Reversible maps in two-degrees of freedom Hamiltonian systemsK. Zare and K. Tanikawa Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 12, 699 (2002); doi: 10.1063/1.1499595 View online: http://dx.doi.org/10.1063/1.1499595 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/12/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertialcoupling Chaos 22, 013118 (2012); 10.1063/1.3683480 Degenerate resonances in Hamiltonian systems with 3/2 degrees of freedom Chaos 12, 539 (2002); 10.1063/1.1484275 Bifurcation analysis of laser systems AIP Conf. Proc. 548, 1 (2000); 10.1063/1.1337756 Observation of structure in the Lorenz map Chaos 9, 206 (1999); 10.1063/1.166391 Symmetries of Hamiltonian systems with two degrees of freedom J. Math. Phys. 40, 210 (1999); 10.1063/1.532769

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Reversible maps in two-degrees of freedom Hamiltonian systemsK. Zare and K. TanikawaNational Astronomical Observatory, Mitaka 181-8588, Japan

~Received 30 November 2000; accepted 20 June 2002; published 22 August 2002!

It has been shown that a sub-class of two-degrees of freedom Hamiltonian systems possesses areversing symmetry discovered by Birkhoff in the restricted problem of three bodies. This mixedspace–time reversing symmetry, which is different from the classical time reversal symmetry, canbe shared by time-reversible as well as time-irreversible systems. Examples of time-irreversiblesystems which possess this reversing symmetry are the restricted problem of three bodies as shownby Birkhoff in 1915, and a special case of the motion of a rigid body with a fixed point discussedin this paper. If a Hamiltonian system possesses this Birkhoff reversing symmetry, then there existsa surface of section for which the corresponding Poincare´ map is Birkhoff-reversible. TheBirkhoff-reversibility of this map may be used to study its global dynamics such as the locations andthe distribution of the stable and unstable periodic points, the distribution of stable and chaoticregions, and the identification of the scattering regions. ©2002 American Institute of Physics.@DOI: 10.1063/1.1499595#

Due to a reversing symmetry „not the classical time-reversal symmetry… discovered by Birkhoff, the Poincaremap for the restricted three-body problem is a Birkhoff-reversible map. The Birkhoff-reversibility of this maphas many uses in the study of its global dynamics. In thepresent study we identify a sub-class of two-degrees offreedom Hamiltonian systems possessing the Birkhoff-reversibility and the Birkhoff-reversible Poincare maps.Some examples from classical mechanics which belong tothis sub-class have been discussed.

I. INTRODUCTION

The global study of the conservative dynamical systemswith two-degrees of freedom was started by Poincare´,1 andBirkhoff2–4 who paid special attention to the restricted prob-lem of three bodies. The reduction of the problem to a two-dimensional map, as well as the demonstration of the areainvariant property are due to Poincare´.1 The decompositionof the mapping function into a composite of two involutionswas recognized by Birkhoff2 and this is a consequence ofwhat we refer to as the Birkhoff-reversibility in this paper.~Usually the Birkhoff-reversibility is simply called the re-versibility. However, this reversibility is frequently confusedwith the time-reversibility. To avoid confusion, we add theheader ‘‘Birkhoff’’ in honor of his discovery.! He used thisproperty to discuss the distribution of symmetric periodicorbits. The Birkhoff-reversibility received renewed attentionin 1958 by DeVogelaere5 who described in more details amethod of searching for the symmetric periodic orbits. Themethod is very efficient, since it requires to search only in asubset of the full phase space. The extension to more generalinvolutory reversing symmetries6–8 has led to the definitionof the reversible dynamical systems~not necessarily Hamil-tonian! which in our terminology are Birkhoff-reversible dy-namical systems. The Birkhoff-reversibility has not onlybeen used to find symmetric periodic orbits, but also to ob-

tain continuous families of periodic orbits and their bifurca-tions as a parameter in the problem varies.9 It has been alsoused to find certain symmetric homoclinic~heteroclinic!orbits.10 For many other applications of the Birkhoff-reversibility and more references, we refer to a recent surveyon this topic.11

In this paper, we show that a sub-class of two-degrees offreedom Hamiltonian systems are reducible to a two-dimensional Birkhoff-reversible map. Then the global dy-namics of this map are investigated utilizing the Birkhoff-reversibility. This sub-class consists of time-reversible andtime-irreversible Hamiltonian systems.

