Homoclinic orbit expansion of arbitrary trajectories in chaotic systems: classical action function and its memory
HHomoclinic orbit expansion of arbitrary trajectories in chaotic systems: classicalaction function and its memory
Jizhou Li
1, 2 and Steven Tomsovic Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA Department of Physics, Tokyo Metropolitan University,Minami-Osawa, Hachioji, Tokyo 192-0397, Japan (Dated: September 28, 2020)Special subsets of orbits in chaotic systems, e.g. periodic orbits, heteroclinic orbits, closed orbits,can be considered as skeletons or scaffolds upon which the full dynamics of the system is built. Inparticular, as demonstrated in previous publications [Phys. Rev. E , 062224 (2017), Phys. Rev. E , 022216 (2018)], the determination of homoclinic orbits is sufficient for the exact calculation ofclassical action functions of unstable periodic orbits, which have potential applications in semiclas-sical trace formulas. Here this previous work is generalized to the calculation of classical actionfunctions of arbitrary trajectory segments in multidimensional chaotic Hamiltonian systems. Theunstable trajectory segments’ actions are expanded into linear combinations of homoclinic orbitactions that shadow them in a piece-wise fashion. The results lend themselves to an approximationwith controllable exponentially small errors, and which demonstrates an exponentially rapid loss ofmemory of a segment’s classical action to its past and future. Furthermore, it does not require anactual construction of the trajectory segment, only its Markov partition sequence. An alternativepoint of view is also proposed which partitions the trajectories into short segments of transient visitsto the neighborhoods of successive periodic orbits, giving rise to a periodic orbit expansion schemewhich is equivalent to the homoclinic orbit expansion. This clearly demonstrates that homoclinicand periodic orbits are equally valid skeletal structures for the tessellation of phase-space dynamics. I. INTRODUCTION
Following some insights and work of Poincar´e [1] andcycle expansions [2, 3], Cvitanovi´c discussed periodic or-bits as a skeleton of classical and quantum chaos [4].Crudely speaking, the idea is that calculating convergentexpressions of dynamical averages in chaotic systems canbe expressed in terms of unstable periodic orbits and bedominated by the shorter ones. In effect, the infinity oforbits in the hyperbolic flow resummed in averages can bereduced to this skeleton. It turns out that other specialsets of orbits, such as closed orbits [5–7] or homoclinic(heteroclinic) orbits [8, 9], can also be thought of as pro-viding a skeleton, depending on the circumstances. Thisis not surprising in the sense that exact relations can beestablished between such special sets. For example, therelations between periodic orbits and homoclinic orbitsfor classical action functions were given in Refs. [10, 11]and stability exponents in Ref. [12]. In a sense, the workin this article is the opposite of that in the calculation ofdynamical averages, i.e. such skeletons can also be usedto predict every microscopic feature of the dynamics. Inparticular, it is shown here that the classical action func-tion for any arbitrary unstable trajectory segment canbe exactly related either to properties of particular ho-moclinic or periodic orbits; this gives a generalizationof Ref. [11].An exact scheme is introduced to expand unstable tra-jectories into sequences of simple homoclinic orbits thatshadow them in a piece-wise fashion, and express theirclassical action functions in terms of sums of the homo-clinic orbit actions plus certain phase-space areas as con-nectors between successive homoclinic orbits. An added benefit is that the action functions of extremely long un-stable trajectory segments can be accurately calculated,even for lengths beyond which the trajectories can be di-rectly followed in detail due to exponential propagationof errors. The method relies on a collection of simplehomoclinic orbits with relatively short transit times [13],which can be computed effortlessly with existing stablealgorithms [14]. Most importantly, the results developedhere may provide a general scheme for the computationof unstable trajectories in chaotic systems from the soleknowledge of homoclinic orbits, which may provide an al-ternative route to the investigation of chaotic phenomenain Hamiltonian systems. An application to the calcula-tion of periodic orbit actions is also introduced, whichmay enable a resummation of the Gutzwiller trace for-mula [15] (or the semiclassical ζ function) in terms ofhomoclinic orbits, although more work in this directionis currently under investigation.An alternative route is also introduced which is thesame in spirit as the cycle expansion [16]. It treats thetrajectories as shadowed by short periodic orbits in apiecewise fashion instead of the homoclinic orbits. Theresulting formulas provide an exact expression for theaction correction to the original cycle expansion, whichis expressed in terms of the same collection of simplehomoclinic orbits. Therefore, periodic and homoclinicorbits are equivalent fundamental structures upon whichthe full dynamics is built.This article is organized as follows. Sec. II introducesthe basic concepts and definitions. The main contentsof the current work are introduced in Sec. III. Exact re-lations are derived along with an asymptotic form thatdoes not require construction of the orbit. This is applied a r X i v : . [ n li n . C D ] S e p to the computation of long periodic orbit actions and astudy of its accuracy. Sec. IV summarizes the article andpoints to the direction of future research. II. BASIC CONCEPTSA. Covariant Lyapunov vectors
Let us consider an ( f + 1)-degree-of-freedom Hamil-tonian system for which the dynamics is highlychaotic. With energy conservation and applying thePoincar´e surface of section technique [1], the Hamil-tonian flow is reduced to a discrete symplectic map M on the 2 f -dimensional phase space z = ( q , p ) =( q , · · · , q f , p , · · · , p f ). The trajectory (or orbit ) of aphase-space point z , denoted by { z } , is the bi-infinitecollection of all M n ( z ): { z } = {· · · , z − , z , z , · · · } where z n = M n ( z ) for all integers n . Assuming the dy-namics is almost everywhere hyperbolic, then the corre-sponding Poincar´e map M is also hyperbolic. Intuitivelyspeaking, the map stretches any given phase-space re-gion along its unstable directions and compresses alongthe stable directions, then folds and remixes it with otherparts of the phase space. The stability matrix at z , de-noted by DM ( z ), is defined by DM ( z ) = ∂M∂z ( z ) (1)where DM ( z ) characterizes the tangent dynamics at z under one iteration of M . The stability matrix of the n -th compound mapping is DM n ( z ) = ∂M n ∂z ( z ) (2)that characterizes the tangent dynamics at z under n iterations of M . Under the hyperbolicity assump-tion, there exists an invariant Oseledec splitting [17]of the tangent space of z into exponentially expand-ing and contracting directions under the asymptotic maplim n →∞ DM n ( z ). These directions are given by the co-variant Lyapunov vectors (CLV) [18–20] at z . Denotethe CLV at z n by v i ( z n ), which are covariant in the sensethat DM ( z n ) v i ( z n ) = λ i ( z n ) v i ( z n +1 ) (3)where v i ( z n ) and v i ( z n +1 ) are vectors with unit norms,and λ i ( z n ) is the local expanding (for 1 ≤ i ≤ f ) orcontracting (for f + 1 ≤ i ≤ f ) factor of z n . v i ( z n )(1 ≤ i ≤ f ) yields the i -th most rapidly expanding di-rection along the unstable manifold of z n , and v f + i ( z n )(1 ≤ i ≤ f ) yields the ( f − i + 1)-th most rapidly con-tracting direction along the stable manifold of z n . Conse-quently, the unstable manifold of z n , denoted by U ( z n ), is the f -dimensional hyper-surface spanned by the stream-lines of [ v ( z n ) , · · · , v f ( z n )], and the stable manifold of z n , denoted by S ( z n ), is the f -dimensional hyper-surfacespanned by the streamlines of [ v f + ( z n ) , · · · , v ( z n )].The Lyapunov spectrum of z , namely µ ( i ) ( z ) (1 ≤ i ≤ f ), is µ ( i ) ( z ) = lim N →∞ N N − (cid:88) n =0 ln | λ i ( z n ) | . (4)For Hamiltonian systems, the Lyapunov exponents comein pairs: µ ( i ) ( z ) = − µ (2 f +1 − i ) ( z ) for i = 1 , . . . , f .Assume the positive Lyapunov spectrum of all relevantpoints z are bounded away from zero: µ (1) ( z ) ≥ · · · ≥ µ ( f ) ( z ) > . (5)Typically, µ ( i ) ( z ) is almost-everywhere independentof z for ergodic trajectories, and therefore the z -dependence can be removed.For periodic orbits: y = M T ( y ), the Lyapunov ex-ponents reduce to µ ( i ) { y } ≡ µ ( i ) ( y ) = 1 T T − (cid:88) n =0 ln | λ i ( y n ) | , (6)which has a dependence on initial conditions since theyare not ergodic. Other special sets may also have thisproperty. To emphasize the difference between periodicorbits and ergodic trajectories, the µ ( i ) { y } are referred toas stability exponents of { y } . B. Symbolic dynamics
