-tories explore, nearly uniformly , most of the available phase space. In such cases, typicalpseudotrajectories should still possess k th-NNDDs that are reasonably well modeled by thetheoretical k th-NNDDs for P (but likely less accurately than in the previously discussedergodic cases). The preceding claims naturally require verification and indeed the intent inthe following sections of the paper is to validate these claims numerically. Before proceeding,the pertinent details concerning the k th-NNDDs for P are briefly reviewed. III. UNIT-MEAN k TH-NEAREST-NEIGHBOR DISTANCE DISTRIBUTIONSFOR P The homogeneous Poisson point process in R (denoted by P ) is essentially the limit of asimpler stochastic model: the binomial point process in R . The latter model consists of N random points uniformly distributed in a compact subset W of R . If the area bounded by W is A ( W ) and we take the limits N → ∞ and A ( W ) → ∞ in such a way that N/A ( W ) ≡ ρ remains constant, then the limiting stochastic point process is P (with intensity ρ ). As aconceptual example, suppose W is a rectangle with side lengths L and H . If we take the6imits N → ∞ , L → ∞ , and H → ∞ in such a way that N/LH ≡ ρ remains constant, thenthe limiting point process is P (with intensity ρ ). More precise and technical definitions of P can be found in the mathematical literature [16, 17], but the technicalities involved arenot relevant to the following developments.The k th-nearest-neighbor distance distribution ( k th-NNDD) for P , denoted here byD( s ; k ), gives the probability D( s ; k )s. of finding the k th-nearest neighbor to a given pointof P at a distance between s and s + s.. It can be shown that the k th-NNDD for P is [16]D( s ; k ) = 2 ( ρπ ) k Γ( k ) s k − exp (cid:0) − ρπs (cid:1) . (1)It is easy to verify that the above distribution is normalized (i.e., R ∞ D( s ; k )s. = 1) and thatthe mean k th-nearest-neighbor distance is¯ s = Z ∞ s D( s ; k )s. = Γ (cid:0) k + (cid:1) Γ( k ) √ ρπ , (2)where Γ( x ) is the standard Gamma function. If we transform to the random variable S = s/ ¯ s , the distribution (1) becomesD( S ; k ) = 2 α k Γ( k ) S k − exp (cid:0) − αS (cid:1) , (3a)where α = " Γ (cid:0) k + (cid:1) Γ( k ) . (3b)Note that the distribution (3) is also normalized (i.e., R ∞ D( S ; k )S. = 1), has unit mean (i.e., R ∞ S D( S ; k )S. = 1), and most importantly, is intensity- independent (i.e., does not explicitlydepend on ρ ). For future reference, the unit-mean k th-NNDDs for k = 1 ,
2, and 3 are:D( S ; 1) = π S exp (cid:16) − π S (cid:17) , (4)D( S ; 2) = 3 π S exp (cid:18) − π S (cid:19) , (5)and D( S ; 3) = 15 π S exp (cid:18) − π S (cid:19) . (6)The distributions (4) and (5), well known to practitioners of random matrix theory, are theWigner and Ginibre distributions, respectively [23–25].7 ± φ φ β φ φ β β FIG. 2: (Left) The two-dimensional cardioid billiard with polar axes included for reference. (Right)The first segments of a typical trajectory specified by a set of polar angles φ and bounce angles β .In this instance, φ = π/ β = π/ IV. EXAMPLE 1: 2D CARDIOID BILLIARD
The first of the three claims put forward in Sec. II can be expressed as follows:
For afully chaotic 2D Hamiltonian system, typical pseudotrajectories of the Poincar´e map possess(Euclidean) k th-nearest-neighbor distance characteristics consistent with those theoreticallypredicted for P . There are many well-known examples of 2D fully chaotic systems, any ofwhich could serve to validate (or invalidate) the preceding statement (henceforth referred toas Claim 1). Among the most celebrated examples are planar hyperbolic billiards such as thecardioid billiard shown in Fig. 2. In the following, the validity of Claim 1 is demonstratedin the representative case of the cardioid billiard.Consider then the motion of a point particle confined in a cardioid billiard. In orderto describe the particle’s motion, it is sufficient to know the points of reflection on theboundary and the corresponding direction afterwards. Particle trajectories in the billiardcan be fully specified using two angles, the polar angle φ , which determines the position ofthe bounce point on the boundary, and the bounce angle β , which is the angle between theoutgoing ray and the local normal vector pointing into the billiard. Individual