We begin in Sec. II with the definition of the two-degrees of freedom Hamiltonian systems and its reversingsymmetries. In Sec. III we present a detailed discussion ofthe reduction to a two-dimensional map and in particular to aBirkhoff-reversible map. Section IV describes the partition-ing of the phase space and the associated symbolic dynam-ics, in particular the partitioning using the symmetry lines.Finally in Sec. V, we introduce some examples from classicalmechanics and physics which belong to the sub-class identi-fied in this paper.

II. DYNAMICAL SYSTEM WITH TWO-DEGREES OFFREEDOM

We consider the dynamical system with two-degrees offreedom described by the following Hamiltonian equations:

dqi

dt5

]H

]pi,

dpi

dt52

]H

]qi, i 51,2, ~1!

where the two-dimensional vectorsq5(q1 ,q2), and p5(p1 ,p2) are, respectively, the coordinates and the mo-menta and the time independent Hamiltonian functionH(q,p) is assumed to be quadratic and convex inp ~i.e.,possessing the positive definite Hessian with respect top!.Since the Hamiltonian is time independent, it is an integral of

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motion. Note that the above assumptions are always satisfiedin the mechanical system for which the Hamiltonian is thesum of kinetic and potential energies. It represents the inte-gral of energy.

Under these assumptions, the Hamiltonian may be writ-ten explicitly as

H5ap2212bp1p21cp1

21ap11bp21g, ~2!

wherea, b, c, a, b, andg are real functions of the coordi-nates (q1 ,q2) satisfying the following restrictions:

a.0, ac2b2.0. ~3!

We note that the quadratic form imposed on H, as well as therestrictions ~3! are invariant under the group of extendedpoint transformations,12

qi5]W

]pi, pi5

]W

]qi, i 51,2, ~4!

where

W5p1w1~ q1 ,q2!1p2w2~ q1 ,q2!.

This follows from the fact that the momenta are transformedlinearly, and therefore the new Hamiltonian remains qua-dratic and convex in the new momenta. In this sense, thesubset defined by Eqs.~2! and ~3!, and invariant under thegroup of transformations~4! is a sub-class of the two-degreesof freedom Hamiltonian systems. We refer to this subset asset A.

Remark: Birkhoff3 has given a normal form for two-degrees of freedom dynamical systems. The advantage of hisnormal form is that only two independent functions are in-volved instead of the six functions in Eq.~2!. He has alsoshown, at least locally, that there exists an extended pointtransformation which transforms Eq.~2! into the normalform. For more details the reader should consult Birkhoff3 orWintner.13 However, finding this transformation globally re-quires the solution of certain nonlinear partial differentialequations. If we use the normal form instead of Eq.~2!, thenonavailability of this transformation limits the applicabilityof our results. Birkhoff2 treated the restricted problem ofthree bodies which is in the normal form to begin with andhe did not have to find a transformation for this purpose.

A. Reduction of the phase space

Equations ~1! describe a vector field in a four-dimensional phase space. Now we use the time and energy toreduce the phase space. If the initial conditions are restrictedto a fixed value of energy, i.e.,

H~q1 ,q2 ,p1 ,p2!5h, ~5!

we may solve Eq.~5! for p2 to obtain

p252K~q1 ,p1 ,q2 ,h!, ~6!

where the function K is given explicitly as

K51

a S bp11b

26AgD , ~7!

with

g5a~h2g!1b2

42~aa2bb!p12~ac2b2!p1

2 . ~8!

From Eqs.~5! and ~6! we obtain

]H

]x2

]H

]p2

]K

]x50, ~9!

along any orbit, wherex5p1 , q1 , or q2 . Equation~9! withx5p1 andq1 leads to the reduced Hamiltonian system

dq1

dq25

]K

]p1,

dp1

dq252

]K

]q1, ~10!

where substitution for the partial derivatives ofH is accord-ing to Eq.~1!. Note that the reduced HamiltonianK dependsexplicitly on the new independent variableq2 and the wellknown equation

dK

dq25

]K

]q2, ~11!

may be obtained from Eq.~9! with x5q2 . This reductionwill be used to establish some of the properties of the mapintroduced in the following section.

