An asymptotic relationship between homoclinic points and periodic orbit stability exponents
AAn asymptotic relationship between homoclinic points and periodic orbit stabilityexponents
Jizhou Li and Steven Tomsovic Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA (Dated: September 4, 2019)The magnitudes of the terms in periodic orbit semiclassical trace formulas are determined by theorbits’ stability exponents. In this paper, we demonstrate a simple asymptotic relationship betweenthose stability exponents and the phase-space positions of particular homoclinic points.
I. INTRODUCTION
A variety of properties of chaotic quantum systems canbe calculated with semiclassical trace formulas, which aresums over certain sets of classical orbits (periodic, het-eroclinic, or closed orbits, etc.) arising in their classicalcounterparts. For example, Gutzwiller’s trace formula [1]is a sum over periodic orbits that determines the spec-trum, and the near-threshold absorption spectra of anatom in a magnetic field involve a sum over closed orbitsthat begin and end at the nucleus [2, 3]. Such orbit sumsproperly account for quantum interferences as each orbitcarries a magnitude determined by its stability exponent,and a phase factor determined by its classical action andMaslov index. The orbits with shorter periods give long-range structure to the quantum spectra, and the longerperiod orbits the finer scale structure. Due to exponen-tial proliferation and instability, explicit construction ofa complete set of orbits with longer and longer periodsrapidly becomes prohibitive.In two previous publications [4, 5], we developed ananalytic scheme to express the classical actions of unsta-ble periodic orbits in terms of action differences betweencertain homoclinic orbits. The homoclinic orbit actiondifferences can then be obtained as phase-space integralsalong the stable and unstable manifolds, which can becalculated stably by efficient numerical techniques [6–9].Thus, the phase factors can be obtained via stable com-putations without the explicit construction of the orbits.Here, we address the magnitudes, i.e. the stability expo-nents of the periodic orbits. A new relationship is devel-oped to link the stability exponents of unstable periodicorbits to the phase-space positions of specific homoclinicpoints. The exponent is determined by the ratio betweenrelative positions from an asymptotic family of homo-clinic points. Thus, the periodic orbit magnitudes canalso be determined without explitic construction. Thisimplies a unified scheme of interchanging the periodicorbits with homoclinic orbits, which may be very benefi-icial depending on the circumstances.The paper is organized as follows: Sec. II introducesthe basic concepts of hyperbolic orbits and the main lan-guage for the description of unstable orbits—symbolicdynamics. A generic model for the symbolic dynamics,the Smale horseshoe [10, 11], is also introduced in thissection. Sec. III is the main content of this work that de-velops the central theorem. Sec. IV provides numerical verification. Sec. V makes a brief conclusion and pointsto directions for future work.
II. BASIC CONCEPTSA. Symbolic dynamics
Let us consider a two-degree-of-freedom chaotic Hamil-tonian system. With energy conservation and applyingthe Poincar´e surface of section technique [12], the Hamil-tonian flow is reduced to a discrete area-preserving map M on the two-dimensional phase space ( q, p ). Assumingthe dynamics of the Hamiltonian systems is hyperbolic,the corresponding Poincar´e map M is also hyperbolic.The orbit of a phase space point z , denoted by { z } , isthe bi-infinite collection of all M n ( z ): { z } = {· · · , M − ( z ) , z , M ( z ) , · · · } = {· · · , z − , z , z , · · · } where z n = M n ( z ) for all n . Generic orbits are hyper-bolic, or exponentially unstable, as two orbits startingfrom nearby initial conditions will typically be separatedexponentially under successive iterations. The exponen-tial rate of a typical orbit is captured by the Lyapunovexponent, µ , which quantifies the mean stretching andcompressing rate of the hyperbolic map. In open sys-tems, such