trajectoriescan then be given as a set of angle pairs: { ( φ , β ) , ( φ , β ) , . . . , ( φ n , β n ) , . . . } (see Fig. 2).This description of the particle motion leads directly to the ‘billiard map’ (i.e., the Poincar´emap for the billiard flow). For a billiard flow, the most convenient SOS is the set of collisionpoints of the flow. Any trajectory of the billiard flow intersects this SOS whenever it reflects8 s = 4 sin ( φ/ p = s i n ( β ) FIG. 3: A typical pseudotrajectory of the Poincar´e map of the cardioid billiard. In this instance,the pseudotrajectory was launched with initial condition ( s = 4 sin( π/ , p = sin( π/ at the boundary. This defines the standard billiard map, and the SOS defined in this wayis the phase space of that map. The standard coordinatization of such a SOS employs thecanonical Birkhoff coordinates ( s, p ), where s = 4 sin( φ/
2) is the arclength position of abounce measured along the boundary, and p = sin( β ) is the tangential momentum, that is,the momentum component parallel to the boundary at the bounce point [26].For the cardioid, the Poincar´e section Σ = { ( s, p ) : s ∈ [ − , , p ∈ [ − , } . The Poincar´emap B specifies completely the evolution of position and momentum from one collision withthe boundary to the next. The map B (sometimes also called the Birkhoff map) is obtainedfrom calculating the image point ξ ′ = ( s ′ , p ′ ) ∈ Σ of a given point ξ = ( s, p ) ∈ Σ (i.e., B : Σ → Σ, ξ = ( s, p ) ( s, p ) = ( s ′ , p ′ ) = ξ ′ ). Details concerning the Poincar´e map forthe cardioid billiard can be found in Ref. [27].In studying ‘typical’ pseudotrajectories of B , the set of initial conditions which startat or will immediately hit the cusp (the one singular point of the billiard) is excluded, asis the set of tangential collision points (the so-called “fixed points” of B ) that result in asliding motion where the moving point particle slides along the boundary wall. These setsare of measure zero, and thus almost all initial conditions produce long pseudotrajectoriessuitable for analysis. A typical pseudotrajectory of B is shown in Fig. 3. Using the initial9ondition ( s , p ) = (4 sin( φ / , sin( β )) = (4 sin( π/ , sin( π/ B wasiterated 16000 times. The resulting pseudotrajectory is shown plotted in the phase spaceof B (i.e., the SOS). The task now is to analyze the distances between the points of thispseudotrajectory.The distance between two points ξ i = ( s i , p i ) and ξ j = ( s j , p j ) on the SOS Σ isdefined using the usual Euclidean metric: ∆( ξ i , ξ j ) = p ( s i − s j ) + ( p i − p j ) . Thedistance between a given point ξ i and its nearest neighbor is then defined by d (1) i =min { ∆ ( ξ i , ξ j ) : i, j = 1 , . . . , N ( j = i ) } , and similarly the distance between ξ i and its fur-thest neighbor is defined by d ( N ) i = max { ∆ ( ξ i , ξ j ) : i, j = 1 , . . . , N ( j = i ) } . If, for agiven point ξ i , the distances { ∆ ( ξ i , ξ j ) : i, j = 1 , . . . , N ( j = i ) } are sorted by size (inascending order), then the k th-nearest-neighbor distance is the k th element of the set { d (1) i , d (2) i , . . . , d ( k ) i , . . . , d ( N ) i } . The set { d ( k ) i : i = 1 , . . . , N } thus defines the (experimental)set of k th-nearest-neighbor distances and density histograms of these interpoint distancescould (barring some details) be compared with model predictions for P [i.e., Eq. (1)]. Amore elegant approach is to remove the inherent dependence on the intensity by construct-ing density histograms of the scaled (and dimensionless) distances S ( k ) i = d ( k ) i / ¯ d ( k ) , where¯ d ( k ) = (1 /N ) P Ni =1 d ( k ) i , is the mean k th-nearest-neighbor distance. Doing so generates (fora given value of k ) an experimental density histogram that can then be directly comparedto the corresponding intensity- independent k th-NNDD given in Sec. III [i.e., Eq. (3)].For the pseudotrajectory shown in Fig. 3, the density histogram of the (scaled) nearest -neighbor distances is shown in Fig. 4(a) and is clearly in accord with the Wigner distribution[Eq. (4)]. The density histograms of the (scaled) second - and third -nearest-neighbor dis-tances are shown in Figs. 4(b) and 4(c), respectively. The correspondence with the Ginibredistribution [Eq. (5)] in the former case and with the distribution