B. Space and time reversing symmetries

The dynamical system possesses a reversing symmetryR, if the equations of motion~1! remain invariant under thetransformationR and the reversal of time. For example, theequations of motion~1! are invariant under the reflection

RT :~q1 ,q2 ,p1 ,p2!→~q1 ,q2 ,2p1 ,2p2!, and t→2t,~12!

if and only if a andb in Eq. ~2! are identically zero.If the dynamical system has this simple time reversing

symmetry it is called time-reversible, and time-irreversibleotherwise. The restricted problem of three bodies does notpossess this simple time-reversibility, and it is an example ofthe time-irreversible systems.

Birkhoff,2 however, discovered a different reversingsymmetry namely,

RB :~q1 ,q2 ,p1 ,p2!→~q1 ,2q2 ,2p1 ,p2!, and t→2t,~13!

which is a mixed space–time reversing symmetry. It shouldbe emphasized that Eqs.~12! and ~13! are completely inde-pendent and the time-reversible as well as the time-irreversible systems may possess the reversing symmetry de-fined by Eq.~13!. Using Eq.~2!, it may be shown that theequations of motion~1! are invariant under the reflection~13! if the following restrictions:

a~q1 ,2q2!5a~q1 ,q2!, b~q1 ,2q2!52b~q1 ,q2!,

c~q1 ,2q2!5c~q1 ,q2!, a~q1 ,2q2!52a~q1 ,q2!, ~14!

b~q1 ,2q2!5b~q1 ,q2!, g~q1 ,2q2!5g~q1 ,q2!,

are imposed in addition to Eq.~3!. We refer to this sub-classas set B. Clearly B is a subset of A, but it is not invariantunder the extended point transformations and therefore not aproper subset under this group. However, it is invariant, if we

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restrict the transformations to the subgroup in which thefunctions defined in Eq.~4! satisfy the following restrictions:

w1~ q1 ,2q2!5w1~ q1 ,q2!,~15!

w2~ q1 ,2q2!52w2~ q1 ,q2!.

For the reduced equations~10!, the Birkhoff-reversing sym-metry ~13! is reduced to the following reversing symmetry:

r1 :~q1 ,p1!→~q1 ,2p1!, and q2→2q2 , ~16!

in which the reversing action is with respect to the new in-dependent variableq2 instead of time. This can be verifiedby observing that Eq.~10! is invariant under Eq.~16!, if Eq.~14! holds. If we usea5b50, instead of Eq.~14!, the equa-tions of motion~10! are not invariant under Eq.~16!. There-fore for the reduced equations~10! to possess the reversingsymmetry~16!, which we call the Birkhoff-reversibility, onlythe reversing symmetry~13! is needed, and the classicaltime-reversibility~12! is neither necessary nor sufficient.

Remark: DeVogelaere5 breaks the Birkhoff-reversibilityinto two steps. In the first step, he assumes a time-reversiblesystem which corresponds to Eq.~2! with a5c5 1

2, b5a5b50, and therefore disregarding the time-irreversible sys-tems. In the second step, he assumes space-symmetry on thepotential function corresponding tog in Eq. ~2!. With thesesteps, he gives an impression that a Birkhoff-reversible mapis a consequence of time-reversibility of the original Hamil-tonian system. This has led to some confusions in the litera-ture.

III. REDUCTION TO A TWO-DIMENSIONAL MAP

The submanifold of the state of motion for a fixed en-ergy is a three-dimensional real manifold embedded in thefour dimensional Euclidean space. This manifold exists~i.e.,K is real! only for the three-dimensional regionV defined by

V5$p1 ,q1 ,q2ug>0%. ~17!

The projection ofV on the configuration space (q1 ,q2) isgiven by

V5$q1 ,q2ug<h%, ~18!

where

g5g2~aa2bb!21b2~ac2b2!