Let x be a hyperbolic fixed point, i.e., M ( x ) = x .Denote the unstable and stable manifolds of x by U ( x )and S ( x ), respectively. Typically, U ( x ) and S ( x ) are f -dimensional invariant hypersurfaces that intersect in-finitely many times and form a complicated patterncalled a homoclinic tangle [1, 13, 21]. It is well-knownthat generating Markov partitions to the phase space[22, 23] exist, under which the mapping M is conju-gate to a subshift of finite type on bi-infinite symbolicstrings [24–27] representing phase-space itineraries of or-bits. The cells of the partition V = [ V , V , · · · , V K ]are 2 f -dimensional curvilinear parallelograms, which aredubbed “vertical slabs” in [28]. Under the mapping M ,the V cells are stretched along the unstable directions,contracted along the stable directions, and mapped intoa set of cells H = [ H , H , · · · , H K ] (where H i = M ( V i ))that intersect with V to create mixing (see upper panel ofFig. 13). Letting s i ∈ { , · · · , K } ( ∀ i ∈ Z ) be the integerdigits that labels the cells, then under successive inversemappings, the intersections V s s ··· s n − ≡ n − (cid:92) i =0 M − i ( V s i ) (7)become a family of cells whose widths along U ( x ) de-crease exponentially with n . Similarly, under successiveforward mappings, the intersections H s − n s − n +1 ··· s − ≡ n (cid:92) i =1 M i − ( H s − i ) (8)become a family of cells whose widths along S ( x ) de-crease exponentially with n . From their definitions it iseasy to see that H γ = M n ( V γ ), where γ = s · · · s n − denotes an arbitrary finite string of length n .For systems with f = 1, both U ( x ) and S ( x ) are 1-dimensional curves. Together they form the boundariesof V i and H i ; see Figs. 12 and 13 for illustrations. The in-tersection H s − n ··· s − ∩ V s ··· s n − therefore localizes an ex-ponentially small region in phase space, as demonstratedby Fig. 14. In the limiting case of n → ∞ , such intersec-tions create a Cantor set of points on which the dynamicsis topologically conjugate to a subshift of finite type onbi-infinite symbolic strings. Each point z from the Can-tor set is assigned a symbolic string z = lim n →∞ H s − n ··· s − ∩ V s ··· s n − ⇒ · · · s − s − · s s s · · · (9)where each character s n in the sequence denotes thecell to which M n ( z ) belongs: M n ( z ) = z n ∈ V s n , s n ∈ { , · · · , K } . The separation dot in the middle in-dicates the current iteration: z ∈ V s . The symboliccode gives an “itinerary” of z under successive forwardand backward iterations, in terms of the Markov cellsin which each iteration lies. The mapping M under thesymbolic dynamics is then reduced to a simple shift ofthe dot in the code: M n ( z ) = z n ⇒ · · · s n − · s n s n +1 · · · . Points along the same trajectory have the same symbolicstrings but shifting separation dots. Therefore, a trajec-tory can be represented by the symbolic string withoutthe dot: { z } ⇒ · · · s − s − s s s · · · (10)For systems with f ≥
2, the partition cells V = [ V , V , · · · , V K ] become “vertical slabs” which aremapped into “horizontal slabs” H = [ H , H , · · · , H K ]that intersect V [28]. As illustrated schematically byFig. 1, some edges of V i are located on S ( x ), and oth-ers located on U ( x ). Under one iteration of M , V i iscontracted along the stable edges and expanded alongthe unstable edges, and deformed into H i = M ( V i ). Tothe authors’ knowledge, there has not been an explicitstudy of the bounding surfaces of V i and H i in multidi-mensional systems. A reasonable conjecture is that theside surfaces of V i and H i are spanned by two portionsof each of the invariant manifolds beginning from theCLVs [ v ( x ) , · · · , v j − ( x ) , v j + ( x ) , · · · , v ( x )], i.e. ex-cluding v j ( x ), and by letting j = 1 , · · · , f . In such away 2 f codimension-1 surfaces can be created, which are M H i V i S(x)U(x)
FIG. 1. Vertical slabs V i and horizontal slabs H i = M ( V i )as generating Markov partition for the symbolic dynamics ofmultidimensional chaotic systems. Notice that V i and H i are actually high-dimensional curvy “parallelograms”, andonly a 3-dimensional illustration is possible. The horizon-tal direction is aligned with U ( x ), and the vertical alignedwith S ( x ). Both U ( x ) and S ( x ) are f -dimensional sur-faces. The expectation is that the bounding side surfacesof V i and H i are spanned by two portions of each of thesurfaces formed by the set of CLVs missing the j th mem-ber, i.e. [ v ( x ) , · · · , v j − ( x ) , v j + ( x ) , · · · , v ( x )], for j =1 , · · · , f , successively. the natural extension of the side surfaces of V i and H i inthe f = 1 case. The contracting and expanding directionsof V i under the mapping are therefore governed by the in-tricate way that the CLV surfaces form its side surfaces,which is left for future studies. As schematically illus- V s s ...s n-1 H s -n ...s -1 H s -n ...s -1 ∩ V s s ...s n-1 FIG. 2. V s s ··· s n − is exponentially thin in the “horizon-tal” (unstable) directions, and H s − n ··· s − is exponentiallythin in the “vertical” (stable) directions. The intersection H s − n ··· s − ∩ V s ··· s n − is an exponentially small region, whichunder the n → ∞ limit gives rise to a unique phase-spacepoint. trated by Fig. 2, the intersections H s − n ··· s − ∩ V s ··· s n − again localize an exponentially small volume in phasespace, which under the n → ∞ limit gives rise to a Can-tor set of points on which the dynamics is topologicallyconjugate to a subshift of finite type on a bi-infinite sym-bolic string. More details on the multidimensional for-mulation can be found in Chap. 2.3 of [28]. Here it isassumed that the symbolic dynamics permits all possibletransitions s i s i +1 for s i , s i +1 ∈ { , , · · · , K } . However,the results derived ahead carry over into more compli-cated systems possessing a pruning front [4, 29].Under the symbolic dynamics, a period- T point y ,where M T ( y ) = y , can always be associated with asymbolic string with infinite repetitions of a substringwith length T : y ⇒ · · · s s · · · s T − · s s · · · s T − · · · = γ · γ (11)where γ = s · · · s T − is the finite substring and γ · γ denotes its infinite repetition (on both sides of the dot).Notice that the cyclic permutations of s · · · s T − can beassociated with the successive mappings of y , generatinga one-to-one mapping to the set of points on the orbit.Since an orbit can be represented by any point on it, theposition of the dot does not matter, therefore we denotethe periodic orbit { y } as { y } ⇒ γ (12)with the dot removed. Correspondingly, the stability ex-ponents µ ( i ) { y } can be written alternatively as µ ( i ) γ .Ahead an approximation scheme is developed to sim-plify the exact action formulas. The scaling relation ofan upper bound for the remainder (error term) is associ-ated with the slowest shrinking dimension of H s − n ··· s − ∩ V s ··· s n − . For V s ··· s n − the slowest shrinking dimensionof the “horizontal” ( U ( x )) direction is governed by thesmallest positive Lyapunov exponent. For H s − n ··· s − thescaling for the “vertical” ( S ( x )) direction is governedby the least negative Lyapunov exponent. A reason-able estimate for this process is provided by the stabil-ity exponents of the periodic orbits γ and γ , where γ = s − n · · · s − and γ = s · · · s n − :scaling of V γ ∼ O ( e − nµ ( f ) γ )scaling of H γ ∼ O ( e − nµ ( f ) γ ) . (13)The scaling of H γ ∩ V γ , as measured by the maxi-mum component-wise phase-space separations betweentwo points z = ( q , p ) and z (cid:48) = ( q (cid:48) , p (cid:48) ) located inside H γ ∩ V γ , i.e.,scaling of H γ ∩ V γ ≡ max z,z (cid:48) ∈ H γ ∩ V γ ;1 ≤ i ≤ f (cid:8) | p i − p (cid:48) i | , | q i − q (cid:48) i | (cid:9) , (14)is then estimated by the greater of two widths in Eq. (13):scaling of H γ ∩ V γ ∼ O (cid:0) max { e − nµ ( f ) γ , e − nµ ( f ) γ } (cid:1) . (15)Here, the hyperbolic fixed point with symbolic code x ⇒ · { x } ⇒ x , S ( x ), and U ( x ) only. The intersections betweenthe f -dimensional S ( x ) and f -dimensional U ( x ) in 2 f -dimensional phase space give rise to homoclinic points, which are asymptotic to x under both M ±∞ . A homo-clinic point h of x has symbolic code of the form [30]: h ⇒ s − m · · · s − · s s · · · s n . (16)Similar to the periodic orbit case, the homoclinic orbitcan be represented as { h } ⇒ s − m · · · s − s s · · · s n C. Generating function and classical action
For any phase space point z n = ( q n , p n ) and its im-age M ( z n ) = z n +1 = ( q n + , p n + ), the mapping M canbe viewed as a canonical transformation that maps z n to z n +1 while preserving the symplectic area, therefore a generating ( action ) function F ( q n , q n + ) can be associ-ated with this process such that [31, 32]: p n = − ∂F/∂ q n p n + = ∂F/∂ q n + . (18)Despite the fact that F is a function of q n and q n + , it isconvenient to denote it as F ( z n , z n +1 ). This should causeno confusion as long as it is kept in mind that it is the q variables of z n and z n +1 that go into the expression of F . The compound mapping M k , which maps z n to z n + k ,then has the generating function: F ( z n , z n + k ) ≡ n + k − (cid:88) i = n F ( z i , z i +1 ) (19)which, strictly speaking, is a function of the q variables.For periodic orbits { y } ⇒ γ with primitive period T ,the primitive period classical action F γ of the orbit is: F γ ≡ T − (cid:88) i =0 F ( y i , y i +1 ) . (20)For the special case of the fixed point x , Eq. (20) reducesto: F = F ( x, x ) (21)where F ( x, x ) is the generating function that maps x intoitself in one iteration.For non-periodic orbits { y } , the classical action is thesum of the generating functions over infinite successivemappings: F { y } ≡ lim N →∞ N − (cid:88) i = − N F ( y i , y i +1 ) = lim N →∞ F ( y − N , y N )(22)and is divergent in general. However, the MacKay-Meiss-Percival action principle [31, 32] can be applied to obtain U(x)S(x)x h U[x,h ]S[h ,x] FIG. 3. Integration paths in Eq. (23). Both U ( x ) and S ( x ) are f -dimensional manifolds in the 2 f -dimensional phase space,and their intersections give rise to homoclinic points such as h . U [ x, h ] is an arbitrary path directed from x to h , andsimilarly for S [ h , x ]. The p · dq integrals along them areindependent of the paths. well-defined action differences for particular pairs of or-bits. Refer to Appendix B for a brief introduction of theaction principle. An important and simple case is the relative action ∆ F { h }{ x } between a fixed point x andits homoclinic orbit { h } , where h ±∞ → x :∆ F { h }{ x } ≡ lim N →∞ N − (cid:88) i = − N [ F ( h i , h i +1 ) − F ( x, x )]= (cid:90) U [ x,h ] f (cid:88) j =1 p j d q j + (cid:90) S [ h ,x ] f (cid:88) j =1 p j d q j = (cid:90) U [ x,h ] p · dq + (cid:90) S [ h ,x ] p · dq = (cid:73) US [ x,h ] p · dq = A ◦ US [ x,h ] (23)where the notation U [ a, b ] is introduced to denote thefinite segment of an arbitrary 1-dimensional curve on U ( x ) extending from a to b , both of which are pointson U ( x ), and similarly for S ( x ). For systems with f = 1, U [ a, b ] and S [ a, b ] are unique segments since both U ( x )and S ( x ) are 1-dimensional curves themselves. For sys-tems with f ≥ U [ a, b ] and S [ a, b ] arenot unique since there are infinitely many 1-dimensionalcurves connecting a and b on multidimensional hyper-surfaces, as demonstrated by Fig. 3. However, due to thefact that both U ( x ) and S ( x ) are Langrangian manifolds,the phase-space integrals (cid:82) U [ a,b ] p · dq and (cid:82) S [ a,b ] p · dq areindependent of the paths and are uniquely determined bythe endpoints a and b . The ◦ superscript on the last lineindicates that the symplectic area is evaluated for a paththat forms a closed loop, and the subscript indicates thepath: U S [ x, h ] = U [ x, h ] + S [ h , x ]. ∆ F { h }{ x } gives the action difference between the ho-moclinic orbit segment [ h − N , · · · , h N ] and the length-(2 N + 1) fixed point orbit segment [ x, · · · , x ] in the limit N → ∞ . In later sections, upon specifying the symboliccode of the homoclinic orbit { h } ⇒ γ