stretching and compressing behaviors of thedynamics leads to certain escaping orbits that tend toinfinity under successive inverse or forward iterations.However, we concentrate on the orbits that do not es-cape to infinity, the nonwandering set (the results applyequally well in closed systems). Denote the set of interestby Ω. The main object of study in this article, namelythe homoclinic and periodic orbits, all belong to Ω.Let x = ( q, p ) be a hyperbolic fixed point from Ω, 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 )intersect infinitely many times and form a complicatedpattern called a homoclinic tangle [12–14]. The notation U [ a, b ] is introduced to denote the finite segment of U ( x )extending from a to b , both of which are points on U ( x ),and similarly for S ( x ). These manifolds are importantskeleton-like structures of the dynamics since the expo-nential stretching and compressing of the map are fully a r X i v : . [ n li n . C D ] S e p captured by the unstable and stable manifolds, respec-tively. Furthermore, the folding of phase-space regionsare also described by the folding of the manifolds.It is well-known that Markov partitions to the phasespace [15, 16] exist that use segments on U ( x ) and S ( x )as boundaries, which are used to assign symbolic dy-namics [17–20] as phase-space itineraries of orbits inΩ. The cells of the partition V = [ V , V , · · · , V L ] areclosed curvilinear parallelograms bounded by the stableand unstable manifolds. The symbolic dynamics assignsa one-to-one correspondence between orbits of the sys-tem and sequences of symbols taken from an alphabet, s i ∈ [0 , , · · · , L ], which are in one-to-one correspondencewith the cells [ V , V , · · · , V L ] [16]. The symbolic code ofa phase-space point z is then a bi-infinite sequence ofalphabets z ⇒ · · · s − s − · s s s · · · (1)where each digit s n in the symbol denotes the cell that M n ( z ) belongs to: M n ( z ) = z n ∈ V s n , s n ∈ { , · · · , L } .The dot in the middle indicates the current iteration: z ∈ V s . In that sense, the symbolic code gives an“itinerary” of z under successive forward and backwarditerations, in terms of the Markov cells in which each iter-ation lies. The mapping M under the symbolic dynamicsis then reduced to a simple shift of the dot in the code: M n ( z ) = z n ⇒ · · · s n − · s n s n +1 · · · . Points along the same orbit have the same symbolicstrings but shifting dots. Therefore, an orbit can be rep-resented by the symbolic string without the dot.
B. The Horseshoe map
Assuming the system is highly chaotic, so the homo-clinic tangle forms a complete horseshoe, part of which isshown in Fig. 1, as this is generic to a significant class ofdynamical systems. In such scenarios, the Markov par-tition is a simple set of two regions [ V , V ], as shownin the upper panel of Fig. 1. Each phase-space point z that never escapes to infinity can be put into anone-to-one correspondence with a bi-infinite symbolicstring in Eq. (1), where each digit s n ∈ , M n ( z ) ∈ V s n .Throughout this paper we use the area-preservingH´enon map [21] with parameter a = 10 for illustrationand numerical implementations: p n +1 = q n q n +1 = a − q n − p n . (2)This parameter is well beyond the first tangency, thusgiving rise to a complete horseshoe-shaped homoclinictangle with highly chaotic dynamics. It serves as a simpleparadigm since the symbolic dynamics permits all possi-ble combinations of binary codes, no “pruning” [22, 23] is x h g g -1 h -1 g -2 V V x h g g -1 h g H H V’ H’