D( S ; k = 3) [Eq. (6)] inthe latter is evident.Similar results were found for several other pseudotrajectories of the same length (sixin total were analyzed). Thus, for the cardioid billiard, typical pseudotrajectories of thePoincar´e map have k th-nearest-neighbor distance characteristics consistent with those the-oretically predicted for P .Equivalent analyses of some typical pseudotrajectories of the stadium billiard (omittedhere) also yield similar results [28]. These results corroborate the validity of Claim 1.10 D ( S ; ) (a)0 1 2 300.511.5 S D ( S ; ) (b)0 1 2 300.511.5 S D ( S ; ) (c) FIG. 4: Near-neighbor distance distributions for the pseudotrajectory shown in Fig. 3. (a) Densityhistogram of the (scaled) nearest -neighbor distances; the smooth curve is the Wigner distribution[Eq. (4)]. (b) Density histogram of the (scaled) second -nearest-neighbor distances; the smoothcurve is the Ginibre distribution [Eq. (5)]. (c) Density histogram of the (scaled) third -nearest-neighbor distances; the smooth curve is the distribution D( S ; k = 3) [Eq. (6)]. V. EXAMPLE 2: 2D SEMI-CIRCULAR MUSHROOM BILLIARD
Chaotic trajectories need not necessarily cover densely all of the available phase spacein order for their point crossings with a SOS to exhibit the properties shown in Fig. 4.Trajectories evolving ergodically in any positive-measure subset of the full phase space shouldhave the same spatial statistical properties. Such dynamical evolution is known to occurbut is relatively rare among non-integrable Hamiltonian systems. The most well-known11 =0s=0.5s=1s=2s=2.5s=3 s= π +3s s s β β FIG. 5: A simple two-dimensional mushroom billiard composed of a semi-circular cap and rectan-gular stem. The cap has unit radius, the stem has unit width, and the stem height is half of thestem width. Also shown are the first three segments of a typical trajectory, which is here specifiedby a set of coordinates ( s, β ), where s ∈ [0 , π + 3) is the arclength position coordinate of a collisionpoint measured along the billiard’s boundary (increasing from zero at the right-most point of themushroom as the boundary is traversed clockwise) and β is the reflection angle at each collision. examples of mixed systems with a so-called sharply-divided phase space are the Bunimovichmushroom billiards [29]. The mushroom billiard shown in Fig. 5 having a semi-circularcap and rectangular stem is the simplest example of this special class of mixed systems.The phase space of this mixed system is unusual in that it has a single regular regionand a single chaotic region, and no bordering Kolmogorov-Arnold-Moser (KAM) hierarchy[29]. (In generic mixed systems, the border between a regular and chaotic region manifestsa complex hierarchical structure of KAM islands and cantori.) Interestingly, mushroomscan be designed so as to have any desired number of positive-measure ergodic componentsand any number of islands of stability. Furthermore, one can controllably alter the relativevolume fractions in phase space containing regular and chaotic trajectories by simply varyingthe dimensions of the mushroom. Aside from these particulars, the main point is that themushroom shown in Fig. 5 has an ergodic region (i.e., there exists a positive-measure subsetof the full phase space wherein the dynamics is ergodic). As argued in Sec. II, the k th-nearest-neighbor distance characteristics of any typical Σ-type DPP initiated in the ergodicregion should be consistent with model predictions for P . The validity of the precedingclaim is demonstrated below. 12 arclength s p = s i n ( β ) FIG. 6: A typical chaotic pseudotrajectory of the Poincar´e map of the 2D semi-circular mushroombilliard (see Fig. 5). The phase space of the map (i.e., the surface-of-section) is coordinatized usingthe Birkhoff coordinates ( s, p = sin( β )). In this instance, the pseudotrajectory was evolved from aninitial point ( s = 5 . , p = sin( π/ As in the case of the cardioid billiard, the SOS is coordinatized using the Birkhoff coordi-nates ( s, p ), where s is the arclength position coordinate of a collision point (measured alongthe boundary starting from zero at the right-most point of the mushroom and increasing toits maximum value as the boundary is traversed clockwise), and p = sin( β ) is the momentumcomponent parallel to