0, we also denote∆ F { h }{ x } alternatively as∆ F { h }{ x } = ∆ F γ , (24)by replacing the orbits in the subscript with their sym-bolic codes.Another case of great interest here is the relative action ∆ F { h (cid:48) }{ h } between a pair of homoclinic orbits { h (cid:48) } and { h } , such that h (cid:48)±∞ = h ±∞ = x :∆ F { h (cid:48) }{ h } ≡ lim N →∞ N − (cid:88) i = − N (cid:2) F ( h (cid:48) i , h (cid:48) i +1 ) − F ( h i , h i +1 ) (cid:3) = (cid:90) U [ h ,h (cid:48) ] p · dq + (cid:90) S [ h (cid:48) ,h ] p · dq = A ◦ US [ h ,h (cid:48) ] . (25)Similar to the notation adopted in Eq. (24), upon thespecification of their symbolic codes { h } ⇒ γ { h (cid:48) } ⇒ γ (cid:48)
0, ∆ F { h (cid:48) }{ h } can also be denoted alterna-tively as ∆ F { h (cid:48) }{ h } = ∆ F γ (cid:48) , γ (26)by replacing the orbits in the subscript with their sym-bolic codes. From the definitions of the relative actionsit is easy to check that∆ F γ (cid:48) , γ = ∆ F γ (cid:48) , − ∆ F γ , . (27)A further useful generalization of Eq. (25) applies tofour arbitrary homoclinic orbits of x , namely { a } , { b } , { c } , and { d } [32]:(∆ F { a }{ x } − ∆ F { b }{ x } ) − (∆ F { c }{ x } − ∆ F { d }{ x } )= A ◦ SUSU [ a ,c ,d ,b ] (28)where A ◦ SUSU [ a ,c ,d ,b ] ≡ (cid:90) S [ a ,c ] p · dq + (cid:90) U [ c ,d ] p · dq + (cid:90) S [ d ,b ] p · dq + (cid:90) U [ b ,a ] p · dq (29)is the symplectic area of a loop formed by alternatingcurve segments from S ( x ) and U ( x ) connecting the fourhomoclinic points. Same with the previous cases, thechoice of the paths does not matter here since the inte-grals are uniquely fixed by the four homoclinic points. III. HOMOCLINIC EXPANSIONA. Exact expansion
This subsection is dedicated to the derivation of a gen-eral formula for the action(generating) function of unsta-ble trajectories. The approach is to cut a long trajectoryinto several segments, and replace each segment with ahomoclinic orbit that has a similar phase-space excur-sion (therefore a similar symbolic code). Consider an ar-bitrary unstable trajectory { y } with the symbolic code y ⇒ α · βδ , where the Greek letters α and δ here denotethe left- and right-infinite strings of digits, respectively: α = · · · s (cid:48)− s (cid:48) δ = s (cid:48)(cid:48) s (cid:48)(cid:48) · · · (30)and β denotes a finite string of digits with length N : β = s s · · · s N − . (31)The main interest here is the actions of long orbit seg-ments, thus the integer N is assumed to be very large.Conceptually, cut the orbit segment of { y } correspond-ing to β into L pieces: β = β β · · · β L (32)where each piece β i has length n i and (cid:80) Li =1 n i = N . Forthe sake of simplicity, define a cumulative index m k ≡ k (cid:88) i =1 n i , (33)note that m = n and m L = N .Setting up the symbolic code of y this way, the codesof forward images of y , namely y m k = M m k ( y ) (1 ≤ k ≤ L − y m k ⇒ αβ β · · · β k · β k +1 · · · β L − β L δ. (34)In particular, two special cases of Eq. (34) under k = 1and k = L − y m ⇒ αβ · β · · · β L δy m ( L − ⇒ αβ β · · · β L − · β L δ. (35)The generating function of interest, F ( y m , y m L − ), cor-responds to the orbit segment { y m , · · · , y m L − } and theaction function F ( y m , y m L − ) is cut in the same way: F ( y m , y m L − ) = L − (cid:88) k =1 F ( y m k , y m k +1 ) (36)where F ( y m k , y m k +1 ) is the generating function thatmaps from the beginning to the ending point of the k -thpiece.The key point of the scheme is to replace the tra-jectory pieces by suitable homoclinic orbits that mimic their phase-space excursions, thereby avoiding the nu-merical construction of the trajectories themselves. To bemore specific, the phase-space behavior of each piece in { y m k , · · · , y m k +1 } is characterized by the substring β k +1 ,which has a similar excursion with a finite segment of anauxiliary homoclinic orbit, namely { h ( k +1)0 } , identified bythe code { h ( k +1)0 } ⇒ β k +1 h ( k +1)0 ⇒ · β k +1 h ( k +1) n ( k +1) ⇒ β k +1 · . (37)This shadowing of the pieces of the homoclinic orbit andthe trajectory piece gives rise to an exact relation be-tween F ( y m k , y m k +1 ) and the homoclinic orbit relativeaction ∆ F { h ( k +1)0 }{ x } , which is the building block of thescheme.To expose this relation, consider the homoclinic orbitrelative action ∆ F { h ( k +1)0 }{ x } (defined in Eq. (23)) splitinto three parts∆ F { h ( k +1)0 }{ x } = lim N →∞ N − (cid:88) i = − N (cid:104) F (cid:16) h ( k +1) i , h ( k +1) i +1 (cid:17) − F (cid:105) = lim N →∞ − (cid:88) i = − N (cid:104) F (cid:16) h ( k +1) i , h ( k +1) i +1 (cid:17) − F (cid:105) + F (cid:16) h ( k +1)0 , h ( k +1) n k +1 (cid:17) − n k +1 F + lim N →∞ N − (cid:88) i = n k +1 (cid:104) F (cid:16) h ( k +1) i , h ( k +1) i +1 (cid:17) − F (cid:105) . (38)The difference between F ( y m k , y m k +1 ) and ∆ F { h ( k +1)0 }{ x } can thus be expressed as F ( y m k , y m k +1 ) − ∆ F { h ( k +1)0 }{ x } = − lim N →∞ − (cid:88) i = − N (cid:104) F (cid:16) h ( k +1) i , h ( k +1) i +1 (cid:17) − F (cid:105) + (cid:104) F ( y m k , y m k +1 ) − F (cid:16) h ( k +1)0 , h ( k +1) n k +1 (cid:17) + n k +1 F (cid:105) − lim N →∞ N − (cid:88) i = n k +1 (cid:104) F (cid:16) h ( k +1) i , h ( k +1) i +1 (cid:17) − F (cid:105) . (39)Invoking the MacKay-Meiss-Percival action principal [31,32], the three terms on the right-hand side of Eq. (39)can be converted into phase-space integrals along certainmanifold segments. For the first term, Eq. (B4) with b i = h ( k +1) i and a i = x , gives − lim N →∞ − (cid:88) i = − N (cid:104) F (cid:16) h ( k +1) i , h ( k +1) i +1 (cid:17) − F (cid:105) = (cid:90) U [ h ( k +1)0 ,x ] p · dq . (40)Similarly, for the third term Eq. (B5) with b i = h ( k +1) i and a i = x gives − lim N →∞ N − (cid:88) i = n k +1 (cid:104) F (cid:16) h ( k +1) i , h ( k +1) i +1 (cid:17) − F (cid:105) = (cid:90) S (cid:104) x,h ( k +1) nk +1 (cid:105) p · dq . (41)The procedure for the second term is less straightfor-ward because there are no stable or unstable manifoldsdirectly connecting points on the { y } orbit with pointson the { h ( k +1)0 } orbit. It is necessary to look for connect-ing curves exclusive of S ( x ) and U ( x ). By construction y m k V β k+1 h ( k+1 ) M n k+1 H β k+1 C h n k+1 ( k+1 ) y m k+1 C’M n k+1 V β k+1 H β k+1 h ( k+1 ) y m k h n k+1 ( k+1 ) y m k+1 C C’
FIG. 4. Dynamics under n k +1 iterations; y m k , h ( k +1)0 ∈ V β k +1 , and y m k +1 , h ( k +1) n k +1 ∈ H β k +1 , where H β k +1 = M n k +1 ( V β k +1 ).The “horizontal” width of V β k +1 and the “ver-tical” width of H β k +1 decrease exponentially with the lengthof the β k +1 string. Under n k +1 iterations of the map, V β k +1 is compressed along its stable directions and stretched alongits unstable directions into H β k +1 . Points inside V β k +1 fol-low a similar dynamics uniformly. C is a curve inside V β k +1 connecting y m k and h ( k +1)0 that is approximately parallel to S ( x ). C (cid:48) = M n k +1 ( C ), similarly, C (cid:48) is a curve inside H β k +1 connecting y m k +1 and h ( k +1) n k +1 that is roughly parallel to U ( x ).Upper panel: the f = 1 case. V β k +1 and H β k +1 are bothphase-space cells bounded by S ( x ) (thick dashed curve) and U ( x ) (thick solid curve). Lower panel: schematic of the f ≥ the symbolic codes of y m k and h ( k +1)0 share a commonsubstring β k +1 on the right-hand sides of the separationdots. This indicates that they are located within thesame “vertical” slab V β k +1 , as illustrated by Fig. 4. Asindicated in Eq. (13), the slowest shrinking scale of V β k +1 contracts as ∼ O ( e − n k +1 µ ( f ) βk +1 ), where µ ( f ) β k +1 is the small-est positive stability exponent of the periodic orbit β k +1 .Therefore, y m k and h ( k +1)0 are confined within the ex-ponentially thin phase-space cell V β k +1 , which is illus-trated schematically in Fig. 4. Under n k +1 iterations ofthe map, V β k +1 is compressed along its stable directionsand stretched along its unstable directions, and eventu-ally mapped into H β k +1 . Points inside V β k +1 follow a sim-ilar dynamics uniformly. Therefore, successive forwardimages of y m k and h ( k +1)0 first approach, then separatefrom each other, making a near fly-by somewhere in themiddle. Just like V β k +1 , the “vertical” width of H β k +1 isalso estimated to be ∼ O ( e − n k +1 µ ( f ) βk +1 ). The initial sep-aration between y m k and h ( k +1)0 is almost entirely alongthe stable manifold direction, while the final separationbetween y m k +1 and h ( k +1) n k +1 is almost entirely along theunstable manifold direction.As shown in Fig. 4, in spite of the fact that y m k and h ( k +1)0 are not directly connected by U ( x ) nor S ( x ), acurve C can be constructed to connect them, where C ischosen to be approximately parallel to S ( x ). Then under n k +1 iterations another curve C (cid:48) = M n k +1 ( C ) is createdthat is approximately parallel to U ( x ). With the helpof C and C (cid:48) , the MacKay-Meiss-Percival action principlecan again be applied to calculate the second term on