Regimes of stability of accelerator modes

Physica D 226 (2007) 1–10 www.elsevier.com/locate/physd

Regimes of stability of accelerator modes Rebecca Hihinashvili a,∗ , Tali Oliker a , Yaniv S. Avizrats a , Alexander Iomin a , Shmuel Fishman a , Italo Guarneri b,c,d a Physics Department, Technion, Haifa 32000, Israel b Center for Nonlinear and Complex Systems, Universit`a dell’Insubria a Como, via Valleggio 11, 22100 Como, Italy c Istituto Nazionale per la Fisica della Materia, via Celoria 16, 20133 Milano, Italy d Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

Received 29 March 2006; received in revised form 13 September 2006; accepted 9 October 2006 Available online 18 December 2006 Communicated by J. Stark

Abstract The phase diagram of a simple area-preserving map, which was motivated by the quantum dynamics of cold atoms, is explored analytically and numerically. Periodic orbits of a given winding ratio are found to exist within wedge-shaped regions in the phase diagrams, which are analogous to the Arnol’d tongues which have been extensively studied for a variety of dynamical systems, mostly dissipative ones. A rich variety of bifurcations of various types are observed, as well as period doubling cascades. Stability of periodic orbits is analyzed in detail. c 2006 Elsevier B.V. All rights reserved.

Keywords: Standart map; Accelerator modes; Involution method; Arnol’d tongues; Bifurcation; Stability analysis

1. Introduction Periodic orbits are the backbone of the description of dynamical systems [1–3], as they enable global understanding of motion in the entire phase space. In the present paper we explore the periodic orbits of the map [4–6]: Jn+1 = Jn + k˜ sin θn+1 + 2π Ω θn+1 = θn + Jn

mod 2π

mod 2π.

(1.1)

This map will be denoted M in this paper. Like all the other maps which are introduced in this paper, it maps the 2-torus onto itself; however, “mod(2π )” will always be left understood in all equations which follow. The map M reduces to the Standard Map [3] for Ω = 0. When k˜ = 0, for each rational value j/ p of Ω the map has periodic orbits of period p. For small k˜ > 0, such orbits still exist in a range of Ω around j/ p. ˜ Ω) This range of Ω decreases with p. The region in the (k, phase diagram, where periodic orbits of period p are found, has the shape of a wedge, with the tip on the Ω axis (k˜ = 0).

∗ Corresponding author.

E-mail address: [email protected] (R. Hihinashvili). c 2006 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2006.10.008

This wedge grows wider as k˜ increases, and intersects other wedges. Furthermore, as k˜ increases more orbits are generated, ˜ Ω ) phase others lose stability, and an extremely complicated (k, diagram is generated (see Fig. 1 of [4]). The purpose of the present paper is to understand this diagram. In the case of dissipative systems, this sort of phase diagrams has been studied extensively [7–9]. The standard sine-circle map [8]: θn+1 = θn − K sin(θn ) − 2π Ω ,

(1.2)

is a paradigm in the study of mode-locking for dissipative maps. If K = 0 and Ω is a rational number j/ p then any trajectory of the sine-circle map returns to its initial value (modulo 2π) after p iterations. For 0 < K < 1, mode-locking is observed; that is, over a range of Ω values around j/ p (the modelocking interval) a periodic trajectory with rational winding number j/ p persists. This periodic orbit attracts all other orbits asymptotically in time, so that all of them eventually acquire this winding number. The widths of the mode-locking intervals are exponentially small in p, and increase as K increases up to K = 1. The regions thus formed in (K , Ω ) parameter space, terminating at K = 0, Ω = j/ p, are known as Arnol’d tongues [7,9]. For K > 1 these tongues intersect, stability

