Transcritical bifurcations in nonintegrable Hamiltonian systems

Transcritical bifurcations in nonintegrable Hamiltonian systems

Matthias Brack¹and Kaori Tanaka^1,2

1Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany

2Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, SK, Canada S7N 5E2 共Received 21 December 2007; revised manuscript received 25 February 2008; published 10 April 2008兲 We report on transcritical bifurcations of periodic orbits in nonintegrable two-dimensional Hamiltonian systems. We discuss their existence criteria and some of their properties using a recent mathematical descrip- tion of transcritical bifurcations in families of symplectic maps. We then present numerical examples of transcritical bifurcations in a class of generalized Hénon-Heiles Hamiltonians and illustrate their stabilities and unfoldings under various perturbations of the Hamiltonians. We demonstrate that for Hamiltonians containing straight-line librating orbits, the transcritical bifurcation of these orbits is the typical case which occurs also in the absence of any discrete symmetries, while their isochronous pitchfork bifurcation is an exception. We determine the normal forms of both types of bifurcations and derive the uniform approximation required to include transcritically bifurcating orbits in the semiclassical trace formula for the density of states of the quantum Hamiltonian. We compute the coarse-grained density of states in a specific example both semi- classically and quantum mechanically and find excellent agreement of the results.

DOI:10.1103/PhysRevE.77.046205 PACS number共s兲: 05.45.⫺a, 03.65.Sq, 82.40.Bj

I. INTRODUCTION

The transcritical bifurcation 共TCB兲, in which a pair of stable and unstable fixed points of a map exchange their stabilities, is a well-known phenomenon in one-dimensional non-Hamiltonian systems. A simple example occurs in the quadratic logistic map共see, e.g.,关1兴兲,

x_n+1=rxn共1 −xn兲, 共1兲 where 兵xn其 are arbitrary real numbers and r is the control parameter. This map has—among others—two fixed points x₁^*= 0 andx₂^*= 1 − 1/r, which exchange their stabilities at the critical valuer= 1. Forr⬍1,x₁^*is stable andx₂^*is unstable, whereas the inverse is true for r⬎1. 共Note that in many textbooks discussing the quadratic map, this bifurcation is not mentioned as the values of the variable x are usually confined to be non-negative, while x

2

*⬍0 for r⬍1.兲 The TCB occurs in many maps used to describe growth or popu- lation phenomena共see关2兴for a recent example兲. TCBs have also been reported to occur in various time-dependent model systems关3–9兴and shown, e.g., to be involved in synchroni- zation mechanisms关5,6兴. In关8,9兴, TCBs have been found to play a crucial role in transitions between low- and high- confinement states in confined plasmas, and their unfoldings have been analyzed.

In this paper we report on the occurrence of such bifurca- tions in a class of two-dimensional nonintegrable Hamil- tonian systems. Since the TCB does not belong to the generic bifurcations in two-dimensional symplectic maps 关10兴, we consider it useful to investigate also the mathematical condi- tions under which it can exist, its stability under perturba- tions of the Hamiltonian, and its unfoldings when it is de- stroyed by a perturbation. For this, we rely on mathematical studies by Jänich关11,12兴, who introduced a class of “cross- ing bifurcations,” to which the TCB belongs, and derived several theorems and useful formulas for crossing bifurca- tions of straight-line librational orbits. Finally, in view of the important role which Gutzwiller’s semiclassical trace for-

mula关13兴plays for investigations of “quantum chaos”共see, e.g.,关14,15兴兲, we study the inclusion of transcritically bifur- cating orbits in the trace formula by an appropriate uniform approximation.

Generic bifurcations of fixed points in two-dimensional symplectic maps have been classified by Meyer关10兴in terms of the number m= 1 , 2 , . . . that corresponds to a period m-tupling occurring at the bifurcation. For an easily readable presentation of this classification of generic bifurcations, and of the corresponding normal forms used in semiclassical ap- plications, we refer to the textbook of Ozorio de Almeida 关16兴. Bifurcations occurring in Hamiltonian systems with discrete symmetries have been investigated in 关17–20兴; the TCB was, however, not mentioned in these papers. In关20兴it has been shown that all other nongeneric bifurcations occur- ring in such systems can be described by the generic normal forms given in关16兴, except for different bookkeeping of the number of fixed points which is connected not only to an m-tupling of the period, but also to degeneracies of the in- volved orbits due to the discrete symmetries. For the TCB this is not the case: it requires a normal form that is not in the generic list of 关16兴. We derive an appropriate normal form for the TCB, starting from the general criteria given in关11兴, and find it to correspond to that given in the literature for non-Hamiltonian systems关21,22兴. We use this normal form to develop the uniform approximation needed to include transcritically bifurcating orbits in the semiclassical trace formula. In a specific example that includes a TCB, we show numerically that our result allows to reproduce the coarse- grained quantum-mechanical density of states with a high precision.

In the nonlinear and semiclassical physics community, there exists an occasional belief that nongeneric bifurcations occur only in systems which exhibit discrete symmetries 共time-reversal symmetry being the most frequently met in physical systems兲. The examples of TCBs which we present in this paper are obtained in a class of autonomous Hamil- tonian systems with mixed dynamics; starting from the fa- mous Hénon-Heiles 共HH兲 Hamiltonian 关23兴 we change the

1539-3755/2008/77共4兲/046205共23兲 046205-1 ©2008 The American Physical Society

coefficient of one of its cubic terms and add other terms destroying some or all of its discrete symmetries. All the TCBs that we have found involve one straight-line libration belonging to the shortest “period one” orbits. Our formal investigations therefore focus on the class of two- dimensional Hamiltonians containing a straight-line libra- tional orbit. In this class of systems the TCB is, in fact, found to be the typical isochronous bifurcation of the librating or- bit. The isochronous pitchfork bifurcation 共PFB兲, however, which in Hamiltonian systems with time-reversal symmetry 共such as the standard HH system兲is the most frequently met nongeneric bifurcation, is the exception here. We show how under a specific perturbation the PFB can unfold into a saddle-node bifurcation共SNB兲followed by a TCB. In a spe- cific example, we demonstrate that the TCB can exist in a system without any discrete共spatial or time-reversal兲 sym- metry, thus proving that the above-mentioned belief is incor- rect.

Our paper is organized as follows. In Sec. II we compile results of Jänich 关11,12兴 relevant for our investigations.

Starting from two-dimensional symplectic maps, we define a class of “crossing bifurcations” to which the TCB and the isochronous PFB belong. We discuss various criteria and properties of these bifurcations and give some useful formu- las for the specific case of a bifurcating straight-line libra- tion. The mathematically less interested reader may skip Sec.