theright-hand side of Eq. (39). This is done with Eq. (B2),setting b = y m k , b (cid:48) = y m k +1 , a = h ( k +1)0 , a (cid:48) = h ( k +1) n k +1 , and c and c (cid:48) to C and C (cid:48) , respectively: F ( y m k , y m k +1 ) − F ( h ( k +1)0 , h ( k +1) n k +1 )= (cid:90) C (cid:48) (cid:104) h ( k +1) nk +1 ,y mk +1 (cid:105) p · dq − (cid:90) C (cid:104) h ( k +1)0 ,y mk (cid:105) p · dq = (cid:90) C (cid:48) (cid:104) h ( k +1) nk +1 ,y mk +1 (cid:105) p · dq + (cid:90) C (cid:104) y mk ,h ( k +1)0 (cid:105) p · dq (42)where the C and C (cid:48) segments are illustrated by Fig. 4.Substituting Eqs. (40), (41), and (42) into Eq. (39) yields F ( y m k , y m k +1 ) = (cid:90) C (cid:104) y mk ,h ( k +1)0 (cid:105) p · dq + (cid:90) U [ h ( k +1)0 ,x ] p · dq + n k +1 F + ∆ F { h ( k +1)0 }{ x } + (cid:90) S (cid:104) x,h ( k +1) nk +1 (cid:105) p · dq + (cid:90) C (cid:48) (cid:104) h ( k +1) nk +1 ,y mk +1 (cid:105) p · dq . (43)Having obtained the expression for the generatingfunction of each piece, the total action function of thetrajectory segment { y m , · · · , y m L − } is then just the sumof all the pieces: F ( y m , y m L − ) = L − (cid:88) k =1 F ( y m k , y m k +1 )= (cid:90) C (cid:104) y m ,h (2)0 (cid:105) p · dq + (cid:90) U (cid:104) h (2)0 ,x (cid:105) p · dq + L − (cid:88) k =1 (cid:104) n k +1 F + ∆ F { h ( k +1)0 }{ x } (cid:105) + L − (cid:88) k =1 A ◦ CUSC (cid:48) (cid:104) y mk +1 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) + (cid:90) S (cid:104) x,h ( L − nL − (cid:105) p · dq + (cid:90) C (cid:48) (cid:104) h ( L − nL − ,y mL − (cid:105) p · dq (44)where A ◦ CUSC (cid:48) (cid:104) y mk +1 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) = (cid:90) C (cid:104) y mk +1 ,h ( k +2)0 (cid:105) p · dq + (cid:90) U (cid:104) h ( k +2)0 ,x (cid:105) p · dq + (cid:90) S (cid:104) x,h ( k +1) nk +1 (cid:105) p · dq + (cid:90) C (cid:48) (cid:104) h ( k +1) nk +1 ,y mk +1 (cid:105) p · dq (45)yields the symplectic area of the loop schematically de-picted in Fig. 5.At this point, Eq. (44) gives an exact expansion of F ( y m , y m L − ) in terms of homoclinic orbit actions andphase-space areas. Although seemingly complicated,terms on the right-hand side of Eq. (44) have explicitgeometric interpretations. The first term, (cid:90) C (cid:104) y m ,h (2)0 (cid:105) p · dq + (cid:90) U (cid:104) h (2)0 ,x (cid:105) p · dq , (46) C’x U(x)S(x) h n k+1 ( k+1 ) h ( k+2 ) y m k+1 CUS
FIG. 5. The A ◦ CUSC (cid:48) (cid:104) y mk +1 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) term in Eq. (44)yields the symplectic area of the loop shown in the figure. Thecurves labeled by U and S are arbitrary paths on U ( x ) and S ( x ), respectively. The curves C and C (cid:48) are defined in thesame way as those in Fig. 4. Therefore, C is nearly parallel tothe nearby S ( x ) that goes through h ( k +2)0 (not shown here),and C (cid:48) is nearly parallel to the nearby U ( x ) that goes through h ( k +1) n k +1 (not shown here). This symplectic area is the actionconnector between { h ( k +1)0 } and { h ( k +2)0 } . y m h ( ) x CU h n L-1 ( L-1 ) y m L-1 C’ S U(x)S(x) x U(x)S(x)
FIG. 6. Phase-space integrals as orbit action connectors. Leftpanel: paths for Eq. (46), which is the connector between y m and h (2)0 , or equivalently, the connector between the β and β segments in Eq. (34). Note that y m is located exponen-tially close to, but not a member of S ( x ). The resulting path C is approximately parallel to, but not contained in S ( x ) ei-ther. Right panel: paths for Eq. (49), which is the connectorbetween h ( L − n L − and y m L − , or equivalently, the connector be-tween the β L − and β L segments in Eq. (34). Similarly, notethat y m L − is located exponentially close to, but not a mem-ber of U ( x ). The resulting path C (cid:48) is approximately parallelto U ( x ) as well. is a phase-space integral along the path shown schemat-ically by the left panel of Fig. 6. It acts as an actionconnector between the β and β segments in Eq. (34).The second term, L − (cid:88) k =1 (cid:104) n k +1 F + ∆ F { h ( k +1)0 }{ x } (cid:105) , (47)is the sum of the contribution of all the auxiliary ho-moclinic orbits { h ( k )0 } (Eq. (37)) that shadow β k for k = 2 , · · · , L −
1. The third term, L − (cid:88) k =1 A ◦ CUSC (cid:48) (cid:104) y mk +1 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) , (48)is the sum of all action connectors between β k +1 and β k +2 , or equivalently, { h ( k +1)0 } and { h ( k +2)0 } , for k =1 , · · · , L −
3. The fourth term, (cid:90) S (cid:104) x,h ( L − nL − (cid:105) p · dq + (cid:90) C (cid:48) (cid:104) h ( L − nL − ,y mL − (cid:105) p · dq , (49)as shown schematically by the right panel of Fig. 6, isthe action connector between the β L − and β L segmentsin Eq. (34).Therefore, Eq. (44) provides an exact expansion of un-stable trajectory actions using homoclinic orbit actionsthat shadow it in a piecewise fashion, and phase-spaceareas as connectors between successive homoclinic or-bits. In applications or numerical implementations, itstill requires the calculation of the trajectory points y m k ( k = 1 , · · · , L − B. Approximate expression
It is possible to give an approximate expression withcontrolled errors for Eq. (44) that does not require thenumerical construction of the trajectory points, whichrepresents a great simplification. The essential spirit ofthe approximation is to replace the y m k ( k = 1 , · · · , L − y m k ( k = 1 , · · · , L −
1) points arereplaced by auxiliary homoclinic points g ( k,k +1)0 , wherethe homoclinic points are identified by symbolic codes g ( k,k +1)0 ⇒ β k · β k +1 , (50)which by design match the forward and backward prop-agated pieces of the { y } symbolic code closest to y m k (Eq. (34)). This implies that y m k , g ( k,k +1)0 ∈ H β k ∩ V β k +1 , (51)where H β k ∩ V β k +1 is the schematically depicted phase-space cell in Fig. 2. The resulting phase-space deviationbetween y m k and g ( k,k +1)0 is thus bounded by the shrink-ing scale of H β k ∩ V β k +1 (defined in Eq. (14)), which is ∼ O (cid:0) max { e − n k µ ( f ) βk , e − n k +1 µ ( f ) βk +1 } (cid:1) , where µ ( f ) β k and µ ( f ) β k +1 are the smallest positive stability exponents of β k and β k +1 , respectively.Starting from Eq. (46), replace y m by an auxiliaryhomoclinic point g (1 , identified by symbolic code g (1 , ⇒ β · β . (52) The two points are necessarily exponentially close to eachother. From Eq. (51) y m , g (1 , ∈ H β ∩ V β . (53)As shown in the left panel of Fig. 7, since the integrationpath C (cid:104) y m , h (2)0 (cid:105) is approximately parallel to the stablemanifold that goes through h (2)0 , it can be replaced by y m h ( ) x CU h n L-1 ( L-1 ) y m L-1 C’ S U(x)S(x) x U(x)S(x) g (1, ) S g ( L- L ) U FIG. 7. The exact integration paths in Fig. 6 are replacedby the approximate integration paths in this figure result-ing in exponentially small errors. Left panel: demonstra-tion for Eq. (54). The path S [ g (1 , , h (2)0 ] is located on S ( x ).The error is estimated to be ∼ O (cid:0) max { e − n µ ( f ) β , e − n µ ( f ) β } (cid:1) .Right panel: demonstration for Eq. (56). The path U [ h ( L − n L − , g ( L − ,L )0 ] is located on U ( x ). The error is estimatedto be ∼ O (cid:0) max { e − n L − µ ( f ) βL − , e − n L µ ( f ) βL } (cid:1) . a simpler integration path S (cid:104) g (1 , , h (2)0 (cid:105) on S ( x ), whichis chosen to be approximately parallel to C (cid:104) y m , h (2)0 (cid:105) .The result is a small error comparable to the symplecticarea of the gap between C (cid:104) y m , h (2)0 (cid:105) and S (cid:104) g (1 , , h (2)0 (cid:105) ,which is also exponentially small. Thus, an excellent ap-proximation for Eq. (46) is (cid:90) C (cid:104) y m ,h (2)0 (cid:105) p · dq + (cid:90) U (cid:104) h (2)0 ,x (cid:105) p · dq = (cid:90) S (cid:104) g (1 , ,h (2)0 (cid:105) p · dq + (cid:90) U (cid:104) h (2)0 ,x (cid:105) p · dq + O (cid:0) max { e − n µ ( f ) β , e − n µ ( f ) β } (cid:1) . (54)The great simplification is that the trajectory point y m no longer enters the calculation, and the integrationpaths are just curves on the stable and unstable manifoldsconnecting simpler homoclinic points. Moreover, since g (1 , ⇒ β · β h (2)0 ⇒ · β
0, it is easy to seethat the integral is uniquely determined by the symbolicsubstring β · β . For the sake of simplicity, denote I ( β k · β k +1 ) ≡ (cid:90) S (cid:104) g ( k,k +1)0 ,h ( k +1)0 (cid:105) p · dq + (cid:90) U (cid:104) h ( k +1)0 ,x (cid:105) p · dq , (55)0which expresses the approximate integral over the S and U paths as I ( β · β ).This procedure applies to Eq. (49) in an identical way.The trajectory point y m L − is replaced by the auxiliaryhomoclinic point g ( L − ,L )0 , where g ( L − ,L )0 ⇒ β L − · β L y m L − . The integration path C (cid:48) (cid:104) h ( L − n L − , y m L − (cid:105) is replaced by the simpler integrationpath U (cid:104) h ( L − n L − , g ( L − ,L )0 (cid:105) , similarly resulting in a expo-nentially small error shown schematically in the rightpanel of Fig. 7. The corresponding approximation forEq. (49) is (cid:90) S (cid:104) x,h ( L − nL − (cid:105) p · dq + (cid:90) C (cid:48) (cid:104) h ( L − nL − ,y mL − (cid:105) p · dq = (cid:90) S (cid:104) x,h ( L − nL − (cid:105) p · dq + (cid:90) U (cid:104) h ( L − nL − ,g ( L − ,L )0 (cid:105) p · dq + O (cid:0) max { e − n L − µ ( f ) βL − , e − n L µ ( f ) βL } (cid:1) . (56)As in the previous case, the integral is uniquely deter-mined by the substring β L − · β L . Denoting I (cid:48) ( β k · β k +1 ) ≡ (cid:90) S [ x,h ( k ) nk ] p · dq + (cid:90) U [ h ( k ) nk ,g ( k,k +1)0 ] p · dq , (57)the approximate form of Eq. (49) is given by I (cid:48) ( β L − · β L ). h n k+1 ( k+1 ) C’ x h ( k+2 ) U y m k+1 U(x)S(x) US S C g ( k+1 , k+2 ) FIG. 8. Illustration of Eq. (58). S ( x ) is plotted as ver-tical surfaces, and U ( x ) as horizontal surfaces. The inte-gration loop A ◦ SUSU (cid:104) g ( k +1 ,k +2)0 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) is marked byarrows. The error term in Eq. (58) is estimated to be ∼ O (cid:18) max { e − n k +1 µ ( f ) βk +1 , e − n k +2 µ ( f ) βk +2 } (cid:19) . Substitutions follow for Eq. (48) in exactly the same way: y m k +1 (cid:55)→ g ( k +1 ,k +2)0 C (cid:104) y m k +1 , h ( k +2)0 (cid:105) (cid:55)→ S (cid:104) g ( k +1 ,k +2)0 , h ( k +2)0 (cid:105) C (cid:48) (cid:104) h ( k +1) n k +1 , y m k +1 (cid:105) (cid:55)→ U (cid:104) h ( k +1) n k +1 , g ( k +1 ,k +2)0 (cid:105) . Equation (48) admits the approximate form A ◦ CUSC (cid:48) (cid:104) y mk +1 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) = A ◦ SUSU (cid:104) g ( k +1 ,k +2)0 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) + O (cid:0) max { e − n k +1 µ ( f ) βk +1 , e − n k +2 µ ( f ) βk +2 } (cid:1) , (58)where the new A ◦ SUSU [ ··· ] symplectic area is shown inFig. 