FIG. 1. 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. Thereare two primary homoclinic orbits { h } and { g } . Let R bethe closed region bounded by manifold segments U [ x, g − ], S [ g − , h ], U [ h , g ], and S [ g , x ]. In the upper panel, R canbe identified as the region composed by V , V (cid:48) , and V . Un-der forward iteration, the vertical strips V and V (includingthe boundaries) from the upper panel are mapped into thehorizontal strips H and H in the lower panel. At the sametime, points in region V (cid:48) are mapped outside R into region H (cid:48) , never to return and escape to infinity. There is a Cantorset of points in V and V that remain inside R for all iter-ations, which is the non-wandering set Ω. The phase-spaceitineraries of points in Ω in terms of V and V give rise tosymbolic dynamics. needed. The results derived ahead mostly carry over intomore complicated systems possessing incomplete horse-shoes, or systems with more than binary symbolic codes,though more work is needed to address such systems.Appendix A of [5] has more details on the partition andsymbolic dynamics relevant here.The intersections between S ( x ) and U ( x ) give rise tohomoclinic orbits, which are asymptotic to x under both M ±∞ . From the infinite families of homoclinic orbits,two special ones { h } and { g } can be identified as pri-mary homoclinic orbits, in the sense that they have thesimplest phase space excursions. The segments S [ x, h ]and U [ x, h ] intersect only at h and x , the same is truefor all its orbit points h i ; this holds for { g } as well.There are only two primary orbits for the horseshoe, butpossibly more for systems with more complicated homo-clinic tangles.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 (3)where γ = s · · · s T − is the finite substring and γ.γ de-notes 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 (4)with the dot removed. Similarly, the finite length- T or-bit segment [ y , y , · · · , y T − ], which composes one fullperiod, is denoted y , y , · · · , y T − ⇒ γ (5)with the overhead bar removed, as compared to Eq. (4).Any cyclic permutation of γ refers to the same periodicorbit.The hyperbolic fixed point has the simplest symboliccode x ⇒ · { x } ⇒ h of x has symbolic code of theform [24]: h ⇒ s − m · · · s − · s s · · · s n
10 (6)along with all possible shifts of the dot, where the 0 onboth ends means the orbit approaches the fixed point(therefore stays in V ) under both M ±∞ . Similar to theperiodic orbit case, the homoclinic orbit can be repre-sented as { h } ⇒ s − m · · · s − s s · · · s n
10 (7)with the dot removed, as compared to Eq. (6).A heteroclinic orbit { h (cid:48) } between the periodic point y ⇒ γ · γ and the fixed-point x ⇒ · h (cid:48) = U ( y ) ∩ S ( x ), and can be represented by { h (cid:48) } ⇒ γγ (cid:48) γ and 0 on the two ends, and the finite symbolic string γ (cid:48) describes the connection from { y } to { x } , which solelydepends on the choice of h (cid:48) . III. PERIODIC ORBIT STABILITY EXPONENT
Consider an arbitrary unstable periodic orbit { y } withsymbolic code { y } ⇒ γ . Let the length of the symbolic string γ be n γ , which is also the periodic of { y } . The pe-riodic point y can also be viewed as a fixed-point underthe n γ -th compound mapping of M : M n γ ( y ) = y . De-note the eigenvalue of the unstable subspace of the tan-gent space of { y } under one full period ( n γ ) by λ γ . Thus λ γ > { y } is hyperbolic without reflection ( λ γ < − { y } , denotedby µ γ , is then n γ µ γ = ln | λ γ | .To help determine λ γ and thus µ γ , choose a family ofauxiliary homoclinic points of the fixed-point x , namely h ( m )0 ( m = 1 , , · · · ), that has the symbolic codes h ( m )0 ⇒ γ m · γ m denotes m repetitions of γ and m = 1 , , · · · .Having identified the auxiliary homoclinic points, let usconsider the homoclinic orbit segments generated by cer-tain numbers of inverse iterations of them, namelySeg(k,m) = { h ( m ) − N ( k,m ) , · · · , h ( m ) − , h ( m )0 } (10)where N ( k, m ) = ( k + m ) n γ is a positive integer deter-mined by k and m ( k, m ≥ k is taken to ∞ ,which yields the limitlim k →∞ h ( m ) − N ( k,m ) = x . (11)The key to the derivation lies in the normal-form trans-formation [25–27] of three orbit segments. For well- D d S ( y ) P Qy U ( y ) C c Y FIG. 2. Schematic visualization of the normal form transfor-mation N . It transforms points from the normal form coor-dinate ( Q, P ) into the phase space coordinate ( q, p ). The Q and P axis are mapped into U ( y ) and S ( y ), respectively. Theadvantage of normal form coordinates is that the dynamicspreserves the QP product [Eq. (12)], thus points are mappedalong invariant hyperbolas, as shown by C in the right panel.The