II and jump directly to Sec. III, where we present numerical examples of the TCB and their characteristic features in the generalized Hénon-Heiles Hamiltonians. We also study there various types of unfoldings of the TCB under perturbations of the Hamiltonian. In Sec. IV we discuss the semiclassical trace formula for the density of states of a quantum Hamil- tonian, and present the uniform approximation by which bi- furcating periodic orbits can be included. In Sec. IV D we present a semiclassical calculation of the density of states in a situation where the TCB occurs between two of the shortest periodic orbits, and demonstrate the validity of the uniform approximation by comparison of the results with those of a fully quantum-mechanical calculation. In Appendix A we de- rive the appropriate normal forms for the TCB and the iso- chronous PFB which are needed in semiclassical applica- tions. In Appendix B we briefly discuss the stability exchange of two orbits in a “false transcritical bifurcation”

which actually consists of a pair of close-lying pitchfork bi- furcations.

II. MATHEMATICAL PREREQUISITES

In this section we present results of Jänich关11,12兴which are relevant for our investigations. We shall only quote theo- rems and other results; for readers interested in the math- ematical proofs or other details, we refer to the explicit con- tents of关11,12兴.

A. Poincaré map and stability matrix

We are investigating bifurcations of periodic orbits in two-dimensional Hamiltonian systems. They are most conve- niently investigated and mathematically described by observ-

ing the fixed points on a suitably chosen projected Poincaré surface of section共PSS兲.¹Since the PSS here is two dimen- sional, we describe it by a pair of canonical variables共q,p兲.

The time evolution of an orbit then corresponds to the two- dimensional Poincaré map

共q,p兲→共Q,P兲, 共2兲

where共q,p兲 is the initial and 共Q,P兲 the final point on the PSS. Fixed points of this map, defined byQ=q, P=p, cor- respond to periodic orbits. We introduce ⑀as a “bifurcation parameter” which in principle may be the conserved energy of the system or any potential parameter, normalized such that a bifurcation occurs at⑀= 0. Here we specialize to the energy variable by defining

⑀=E−E₀, 共3兲

whereE₀ is the energy at which the considered bifurcation takes place. We assume that the bifurcating orbit returns to the same point on the PSS after one map共2兲, so that Q=q, P=p; in this paper this will be called a “period one” orbit.

We shall only study its isochronous bifurcations and hence only consider the noniterated Poincaré map.

The map共2兲is symplectic and thus area conserving in the 共q,p兲plane, and may be understood as a canonical transfor- mation,

Q=Q共q,p,⑀兲, P=P共q,p,⑀兲. 共4兲 Jänich关11兴has given a classification of bifurcations of fixed points in two-dimensional symplectic maps, which we shall summarize in the following. We use his notationQu,Pufor partial derivatives of the functions Q and P, respectively, with respect tou,

Qu=⳵^Q

⳵^{u, Pu=}

⳵^P

⳵^{u, 共5兲}

whereuis any of the three variablesq,p, or⑀. Analogously Qqq,Pqp⑀, etc., denote second and higher partial derivatives.

Due to the symplectic nature of Eq.共4兲, the determinant of the first derivatives ofQ and P with respect to q and p is unity

det

冉

^QPqq共q,p,^共q,p,⑀^⑀兲^{兲 Q}P^pp共q,p,^共q,p,⑀^⑀兲^兲

冊

^{= 1. 共6兲}

We consider an isolated “period one” orbit with fixed point 共q,p兲=共0 , 0兲, for all values of ⑀ where it exists, and denote it as theAorbit. Its stability matrix is then given by

1With “projected” we mean the fact that we ignore the value of the canonically conjugate variable共e.g.,p_y兲, to the variable共e.g.,y兲that has been fixed共e.g., byy=y₀兲to define the true mathematical PSS which lies in the energy shell. In the physics literature, it is standard to call its projection共withp_y= 0兲the PSS. Due to energy conserva- tion, the value of p_y on the unprojected PSS can be calculated uniquely, up to its sign which usually is chosen to be positive, from the knowledge of q,p,y₀ and the energy E through the implicit equationE=H共q,y₀,p,p_y兲, whereH共x,y,p_x,p_y兲is the Hamiltonian in Cartesian coordinates.

MA共⑀兲=

冉

^QPq^q共0,0,共0,0,⑀^⑀兲^{兲 Q}P^pp共0,0,^共0,0,⑀^⑀兲^兲

冊

^{. 共7兲}

At⑀= 0, where the orbit undergoes an isochronous bifurca- tion, MA共0兲 has two degenerate eigenvalues +1, so that trMA共0兲= 2.

Henceforth we shall omit the arguments 共0,0,0兲 in the partial derivatives ofQandPwhich—unless explicitly men- tioned otherwise—will always be evaluated at the bifurcation point. When we need some of these partial derivatives at p

=q= 0, but at arbitrary values of⑀, we shall denote them by Q_p共⑀兲, etc. When no argument is given,⑀= 0 is assumed. We thus write

MA共⑀兲=

冉

^QPqq共^共⑀⑀^兲兲 ^QP_p^p共^共⑀⑀^兲兲

冊

^{, MA共0兲=}

冉

^QPq^{q Q}P_p^p

冊

^{. 共8兲}

The slope of the function trMA共⑀兲 at ⑀^{= 0} 共coming from a side where the orbitAexists兲becomes, in this notation,

trM_A⬘共0兲=Q_q⑀+P_p⑀. 共9兲 By a rotation of the canonical coordinatesq,p it is always possible to bringMA共0兲into the form

MA共0兲=

冉

^{10 Q1p}

冊

^{, Qp⫽0. 共10兲}

We shall henceforth assume that the coordinates have been chosen such that Eq.共10兲 is true.² Then, with Eq. 共6兲 one finds easily thedeterminant derivative formula关11兴

Q_qu+P_pu=Q_pP_qu 共u=q,p,⑀兲, 共11兲 and Eq.共9兲takes the simpler form

trM_A⬘共0兲=QpPq⑀. 共12兲 The total fixed point set

Fª兩兵共q,p,⑀兲兩Q共q,p,⑀兲=q,P共q,p,⑀兲=p其 共13兲 is the inverse image of the origin共0 , 0兲inR²under the map 共Q−q,P−p兲whose Jacobian matrix at 共0 , 0 , 0兲is

J=

冉

^{QqP− 1q PpQ− 1p QP⑀}⑀

冊

⁼

冉

^{00 Q0p QP}⑀^⑀

冊

^{. 共14兲}

In the generic case,P_⑀⫽0 andJhas rank 2. This leads to the only generic isochronous bifurcation according to Meyer 关10兴, thesaddle-node bifurcation共SNB兲 共also called “tangent bifurcation”兲. For this bifurcation, the fixed-point setF Eq.