8. It is important to note that the approximate sym-plectic area relies only on homoclinic orbits of relativelyshort excursions and their stable/unstable manifolds andthe explicit dependence on the trajectory points y m k +1 isgone. Furthermore, the symbolic codes of the four cor-ners of the loop have a particular simple form: g ( k +1 ,k +2)0 ⇒ β k +1 · β k +2 h ( k +1) n ( k +1) ⇒ β k +1 · h ( k +2)0 ⇒ · β k +2 x ⇒ · A ◦ SUSU (cid:104) g ( k +1 ,k +2)0 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) isuniquely determined by the symbolic string β k +1 · β k +2 .Therefore, to simplify notation let A ◦ ( β k +1 · β k +2 ) ≡ A ◦ SUSU (cid:104) g ( k +1 ,k +2)0 ,h ( k +2)0 ,x,h ( k +1) nk +1 (cid:105) . (59)It turns out that A ◦ ( β k +1 · β k +2 ) = I ( β k +1 · β k +2 ) + I (cid:48) ( β k +1 · β k +2 )= ∆ F β k +1 β k +2 , β k +1 − ∆ F β k +2 , , (60)where the last equality comes from Eq. (28).Substituting Eqs. (54-59) into Eq. (44) leads to theapproximate expansion for trajectory actions in general: F (cid:0) y m , y m L − (cid:1) = I ( β · β ) + I (cid:48) ( β L − · β L )+ L − (cid:88) k =1 (cid:104) n k +1 F + ∆ F β k +1 , (cid:105) + L − (cid:88) k =1 A ◦ ( β k +1 · β k +2 )+ O (cid:18) max k ∈ [1 ,L ] e − n k µ ( f ) βk (cid:19) (61)where ∆ F β k +1 , = ∆ F { h ( k +1)0 }{ x } [Eq. (24)].1 C. Loss of memory
Compared to the exact expansion in Eq. (44) that re-quires the knowledge of all the trajectory points y m k ( k = 1 , · · · , L − { h ( k )0 } (for k = 2 , · · · , L − { g ( k,k +1)0 } (for k = 1 , · · · , L − n k values, thus onecan choose any desired level of accuracy for the actioncalculation of long trajectory segments in general. Per-haps most importantly though, it reveals an important“exponential memory decay” property for long trajec-tory segments in chaotic systems that is presumably ex-pected, but is proven here. Notice that the left-hand sideof Eq. (61) is the action function evaluated from y m ⇒ αβ · β · · · β L δ to y m ( L − ⇒ αβ β · · · β L − · β L δ . Tospecify this trajectory uniquely, one must either know itsentire symbolic history { y } ⇒ αβδ (where y ⇒ α · βδ )or know y and its iterates with infinite precision. How-ever, the approximate classical action with controllableexponentially small errors depends only on the β symbolsequence. Not a single value of y or its iterates is neces-sary to calculate the approximation. Thus all trajectorieswith the same β sequence, independent of α and δ givethe same exponentially accurate classical action functionin shifting the present from just after β to just after β L − . The “memory” of the past α and future δ fadesexponentially away in the action function depending onlyon the lengths of the β and β L pieces. Such memory lossshould enable an improved matrix-product approach tosemiclassical trace formulas, which is currently under in-vestigation by the authors. D. Periodic orbits
Although the exact and approximate expressions givenabove apply to any trajectory segment, due to their greatinterest in semiclassical theories [36], it is worthwhile ap-plying the approximation procedure to long periodic or-bits. Let { y } ⇒ γ with period N , where γ is a symbolicstring of N digits. Partition γ into L − γ = γ γ · · · γ L − , and denote the length of each γ k by n k . Placing the separation dot such that y ⇒ γ · γ, (62)then y = M N ( y ) ⇒ γ · γ, (63)i.e., the mapping from y back to itself corresponds to ashift of the dot for N digits, thereby leading to identical symbolic codes. The classical action of interest is F γ = N − (cid:88) i =0 F ( y i , y i +1 ) = F ( y , y N ) . (64)The connection to the notation for a general trajectorysegment is α = γγ γ · · · γ L − β = γ L − β k = γ k − ( k = 2 , · · · , L − β L = γ δ = γ γ · · · γ L − γ . (65)The approximation, Eq. (61), yields the periodic orbitaction F γ = F ( y , y N ) = I ( γ L − · γ ) + I (cid:48) ( γ L − · γ )+ L − (cid:88) k =1 (cid:2) n k F + ∆ F γ k , (cid:3) + L − (cid:88) k =1 A ◦ ( γ k · γ k +1 )+ O (cid:18) max k ∈ [1 ,L − e − n k µ ( f ) γk (cid:19) , (66)where µ ( f ) γ k is the smallest positive stability exponent ofthe periodic orbit γ k .Notice that I ( γ L − · γ ) + I (cid:48) ( γ L − · γ ) = A ◦ ( γ L − · γ ) . (67)Equation (66) can be simplified: F γ = N F + L − (cid:88) k =1 (cid:2) ∆ F γ k , + A ◦ ( γ k · γ k +1 ) (cid:3) + O (cid:18) max k ∈ [1 ,L − e − n k µ ( f ) γk (cid:19) (68)where the γ k subscript is understood to be cyclic in L − γ L − = γ . This equation provides an expansion of longperiodic orbit actions in terms of homoclinic orbits 0 γ k A ◦ ( γ k · γ k +1 ) as action connectors between successivehomoclinic orbits. Just like Eq. (61), it does not requireprior numerical construction of the periodic orbits them-selves.With the help of Eq. (60), an alternative form equiva-lent to Eq. (68) can be given. Taking into account that A ◦ ( γ k · γ k +1 ) = ∆ F γ k γ k +1 , γ k − ∆ F γ k +1 , = ∆ F γ k γ k +1 , − ∆ F γ k , − ∆ F γ k +1 , (69)and substituting into Eq.(68) gives F γ = N F + L − (cid:88) k =1 ∆ F γ k γ k +1 , γ k +1 + O (cid:18) max k ∈ [1 ,L − e − n k µ ( f ) γk (cid:19) (70)2where the index k is also cyclic in L − γ L − = γ .Eq. (70) is equivalent to Eq. (68), and changes the evalu-ation of phase-space areas into action differences betweencertain auxiliary homoclinic orbits. E. Alternative view: periodic orbit expansion
To this point, homoclinic orbits have been used as thebuilding blocks to generate the full dynamics of arbitrarytrajectories. Due to the intimate relationship betweenhomoclinic and periodic orbits, an alternative approachcan be established by using unstable periodic orbits asthe building blocks. The resulting periodic orbit expan-sion works equivalently well as the homoclinic orbit ex-pansion, therefore putting periodic and homoclinic orbitson an equal footing as a scaffolding for the dynamics ofchaotic systems. The original idea of a periodic-orbit ex-pansion was pioneered by Cvitanovi´c and coauthors intheir studies of dynamical ζ functions in classical andquantum chaos [2, 3, 37], and has been widely know asthe cycle expansion where the term “cycle” stands forperiodic orbits. It has since been generalized into a widerange of systems [38], in particular recent applications tothe state space of turbulent flows [39–41].The scheme begins by specifying a sequence of peri-odic orbits { z ( k )0 } with k = 1 , . . . , L , each identified bysymbolic code z ( k )0 ⇒ β k · β k , (71)where M n k ( z ( k )0 ) = z ( k )0 . The trajectory segment { y m , . . . , y m L − } is divided into short segments of tran-sient visits to the neighborhoods of successive periodicorbits { z ( k )0 } for k = 2 , . . . , L −
1. As illustrated by z ( k ) S(z ) ( k ) U(z ) ( k ) y m k-1 y m k FIG. 9. The trajectory segment { y m k − , . . . , y m k } enters andexits the neighborhood of { z ( k )0 } via a region exponentiallyclose to S ( z ( k )0 ) and U ( z ( k )0 ), respectively. Note that y m k − is O ( e − n k µ ( f ) βk )-close to S ( z ( k )0 ), and y m k is O ( e − n k µ ( f ) βk )-close to U ( z ( k )0 ). Fig. 9, under n k iterations of the map, the trajectory seg-ment { y m k − , . . . , y m k } enters the neighborhood of each { z ( k )0 } via a region exponentially close to its stable man-ifold S ( z ( k )0 ), makes a near fly-by with { z ( k )0 } , and exitsvia a region exponentially close to its unstable manifold U ( z ( k )0 ). The action F ( y m , y m L − ) can be built up fromthe sum of the periodic-orbit actions (cid:80) L − k =2 F β k , plus cor-rection terms J ( β k · β k +1 ) as action connectors between { z ( k )0 } and { z ( k +1)0 } . The main purpose of this subsectionis to derive an explicit expression of J ( β k · β k +1 ).Additional auxiliary homoclinic points g ( i,j )0 are neededfor the process, which are identified by generalizingEq. (50) into arbitrary combinations of β i and β j : g ( i,j )0 ⇒ β i · β j , (72)and in particular, g ( k,k )0 ⇒ β k · β k