4a~ac2b2!. ~19!

This follows from Eq.~8!, whereg is quadratic inp1 and wehaveg>0 if and only if the discriminant is non-negative. Weremark thatV may also be obtained more directly by mini-mizing the Hamiltonian~2! with respect to the momentaonly, as it has been shown for the system with n-degrees offreedom in general.14 Note thatV represents the permissibleregion of motion in the configuration space for a fixed valueof energy. This region may not be connected or bounded andit may have a rather complicated structure. However, there isno loss of generality to assume it to be connected but notnecessarily bounded. In the general case where there is morethan one connected component, we can treat each componentseparately.

For a fixed value of energy, the motion of a trajectory~or, more precisely, its projection! is restricted toV. UsingEqs.~6! and ~7!, we obtain

dq2

dt UV

572Ag. ~20!

This implies that inV, q2 is monotonic and it changes fromincreasing to decreasing and vice versa only when the trajec-tory is on the boundary]V. Between two consecutive pointson the boundary the trajectory can be represented uniquelyby the vector-field~10! for which the reduced Hamiltonian Kis only one of the two branches given by Eq.~7!, i.e., thepositive sign ifq2 is decreasing and the negative sign ifq2 isincreasing.

In V, q250 defines a surface of section and

VS5$Vuq250%, ~21!

is the permissible region on this surface. Every point (q1 ,p1)in VS represents a unique initial condition for Eq.~10!, orwith q250, andp252K, a unique initial condition for Eq.~1!. Note that by Eq.~20!, every point on the boundary]V isa critical point ofq2(t). If the points on]V are maximumwith q2.0 and minimum withq2,0, then every solutionintersectsVS at least once, and repeatedly ifq2(t) remainsbounded. This condition can be written as

G52q2

d2q2

dt2 U]V

.0, ~22!

and it implies thatVS is a global section. However, Eq.~22!is very restrictive and it is not satisfied in general for manyproblems. Therefore we do not demand thatVS to be a glo-bal section. In other words we allow some of the trajectoriesnot to intersectVS . We remark that in some problems it ispossible to identify these exceptional trajectories which areeliminated from the analysis.15

Now let us denote the solution of Eq.~10! by w(p,q2)with the initial conditionp5(q1 ,p1). A solution started at apoint pPVS may or may not return to another point inVS .Let V1,VS andV2,VS denote the set of points that re-turn to VS in forward and backward direction, respectively,then a two dimensional mapf can be defined as follows:

For anypPV1 there exists a smallest later timet1 suchthat q2(t1)50 and the forward image ofp under f is de-fined by

f ~p!5w~p,q2~ t1!!. ~23!

Similarly, for anypPV2 there exits a largest earlier timet2

such thatq2(t2)50 and the backward image ofp underf isdefined by

f 21~p!5w~p,q2~ t2!!. ~24!

With these definitions, we may denote the Poincare´ returnmap by

f :V1→VS , and f 21:V2→VS . ~25!

Note that this restriction of the domain is necessary to in-clude systems in whichV does not impose any bound onq2 .Under iteration off @i.e., f n(p), n51,2,3,...#, the forward

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orbit of any point pPVS is either a finite or an infinitesequence of points inVS . In the case of a finite sequence thelast point is not inV1 . Similarly under iteration off 21, thebackward orbit is defined.

From the Hamiltonian form of the reduced equations~10!, it follows that the area of any local region in the phaseplane (q1 ,p1) remains invariant asq2 varies and in particu-lar for anyq250. Denoting the derivative of the map byD f ,this implies thatf is locally area preserving (detDf51) andthat the eigenvalues ofD f are a reciprocal pair (l1l251).

We remark that in general, such Poincare´ maps are notdefined analytically and they must be obtained by numericalintegration of the equations of motion~1!.

Reversible maps

If we consider only the Hamiltonians in set B, then thereduced system~10! possesses the Birkhoff-reversibilitygiven by Eq.~16!. In terms of the trajectoryw(p,q2), Eq.~16! implies

w~r1~p!,2q2!5r1~w~p,q2!!, ~26!

wherer1(p)5(q1 ,2p1) is the reflection with respect to theq1-axis.