共13兲 is a smooth one-dimensional manifold, consisting of two half-branches tangent to the q axis at the bifurcation point with slopes trM_A⬘共0兲=⫾⬁. The orbitAthen exists ei- ther only for⑀ⱕ0 or only for⑀ⱖ0; no other orbit takes part in such a bifurcation.

Following Jänich关11兴, we speak of a rank 1 bifurcation, when the JacobianJin Eq.共14兲has rank 1, which is the case for

P_⑀= 0. 共15兲

Then, after a suitable共⑀-dependent兲translation of thep vari- able,Jcan always be brought into the form

J=

冉

^{QqP− 1}q P^Qp− 1^{p Q}P_⑀^⑀

冊

⁼

冉

⁰0 ^Q0^{p 0}0

冊

^{. 共16兲}

We shall formulate all following developments in the suit- ably adapted coordinates 共q,p兲, for which the form 共16兲 holds, and discuss only rank 1 bifurcations.

B. Crossing bifurcations of isolated periodic orbits A rank 1 bifurcation for which the Hessian

K=

冉

^PPqqq⑀ ^PP^q_⑀⑀^⑀

冊

^共17兲

at 共0,0,0兲 is regular and indefinite, i.e., for which detK

=PqqP_⑀⑀−P_q²_⑀⬍0, shall be called a crossing bifurcation.

Jänich showed关11兴 that a necessary and sufficient criterion for an orbitAto undergo a crossing bifurcation at⑀= 0 is for the slope trM_A⬘共⑀= 0兲to be finite and nonzero. With Eqs.共10兲 and共12兲we see that

Pq⑀⫽0 共18兲

for crossing bifurcations. It follows that if the orbitAunder- goes a crossing bifurcation at⑀= 0, it exists on both sides of a finite two-sided neighborhood of⑀= 0. Jänich also showed that for such a bifurcation, the total fixed-point setF共13兲is the union A艛B of two smooth one-dimensional submani- folds intersecting at the bifurcation point. The setAis the set of fixed points 共0 , 0 ,⑀兲 of the A orbit; we shall call it the fixed-point branch A. The set B is the fixed-point set of a second orbitBwhich takes part in the crossing bifurcation.

We shall discuss here only two types of crossing bifurca- tions: transcritical and forklike bifurcations. Their properties are specified in the following two sections. A rank 1 bifurca- tion with a regular and definite Hessian K, i.e., with detK

⬎0, is sometimes called an “isola center” 共cf. the normal form for the isola center in one-dimensional Hamiltonians at the end of Appendix A 1兲. Here the total fixed-point setF consists of the single isolated point共q,p,⑀兲=共0 , 0 , 0兲.

1. Transcritical bifurcation (TCB)

A transcritical bifurcation 共TCB兲 occurs when, in the adapted coordinates共q,p兲for which Eq.共10兲holds, one has

Pqq⫽0. 共19兲

Then, there exists another isolated periodic orbitB on both sides of ⑀= 0, forming a fixed-point branch B intersecting that of the orbitAat⑀= 0 with a finite angle. The functions trM_A共⑀兲and trM_B共⑀兲have opposite slopes at the bifurcation trM_A⬘共0兲= − trM_B⬘共0兲 共“TCB slope theorem”兲. 共20兲

2In some cases one may find thatM_A共0兲has the transposed simple form in whichQ_p= 0 and P_q⫽0. In this case one may simply ex- change the coordinates by a canonical rotationQ→P,P→−Q共and q→p,p→−q兲in all formulas below and in Appendix A 1. The case Q_p=P_q= 0 is exceptional and occurs only for harmonic potentials 共cf.关24兴兲.

In the scenario of a TCB, the orbitsA andBsimply ex- change their stabilities and no new orbit appears共or no old orbit disappears兲at the bifurcation.

Note.Assume that the orbitA is a straight-line libration, chosen to lie on theyaxis, so that the Poincaré variables are q=x, p=px 共see Sec. II B 2 below兲. Then, if the system is invariant under reflexion at theyaxis, such a reflexion leads to P共q,p,⑀兲= −P共−q, −p,⑀兲. Therefore, Pqq共q= 0 ,p= 0 ,⑀

= 0兲=Pqq= 0, and the bifurcation cannot be transcritical. The simplest possible crossing bifurcation then is forklike 共see next item兲. In short:Straight-line librations along symmetry axes cannot undergo transcritical bifurcations.

2. Forklike bifurcation (FLB)

A forklike bifurcation共FLB兲occurs when one has Pqq= 0, Pqqq⫽0. 共21兲 Then, there exists another isolated periodic orbit B, either only for ⑀ⱖ0 or only for ⑀ⱕ0. The fixed-point set of B consists of two half-branches intersecting the setAat⑀= 0 at a right angle. In the adapted coordinates corresponding to Eq.共10兲, one may parametrize the setBby(q,pB共q兲,⑀B共q兲) and finds

p_B⬘共0兲=⑀B⬘共0兲= 0, ⑀B⬙共0兲⫽0. 共22兲 Although trMB共⑀兲 is not a proper function of ⑀, a limiting slope trM_B⬘共0兲⫽0 can be defined for both half-branches of the setBin the limit⑀→0, coming from that side where they exist, and be shown关11兴to fulfill the relation

trM_B⬘共0兲= − 2 trM_A⬘共0兲 共“FLB slope theorem”兲. 共23兲 In the same limit, the curvature of the setBat the bifurcation point is given by

⑀B⬙共0兲=3Q_qqP_qp−Q_pP_qqq

3Q_pP_q⑀ . 共24兲

In the pertinent physics literature, this bifurcation is often called the 共nongeneric兲 isochronous pitchfork bifurcation (PFB). Note that here the two half-branches of the set B correspond to two different periodic orbits. They can be ei- ther locally degenerate共to first order in ⑀兲, or globally de- generate due to a discrete symmetry共reflexion at a symmetry axis or time reversal兲.

In thegeneric PFB corresponding to Meyer’s classifica- tion关10兴, the fixed point scenario near⑀= 0 is identical with that of the FLB. However, here the two fixed points of the set Bcorrespond to one single orbit B which has twice the pe- riod of the primitive orbitA. In fact, the fixed-point branchA crossing the line trMA= 2 is that of the iterated Poincaré map: the generic PFB isperiod doubling. The existence cri- terion共21兲 and the relations 共22兲–共24兲 for the B orbit hold here also关25兴.