0, which will be usedextensively later. Starting from Eq. (27) of Ref. [11] andby replacing γ with β k , one obtains an expression for theaction of each β k : F β k = n k F + ∆ F β k β k , β k + O ( e − n k µ ( f ) βk )= n k F + ∆ F { g ( k,k )0 }{ h ( k )0 } + O ( e − n k µ ( f ) βk ) . (73)With the help of Eq. (28) and replacing a (cid:55)→ g ( k,k )0 c (cid:55)→ h ( k )0 d (cid:55)→ xb (cid:55)→ h ( k ) n k , (74)gives∆ F { g ( k,k )0 }{ h ( k )0 } − ∆ F { h ( k )0 }{ x } = A ◦ SUSU [ g ( k,k )0 ,h ( k )0 ,x,h ( k ) nk ] . (75)Substituting Eq. (75) into Eq. (73) yields F β k = n k F + ∆ F { h ( k )0 }{ x } + A ◦ SUSU [ g ( k,k )0 ,h ( k )0 ,x,h ( k ) nk ] + O ( e − n k µ ( f ) βk )= n k F + ∆ F β k , + A ◦ SUSU [ g ( k,k )0 ,h ( k )0 ,x,h ( k ) nk ] + O ( e − n k µ ( f ) βk ) . (76)Substituting Eq. (76) into Eq. (61) and accounting for3cancellations between common integration paths leads to F ( y m , y m L − ) = (cid:90) S (cid:104) g (1 , ,g (2 , (cid:105) p · dq + L − (cid:88) k =2 F β k + L − (cid:88) k =2 (cid:90) U (cid:104) g ( k,k )0 ,g ( k,k +1)0 (cid:105) p · dq + L − (cid:88) k =2 (cid:90) S (cid:104) g ( k,k +1)0 ,g ( k +1 ,k +1)0 (cid:105) p · dq + (cid:90) U (cid:104) g ( L − ,L − ,g ( L − ,L )0 (cid:105) p · dq + O (cid:18) max k ∈ [1 ,L ] e − n k µ ( f ) βk (cid:19) . (77)To simplify the notations, define the action connectors J S ( β k · β k +1 ), J U ( β k · β k +1 ), and J ( β k · β k +1 ) as J S ( β k · β k +1 ) ≡ (cid:90) S (cid:104) g ( k,k +1)0 ,g ( k +1 ,k +1)0 (cid:105) p · dq J U ( β k · β k +1 ) ≡ (cid:90) U (cid:104) g ( k,k )0 ,g ( k,k +1)0 (cid:105) p · dq J ( β k · β k +1 ) ≡ J U ( β k · β k +1 ) + J S ( β k · β k +1 ) . (78)Since the points g ( i,j )0 are uniquely specified by their sym-bolic codes 0 β i · β j J S ( β k · β k +1 ), J U ( β k · β k +1 ),and J ( β k · β k +1 ) are also uniquely defined by substrings β k and β k +1 . Equation (77) simplifies to the form F ( y m , y m L − )= J S ( β · β ) + L − (cid:88) k =2 F β k + L − (cid:88) k =2 J ( β k · β k +1 ) + J U ( β L − · β L )+ O (cid:18) max k ∈ [1 ,L ] e − n k µ ( f ) βk (cid:19) , (79)which represents the periodic-orbit expansion of F ( y m , y m L − ). The first term on the RHS of Eq. (79), J S ( β · β ), is the action connector between the β and β segments. The second term, (cid:80) L − k =2 F β k , is the sumof the contributions from all periodic orbits β k ( k =2 , . . . , L − (cid:80) L − k =2 J ( β k · β k +1 ), asillustrated in Fig. 10, is the sum of the action connec-tors between β k and β k +1 segments. The fourth term, J U ( β L − · β L ), is the final action connector between β L − and β L . The formula only requires the construction of z ( k ) U(z ) ( k ) z ( k + ) ( k ) S(z ) ( k+1 ) S(z ) ( k+1 ) U(z ) g ( k , k ) U g ( k , k+1 ) g ( k+1 , k+1 ) S f ( k , k+1 ) FIG. 10. The action connector J ( β k · β k +1 ) is indicatedby arrows. f ( k,k +1)0 ∈ U ( z ( k )0 ) ∩ S ( z ( k +1)0 ) is a hete-roclinic point between z ( k )0 and z ( k +1)0 , defined by code f ( k,k +1)0 ⇒ β k · β k +1 . Due to the special choice of sym-bolic codes, z ( k )0 and g ( k,k )0 are O ( e − n k µ ( f ) βk )-close, while f ( k,k +1)0 and g ( k,k +1)0 are O (cid:18) max { e − n k µ ( f ) βk , e − n k +1 µ ( f ) βk +1 } (cid:19) -close. Notice that g ( k,k +1)0 / ∈ U ( z ( k )0 ). The integration paths U (cid:104) g ( k,k )0 , g ( k,k +1)0 (cid:105) and S (cid:104) g ( k,k +1)0 , g ( k +1 ,k +1)0 (cid:105) are arbitrarycurves on U ( x ) and S ( x ), respectively, connecting the cor-responding endpoints. Although not plotted here, the lo-cal U ( x ) is approximately parallel to U ( z ( k )0 ), and the local S ( x ) is approximately parallel to S ( z ( k +1)0 ). Therefore, thepaths U (cid:104) g ( k,k )0 , g ( k,k +1)0 (cid:105) and S (cid:104) g ( k,k +1)0 , g ( k +1 ,k +1)0 (cid:105) are alsoapproximately parallel to U ( z ( k )0 ) and S ( z ( k +1)0 ), respectively. simple periodic orbits β k and homoclinic points g ( k,k )0 and g ( k,k +1)0 , which can be done by stable numerical tech-niques.Similar to Sec. III D, Eq. (79) can be used in the spe-cial case in which { y } is a periodic orbit. For a longperiodic orbit { y } ⇒ γ , where y ⇒ γ · γ , we parti-tion it in the same way as Sec. III D into L − γ = γ · · · γ L − . This indicates during one full periodthe orbit visits the neighborhoods of simpler periodic or-bits γ k successively for k = 1 , . . . , L −
2. Upon usingthe same substitutions as Eq. (65), Eq. (79) yields theperiodic-orbit action expansion F γ = L − (cid:88) k =1 (cid:2) F γ k + J ( γ k · γ k +1 ) (cid:3) + O (cid:18) max k ∈ [1 ,L − e − n k µ ( f ) γk (cid:19) (80)where the index k of γ k is cyclic in L − γ L − ≡ γ .Eq. (80) gives the action expansion of a long periodicorbit γ in terms of short periodic orbits γ k constructedfrom its substrings, and J ( γ k · γ k +1 ) as the action con-nector between γ k and γ k +1 . An interesting fact is thatthe sum of the connectors, L − (cid:88) k =1 J ( γ k · γ k +1 ) , (81)yields the symplectic area of the loop shown by Fig. 11.4 g ( , ) g ( , ) g ( , ) g ( , ) g ( , ) g ( L-1 , L-1 ) g ( L-1 , L ) U SUU SS U
FIG. 11. (Schematic) (cid:80) L − k =1 J ( γ k · γ k +1 ) is the symplectic areaof the loop marked by arrows. Note that g ( i +1 ,j +1)0 ⇒ γ i · γ j g ( L,L )0 = g (2 , . This symplectic area is thus the main action correctionbetween F γ and (cid:80) L − k =1 F γ k .Equations (79) and (80), which are based on periodic-orbit expansions, are equivalent to Eqs. (61) and (68),respectively, which are based on homoclinic-orbit ex-pansions. Therefore, homoclinic and periodic orbits areequally ideal skeletal structures for the phase space dy-namics. F. Numerical examples: approximation accuracy
The accuracy of the procedure, Eqs. (61) and (68), andits dependence on substring length is demonstrated witha numerical example from the H´enon map [Eq. (A1)] atparameter value a = 10. This parameter value gives riseto a complete Smale horseshoe-shaped homoclinic tan-gle [42, 43] (see Appendix. A) with highly chaotic dy-namics as it is well beyond the first tangency [44]. Thegenerating Markov partition is a simple set of two regions[ V , V ]. The trajectory of a non-escaping initial point z is then described by a symbolic string of binary digits,where each digit s n ∈ { , } such that M n ( z ) ∈ V s n .
1. Accuracy expectations
In this two-degree-of-freedom example, all periodic or-bits γ have one positive stability exponent, µ γ > x ⇒ · x (cid:48) ⇒ ·
1. Their stability exponentsare calculated to be µ = 2 . µ = 1 . . (82) There exists a periodic orbit for any combination of zerosand ones as a symbolic string; i.e. no “pruning front”exists in the symbolic plane [4, 29]. Traversing a periodicorbit for a full period, if the current iteration is at digit 0,it indicates that the current point belongs to the V strip(Fig. 12). As a rough estimate, the tangent dynamicsunder one iteration is approximately uniform everywhereinside V . The exponential stretching and compressingrate for the current iteration of the orbit will be closeto µ . Following the same reasoning, tangent dynamicsalong the periodic orbit at digits 1 can be characterizedby µ . Because of this, given any periodic orbit { y } ⇒ γ ,its stability exponent µ γ can be estimated roughly by µ γ ∼ ( N µ + N µ ) /N (83)where N and N are the numbers of 0s and 1s, respec-tively in the string γ , and N = N + N is the length of γ .We emphasize here that Eq. (83) only serves as a practi-cal estimate for the error terms in Eqs. (61) and (68), andis not intended to provide an accurate calculation of thestability exponents of the periodic orbits themselves. Asa demonstration, consider three periodic orbits, namely1011, 0001, and 00011, and calculated their stability ex-ponents: µ = 1 . ≈ ( µ + 3 µ ) / . µ = 1 . ≈ (3 µ + µ ) / . µ = 1 . ≈ (3 µ + 2 µ ) / . . (84)The values roughly agree. The key insight of Eq. (83)is for two different periodic orbits of the same lengths,the one that has more 0s in its symbolic code tends tohave the larger stability exponent. This is simply because µ > µ . Therefore, the magnitude of the estimated errorin using the approximate form to calculate the classicalaction is dominated by the shortest string with the fewest0s in it. The effect on the approximation accuracy ofdifferent length partitions and proportion of 0 symbols isillustrated ahead.
2. Partition length
Consider the exact trajectory { y } = αβδ whose sym-bolic sequence is given by α = 0 β = 011110111011110 δ = 0 . (85)The symbol length of β has 15 characters, which is con-veniently partitioned into 5 pieces as β = 011 β = 110 β = 111 β = 011 β = 110 (86)5The classical action of interest for this partition is the onegiven by 9 iterations of the mapping taking the trajectoryfrom an initial condition y to y , i.e. y m = y ⇒ · y m = y ⇒ · . (87)The exact classical action for this trajectory segmentturns out to be F (exact) ( y , y ) = − . F (approx) ( y , y ) = − . , (89)and thus the absolute error is given by F (exact) ( y , y ) − F (approx) ( y , y ) = 0 . , (90)which is quite accurate relatively speaking with suchshort partition lengths. Nevertheless, in a semiclassicaltheory where (cid:126) divides the actions, small differences canlead to unwanted large phase changes.To increase the partition length used for the approx-imation scheme to test how the accuracy changes, thefirst step is to borrow a character each from the α and δ codes. This increases the length of β to 17 characters(an extra 0 on the left and right, but of course the orbitis still the same). Now consider the partition β = 0011 β = 1101 β = 11011 β = 1100 , (91)mostly of symbol length 4 except β which has length5. For this new partition, the point previously denoted y ⇒ y (cid:48) and y ⇒ y (cid:48) (the index shifted by 1 due to the0 symbol taken from α ). The new approximate actionturns out to be F (approx) ( y (cid:48) , y (cid:48) ) = − . . (92)which gives an absolute error of F (exact) ( y (cid:48) , y (cid:48) ) − F (approx) ( y (cid:48) , y (cid:48) ) = − . , (93)which is nearly two orders of magnitude more accurate.This illustrates the surprising rapidity of exponential con-vergence rates with partition length. Finally, we mention that we have constructed other ex-amples (not given here) that illustrate another feature ofthe accuracy expectations. Note that if one keeps the tra-jectory segment fixed, but increases the mean partitionlength as in the previous example, there are necessarilyfewer partitions. If instead, one allows the trajectory seg-ment to change, but instead one fixes the number of par-titions, one can generally expect the accuracy to increaseexponentially with increasing partition lengths. This isborn out although some variation is expected dependingon the relative proportion of 0 and 1 symbols.
3. Relative proportion of symbols
A final meaningful comparison with the partition inEq. (86) is concerned with the relative proportion of 1and 0 symbols in β . A trajectory segment whose β hasa greater proportion of 0s is expected to have less errorthan a trajectory with a smaller proportion because µ >µ . Swapping several of the 1s for 0s in the example givenby Eq. (86) leads to β = 001 β = 010 β = 100 β = 001 β = 100 (94)which should result in somewhat smaller approximationerrors. The exact classical action for this case is F (exact) ( y , y ) = 59 . F (approx) ( y , y ) = 59 . , (96)giving F (exact) ( y , y ) − F (approx) ( y , y ) = − .