Therefore the backward orbit of the reflection ofp is thesame as the reflection of the forward orbit ofp. In particularif the forward orbit ofp returns toVS , the backward orbit ofr1(p) also returns toVS , and we have

V25r1~V1!. ~27!

Using the definitions~23! and ~24!, Eq. ~26! implies

f 21r15r1f , ~28!

from which we have by induction

f 2nr15r1f n. ~29!

Therefore if the Hamiltonian system has a Birkhoff reversingsymmetry~13!, then by Eq.~28!, its corresponding Poincare´map f defined by Eq.~25! is a Birkhoff-reversible map witha reversing symmetryr1 . Sincer1 is an involutory reversingsymmetry, it follows from Eq.~28! that f can be written ascomposition of two involutions,

f 5r1r2 , r125r2

25 id. ~30!

We also obtainr25r1f . These properties were first noted byBirkhoff in the restricted three-body problem.2

The fundamental symmetry lines of the two involutionsin Eq. ~30! are defined as

L15$pur1~p!5p%, ~31!

L25$pur2~p!5p%. ~32!

HereL1 is the set of fixed points ofr1 and it is simply theq1-axis, butL2 is the set of fixed points ofr2 which is not sosimple. To obtainL2 , we use the following theorem.

Theorem: Let r:V1→VS be an involutory map~that isr25 id! and denote its derivative byDr. If ~i! p is a fixedpoint of r, and~ii ! detDr(p)521, then trDr(p)50.

Proof: Since r is an involution, we haveDr(r(p))Dr(p)5I . Sincep is a fixed point, this leads to

~iii ! Dr(p)5@Dr(p)#21. The eigenvalues ofDr(p) are(l,2 1/l) by ~ii !, so the eigenvalues of@Dr(p)#21 are(1/l ,2l). These eigenvalues are equal by~iii ! which im-plies l251, and trDr(p)50. h

SinceDr25Dr1D f , we have detDr2521. Then forr5r2 , the above conditions are satisfied and the Theoremimplies L2,T2 , whereT25$putr Dr2(p)50%.

Using Eqs.~29! and ~30!, we obtain

f 2n~p!5 f 2nr1~p!5r1f n~p!, for pPL1

~33!f 2n~p!5 f 2nr2~p!5 f 2n21r1~p!5r1f n11~p! for pPL2 .

Therefore the points on the fundamental symmetry lineshave symmetric orbits with respect to theq1-axis, and the setof all symmetry lines~i.e., the set of all points which havesymmetric orbits! can be generated by iterations.

~L1!n5 f n~L1!, ~L2!n5 f n~L2!. ~34!

In fact only the forward iteration is needed, since the back-ward iteration can be obtained by reflection.

One of the important properties of the symmetry lines isthat their intersections are symmetric periodic points. Thishas been studied in detail by DeVogelaere5 followingBirkhoff’s work on this subject. Note that any intersectionpof the symmetry lines can be mapped by iterations into anintersection with a fundamental symmetry line. Thereforewithout loss of generality we may show the periodicity of theintersections by considering only the following three cases:

~i! pPL1ù(L1)n

Sincef 2n(p)PL1 , we havef 22n(p)5r1(p) by the firstequation in Eq.~33!. Since pPL1 , this implies f 22n(p)5p. So in this casep is a periodic point with even period.

~ii ! pPL2ù(L2)n

Since f 2n(p)PL2 , we have f 22n(p)5r1f (p)5r2(p)by the second equation in Eq.~33! and Eq.~30!. Since pPL2 , this impliesf 22n(p)5p. Sop is a periodic point witheven period in this case also.

~iii ! pPL1ù(L2)n

Since f 2n(p)PL2 , we have f 22n(p)5r1f (p) by thesecond equation in Eq.~33!. Since pPL1 , this impliesf 2n21(p)5p. Therefore in this casep is a periodic pointwith odd period.