the boundary at the collision point (see Fig. 5). For the mushroombilliard shown in Fig. 5, the SOS Σ = { ( s, p ) : s ∈ [0 , π + 3) , p ∈ [ − , } . Recall from Sec. IVthat the SOS is the phase space of the Poincar´e map and that the conventional Poincar´emap B for a billiard flow specifies the evolution of position and momentum from one collisionwith the boundary to the next: B : Σ → Σ, ξ = ( s, p ) ( s, p ) = ( s ′ , p ′ ) = ξ ′ . A typi-cal chaotic pseudotrajectory of B is shown in Fig. 6. This particular pseudotrajectory wasgenerated by iterating the Poincar´e map B of the mushroom billiard 25000 times startingfrom the initial condition ( s , p ) = (5 . , sin( π/ k th-nearest-neighbor distances between the points of thispseudotrajectory is the same as the one detailed in Sec. IV.13 S D ( S ; ) (a) S D ( S ; ) (b) S D ( S ; ) (c) FIG. 7: Near-neighbor distance distributions for the pseudotrajectory shown in Fig. 6. (a) Densityhistogram of the (scaled) nearest -neighbor distances; the smooth curve is the Wigner distribution[Eq. (4)]. (b) Density histogram of the (scaled) second -nearest-neighbor distances; the smoothcurve is the Ginibre distribution [Eq. (5)]. (c) Density histogram of the (scaled) third -nearest-neighbor distances; the smooth curve is the distribution D( S ; k = 3) [Eq. (6)]. Density histograms of the k th-nearest-neighbor distances (for k = 1 ,
2, and 3) are shownin Fig. 7. In each case, the sample histogram is consistent with model predictions for P [see Eqs. (4), (5), and (6) of Sec. III]. Similar results were found for several other pseudo-trajectories of the similar length. Thus, in the case of this simple mushroom billiard (whichhas a sharply-divided phase space with one ergodic region), typical pseudotrajectories of B from the ergodic region of the SOS possess k th-nearest-neighbor distance characteristicsconsistent with those of P . 14 I. EXAMPLE 3: 2D H´ENON-HEILES POTENTIAL
Corresponding analyses of Σ-type DPPs in generic mixed systems lie outside the scopeof the present paper. There are however two important limiting cases where typical Σ-type DPPs can be expected to have properties similar to those observed in the prior twoexamples: (i) generic systems above the energy threshold; and (ii) KAM-type systems (i.e.,Hamiltonian systems to which the KAM theorem applies) above the stochasticity threshold.An explicit example of the former is considered in this section.In generic mixed systems, there are no ergodic components (i.e., no positive-measureregions of phase space completely devoid of islands). Nevertheless, there generally existvalues of a system parameter at which there are only a few observable islands and thefraction of the phase space volume occupied by these islands is small (e.g., < . nearly uniformly most of the available phase space, and thusby extension, the k th-nearest-neighbor distance characteristics of any typical Σ-type DPPinitiated in the chaotic sea should be very similar to those that would be observed for atypical Σ-type DPP evolved in a strictly ergodic region.The system parameter referred to above could be a coupling or perturbation parameter(e.g., kicked rotor [31]), a shape parameter defining a family of billiards (e.g., the δ parameterdefining the family of lemon billiards [32]), or even simply the total system energy in time-independent potential systems (e.g., H´enon-Heiles [33], Pullen-Edmonds [34] etc.). Thetypical scenario in the last case (the most fundamental case for smooth Hamiltonian systems)is a phase space dominated by regular trajectories at low energies and chaotic trajectoriesat high energies with a mix of regular and chaotic trajectories at intermediate values ofthe energy. In the well-known H´enon-Heiles system, for example, most trajectories arequasi-periodic at E = 1 /
12 and chaotic at E = 1 / E = 1 / E = 1 /