4. Periodic orbits
It is worth giving an example of the application ofEq. (68), which is the equivalent of Eq. (61) for peri-odic orbits. A period-12 orbit { y } ⇒ γ with symboliccode y = y ⇒ γ · γγ = 111111011110 (98)has a classical action given by F (exact) γ = F ( y , y ) = − . . (99)6Consider the partition into 3 length-4 substrings γ = γ γ γ (100)where γ = 1111 γ = 1101 γ = 1110 . (101)The substrings γ i are chosen to be dominantly 1s ratherthan 0s so that the size of the V γ i regions will be rela-tively large. So this example is expected to be a nearlyworst case scenario or a nearly upper bound for the errorterms in Eq. (68) under three length-4 partitions. Theapproximate result is F (approx) γ = − . F (exact) γ − F (approx) γ = − . γ into two length-6 sub-strings: γ = 111111 γ = 011110 , (104)yields F (exact) γ − F (approx) γ = 0 . , (105)which is more than an order of magnitude more accurate. IV. CONCLUSIONS
Special classical trajectory sets play important rolesin both classical and quantum chaotic dynamics throughtheir use in trace formulas. It has long been known thatone of these special sets, i.e. homoclinic orbits, periodicorbits, etc., can be relevant for the calculation of dynam-ical averages, depending on the quantity of interest. Infact, shown here is that all the details of the dynamicsof individual trajectories can be captured by these spe-cial sets. The results apply quite generally and are notrestricted to low-dimensional chaotic dynamics. In par-ticular, exact formulas are given that express the classicalaction of any trajectory segment in terms of simpler ho-moclinic or periodic orbits and certain symplectic areas.Whereas the exact formulas require construction of thetrajectory segments, approximation schemes are givenwith controllable exponentially small errors in which theconstruction is not required, only a section of its sym-bolic sequence corresponding to the segment. This is agreat simplification.The total number of relevant trajectories that areneeded for the semiclassical trace formulas proliferatesexponentially fast with increasing propagation times (or iteration numbers), rendering exponentially demandingcomputation times and storage spaces for standard nu-merical procedures. On the contrary, the relations givenhere make use of a much smaller set of simple homoclinic(periodic) orbits, and provides exact or extremely accu-rate approximate expressions of generic unstable trajec-tory actions. They can be used as a starting point for un-derstanding the action correlations in Hamiltonian chaos,corrections to cycle expansions, or the role of Richter-Sieber [45] pairs in time reversal invariant systems.The main results in this article are expressed in termsof symbolic dynamics. Since each symbolic code corre-sponds to a unique phase-space trajectory, the formu-las derived here will hold true for systems without aknown symbolic dynamics, although more work is neededto identify the one-to-one correspondences between thetrajectory segments and the auxiliary homoclinic orbits,without the help of their symbolic codes.Another fascinating issue is the identification of gen-erating Markov partitions in multidimensional systems.Although the theory for their existence criteria and themechanisms for the creation of symbolic dynamics inhigher dimensions are sophisticated [28, 46], more workwould be desirable on the practical identifications of theMarkov partitions in such systems. However, new meth-ods have been developed in recent years [47, 48], whichprovides promising instruments for finding Markov par-titions in multidimensional systems.
ACKNOWLEDGMENTS
JL acknowledges financial support from Japan Soci-ety for the Promotion of Science (JSPS) in the formof JSPS International Fellowship for Research in Japan(Standard).
Appendix A: HORSESHOE, MARKOVPARTITIONS AND SYMBOLIC DYNAMICS
Symbolic dynamics provides a powerful technique,i.e. the topological description of orbits in chaotic sys-tems [24–27]. Perhaps the most famous model thatdemonstrates its elegance is the horseshoe map [42, 43],a two-dimensional diffeomorphism possessing an invari-ant Cantor set Ω, which is topologically conjugate to aBernoulli shift on symbolic strings composed by “0”s and“1”s. In such scenarios, the Markov partition is a simpleset of two regions [ V , V ], as shown in the upper panel ofFig. 12. Each phase-space point z ∈ Ω can be put intoan one-to-one correspondence with a bi-infinite symbolicstring in Eq. (9), where each digit s n ∈ , M n ( z ) ∈ V s n . A numerical realization of the horse-shoe is the area-preserving H´enon map [49] defined onthe phase plane ( q, p ), which is the simplest polynomial7 x h g g -1 h -1 g -2 a (0) b (0) V V x h g g -1 h g H H a’ (0) b’ (0) V’ H’
FIG. 12. Example partial homoclinic tangle from the H´enonmap, which forms a complete horseshoe structure. The unsta-ble (stable) manifold of x is the solid (dashed) curve. Underforward iteration, the vertical strips V and V (including theboundaries) from the upper panel are mapped into the hori-zontal strips H and H in the lower panel. automorphism giving rise to chaotic dynamics [50]: p n +1 = q n q n +1 = a − q n − p n . (A1)It follows from the work in Ref. [44] that for sufficientlylarge parameter values of a the H´enon map is topolog-ically conjugate to a horseshoe map, therefore possess-ing a hyperbolic invariant set of orbits labeled by binarysymbolic codes; see Chapters 23 and 24 of Ref. [51] for abrief review of the Smale horseshoe and the correspond-ing symbolic dynamics.To visualize the action of the mapping M (e.g.Eq.(A1)) on the homoclinic tangle, let us considerthe closed region R in Fig. 12, bounded by loop L USUS [ x,g − ,h ,g ] , where L USUS [ x,g − ,h ,g ] = U [ x, g − ]+ S [ g − , h ] + U [ h , g ] + S [ g , x ]. Under the mapping M ,the trapezoid-shaped R is compressed along the stabledirection and stretched along the unstable direction, andfolded back to partially overlap with itself, with the ver-tical strips V and V mapped into the horizontal strips H and H , respectively. Similarly, the inverse mapping M − stretches R along the stable direction and fold back,with the horizontal strips H and H mapped into V and V , respectively. Therefore, points in region E bounded by L USUS [ g − ,h − ,h (cid:48)− ,g (cid:48)− ] are mapped outside R into E bounded by L USUS [ g − ,h ,h (cid:48) ,g (cid:48) ] under one iteration. Foropen systems such as the H´enon map, any point outside R never returns and escapes to infinity; there is a similarconstruction for inverse time. Of great structural signif-icance is the non-wandering set Ω of phase-space points z that stay inside R for all iterations [51, 52]:Ω = (cid:8) z : z ∈ ∞ (cid:92) n = −∞ M n ( R ) (cid:9) . (A2)In particular, we focus on the homoclinic and periodicpoints that belong to Ω.Using the closed regions V and V in Fig. 12 as Markovgenerating partition for the symbolic dynamics, everypoint z in Ω can be labeled by an infinite symbolic stringof 0’s and 1’s: z ⇒ · · · s − s − · s s s · · · (A3)where each digit s n in the symbol denotes the region that M n ( z ) lies in: M n ( z ) = z n ∈ V s n , s n ∈ { , } . In thatsense, the symbolic code gives an “itinerary” of z undersuccessive forward and backward iterations, in terms ofthe regions V and V that each iteration lies in. Thesemi-infinite segment “ s s s · · · ” (resp. “ · · · s − s − ”)from the symbolic code is referred to as the head (resp. tail ) of the orbit with initial condition z [30], and thedot separating the head and the tail denotes the region( V s ) that the current iteration z belongs to. Let Σ de-note the symbolic space of all such bi-infinite symbolicstrings. Strings in Σ are then in 1-to-1 correspondencewith points in Ω, and the mapping M in phase space istopological conjugate to a Bernoulli shift in the symbolicspace. Therefore, forward iterations of z move its dottowards the right side of the symbolic string, and back-ward iterations move it towards the left side.Besides elegant topological conjugacy, the symbolicstrings also contain information about the location ofpoints in phase space. Following a standard procedure[28], subsequent Markov partitions [22, 23] can be con-structed from the generating partitions [ V , V ], whichspecifies the phase-space regions that points with certaincentral blocks of fixed lengths must locate within. Start-ing from V and V , define recursively an ever-shrinkingfamily of vertical strips V s ··· s n − in phase space, suchthat: V s ··· s n − ≡ V s (cid:92) M − ( V s ··· s n − ) (A4)where s i ∈ { , } for i = 0 , · · · , n −
1. Similarly, start-ing from H and H , an ever-shrink family of horizontalstrips H s − n ··· s − can be defined: H s − n ··· s − ≡ M ( H s − n ··· s − ) (cid:92) H s − (A5)where s − j ∈ { , } for j = 1 , · · · , n . The horizontalstrips are just forward images of the corresponding ver-8 H H V V H H H H V V V V FIG. 13. Markov partitions constructed in the H´enon map.Upper panel: The V s and H s − regions corresponds to thesame regions in Fig. 12. The four cells H s − ∩ V s ⇒ s − · s are the Markov partitions of lengths 2. Lower panel: Markovpartitions of length 4. The horizontal and vertical stripsare created as H s − s − = M ( H s − ) ∩ H s − and V s s = V s ∩ M − ( V s ). The H and V strips intersect at sixteen cells H s − s − ∩ V s s ⇒ s − s − · s s , as indicated by a black dot in-side each of them. For the sake of clarity, we only explicitly la-beled four cells in the lower left corner. Any point from Ω withsymbolic string of fixed central block · · · s − s − · s s · · · musteither locate inside or on the boundary of the s − s − · s s cell. The sizes of the cells shrink exponentially with increasingstring lengths. tical strips: H s ··· s n − = M n ( V s ··· s n − ). Under n itera-tions of the map, V s ··· s n − is compressed along the sta-ble manifold, at the meantime stretched along the unsta-ble manifold while keeping its total area unchanged, andeventually deformed into H s ··· s n − . Denoting the sym-bolic string s · · · s n − by Greek letter γ : γ = s · · · s n − ,the exponential stretching rate for the entire process canbe estimated using the stability exponent of the periodicorbit γ , namely µ γ , which leads to an estimate for thesize of the areas of V γ and H γ : V γ , H γ ∼ O ( e − n γ µ γ ) (A6)where n γ is the length of γ . Typical periodic orbits γ in chaotic systems will have positive µ γ , and thus thesizes of V γ and H γ shrink exponentially rapidly with thelength of γ .The horizontal and vertical strips intersect at curvy“rectangular” cells, which can be labeled by a finite stringof symbols: H γ (cid:92) V γ ⇒ γ · γ (A7) where γ = s − n · · · s − and γ = s · · · s n − denote thelength- n symbolic strings. These cells are Markov parti-tions of central block lengths 2 n , in the sense that anypoint from Ω with coinciding central blocks γ · γ mustlocate inside (or on the boundary of) the correspondingcell. Shown in the upper and lower panels of Fig. 13are two examples of Markov partitions of lengths 2 and4, respectively, numerically generated from the H´enonmap. Take the cell 10 ·