Another important property of the symmetry lines is thatthey contain certain symmetric homoclinic and heteroclinicpoints. For example, ifpPL1 is a hyperbolic fixed point forf , and if qPwu(p)ùL1 , wherewu(p) denotes the unstablemanifold of p, thenq is a symmetric homoclinic point. Formore details on this property, we refer to Devaney.10

In the case thatf depends on some parameter, the sym-metry lines can be used to obtain continuous families ofperiodic orbits and their bifurcations. As the parameter valuechanges, at some special value, two symmetry lines make atangent inflection, and a new family bifurcates from the in-tersection point. For a higher value of the parameter, thetangent inflection becomes a cubic intersection which leadsto a pitchfork bifurcation with two new families on bothsides of the original family. For a recent application to theSitnikov problem, see Ref. 16.

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In the next section, we use the fundamental symmetrylines as natural boundaries to partitionVS into subregionswith an associated symbolic dynamics.

IV. GLOBAL DESCRIPTION OF DYNAMICS IN VS

In this section, we offer a global description off basedon the mapping of areas~i.e., lattice of points! in VS ratherthan points. The basic idea is to partitionVS into subregionsaccording to symbolic dynamics and consider the images ofthese subregions and their intersections with the preimages.In principle the partition can be arbitrarily selected. How-ever, only the dynamically meaningful partitions lead to re-sults with dynamic interpretations.

Let (V1 ,...,Vk ,...,VN) be a given partition, then theorbit of any pointpPVS may also be represented by a se-quence ofk’s

s5km ,km11 ,...,k21 .k0 ,k1 ,...,kn21 ,kn , ~35!

obtained by iterating the map in both forward and backwarddirections. Under this notation the subscript immediately af-ter the decimal point~k0 in the definition! refers to thepresent location. Subscripts further to the right describe thesequence ofVk traversed under forward mapping off whilesubscripts to the left refer to backward iterations. Thereforetheki are determined from an alphabet (1,2,...,N) accordingto

f i~p!PVki. ~36!

Note thatm, andn can be either finite, or infinite. Sos isrepresenting a finite, infinite, or bi-infinite sequence of inte-gers (1,2...,N). If we denote the complements ofV2 andV1 by Vc , andVe , then we may specify the terminatingsequences as follows. In the sequences that terminate on theleft, the first point of the orbit is inVkm

ùVc . Similarly, inthe sequences that terminate on the right, the last point of theorbit is in Vkn

ùVe .Denoting the set of all possibles by S, the above corre-

spondence between the orbit of a pointpPVS underf and asequence inS can be defined as a mapping,

f:VS→S. ~37!

Clearly f(p) is simply a sequence relating the locationwithin VS that succeeding~and preceding! iterates ofp un-der f falls. The decimal point in a sequence serves as a placemarker for the present location. Many points inVS may havethe same representation inS. Also some possible sequencesin S, called forbidden blocks, do not represent an orbit foranypPVS , and they cannot occur in the image off. There-fore, in general,f may be neither one-to-one nor ‘‘onto.’’

Now we define the shift maps:S1→S, whereS1,Sis the subset for which there is more than one symbol to theright of the decimal point. The shift maps moves the deci-mal point in a sequence one place to the right. Similarly, wedefine a subsetS2,S for which s21:S2→S is a left shiftof the decimal point. From the definition off it follows thatsnf(p)5f f n(p) for all n for which f n(p) is defined. Thisimplies that a trajectory in the new spaceS is determined bysimply moving the decimal point on a fixed sequence.

Sincef is not one-to-one, for everysPS, there is anassociated set of points inVS , defined by

As5ù i 5mn f 2 i~Vki

!, ~38!

which traces that particular sequence. We define

m~As!5area of As

area of VS, ~39!

as a probability measure associated with the sequence s, pro-vided that the area ofVS is finite. If the area is infinite, arelative probability measure may be introduced using a partof VS .

The permissible sequence blocks may be obtained bymapping the partition areas forward and backward, and in-specting various intersections. For example for blocks withtwo entries, the sequence block (i . j ) is permissible if theintersectionAi . j5 f (V i)ùV jÞB, and forbidden otherwise.The measure of this intersection is the probability measureassociated with the block (i . j ). Similarly, the intersectionAj .i5 f (V j )ùV i determines whether the reverse sequenceblock (j .i ) is permissible or not, and its associated measure.