C. Some explicit formulas for straight-line librations 1. Definition of the librational A orbit

We now specialize to straight-line librational orbits in two-dimensional autonomous Hamiltonian systems, defined by Hamiltonian functions

H₀共x,y,px,py兲=¹₂共px2+p_y²兲+V共x,y兲 共25兲 with a smooth potentialV共x,y兲. Straight-line librations form the simplest type 共and so far the only one known to us兲of periodic orbits in Hamiltonian systems that undergo tran- scritical bifurcations. Let us choose the direction of the libra- tion to be theyaxis and call it theAorbit. The potential then must have the property

⳵^V

⳵^{x共0,y兲= 0 共26兲}

for all y reached by the libration. The A orbit, which we assume to be bound at all energies, then hasx共t兲=x˙共t兲= 0 for all timest, and itsymotion is given by the Newton equation

y¨共t兲+⳵^V

⳵^{y„0,y共t兲…= 0⇒y共t兲=yA共t,}⑀兲, 共27兲 whereyA共t,⑀兲is henceforth assumed to be a known periodic function oft with periodTA共⑀兲. For theAorbits in the共gen- eralized兲 HH potentials discussed in the following section, the functiony_A共t,⑀兲 can be expressed in terms of a Jacobi- elliptic function 关26兴. We choose the time scale such that yA共0 ,⑀兲 is maximum with

y˙A共0,⑀兲= 0 ∀ ⑀. 共28兲 A suitable choice of Poincaré variables is to use the surface of section defined byy= 0, and the projected PSS becomes the 共x,px兲 plane, so that we define q=x, p=px. We again assume that the orbitAis isolated and exists in a finite inter- val of⑀around zero. The fixed-point branchAis thus again given by the straight line共qA,pA,⑀兲=共0 , 0 ,⑀兲in the共q,p,⑀兲 space.

In 关12兴 Jänich has given an iterative scheme to calculate the partial derivativesQq,Qp, etc. for this situation for any given共analytical兲potentialV共x,y兲with the above properties.

To this purpose, one has first to determine the fundamental systems of solutions 共␰1,␰2兲 and 共␩1,␩2兲 of the linearized equations of motion in thexandy directions, respectively,

␰^¨共t兲+Vxx„0,yA共t,⑀兲…␰共t兲= 0, 共29兲

␩^¨共t兲+Vyy„0,yA共t,⑀兲…␩共t兲= 0, 共30兲 with the initial conditions

冉

^␰␰^˙11^共0兲共0兲 ^␰␰^˙22^共0兲共0兲

冊

⁼

^冉

^{␩␩˙11共0兲共0兲 ␩␩˙22共0兲共0兲}

^冊

⁼

^冉

^{1 00 1}

^冊

^{∀ ⑀.}

共31兲 For simplicity, we do not give the argument⑀^{of the}␰i共t兲and

␩i共t兲, but we should keep in mind that they are all functions of ⑀^{. In Eqs.} 共29兲 and 共30兲 the subscripts on the function V共x,y兲 denote its second partial derivatives with respect to

the corresponding coordinates. In the formulas given below, we denote by V_i共t兲, V_ij共t兲, etc., with i,j苸共x,y兲 the partial derivatives taken along theAorbit, i.e., atx= 0,y=yA共t,⑀兲as in Eqs. 共29兲 and 共30兲, evaluated at the bifurcation point ⑀

= 0. If the partial derivatives have no argument, they are taken at the periodTA共⑀0兲, i.e.,Vy=Vy(TA共⑀0兲), etc.

Knowing the five functions y_A共t,⑀^{= 0}兲 and ␰i共t兲,␩i共t兲 共i

= 1 , 2兲at⑀= 0, all desired partial derivatives ofQ共q,p,⑀兲and P共q,p,⑀兲 at共q,p,⑀兲=共0 , 0 , 0兲 can be obtained by 共progres- sively repeated兲 quadratures, i.e., by finite integrals over known expressions including these five functions, partial de- rivatives of V共x,y兲, and the functions obtained at earlier steps of the scheme 共whereby the progression comes from increasing degrees of the desired partial derivatives兲.

2. Stability matrix of the A orbit

We note that the equation共29兲is nothing but the stability equation of theA orbit, since the ␰i by definition are small variations transverse to the orbit. In the standard literature, Eq.共29兲is also called the “Hill equation”共cf., e.g.,关14,27兴兲.

The stability matrix MA at the bifurcation of theA orbit is therefore simply given by

MA共0兲=

冉

^QP^qq ^QPp^p

冊

⁼

冉

^␰␰^˙11^共T共T^AA^兲兲 ^␰␰^˙22^共T共T^AA^兲兲

冊

^{, 共32兲}

withTA=TA共⑀= 0兲. Its eigenvalues must be ␭1=␭2= + 1, as seen directly from Eq.共10兲. The solutions␰i共t,⑀兲of Eq.共29兲 are in general not periodic. But at the bifurcations of theA orbit, where trM_A= + 2, one of the␰i共t,⑀^{= 0}兲is always peri- odic with periodTA 共or an integer multiplem thereof兲 关27兴 and describes, up to a normalization constant depending on

⑀, the transverse motion of the bifurcated orbit at an infini- tesimal distance⑀from the bifurcation共cf.关26,28,29兴兲.

3. Slope of the functiontrM_A(⑀)at⑀= 0

Here we give the explicit formulas, obtained from 关12兴, for the slope trM_A⬘共0兲=Qq⑀+Pp⑀, see Eq.共9兲, of the function trMA共⑀兲 at the bifurcation. The quantities Qq⑀ and Pp⑀ are given, in terms of the potentialV共x,y兲 in Eq. 共25兲 and the other ingredients defined above, by

Qq⑀= 1

共Vy兲²Pq␩^˙1共TA兲− 1 Vy

冕

0

T_A

Vxxy共t兲关Qp␰1共t兲

−Qq␰2共t兲兴␰1共t兲␩1共t兲dt, 共33兲

Pp⑀=共−V_xx兲

共V_y兲² Qp␩^˙1共TA兲− 1 V_y

冕

0

T_A

Vxxy共t兲关Pp␰1共t兲

−P_q␰2共t兲兴␰2共t兲␩1共t兲dt. 共34兲 In the adapted coordinates where trMA共0兲has the form共10兲 withQ_q=P_p= 1 and P_q= 0, the slope becomes

trM_A⬘共0兲=Q_pP_q⑀=共−Vxx兲

共Vy兲² Q_p␩^˙1共TA兲

− 1 Vy

Qp

冕

0 T_A

Vxxy共t兲␰1

2共t兲␩1共t兲dt. 共35兲

For the case that trMA共0兲has the transposed tridiagonal form withQp= 0 and Pq⫽0, the formula becomes

trM_A⬘共0兲=PqQp⑀= 1

共Vy兲²Pq␩^˙1共TA兲 + 1

V_yP_q

冕

0 T_A

V_xxy共t兲␰22共t兲␩1共t兲dt. 共36兲 An independent derivation of Eqs. 共33兲–共36兲 is given in 关30兴, where it is shown that the first terms are due to the variation of theAorbit’s periodTAwith⑀, whereas the inte- gral terms are due to the⑀dependence of the functions␰i共t兲.