6, the interpoint distance characteristics of any typical Σ-type DPP should beadequately modeled by those of P . In particular, good agreement between the sample k th-NNDDs of the DPP and the theoretical k th-NNDDs predicted for P should be observed.In the following, this claim is numerically confirmed.The motion of a point particle of unit mass in a smooth 2D time-independent potential15 p y FIG. 8: Successive intersections of a typical pseudotrajectory of (9) with the x = 0 plane in theupward direction ( p x > x (0) = 0 , p x (0) = p / , y (0) = 0 , p y (0) = 0) in the E = 1 / V ( x, y ) is governed by the Hamilton equations of motion˙ x = p x , ˙ p x = − ∂V∂x , ˙ y = p y , ˙ p y = − ∂V∂y . (7)In the case of the 2D H´enon-Heiles potential [33] V ( x, y ) = 12 (cid:0) x + y (cid:1) + x y − y , (8)the system (7) becomes ˙ x = p x , (9a)˙ p x = − ( x + 2 xy ) , (9b)˙ y = p y , (9c)˙ p y = − ( y + x − y ) . (9d)The most common and convenient Poincar´e surfaces for a system such as (8) are planeswith either x = 0 or y = 0. The former will be employed here. Successive intersections of16 D ( S ; ) (a)0 1 2 300.511.5 S D ( S ; ) (b)0 1 2 300.511.5 S D ( S ; ) (c) FIG. 9: Near-neighbor distance distributions for the Σ-type DPP shown in Fig. 8. (a) Densityhistogram of the (scaled) nearest -neighbor distances; the smooth curve is the Wigner distribution[Eq. (4)]. (b) Density histogram of the (scaled) second -nearest-neighbor distances; the smoothcurve is the Ginibre distribution [Eq. (5)]. (c) Density histogram of the (scaled) third -nearest-neighbor distances; the smooth curve is the distribution D( S ; k = 3) [Eq. (6)]. a pseudotrajectory with the plane x = 0 can occur in two different directions depending onthe sign of ˙ x when the pseudotrajectory crosses the plane. The Poincar´e map can be fullyspecified by considering only crossings in the positive direction (i.e., ˙ x > x = 0 and ˙ x >
0. Since y and ˙ y ≡ p y remainarbitrary, the SOS is thus coordinatized by ( y, ˙ y ).A simple procedure for generating the DPPs on the SOS (i.e., the pseudotrajectories ofthe Poincar´e map) was employed here which can be briefly summarized as follows. First,17he energy was fixed at E = 1 / x (0) = 0 , p x (0) , y (0) =0 , p y (0)) were specified subject to the condition p x (0) + p y (0) = 2 E . The equations of motion(9) were then solved numerically in MATLAB using the ODE solver ode45 , which utilizesfourth- and fifth-order Runge-Kutta formulas [35]. Finally, the event location property of theODE solver was employed to determine when x ( t ) = 0 and p x ( t ) > y ( t ) and p y ( t ) at these instances.A typical chaotic pseudotrajectory of the Poincar´e map is shown in Fig. 8 (see cap-tion of Fig. 8 for more details). Density histograms of the k th-nearest-neighbor distances(again computed using the procedure detailed in Sec. IV) for k = 1 ,
2, and 3 are shownin Fig. 9. Even for this relatively short pseudotrajectory (the pseudotrajectory analyzedin the previous example was roughly two and a half times longer) the overall agreementwith model predictions for P is quite satisfactory. Similar results were found for severalother pseudotrajectories of similar length. While the expectation is that substantially longerand more accurate pseudotrajectories will yield better agreement with P predictions thanthose obtained from the simple numerical techniques employed here, the present results al-ready substantiate the general claim, namely, that for a 2D generic mixed system abovethe energy or stochasticity threshold, the Euclidean k th-nearest-neighbor distance charac-teristics of any typical Σ-type DPP are well modeled by the corresponding characteristicstheoretically predicted for P . VII. DISCUSSIONA. Zaslavsky’s Dictum on Chaotic Trajectories and Stochastic Processes
To contextualize the numerical results of Secs. IV - VI, it is useful to quote here a certainremark (pertaining to chaotic particle trajectories in generic Hamiltonian systems) froma review paper by Zaslavsky [14]: “Trajectories, being considered as a kind of stochasticprocess, do not behave like well-known Gaussian, Poissonian, Weiner, or other processes.”
While Zaslavsky’s remark may be true for chaotic pseudotrajectories evolving in a genericallymixed phase space, chaotic pseudotrajectories evolving in an ergodic region do in fact behavelike one of the well-known Poissonian processes. In the latter case, and in the specificcontext of 2D non-integrable systems, chaotic pseudotrajectories of the Poincar´e map, “being18onsidered as a kind of stochastic process” (to use Zaslavsky’s phrasing), behave like a 2DPoisson point process. Under certain conditions, the preceding holds (at least approximately)even in generic mixed systems, as demonstrated in Sec. VI.