For problems in whichVc and Ve are not empty, it isimportant to identify the setAt,VS which represents allterminated sequences. Denoting the subsets associated, re-spectively, with the left and right terminated sequences byAc

andAe , we have

Ac5ø i 50` f i~Vc!, Ae5ø i 50

` f 2 i~Ve!,

and At5AcøAe .

Note that the subset associated with the finite sequences issimply Af5AcùAe , and the complement ofAt is the set ofpointsAb corresponding to the bi-infinite sequences.

The importance ofAt can be explained as follows. Forany initial conditionpPAt,VS , the orbit of p defined byw(p,q2) does not return toVS before some finite time in thepast, after some finite time in the future, or both. For systemsin which q2 is not bounded inV, this usually means thatq2

becomes unbounded which implies some kind of scattering~e.g., capture and escape of a particle!. Moreover, the struc-ture ofAt allows us to distinguish between the so-called fastand chaotic scattering regions. For an illustrative example ofthe scattering regions in the planar isosceles three-bodyproblem see Fig. 11 in Ref. 15.

However, one should be careful with interpretation,since there may exist solutions with boundedq2 which donot return toVS also. For example, solutions which termi-nate in a finite time, or solutions which oscillate on one sideof VS after some finite time. The first may occur due to anonremovable singularity~e.g., triple collision in the threebody problem!, and the second is possible ifVS is not aglobal section and Eq.~22! is not satisfied.

Global description for reversible maps

If f is a Birkhoff-reversible map, then the fundamentalsymmetry lines can be used to define the initial partition.This is a dynamically meaningful and generic partition forthe reversible maps. Under iterations off , the successiveimages of these initial subregions may be obtained. These

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images are new subregions with symmetry lines (L1)n ,(L2)n

on the boundaries. One can obtain few iterations and con-sider the intersections to obtain the permissible and forbid-den blocks with few entries. The intersections of the bound-aries are the intersections of the symmetry lines whichcorrespond to the symmetric periodic points. This is equiva-lent to the usual procedure of generating@(L1) i ,(L2) i ,i51,2,...,n# and their intersections to obtain the symmetricperiodic points~up to period 2n! and their distribution.

Using the fundamental symmetry lines for the initial par-titioning of VS , we define the global structure ofVS by thepartitioning obtained fromf n(VS) asn→`. Since symmetrylines are always on the boundaries, the global structure asdefined above is related to the limiting behavior of (L1)n and(L2)n .

As n increases, the iterates (L1)n and (L2)n of symmetrylines in the neighborhood of elliptic and hyperbolic pointsbehave in a completely different way. In the first case theyspiral out of the elliptic point toward the boundary of stableregion, partitioning the neighborhood into spiral shape re-gions, with more and more turns asn increases. In the sec-ond case and in the presence of a homoclinic~heteroclinic!point the symmery lines fold infinitely often in the neighbor-hood of the hyperbolic point asn→`, leading to destructionof the region boundaries and creation of chaotic regions. Inpractice a distinct global portrait appears at some finiten,and further increasing ofn adds only details.

V. EXAMPLES FROM CLASSICAL MECHANICS

In this section, we list some problems from classicalmechanics and physics which their Hamiltonian is in the setB, and therefore they can be subjected to the analysis pre-sented in this paper.

A. Special cases of the problem of three bodies

The problem of three bodies is the most celeberated ofall dynamical problems.12 The general case cannot be re-duced to two degrees of freedom. However, the followingspecial cases of this problem have two-degrees of freedom.

1. The restricted problem of three bodies

This problem has been treated extensively by Birkhoff,2

where he has introduced the reversing symmetry~13! fol-lowed by a discussion of the distribution of periodic orbits.The Hamiltonian for this problem is given by Eq.~2! where

a5c5 12 , b50, a5q2 , b52q1 , and

g5212m

A~q12m!21q22

2m

A~q11m!21q22

.