4. Criterion for the TCB

For a bifurcation to be transcritical, we need P_qq⫽0.

From关12兴we find the following explicit formula forPqq

Pqq= −

冕

0 T_A

Vxxx共t兲␰1

3共t兲dt, 共37兲

which also yields explicitly the parameter b in its normal form given in Eq.共A24兲.

If the potential is symmetric about theyaxis, thenVxxx共t兲 is identically zero and the TCB cannot occur, as already stated in Sec. II B 1. However, even if Vxxx共t兲 is not zero, special symmetries of the function␰1共t兲, in combination with that ofVxxx共t兲, can make the integral in Eq.共37兲 vanish. An example of this is discussed in Sec. III C 5.

III. TCBS IN THE GENERALIZED HÉNON-HEILES POTENTIAL

A. The generalized Hénon-Heiles potential

For our numerical studies, we have investigated the fol- lowing family of generalized Hénon-Heiles共GHH兲Hamilto- nians:

H共x,y,px,py兲=¹₂共px

2+p_y²兲+¹₂共x²+y²兲

+␣

共

⁻¹3y³+␥x²y+␤y²x

兲

^{. 共38兲}

Here␣is the control parameter that regulates the nonlinear- ity of the system, and␥,␤are parameters that define various members of the family. The standard HH potential关23兴cor- responds to␥^{= 1,}␤= 0. It has three types of discrete symme- tries: 共i兲 rotations about 2␲/3 and 4␲/3, 共ii兲 reflections at three corresponding symmetry lines, which together define the C_3v symmetry, and 共iii兲 time-reversal symmetry. There exist three saddles at the critical energyE^*= 1/6␣^{2, so that} the system is unbound and a particle can escape if its energy is E⬎E^*. For ␥⫽1, ␤⫽0, the spatial symmetries are in general broken共except for particular values of␥and␤兲and only the time-reversal symmetry is left. There still exist three

saddles, but in general they lie at different energies. There is always a stable minimum atx=y= 0.

It is convenient to scale away the nonlinearity parameter

␣ ^{in Eq.} 共38兲 by introducing scaled variables x⬘⁼␣^{x, y}⬘

=␣y, and a scaled energy e=E/E^*= 6␣²E. Then Eq. 共38兲 becomes

h=e= 6␣²E= 3共p_x²_⬘+p_y²_⬘+x⬘^2+y⬘²兲− 2y⬘³

+ 6共␥^x⬘^2y⬘⁺␤^y⬘^2x⬘^{兲, 共39兲} so that one has to vary one parameter less to discuss the classical dynamics. 共For the standard HH potential with ␥

= 1,␤= 0, the scaled energyethen is the only parameter.兲For simplicity, we omit henceforth the primes of the scaled vari- ablesx⬘^,y⬘^.

Before we discuss the periodic orbits in the system共39兲, let us briefly recall the situation in the standard HH system in which all three saddles lie at the scaled energye= 1.

1. Periodic orbits in the standard HH potential

The periodic orbits of the standard HH system have been studied in 关26,28,31,32兴, and their use in connection with semiclassical trace formulas in关33–38兴. We also refer to关24兴 共Sec. 5.6.4兲 for a short introduction into this system, which represents a paradigm of a mixed Hamiltonian system cov- ering the transition from integrability 共e= 0兲 to near chaos 共e⬎1兲.

In Fig.1we show the trace trMof the stability matrixM, henceforth called “stability trace,” of the shortest orbits as a function ofe. Up to energy e⯝0.97, there exist 关31兴 only three types of “period one” orbits关in the sense defined after Eq. 共3兲兴: 共1兲 straight-line librations A along the three sym- metry axes, oscillating towards the saddles;共2兲curved libra- tions B which intersect the symmetry lines at right angles and are hyperbolically unstable at all energies; and共3兲rota-

tional orbitsC in the two time-reversed versions which are stable up toe⯝0.89 and then become inverse-hyperbolically unstable. While theB andCorbits exist at all energies, the orbits A cease to exist at the critical saddle energy e= 1 where their period becomes infinite.

When兩trM兩⬎2 or ⬍2, an orbit is unstable or stable, re- spectively. When trM= 2 it either undergoes a bifurcation if the orbit is isolated, or it belongs to a family of degenerate orbits in the presence of a continuous symmetry. The latter is seen to occur in the limite→0, where the orbitsA,B, andC all converge to the family of orbits of the isotropic two- dimensional harmonic oscillator withU共2兲symmetry. TheA orbits undergo an infinite sequence of共nongeneric兲isochro- nous PFBs, starting at e⯝0.97 and cumulating at e= 1. At these bifurcations an alternating sequence of rotational orbits 共labeled R兲 and librational orbits共labeled L兲 are born. This bifurcation cascade, theR and L type orbits, and their self- similarity have been discussed extensively in关26,28兴. In Fig.