family of invariant hyperbolas then give rise to a family ofMoser invariant curves in phase space via the transformation N ( C ) = c , shown in the left panel. behaved (invertible and analytic) Poincar´e maps, thenonlinear dynamics near the stable and unstable man-ifolds can be linearized via a common technique called normal-form transformation , denoted by N , which trans-forms points from the normal form coordinates ( Q, P ) tothe neighborhood of stable and unstable manifolds of thehyperbolic fixed point y : N : ( Q, P ) (cid:55)→ ( q, p ), as shownin Fig. 2. In the normal form coordinates of y , the com-pound mapping M n γ takes a simple form: Q n +1 = Λ( Q n P n ) · Q n P n +1 = [Λ( Q n P n )] − · P n (12)where Λ( Q n P n ) is a polynomial function of the product Q n P n [28]:Λ( QP ) = λ γ + w · ( QP ) + w · ( QP ) + · · · (13)The normal form convergence zone was first proved byMoser [26] to be a small disk-shaped region centered atthe fixed point ( D and its image d in Fig. 2), and laterproved by da Silva Ritter et . al . [27] to extend alongthe stable and unstable manifolds to infinity. The ex-tended convergence zone follows hyperbolas to the man-ifolds (“gets exponentially close” the further out alongthe manifolds). The stable and unstable manifolds arejust images of the P and Q axes respectively under thenormal form transformation.All points inside the extended convergence zone nearthe Q or P axis move along invariant hyperbolas, whichare mapped to Moser invariant curves in phase space. Aschematic example is shown in Fig. 2, where the hyper-bola C in the normal form coordinates is transformed intoa Moser curve c in phase space. Being confined in the ex-tended convergence zone, the Moser invariant curves alsoget exponentially close to the stable and unstable mani-folds while extending along them outward to infinity. Infact, as shown by [28], the convergence zone can be quan-tified using the outermost Moser curve with the largest QP product. S(x)H (m) P QY U(x)H H C m C m+1 C m+2 H -N(k,m) (m)(m+1)(m+2)(m+2) H -N(k,m+2) (m+1) H -N(k,m+1) XH’
FIG. 3. (Schematic) Normal-form coordinate picture of theauxiliary homoclinic orbit segments. Y is the image of y , andthe P , Q axis are the images of S ( y ) and U ( y ), respectively, inthe normal-form coordinate. X is the image of the fixed-point x . Three auxiliary homoclinic orbit segments, correspondingto m , m + 1, and m + 2 in Eq. (14), lie on the hyperbolas C m , C m +1 , and C m +2 , respectively. Note that only the first andthe last points of each orbit segment are drawn here. Let the image of Seg( k, m ) in the normal-form coordi-nate of y beSeg(k,m) = { H ( m ) − N ( k,m ) , · · · , H ( m ) − , H ( m )0 } (14)where N ( H ( m ) n ) = h ( m ) n . In the normal-form coordinates,every Seg( k, m ) lies on a hyperbola, labeled by C m inFig. 3. This figure shows Seg( k, m ), Seg( k, m + 1), andSeg( k, m + 2), in the normal-form coordinate of the pe-riodic point y . Letting k → ∞ , because of Eq. (11), theinitial points of the three segments, namely H ( m ) − N ( k,m ) , H ( m +1) − N ( k,m +1) , and H ( m +2) − N ( k,m +2) , are all located infinitesi-mally close to X along U ( x ):lim k →∞ H ( m ) − N ( k,m ) = lim k →∞ H ( m +1) − N ( k,m +1) = lim k →∞ H ( m +2) − N ( k,m +2) = X . (15)Then, under N ( k, m ), N ( k, m + 1), and N ( k, m + 2) it-erations, respectively, they are mapped to the final points H ( m )0 , H ( m +1)0 , and H ( m +2)0 , as shown near the hetero-clinic point H (cid:48) on the Q axis. This heteroclinic point hassymbolic code N ( H (cid:48) ) = h (cid:48) ⇒ γ · { y } and { x } . Recall that N ( H ( m )0 ) = h ( m )0 ⇒ γ m · . (17)Comparing the symbolic strings in Eqs. (16) and (17), itfollows that the codes of h (cid:48) and h ( m )0 to the right of thedot are identical, and to the left of the dot they matchup to γ m (which has length mn γ ). This indicates that h ( m )0 is ∼ O ( e − mn γ µ γ ) close to h (cid:48) along S ( x ) (for moredetails, see Appendix A of Ref. [5]). Due to the samereason, h ( m +1)0 and h ( m +2)0 are ∼ O ( e − ( m +1) n γ µ γ ) and ∼ O ( e − ( m +2) n γ µ γ ) close to h (cid:48) , respectively, along