01 from the lower panel asexample, any point with symbolic string of the form: · · · s − s − · s s · · · must either locate inside or onthe boundary of 10 · H s -n ...s -1 V s ...s n-1 s -n ... s -1 . s ... s n-1 FIG. 14. (Schematic) γ = s − n · · · s − and γ = s · · · s n − .The width of H γ is ∼ O ( e − nµ γ ), and the width of V γ is ∼ O ( e − nµ γ ), so the cell area γ · γ is ∼ O ( e − ( nµ γ + nµ γ ) ). Closeness between two symbolic strings imply close-ness between the corresponding points in phase space.Because of the compressing and stretching nature of thehorseshoe map, the widths of the horizontal and verticalstrips becomes exponentially small with increasing blocklengths, and so do the cell areas they intersect. Withoutloss of generality, we assume, in Fig. 12, that the area A ◦ SUSU [ x,g ,h ,g − ] is of order ∼ O (1). Then the resultingarea of the cell γ · γ is of order ∼ O ( e − ( nµ γ + nµ γ ) ),where µ γ is the stability exponent of the periodic orbit γ , and µ γ is the stability exponent of the periodic or-bit γ . Averaging over all possible combinations of γ and γ , the area of the cell γ · γ can be estimatedas ∼ O ( e − nµ ), where µ is the Lyapunov exponent ofthe system, an exponentially small area for large n val-ues. This geometry [28] is shown by Fig. 14. Therefore,any two points from Ω with identical central blocks oflength 2 n must locate in the same exponentially smallcell. Consider two points h ⇒ · · · s − n · · · s − · s · · · s n · · · and h (cid:48) ⇒ · · · s (cid:48)− n · · · s (cid:48)− · s (cid:48) · · · s (cid:48) n · · · , if h and h (cid:48) agree ona central block of length 2 n , i.e., s (cid:48)− n · · · s (cid:48)− · s (cid:48) · · · s (cid:48) n − = s − n · · · s − · s · · · s n − , they must both located in samecell labeled by s − n · · · s − · s · · · s n − h, h (cid:48) ∈ H s − n ··· s − (cid:92) V s ··· s n − ⇒ s − n · · · s − · s · · · s n − (A8)the area of which is ∼ O ( e − nµ ). Therefore, by specify-ing longer and longer central block lengths of a point’ssymbolic string, we can narrow down its possible loca-9tion in phase space with smaller and smaller cells fromthe Markov partition. Appendix B: MACKAY-MEISS-PERCIVALACTION PRINCIPLE
Aa qp pq a q b q a’ q b’ Mqc b a’ b’c’A’Mp pq qc c’a b a’ b’q a q b q a’ q b’ FIG. 15. (Schematic) a and b are arbitrary points and c is acurve connecting them. a (cid:48) = M ( a ), b (cid:48) = M ( b ) and c (cid:48) = M ( c ).Upper panel: two-dimensional version. A (cid:48) − A = F ( q b , q b (cid:48) ) − F ( q a , q a (cid:48) ). Lower panel: multidimensional version. The MacKay-Meiss-Percival action principle discussedin this section was first developed in [31] for trans-port theory. A comprehensive review can be found in[32]. Generalization of the original principle beyond the“twist” and area-preserving conditions is discussed in[53], and we only give a brief outline of the theory inthis appendix. Shown in Fig. 15 are two arbitrary points a = ( q a , p a ), b = ( q b , p b ) and their images a (cid:48) = M ( a ), b (cid:48) = M ( b ). Let c be an arbitrary curve connecting a and b , which is mapped to a curve c (cid:48) = M ( c ) connecting a (cid:48) and b (cid:48) . Shown in Fig. 15 are the two-dimensional (up-per panel) and multidimensional (lower panel) scenariosof the action principle. For two-dimensional cases, let A and A (cid:48) denote the algebraic area under c and c (cid:48) respec-tively. Then the difference between these areas is A (cid:48) − A = (cid:90) c (cid:48) p d q − (cid:90) c p d q = F ( q b , q b (cid:48) ) − F ( q a , q a (cid:48) ) (B1)i.e., the difference between the two algebraic areas givesthe difference between the action functions for one itera-tion of the map. For 2 f -dimensional phase space we have similarly F ( q b , q b (cid:48) ) − F ( q a , q a (cid:48) )= (cid:90) c (cid:48) [ a (cid:48) ,b (cid:48) ] f (cid:88) j =1 p j d q j − (cid:90) c [ a,b ] f (cid:88) j =1 p j d q j = (cid:90) c (cid:48) [ a (cid:48) ,b (cid:48) ] p · dq − (cid:90) c [ a,b ] p · dq . (B2)Starting from this, MacKay et al . [31] derived a for-mula on the action difference between a pair of homo-clinic orbits, namely { a } and { b } , for which a b A U(x)S(x)U[a ,b ]S[b ,a ]a b FIG. 16. Homoclinic orbit pair a and b . Left panel: two-dimensional phase space. They are connected by an unsta-ble segment U [ a , b ] (solid) and a stable segment S [ b , a ](dashed). Then the action difference between the homoclinicorbit pair is ∆ F { b }{ a } = A . Right panel: 2 f -dimensionalphase space. U ( x ) and S ( x ) are f -dimensional surfaces. U [ a , b ] ⊂ U ( x ) and S [ b , a ] ⊂ S ( x ) are arbitrary paths be-tween a and b . Together they form a loop US [ a , b ] whichgives rise to the symplectic area in Eq. (B6). a ±∞ = b ±∞ = x, (B3)where x is a hyperbolic fixed point. Then as shown byFig. 16, a and b are connected by U ( x ) and S ( x ). Let U [ a , b ] ⊂ U ( x ) and S [ b , a ] ⊂ S ( x ) be arbitrary pathsbetween the two points, we first apply Eq. (B2) repeat-edly to the semi-infinite pair of homoclinic orbit segments { a −∞ , · · · , a } and { b −∞ , · · · , b } , and get:lim N →∞ − (cid:88) i = − N [ F ( b i , b i +1 ) − F ( a i , a i +1 )]= (cid:90) U [ a ,b ] p · dq − (cid:90) U [ a −∞ ,b −∞ ] p · dq = (cid:90) U [ a ,b ] p · dq (B4)where (cid:82) U [ a −∞ ,b −∞ ] p · dq = 0 since a −∞ → b −∞ . Similarlyfor the semi-infinite pairs { a , · · · , a ∞ } and { b , · · · , b ∞ } we have:0lim N →∞ N − (cid:88) i =0 [ F ( b i , b i +1 ) − F ( a i , a i +1 )]= (cid:90) S [ a ∞ ,b ∞ ] p · dq − (cid:90) S [ a ,b ] p · dq = (cid:90) S [ b ,a ] p · dq . (B5)Adding up Eqs. (B4) and (B5) we have: ∆ F { b }{ a } = lim N →∞ N − (cid:88) i = − N [ F ( b i , b i +1 ) − F ( a i , a i +1 )]= (cid:90) U [ a ,b ] p · dq + (cid:90) S [ b ,a ] p · dq = A ◦ US [ a ,b ] . (B6)For two-dimensional systems, A ◦ US [ a ,b ] reduces to thearea A shown in the left panel of Fig. 16. For systemswith 2 f -dimensional phase space ( f ≥ A ◦ US [ a ,b ] isthe symplectic area of the loop shown in the right panelof Fig. 16. [1] H. Poincar´e, Les m´ethodes nouvelles de la m´ecaniquec´eleste , Vol. 3 (Gauthier-Villars et fils, Paris, 1899).[2] R. Artuso, E. Aurell, and P. Cvitanovi´c, Nonlinearity ,325 (1990).[3] R. Artuso, E. Aurell, and P. Cvitanovi´c, Nonlinearity ,361 (1990).[4] P. Cvitanovi´c, Physica D , 138 (1991).[5] M. L. Du and J. B. Delos, Phys. Rev. A , 1896 (1988).[6] M. L. Du and J. B. Delos, Phys. Rev. A , 1913 (1988).[7] H. Friedrich and D. Wintgen, Phys. Rep. , 37 (1989).[8] S. Tomsovic and E. J. Heller, Phys. Rev. Lett. , 664(1991).[9] S. Tomsovic and E. J. Heller, Phys. Rev. E , 282(1993).[10] J. Li and S. Tomsovic, Phys. Rev. E , 062224 (2017),arXiv:1703.07045 [nlin.CD].[11] J. Li and S. Tomsovic, Phys. Rev. E , 022216 (2018),arXiv:1712.05568 [nlin.CD].[12] J. Li and S. Tomsovic, Phys. Rev. E , 052202 (2019),arXiv:1909.00544 [nlin.CD].[13] V. Rom-Kedar, Physica D , 229 (1990).[14] J. Li and S. Tomsovic, J. Phys. A: Math. Theor. ,135101 (2017), arXiv:1507.06455 [nlin.CD].[15] M. C. Gutzwiller, J. Math. Phys. , 343 (1971), andreferences therein.[16] P. Cvitanovi´c, R. Artuso, P. Dahlqvist, R. Mainieri,G. Tanner, G. Vattay, N. Whelan, and A. Wirzba, chaos-book.org , 1.[17] V. I. Oseledec, Trudy Moskov. Mat. Obˇsˇc. , 197 (1968).[18] F. Ginelli, P. Poggi, A. Turchi, H. Chat´e, R. Livi, andA. Politi, Phys. Rev. Lett. , 130601 (2007).[19] F. Ginelli, H. Chat´e, R. Livi, and A. Politi, J. Phys. A:Math. Theor. , 254005 (2013).[20] P. V. Kuptsov and U. Parlitz, J. Nonlinear. Sci. , 727(2012).[21] R. W. Easton, Trans. Am. Math. Soc. , 719 (1986).[22] R. Bowen, Lect. Notes in Math. Vol. 470. (Springer-Verlag, Berlin, 1975).[23] P. Gaspard,
Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, UK, 1998).[24] J. Hadamard, J. Math. Pures Appl. series 5 , 27 (1898). [25] G. D. Birkhoff, A.M.S. Coll. Publications, vol. 9 (Amer-ican Mathematical Society, Providence, 1927).[26] G. D. Birkhoff, Mem. Pont. Acad. Sci. Novi Lyncaei ,85 (1935).[27] M. Morse and G. A. Hedlund, Amer. J. Math. , 815(1938).[28] S. Wiggins, Global Bifurcations and Chaos (Springer-Verlag, New York, Berlin, Heidelberg, 1988).[29] P. Cvitanovi´c, G. Gunaratne, and I. Procaccia,Phys. Rev. A , 1503 (1988).[30] R. Hagiwara and A. Shudo, J. Phys. A: Math. Gen. ,1052110543 (2004).[31] R. S. MacKay, J. D. Meiss, and I. C. Percival, Physica D , 55 (1984).[32] J. D. Meiss, Rev. Mod. Phys. , 795 (1992).[33] A. M. O. de Almeida and M. Saraceno, Ann. Phys. ,1 (1991).[34] P. W. O’Connor and S. Tomsovic, Ann. Phys. (N.Y.) , 218 (1991).[35] P. W. O’Connor, S. Tomsovic, and E. J. Heller, PhysicaD , 340 (1992).[36] M. C. Gutzwiller, Chaos in Classical and Quantum Me-chanics (Springer-Verlag, New York, 1990).[37] P. Cvitanovi´c, Phys. Rev. Lett. , 2729 (1988).[38] Y. Lan, Commun. Nonlinear. Sci. Numer. Simulat ,502 (2010).[39] B. Suri, J. Tithof, R. O. Grigoriev, and M. F. Schatz,Phys. Rev. Lett. , 114501 (2017).[40] B. Suri, L. Kageorge, R. O. Grigoriev, and M. F. Schatz,Phys. Rev. Lett. , 064501 (2020).[41] G. Yalnız, B. Hof, and N. B. Budanur, arXiv:2007.02584[physics.flu-dyn] (2020).[42] S. Smale, Differential and Combinatorial Topology , editedby S. S. Cairns (Princeton University Press, Princeton,1963).[43] S. Smale,
The Mathematics of Time: Essays on Dy-namical Systems, Economic Processes and Related Topics (Springer-Verlag, New York, Heidelberg, Berlin, 1980).[44] R. Devaney and Z. Nitecki, Comm. Math. Phys. , 137(1979).[45] M. Sieber and K. Richter, Physica Scripta T90 , 128 (2001).[46] A. Katok and B. Hasselblatt, Introduction to the Mod-ern Theory of Dynamical Systems (Cambridge UniversityPress, Cambridge, 1995).[47] N. Rubido, C. Grebogi, and M. S. Baptista, Chaos ,033611 (2018).[48] C. Zhang and Y. Lan, arXiv:2007.11236 [nlin.CD](2020).[49] M. H´enon, Comm. Math. Phys. , 69 (1976).[50] S. Friedland and J. Milnor, Ergod. Th. & Dynam. Sys. , 67 (1989).[51] S. Wiggins, Introduction to Applied Nonlinear Dynami-cal Systems and Chaos, Second Edition (Springer-Verlag,New York, Berlin, Heidelberg, 2003).[52] P. Cvitanovi´c, R. Artuso, R. Mainieri, G. Tanner, andG. Vattay,
Chaos: Classical and Quantum (Niels BohrInst., Copenhagen, 2016).[53] R. Easton, Nonlinearity4