1 and in the text below, we indicate their Maslov indices 共needed in semiclassical trace formulas, see Sec. IV兲by suf- fixes to their labels, which allows for unique book keeping of all orbits.共Only the first two representativesR₅andL₆of the orbits born along the bifurcation cascade are shown in Fig.1 by the dashed lines.兲 At each bifurcation, the orbit A in- creases its Maslov index共which is 5 up to the first bifurca- tion兲by one unit. Only the first three bifurcations can be seen in the figure; the others are all compressed into a tiny interval belowe= 1. As has been observed numerically in关26,28,32兴, trMA becomes a periodic function of the period TA in the limite→1.关It can actually be rigorously shown that, asymp- totically, trM_A共T_A兲→−2.68044 sin共

冑

3T_A兲in this limit关30兴.兴 As is characteristic of isochronous PFBs共cf. Sec. II B 2兲, the newborn orbits come in degenerate pairs due to the dis- crete symmetries: the two librational L orbits are mapped onto each other under reflection at the axis containing theA orbit, and the two rotationalRorbits are connected by time

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

e

-2 0 2 4

tr M

B4

A5

C3

R5

L6

A6

FIG. 1. Trace of the stability matrixM of the “period one” orbits in the standard HH potential, plotted versus scaled energye. The suffixes indicate their Maslov indices. Only the first two共R₅andL₆兲of the orbits born at an infinite sequence of isochronous PFBs of the Aorbit, cumulating at the saddle energye= 1, are shown共dashed lines兲.

reversal. Note that although the A orbit ceases to exist for eⱖ1, all R andL type orbits bifurcated from it exist at all energieseⱖ1. For some new orbits appearing there, we refer to the literature关32,38兴; in this paper we shall not be con- cerned with them.

2. Periodic orbits in the generalized HH potential For ␥⫽0, ␤⫽0, there exist in general three different saddles at scaled energiese₀,e₁, ande₂, and three different straight-line periodic orbits, labeledA,A⬘^{, andA}⬙, oscillating towards the saddles. In general, there are three curved libra- tional orbits B, B⬘^{, and B}⬙ 共not necessarily existing at all energies兲intersecting the threeA type orbits at right angles, and there is always a time-reversally degenerate pair of ro- tational orbitsCgoing around the origin. It is rather easy to see that the threeAtype orbits always intersect each other at the minimum of the potential located at the origin 共x,y兲

=共0 , 0兲. The equations of motion for the Hamiltonian共38兲in the Newton form are共in the scaled variables corresponding to␣^{= 1}兲

x¨+x共1 + 2␥^y兲+␤^{y2= 0,}

y¨+y共1 + 2␤x兲+␥^{x2−y2= 0.} 共40兲 For a straight-line orbit librating through the origin we have y=ax which, inserted into Eq. 共40兲, yields a cubic equation for the slopea,

␤^a3+共2␥^{+ 1}兲a²− 2␤^a−␥^{= 0.} 共41兲 For the rest of this paper, we limit the parameters to the range␥⬎0 and␤ⱖ0. Then, Eq.共41兲has always real roots that are in general different. In the right panel of Fig. 2 below, we have shown the six shortest共“period one”兲libra- tions obtained numerically for␥^{= 0.6,}␤= 0.07, including the three straight-line orbitsA,A⬘^,A⬙ intersecting at the origin.

Further analytical analysis is cumbersome except for the following special cases:

共i兲␤^{= 0,}␥^{= 1}共standard HH兲. Two of the slopes area_1,2

=⫾1/

冑

3; the third isa₀=⬁corresponding to the orbit along the y axis with x共t兲= 0. The three saddles lie at 共x,y兲

=共0 , 1兲,共−

冑

3/2 , −1/2兲, and共

冑

3/2 , −1/2兲, forming an equi- lateral triangle with side length

冑

3; its sides共and their exten- sions兲 form the equipotential lines for e= 1. The periodic orbits are those discussed in Sec. III A 1.

共ii兲 ␤= 0,␥⫽1. The rotational C_3v symmetry is broken, but the reflection symmetry at theyaxis is kept. Correspond- ingly, we find two degenerate orbits A⬘^{, A}⬙ with opposite slopes a_1,2=⫾

冑

␥/共2␥+ 1兲. There is a horizontal equipoten- tial line at y₁=y₂= −1/2␥ with scaled energy e₁=e₂=共3 + 1/␥兲/4␥² that contains two saddle points symmetrically positioned atx_1,2=⫾

冑

共2 + 1/␥兲/2␥. At low energies, there is only oneBtype orbit intersecting theyaxis at a right angle;

two further orbits B⬘ ^{and B}⬙ appear through bifurcations at higher energies共see examples in Sec. III B兲. For␥⬎0 there is a thirdAorbit librating along theyaxis共a0=⬁兲towards a third saddle at 共0 , 1兲 with energy e₀= 1. The equipotential line for e=e_1,2 consists of the horizontal line at y_1,2

= −1/2␥ and two branches of a hyperbola. For ␥⬎1 the hyperbola branches lie symmetrically about they axis, each intersecting the horizontal line at one of the two symmetric saddle points. For 0⬍␥⬍1, they lie symmetrically about a horizontal line at y^*=共1 + 3␥兲/4␥, the lower of them inter- secting the liney=y_1,2at the two symmetric saddle points.

The limiting case␤= 0,␥= 0 yields a separable and hence integrable system with only one saddle at 共0 , 1兲 at energy e₀= 1 and one A orbit 共with a=⬁兲. We do not discuss this system here, but refer to关37兴in which it is investigated both classically and semiclassically in full detail.

B. Examples of transcritical bifurcations and their properties As mentioned above, we have restricted the parameters␥ and␤in the GHH potential共38兲to be positive共or␤^{= 0}兲. We find that, depending on the values of␤and␥, at least one or two of the straight-line orbits A, A⬘^{, or A}⬙ can undergo a

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

e

-4 -3 -2 -1 0 1 2 3 4

trM

A₅ A’₅ A’’₅

A’₄ A’’₄ B₅

B₄

C₃ B’’₄

B’₄ B’₅

-1.0 -0.5 0.0 0.5 1.0

x

-0.5 0.0 0.5 1.0

y

e = 2.0

A(e=0.99) B A’’ A’

B’’

B’ C

FIG. 2. Left panel: Stability traces ofAandBtype orbits in the GHH potential with␥= 0.6,␤= 0.07, plotted versus scaled energye. The three saddles are ate₀= 0.993,e₁= 2.81, ande₂= 3.74. Right panel: Shortest orbits, projected on the共x,y兲 plane. OrbitAis evaluated ate

= 0.99 just below its saddle共e₀= 0.993兲; all other orbits are taken ate= 2. The line types correspond to those in the left panel.

TCB with a partner of the curved librational orbitsB,B⬘^{, or} B⬙. In the following, we shall first show two examples and then discuss characteristic properties of the TCB. In Sec.

III C we shall study its stability and its unfoldings. Some of the numerical results can easily be understood analytically in terms of the normal forms of the various bifurcations and their unfoldings. These are discussed in detail in Appendix A and shall be referred to in the following text.