B. Exact Trajectories and Shadowing
The numerical results describe the k th-nearest-neighbor distance characteristics of pseudo trajectories. This begs the question: Do the same results hold for the exact trajec-tories of a Poincar´e map? This is a question that is difficult to answer definitively withoutdetailed analysis. Chaotic pseudotrajectories emulate the true dynamics of a given systemonly when ‘shadowed’ by exact trajectories of the system, and are otherwise (individually)meaningless. Shadowing of numerical trajectories is a fundamental issue, in particular, forstrictly non-hyperbolic chaotic systems. In hyperbolic systems (i.e., ‘hard chaotic’ systems),which is an extreme and rather exceptional case, the existence of shadowing trajectories forall pseudotrajectories is guaranteed (see, for example, Ref. [36]). For such systems, con-clusions about the statistical properties of numerical trajectories will also generally applyto exact trajectories. Long shadowing trajectories are not precluded for all non-hyperbolicchaotic systems (for example, the shadowing property has been established for the standardmap [37]), but their consideration introduces technical questions about how accurate andfor how long numerical trajectories are valid.The shadowing property also holds for pseudotrajectories of chaotic billiards, under cer-tain conditions [38]. (“Chaotic billiards” includes all the famous examples, such as the sta-dium billiard, the semi-dispersing Sinai billiard, the diamond billiards, and many others.) Inapplying shadowing arguments to chaotic billiards, it is necessary to exclude the vicinity ofsingularities (i.e., discontinuities, corners, cusps, etc.), where hyperbolicity can deteriorateand shadowing arguments may fail. The presence of singularities actually makes a detailedquantitative analysis of shadowing highly nontrivial. Consider again the cardioid billiard.The cardioid is an example of a non-uniformly hyperbolic system with a cusp singularity.For such a system, shadowing of pseudotrajectories is assured except in small neighborhoodsaround each point of the singularity set of the Poincar´e map [39]. In the case of the cardioidbilliard, the singularity set S = C ∪ F , where C is the set of initial conditions that start ator will straightaway hit the cusp (i.e., C = { ξ ∈ Σ : s = ± } ∪ { ξ ∈ Σ : p = s/ } ), and19 = { ξ ∈ Σ : p = ± } is the set of tangential collision points (“fixed points”) mentioned inSec. IV. Roughly speaking, as long as a typical pseudotrajectory does not come too close toany point of the singularity set, the existence of a shadowing trajectory is assured, and thestatistical properties of the pseudotrajectory will be very similar to those of the (exact) shad-owing trajectory. If and when a pseudotrajectory enters a sufficiently small neighborhoodof any point of S (even only once), shadowing of the whole pseudotrajectory is no longerassured. The main issue then, in the case of the cardioid, is to determine how close to thecusp the particle can come before shadowing breaks down. This is a difficult mathematicalproblem requiring sophisticated analysis. In the absence of more rigorous analysis, it is notpossible to make any quantitative statements about how closely typical pseudotrajectoriesare shadowed by true trajectories and for how long.Similar comments apply to the two other billiard systems that were studied. In the case ofthe stadium billiard, questions concerning shadowability arise when pseudotrajectories passthrough sufficiently small neighborhoods of the critical points of the stadium (where theboundary curvature changes discontinuously). In the case of the mushroom, the restrictionof the billiard flow to the chaotic region is hyperbolic [29] but shadowing may neverthelessbreak down near any corner of the mushroom. As before, determining how close to a cornera trajectory must pass before shadowing begins to break down is a difficult mathematicalproblem.To the author’s knowledge, shadowing dynamics in the H´enon-Heiles system (which isnon-hyperbolic) have not been studied. It has been suggested on the basis of detailednumerical studies (see Ref. [40]) that any asymptotically-evolved pseudotrajectory of theH´enon-Heiles system at the critical energy does not represent an accurate dynamical historyof any single trajectory on the E = 1 / k th-nearest-neighbor distance characteristics of the exact orbitsof the Poincar´e map can be inferred from the numerical experiments of Sec. VI. The validityof this interpretation, however, has not yet been firmly established. Thus, the question ofwhether typical chaotic trajectories on the E = 1 / For any ergodic componentof a 2D non-integrable Hamiltonian flow, typical chaotic trajectories of the Poincar´e maphave the same (Euclidean) k th-nearest-neighbor distance characteristics as P . It is impor-tant to emphasize that the interpoint distances being referred to here are measured usingthe usual Euclidean metric.