S ( x ).Therefore, the above four points, h (cid:48) , h ( m )0 , h ( m +1)0 , and h ( m +2)0 , are all within exponentially small neighborhoodsof each other.The same conclusion holds true in the normal-formcoordinates. In fact, as shown by Fig. 3, in the nor-mal coordinate of y , the proportionality factors of thedistances between them can be determined analytically.Plotted in the figure are the initial and final points ofSeg( k, m ), Seg( k, m +1), and Seg( k, m +2). The k here isassumed to be a large integer, so H ( m ) − N ( k,m ) , H ( m +1) − N ( k,m +1) ,and H ( m +2) − N ( k,m +2) are exponentially close to X . Under suc-cessive forward iterations, they are mapped along the hy-perbolas C m , C m +1 , and C m +2 , respectively, into H ( m )0 , H ( m +1)0 , and H ( m +2)0 : M N ( k,m + i ) ( H ( m + i ) − N ( k,m + i ) ) = H ( m + i )0 (18)where i = 0 , ,
2. The mapping equations take the simpleform of Eq. (12), with the stability factor Λ given byΛ( QP ) = λ γ + w · ( QP ) + w · ( QP ) + · · · . (19)Under the limit m → ∞ , H ( m + i )0 → H (cid:48) ( i = 0 , , QP products along the hyperbolas →
0. Cor-respondingly, the stability factor Λ( QP ) → λ γ , and the C m curve becomes infinitely close to the P and Q axiswhen m → ∞ .Consequently, the P coordinate values of H ( m + i )0 ,namely P ( H ( m + i )0 ), are determined asymptotically bylim m →∞ P ( H ( m + i )0 ) = lim k →∞ m →∞ P ( X ) · λ − ( k + m + i ) γ (20)for i = 0 , , , · · · , where P ( X ) denotes the P -coordinatevalue of X . Furthermore, notice that N ( k, m + j ) − N ( k, m ) = j · n γ , thus using Eq. (20) we getlim m →∞ P ( H ( m + j )0 ) = lim m →∞ P ( H ( m )0 ) · λ − jγ (21)for j = 0 , , , · · · . Therefore, the family of homoclinicpoints, H ( m + j )0 ( j = 0 , , , · · · ), converges to H (cid:48) underconvergence factor λ − γ . Since the normal-form transfor-mation preserves the convergence factor of asymptoticseries of points (see Appendix B.2 of Ref. [29] for a de-tailed proof), the family h ( m + j )0 ( j = 0 , , , · · · ) also con-verge to h (cid:48) in phase space. Therefore, their phase-spacepositions satisfylim m →∞ p ( h ( m )0 ) − p ( h (cid:48) ) p ( h ( m + j )0 ) − p ( h (cid:48) ) = λ jγ lim m →∞ q ( h ( m )0 ) − q ( h (cid:48) ) q ( h ( m + j )0 ) − q ( h (cid:48) ) = λ jγ (22)where p ( a ) and q ( a ) denotes the p - and q - coordinatevalues, respectively, of the point a . Here it is assumedthe generic case that the local direction of S ( x ) at h (cid:48) is not strictly vertical or horizontal, so the differencesbetween the p and q values of successive members do notvanish. The distances between successive members of thefamily are also in scale:lim m →∞ p ( h ( m )0 ) − p ( h ( m +1)0 ) p ( h ( m +1)0 ) − p ( h ( m +2)0 ) = λ γ (23)and the same is true for the q coordinate values as well.Therefore, the stability exponent n γ µ γ = ln | λ γ | of theperiodic orbit { y } ⇒ γ can be determined using Eq. (23)from the family of auxiliary homoclinic points h ( m )0 ⇒ γ m ·
0, which does not require the numerical constructionof the periodic orbit. In practice, for long periodic orbitswith large periods ( n γ ), the leading terms in the h ( m )0 family should provide an accurately enough calculationof λ γ : λ γ ≈ p ( h (1)0 ) − p ( h (2)0 ) p ( h (2)0 ) − p ( h (3)0 ) . (24) IV. EXPLICIT EXAMPLE
To verify Eq. (24), we have numerically constructedfour different periodic orbits in the H´enon map (Eq. (2)),namely { v } , { y } , { w } , and { z } , with symbolic codes { v } ⇒ { y } ⇒ { w } ⇒ { z } ⇒ . (25)The phase-space positions of one of their orbit points are v = (3 . , . y = ( − . , . w = ( − . , − . z = ( − . , . λ γ and exponents µ γ havebeen calculated. In addition, by constructing the respec-tive auxiliary homoclinic points in Eq. (9) for each orbit,the same stability eigenvalues λ (cid:48) γ and exponents µ (cid:48) γ havebeen approximated with Eq. (24). The results are listedin Table I. Although only using the leading terms in eachauxiliary homoclinic family, the agreement is excellent. γ λ γ λ (cid:48) γ n γ µ γ n γ µ (cid:48) γ − . − .
741 6 . . .
00 1602 .
20 7 . . − . − .