1. Two examples

As a numerical example, we choose␥^{= 0.6,}␤= 0.07. The three saddle energies aree₀= 0.993 for the Aorbit, e₁= 2.81 for the A⬘ ^{orbit, and e}3= 3.74 for the A⬙ orbit. In the left panel of Fig. 2 we show the stability traces trM共e兲 of the shortest orbits. In the right panel we display the shapes of these orbits in the 共x,y兲 plane. The orbits B⬘ ^{and B}⬙ ^are created in a SNB atetb⯝1.533 and do not exist below this energy; at high energies they are hyperbolically unstable with increasing Lyapunov exponents. Contrary to the stan- dard HH system, only the Aorbit is stable at low energies, while the orbitsA⬘^andA⬙ ^{leave the}e= 0 limit unstable and cross the critical line trM= + 2 at some finite energiese_TCB ande_TCB⬘ to become stable. At higher energies, all three A type orbits undergo an infinite PFB cascade as in Fig.1, each of them converging at its saddle energy. 共We do not show here theRandLtype orbits born at these bifurcations.兲

It is between the pairs of orbitsA⬘^{,B andA}⬙^,B⬘ ^{that we} here observe two TCBs. They occur at the energy e_TCB

= 0.854 447 between the orbits A⬘ ^{and B, and at e}TCB⬘

= 1.644 between the orbitsA⬙^andB⬘. The situation neare_TCB⬘ actually displays an example of a slightly broken PFB which will be discussed in Sec. III C 5.

Another example of a TCB is shown in Fig.3, obtained for the GHH potential with␥^{= 0.75 and}␤= 0. This potential is symmetric about theyaxis and therefore the pairs of orbits A⬘^{, A}⬙ ^{and B}⬘^{, B}⬙ are degenerate, lying opposite to each other with respect to the y axis. The crossing happens at e_TCB= 0.4889 and exhibits the same features as those dis- cussed in the first example.

2. Characteristic properties of the TCB

We now discuss some of the properties of a TCB and compare our numerical results to their analytical predictions

from the normal form of the TCB. For this purpose, we take the example at e_TCB= 0.854 447 seen in Fig. 2, where the orbits A⬘ ^{and B} bifurcate transcritically. Their crossing is shown in Fig.4on an enlarged scale in the upper left panel, where the numerical results for trM共e兲 are displayed by crosses 共orbit A⬘^兲 and circles 共orbit B兲. We see that the graphs of trM共e兲cross the critical line trM= 2 with opposite slopes. Their Maslov indices, differing by one unit, are ex- changed at the bifurcation共see Secs. IV and IV B兲. The up- per right panel displays the numerical action difference ⌬S

=SB−SA⬘ 共circles兲, where the action of each periodic orbit 共PO兲is, as usual, given by

S_PO=

冖

^{p·dq. 共42兲}

In the lower panels, we show the shapes of the orbits in the 共x,y兲 plane below 共left兲 and above 共right兲 the TCB. TheB orbit is seen to have passed through theA⬘orbit at the bifur- cation. The lengths of both orbits increase with energye.

The normal form of the TCB is derived and discussed in Appendix A 2, Sec. C. From it, one can derive the local behavior of the actions, periods, and stability traces of the two orbits in the neighborhood of a TCB. For small deviations ⑀=c共e−e_TCB兲 共with c⬎0兲 from the bifurcation energy, the stability traces go similar to trM共⑀兲= 2⫾2␴⑀^, and the action difference of the two orbits similar to⌬S共⑀兲

= −⑀³/6b²共see Appendix A 2, Sec. C for the meaning of the other parameters兲. These local predictions, given in Fig.4by the solid lines, can be seen to be well followed by the nu- merical results.

The crossing of the graphs trM共e兲of the two orbits at the bifurcation energye_TCB with opposite slopes is a character- istic feature of the TCB 共see Sec. II B 1兲. Since the fixed points of the two orbits coincide at the bifurcation point, their shapes must be identical there. In the present example, the orbitB is a curved libration; the sign of its curvature is changed at the bifurcation, as illustrated in the two lower panels of Fig.4.

We note that a completely different mechanism of stabil- ity exchange of two orbits, which happens through two close-lying PFBs, has been described in 关39兴. The stability diagram may then appear similar to that in the upper left of Fig.4, if the crossing point is not analyzed with sufficient

0.0 0.1 0.2 0.3 0.4 0.5

e

1.99 2.0 2.01

tr M

A’/A’’

B

B’/B’’ A’/A’’

B

B

B’/B’’

FIG. 3. TCB of a degenerate pair of orbitsA⬘^,A⬙^andB⬘^,B⬙^ateTCB= 0.4889 in the GHH potential with␥= 0.75,␤= 0. The degeneracy is due to the reflection symmetry about theyaxis.

numerical resolution. Such a “false transcritical bifurcation”

will be briefly discussed and illustrated in Appendix B.

C. Stability and unfoldings of the TCB

Since the TCB is not a generic bifurcation according to Meyer’s list关10兴, we now address the question under which circumstances it can exist and what its structural stability is.

The GHH systems discussed here have time-reversal symme- try, and it is therefore of interest to study the stability of the TCB under perturbations of the Hamiltonians that destroy this symmetry. In this context, it is important to note that a detailed mathematical study关17兴, in which all generic bifur- cations in systems with time-reversal symmetry are classi- fied, does not mention the TCB; the same holds also for关20兴.

So far we have only found TCBs which involve a straight- line libration. On the basis of the results presented in Sec. II, we believe that in the class of all Hamiltonian systems con- taining a straight-line librating orbit, the TCB is actually the generic isochronous bifurcation of the librating orbit. There- fore, if we find a perturbation of the GHH system that de- stroys the time-reversal symmetry but preserves a straight- line libration, the TCB should also exist there. This will be demonstrated in Sec. III C 4 for a specific example.

A general Hamiltonian H共x,y,p_x,p_y兲 supports the exis- tence of a straight-line libration—which, without loss of gen- erality, may be chosen to lie on theyaxis—if the following conditions are fulfilled:

⳵^H

⳵^{x共0,y,0,py兲= 0,}

⳵^H

⳵^px

共0,y,0,py兲= 0. 共43兲

In the following we will first show how some TCBs are destroyed under perturbations that violate the conditions 共43兲, and how they unfold. We find two types of unfoldings which are also discussed in 关8,21兴 for TCBs in non- Hamiltonian systems. In the first scenario, the TCB breaks up into SNBs. In the second scenario, no bifurcation is left in the presence of the perturbation and the functions trM共⑀兲 approach the critical line trM= 2 without reaching it, so that one may speak of an avoided bifurcation. These scenarios can be described by the extended normal forms given in Appendix A 2, Sec. D. We then also investigate perturbations that fulfill the criteria 共43兲, allowing for the existence of TCBs in systems with or without any discrete symmetries.