Clearly these coefficients satisfy Eq.~14! which implies thatthe restricted problem of three bodies possesses the reversingsymmetry~13!. Note that with these coefficients~18! reducesto

V5$q1 ,q2ug5g2 12 ~q1

21q22!<h%,

which is the well known permissible regions of motion in therestricted problem of three bodies.17

2. The planar isosceles three-body problem

This is a special case of the three body problem, inwhich two equal primary masses move on a line, and thethird mass moves on a line that is perpendicular to the pri-maries line and through their center of mass. The Hamil-tonian for this problem15 is specified by the following coef-ficients:

a51

2m, b50, c5

1

2, a5b50, and

g52Gm1

4q12

Gm3

Aq121q2

2,

which satisfy Eq.~14!. Therefore this problem also possessesthe reversing symmetry~13!. The role of this symmetry inthe global description of the system has been already shownin Refs. 15 and 18. However, in these papers, the set ofpoints leading to triple collisions have been used to partitionthe phase space. The use of symmetry lines, instead, willlead to additional information regarding the distribution ofstable regions.

3. The collinear three-body problem

This is the special case, in which, all three masses moveon a fixed line. In the particular case of three equal masses,this problem has been studied via symbolic dynamics usingthe triple collision curves to partition the phase space.19,20 Inthis case the Hamiltonian19 is given by

H5p222p1p21p1

221

q12

1

q22

1

q11q2.

The coefficients in this Hamiltonian do not satisfy Eq.~14!.However, if we use the transformation~4! with

w1~ q1 ,q2!5q11q2

2, w2~ q1 ,q2!5

q12q2

2,

the new Hamiltonian becomes

H53p221 p1

222

q11q22

2

q12q22

1

q1,

which satisfies Eq.~14!, and therefore it possesses theBirkhoff reversing symmetry. The use of symmetry lines inthis problem should also lead to additional information.

B. The problem of the motion of a rigid body with afixed point

Few problems in dynamics have received so much atten-tion as that of the motion of a rigid body with a fixed point.21

The problem has three-degrees of freedom, however, one ofthe coordinates~i.e., the precession angle! is ignorable, so itcan be reduced to two-degrees of freedom. Coefficients ofthe reduced Hamiltonian are given by22

704 Chaos, Vol. 12, No. 3, 2002 K. Zare and K. Tanikawa

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a5S cosq1

sinq1D 2S cos2 q2

2A1

sin2 q2

2BD 1

1

2C,

b5S cosq1

sinq1D S 1

2A2

1

2BD sin 2q2 ,

c51

2A1

1

2B, a52

k

cosq1

b, b522k cosq1

sin2 q1

c,

g5Mg~2a sinq1 cosq21b sinq1 sinq21 c cosq1!

1k2

sin2 q1

c.

These coefficients satisfy Eq.~14!, if the physical parameterb50. This implies that the center of mass of the rigid body isin the plane formed by the principal axes corresponding tothe principal moments of inertiaA andC. So for this specialcase, the problem possesses the Birkhoff reversing symme-try. Note that with these coefficients, Eq.~18! reduces to

V5H q1 ,q2Ug5Mgd2k2

2I<hJ ,

where

d52a sinq1 cosq21b sinq1 sinq21 c cosq1 ,

and

I 5A sin2 q1 cos2 q21B sin2 q1 sin2 q21C cos2 q1 .

This is the known permissible regions of motion for thisproblem.22

C. The problem of the motion of an electric charge inthe magnetic field of a dipole

This problem also referred to as Sto¨rmer’s problem hasbeen considered by DeVogelaere.5 The Hamiltonian in thiscase has the following coefficients:

a51

2mq12 , b50, c5

1

2m, a5b50, and

g5S k

q12 cos2 q2

2Me

q1cosq2D 2

.

Once again, these coefficients satisfy Eq.~14! and the prob-lem possesses the Birkhoff reversing symmetry.

Finally, the forced pendulum23 is one of simplest ex-amples.

ACKNOWLEDGMENT

The first author would like to acknowledge the hospital-ity and support of NAO during his stay in Japan where thisresearch was completed.

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705Chaos, Vol. 12, No. 3, 2002 Hamiltonian systems

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