72 7 . . . . . . λ γ are calcu-lated from the numerical orbits, and λ (cid:48) γ are determined fromEq. (24). The exponents are obtained as n γ µ γ = ln | λ γ | and n γ µ (cid:48) γ = ln | λ (cid:48) γ | . V. CONCLUSION
An exact formula (Eq. (23)) is introduced that linksthe stability properties of unstable periodic orbits to thephase space locations of certain homoclinic points. Al-though the formula is asymptotic in nature, the numer-ical results from using the leading term already repro-duces the actual exponents quite accurately in the nu-merical model used. Since the numerical computationof long periodic orbits suffers from an exponential in-stability problem, whereas the positions of homoclinicpoints can be determined relatively easily as the intersec-tions between the invariant manifolds [6], this approachmay provide an efficient alternative to direct calculations.Furthermore, in previous work [4, 5], the classical ac-tions of periodic orbits are expressed in terms of certainhomoclinic orbit action differences. Combined with thecurrent results, they provide a unified scheme of replac-ing the periodic orbits in the trace formula by homoclinicorbits, which may lead to new resummation techniquesin semiclassical methods.An important generalization of the current theorywould be to extend it to higher-dimensional symplec-tic maps with chaotic dynamics. For instance, in 4-dimensional maps, the stable and unstable manifolds of hyperbolic fixed points will each be 2-dimensional sur-faces. They intersect in the 4D phase-space generatinghomoclinic points. It may be the case that the relativepositions of certain homoclinic points distributed alongthe dominant contraction direction in the stable mani-folds yield the dominant stability exponent, and the rel-ative positions of the homoclinic points along the sub-dominant contraction direction of the stable manifoldsyield the sub-dominant stability exponent. However, thegeneralization of the symbolic code description to higher-dimensions is a challenging issue. [1] M. C. Gutzwiller, J. Math. Phys. , 343 (1971), andreferences therein.[2] M. L. Du and J. B. Delos, Phys. Rev. A , 1896 (1988).[3] M. L. Du and J. B. Delos, Phys. Rev. A , 1913 (1988).[4] J. Li and S. Tomsovic, Phys. Rev. E , 062224 (2017),arXiv:1703.07045 [nlin.CD].[5] J. Li and S. Tomsovic, Phys. Rev. E , 022216 (2018),arXiv:1712.05568 [nlin.CD].[6] J. Li and S. Tomsovic, J. Phys. A: Math. Theor. ,135101 (2017), arXiv:1507.06455 [nlin.CD].[7] B. Krauskopf and H. M. Osinga, J. Comput. Phys. ,404 (1998).[8] A. M. Mancho, D. Small, S. Wiggins, and K. Ide, Physica D 182 , 188 (2003).[9] B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Hender-son, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, andO. Junge, Int. J. Bifurcation and Chaos , 763 (2005).[10] S. Smale, Differential and Combinatorial Topology , editedby S. S. Cairns (Princeton University Press, Princeton,1963).[11] S. Smale,
The Mathematics of Time: Essays on Dy-namical Systems, Economic Processes and Related Topics (Springer-Verlag, New York, Heidelberg, Berlin, 1980).[12] H. Poincar´e,
Les m´ethodes nouvelles de la m´ecaniquec´eleste , Vol. 3 (Gauthier-Villars et fils, Paris, 1899).[13] R. W. Easton, Trans. Am. Math. Soc. , 719 (1986). [14] V. Rom-Kedar, Physica D , 229 (1990).[15] R. Bowen, Lect. Notes in Math. Vol. 470. (Springer-Verlag, Berlin, 1975).[16] P. Gaspard,
Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, UK, 1998).[17] J. Hadamard, J. Math. Pures Appl. series 5 , 27 (1898).[18] G. D. Birkhoff, A.M.S. Coll. Publications, vol. 9 (Amer-ican Mathematical Society, Providence, 1927).[19] G. D. Birkhoff, Mem. Pont. Acad. Sci. Novi Lyncaei ,85 (1935).[20] M. Morse and G. A. Hedlund, Amer. J. Math. , 815(1938).[21] M. H´enon, Comm. Math. Phys. , 69 (1976).[22] P. Cvitanovi´c, G. Gunaratne, and I. Procaccia,Phys. Rev. A , 1503 (1988).[23] P. Cvitanovi´c, Physica D , 138 (1991).[24] R. Hagiwara and A. Shudo, J. Phys. A: Math. Gen. ,1052110543 (2004).[25] G. D. Birkhoff, Acta Math. , 359 (1927).[26] J. Moser, Commun. Pure Appl. Math. , 673 (1956).[27] G. L. da Silva Ritter, A. M. Ozorio de Almeida, andR. Douady, Physica D , 181 (1987).[28] M. Harsoula, G. Contopoulos, and C. Efthymiopou-los, J. Phys. A: Math. Theor. , 135102 (2015),arXiv:1502.00664 [nlin.CD].[29] K. A. Mitchell, J. P. Handley, B. Tighe, J. B. Delos, andS. K. Knudson, Chaos13