1. Addition of a homogeneous transverse magnetic field We first discuss the addition of a homogeneous magnetic fieldB=ezB₀to the Hamiltonian共38兲which is transverse to the 共x,y兲 plane of motion. This is a situation that is fre- quently set up in experimental physics and gives us one im- portant way of breaking the time-reversal symmetry. The momentapi共i=x,y兲in Eq.共38兲are replaced by the standard

“minimal coupling,”

0.84 0.85 0.86 0.87

e

1.995 2.0 2.005

trM

2 - 0.29 (e-0.854447) 2 + 0.29 (e-0.854447)

0.84 0.85 0.86 0.87

e

-1.e-08 0.0 1.e-08

S

- 0.00335 (x-0.854447)³ SB-SA’

-0.5 0.0 0.5

x

-0.3 0.0 0.3

y

-0.5 0.0 0.5

x

-0.3 0.0 0.3

y

A’₄ B₅

B₄ A’₅

e=0.6 e=1.0

A’₄

B₅ A’₅

B₄

FIG. 4. TCB in the GHH potential with␥= 0.6,␤= 0.07. Orbits A⬘ ^{and B}exchange their stabilities 共and Maslov indices ␴= 4 , 5兲 at e_TCB= 0.854 447. Upper left: trMversus energye; crosses共A⬘兲and circles共B兲are numerical results, solid lines the local prediction共A26兲. Upper right: Action difference⌬Sversuse; circles are numerical results and the solid line the local prediction共A27兲. Lower panels: Shapes of the crossing orbits in the共x,y兲 plane before and after the bifurcation.

p_i→p_i−e

cA_i, A=1

2共r⫻B兲, 共44兲 whereAis the vector potential and ethe charge of the par- ticle. This adds the following perturbation to the Hamil- tonian:

␦^H共x,y,p_x,p_y兲=eB₀

2c共xp_y−yp_x兲+1

2

冉

^eB2c0

冊

^{2共x2+y2兲,}

共45兲 which breaks the time-reversal symmetry of Eq.共38兲due to the linear terms inpxandpy, but also breaks the straight-line libration condition共43兲.

As an example, we choose the GHH potential with ␥

= 0.5,␤= 0.1. Here the saddle energy for theA⬘ ^{orbit is e}1

= 3.83; the other saddles are ate₀= 0.9852 and e₂= 6.35. In Fig.5 we show the stability traces trM共e兲 of the orbits A⬘ andB⬘with and without magnetic field. ForB₀= 0共triangles and dashed-dotted lines兲, these orbits A⬘ ^{andB cross ate}bif

= 1.426 65 in a TCB similar to the examples discussed above.

For B₀⫽0 共circles and solid lines兲, they rearrange them- selves into pairsA₄⬘^-B5 andB₄-A₅⬘ colliding in SNBs accord-

ing to the prediction 共A32兲 of the normal form 共A31兲, in which␬ is taken proportional to the value ofB₀.

2. Destruction of the TCB by a perturbation of the potential Another example of the same unfolding of a destroyed TCB is shown in Fig.6. Here the unperturbed GHH potential is the same as that used in Fig.3above, which is symmetric about the y axis. This time we apply a perturbation of the potential alone

␦^V共x,y兲=␬^x⬘^y⬘^3, 共46兲 wherebyx⬘^,y⬘are rotated Cartesian coordinates such that the bifurcatingA⬘ orbit lies on they⬘ axis. Clearly, this pertur- bation does not fulfill the conditions 共43兲 共expressed in the rotated coordinates兲and in fact destroys the original TCB of the orbitsA⬘ ^andB⬘ shown in Fig.3; the same fate happens also to the pairA⬙ ^{and B}⬙ of orbits. We see in Fig. 6 that, again, the original pairs of orbits on either side of the unper- turbed TCB rearrange themselves such as to destroy each other in two pairs of SNBs, each according to the prediction 共A32兲of the corresponding normal form. Since the effective perturbation strengths are different in the two original direc- tions of theA⬘ ^andA⬙ orbits, the splitting between the two pairs of SNBs is slightly different. A problem arises with the nomenclature of the perturbed orbits, which is somewhatad hoc, since all perturbed orbits have become rotations. In the square brackets in the figure we indicate the names of the unperturbed orbits, of whichA⬘^,A⬙are straight line andB⬘^, B⬙curved librations共their stability traces are shown in Fig.3 above兲. The stability traces of the perturbed orbits change drastically at the original bifurcations, but approach those of the unperturbed orbits sufficiently far from the bifurcations.

The inset in the upper left of Fig.6 illustrates one possible unfolding of a destroyed isochronous PFB 共that seen at e

= 0.34 between the orbitsBandB⬘^-B⬙^{in Fig.3兲}and will be commented on in Sec. III C 5 below.

3. An avoided TCB

In Fig.7 we give an example of an avoided bifurcation.

We start again from the same example as in Fig.3, but now we apply the following perturbation:

␦H共x,y,px,py兲=␬⬘^x⬘^2py_⬘^{, 共47兲} again in the same rotated coordinates as for the perturbation 共46兲above. By construction, this perturbation does fulfill the

1.422 1.425 1.428 1.431

e

1.998 2.0 2.002

trM

2 0.39 ((e-1.42665)²-0.00000285)^1/2 2 0.39 (e-1.42665)

B=0.0001 B=0

A’₄

B₅ A’₅

B₄

FIG. 5. Unfolding of a TCB by a transverse magnetic field in the GHH potential with ␥= 0.5, ␤= 0.1. Shown is trM versus scaled energye. Dashed lines and triangles: Prediction共A26兲and numeri- cal results for field strengthB₀= 0 of the unperturbed TCB; Solid lines and circles: Local prediction共A32兲 共with an adjusted value of

␬兲and numerical results forB₀= 0.0001.

0.0 0.1 0.2 0.3 0.4 0.5

e

1.99 2.0 2.01

trM

A’, B’

A’’, B’’

0.336 0.34 0.344

1.9996 2.0 2.0004

A’

A’’

A’

A’’ [B]

A’’

[B]

A’

B’’ B’

[A’’] [A’]

[B’’]

[B’]

A’

A’’

FIG. 6. Unfolding of the TCB shown in Fig.3 under the perturbation共46兲 with ␬= 0.0001共see text for details兲. The labels in brackets关 兴corre- spond to the orbits of the unperturbed system in Fig.3. For the inset, see Sec. III C 5.