C. Quantum Chaos
Strongly-chaotic bounded conservative systems typically possess quantum energy-levelnearest-neighbor spacing distributions that are well modeled by the Wigner distribution(see Refs. [24, 41–45] for examples and discussions). A “Wigner-like” energy-level nearest-neighbor spacing distribution (NNSD) has long been widely regarded to be a “generic”property of time-reversal-invariant quantum systems having strongly-chaotic classical limits[46]. The underlying reasons for it being so nevertheless remain elusive. It is presumed that, in “generic” cases, the Wignerian shape of the NNSD (a property of the quantumeigenvalues) derives solely from the chaoticity of the classical dynamics (a property of theclassical trajectories). It is however not fundamentally understood how classical chaos isitself responsible for producing the observed Wignerian shape of the energy-level spacingdistribution (a point that is often glossed over in the “quantum chaos” literature).As observed in analyses of the cardioid and stadium billiards (which are strongly-chaoticbounded conservative systems), typical pseudotrajectories of the Poincar´e map have aWignerian nearest-neighbor distance distribution (NNDD). One of the premises of this pa-per is that the preceding is a “generic” property of strongly-chaotic 2D conservative systems(i.e., a spatial statistical property of the chaotic pseudotrajectories that one expects toobserve more generally in numerical simulations of such systems) [48]. That the Wigner dis-tribution should have any fundamental significance whatsoever in the description of classical chaos is interesting given its significance (albeit as an approximation) in the description ofenergy-level fluctuations in quantum chaos.The spectrum of the quantized cardioid billiard is generic in the sense that any sufficientlylarge sample histogram of the nearest-neighbor spacings is well modeled by the Wignerdistribution (see, for example, Ref. [49]), in accordance with the GOE hypothesis for chaoticsystems [50]. Incidentally, the higher-order spacings of the cardioid spectrum exhibit clear21nd significant deviations from GOE predictions (which is expected), but these higher-orderspacings likewise do not follow the higher-order “spacing” distributions predicted for P (asgiven by Eq. (3) with k ≥ VIII. CONCLUSION
A fundamental chaotic process is the deterministic point process (DPP) generated fromthe successive intersections of a chaotic pseudotrajectory of the flow with a canonical SOS(i.e., a Σ-type DPP). In the specific context of 2D non-integrable Hamiltonian flows possess-ing ergodic components, the fundamental question raised in this paper is the following: Whatare the spatial statistical (or geometrical-statistical ) characteristics of a typical Σ-type DPPin an ergodic region of phase space? The hypothesis that the (Euclidean) distance character-istics of such a DPP are consistent with those theoretically predicted for the two-dimensionalhomogeneous Poisson point process ( P ), was put forward and tested. Employing the car-dioid and semi-circular mushroom billiards as generic test cases, it was shown that typicalchaotic pseudotrajectories of the Poincar´e map have k th-nearest-neighbor distance distribu-tions that are well modeled by the corresponding distributions theoretically predicted for P . In this respect, the (deterministic) discrete pseudo -dynamics generated by the Poincar´emap behave like a Poisson point process in W (the corresponding ergodic component of theSOS), and typical chaotic pseudotrajectories of the map can be viewed as realizations ofa Poisson point process in W [51]. An open problem is to determine whether or not the exact trajectories of these and other chaotic billiard maps possess spatial statistical proper-ties consistent with those of the pseudo trajectories. In principle, a rigorous analysis of theshadowing dynamics is one way of attacking this problem but the presence of singularitiesintroduces highly nontrivial technicalities.The interpoint distance characteristics of a Σ-type DPP initiated and evolved in a non-ergodic region U will, in general, deviate from those of a Poisson point process in U . Ifa Σ-type DPP is initiated in the chaotic sea, then in certain special cases (e.g., in param-eter regimes where the dynamics is strongly, but not fully, chaotic), the deviation will beinconsiderably small, as demonstrated in the case of the H´enon-Heiles system at the critical22nergy. In general, however, deviations from Poissonian behavior will be significant and adrastically different model is needed to understand the distance characteristics of the chaoticpseudotrajectories. Regular pseudotrajectories will furthermore require the introduction ofa different type of stochastic geometric model to understand their corresponding properties.The distance characteristics of Σ-type DPPs in 2D generic systems is a subject that shallhopefully be explored more extensively in a future publication. Acknowledgments
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