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of orbits is lost and new orbits are generated. In the same way as the sine-circle map (1.2) is representative of dissipative systems, the map (1.1) is representative of area preserving ones. The wedges in the phase space of (1.1), where the periodic orbits of period p are found, will again be referred to as Arnol’d tongues; however, it will be shown that the phase space structure, and the structure of the tongues in particular, is significantly different. This is expected, because orbits of areapreserving maps cannot attract. The map (1.1) was obtained in studies of the quantum dynamics of cold atoms, which are kicked by a standing light wave, and are at the same time accelerated by gravity [5,6, 11,12]. Its periodic orbits, and the stable islands around them, provide a theoretical explanation of the Quantum Accelerator Modes [5,6] which were discovered experimentally [11–13]. Compared to that of dissipative systems, the phase diagram of (1.1) is much more complicated, mainly because overlaps ˜ of different tongues occur at arbitrarily small values of k. Therefore, its numerical exploration demands high precision in determining periodic orbits. A property of (1.1) which proves of decisive help in this task is that (1.1), like the Standard Map, can be written as a product: M = IB I A,

(1.3)

where the maps I A , I B are defined as: I A : (J, θ ) → (−J, θ + J ) I B : (J, θ ) → (−J + k˜ sin(θ ) + 2π Ω , θ ).

(1.4)

and are involutions, that is I2A = I2B = 1 (the identity map). If a map has property (1.3), then periodic orbits which include fixed points of either involution are invariant under both involutions [14,15]. This class does not include all periodic orbits, and we give examples; still, finding orbits in this class reduces to finding zeroes of a function of a single variable, which affords decisive simplification. This method was introduced by Greene [16] for the calculation of the last separating KAM torus of the Standard Map. In the present work this method is used to calculate periodic orbits of (1.1), both analytically, in the perturbative regime in Section 2, and numerically in other regimes in Section 3. Also useful in our study was the time-reversal symmetry [15] of map (1.1), which is expressed by the equation: RMR = M−1 ,

2. Perturbative calculation of periodic orbits In this section the periodic orbits of the map (1.1) are ˜ Throughout this section Jn and calculated to the first order in k. θn are understood to be on the 2π torus, notably they are taken mod 2π ; moreover, Ω is also taken mod 1. First we note that the n-th iterate of the map (1.1) at zeroth order in k˜ is: Jn = J0 + 2πΩ n θn = θ0 +

n−1 X

Jn 0 = θ0 + J0 n + πΩ n(n − 1).

(2.1)

n 0 =0

(1.5)

˜ only the 0th order In order to determine Jn at the 1st order in k, in θ is required, and so, on replacing the 2nd equation of (2.1) in (1.1), and iterating n times, we find:

(1.6)

Jn = J0 + k˜

where R is the area-preserving involution: R : (J, θ ) → (−J, θ + J ).

orbits have j and p coprime. As k˜ is increased these become unstable at some stability border where a period doubling bifurcation takes place. The ratio j/ p does not change as a result of continuity. This period doubling is followed by a period doubling cascade, that satisfies a characteristic scaling rule of such a route to chaos for area preserving systems, and is universal [17]. Also, below the stability border, a pitchfork bifurcation of an orbit of period p to two orbits of the same period takes place, resulting in a cusp in the ˜ Ω ) phase diagram (compare with [7]). The analysis is (k, performed analytically for small k˜ in Section 2 and mostly numerically in Section 3. Examples of periodic orbits that cannot be calculated with the involution method are presented in Section 4. Properties of Gauss sums and the relation to points on periodic orbits are outlined in Appendix A. Our method of numerically implementing the involution method is described in Appendix B. ˜ Ω ) phase As tongues intersect, there are regions in the (k, diagram where orbits characterized by different j/ p coexist. These orbits are found in different parts of the (θ, J ) phase space, and typically do not “interact”. Here we are interested mainly in stable periodic orbits. If the issue of stability is ignored, strong rigorous results can be obtained. In particular the work by Oliker and Wajnryb [18] implies that in the center of the tongue ( j/ p) all the orbits of period p · l (l integer) exist, and their winding number is j/ p. If a line in the tongue ( p1 , j1 ) starts from its tip (vertex) and intersects the boundary of a tongue ( p2 , j2 ), then it intersects ˜ such that j/ ˜ p˜ are in the Farey interval all the tongues ( p, ˜ j) j1 j2 [ p1 , p2 ) [4,19–21], up to some resolution 1/ p, ¯ that is p˜ ≤ p. ¯

Many classic items of the theory of dynamical systems, such as period-doubling cascades, and other bifurcations [1–3] are met in the study of map (1.1). In our analysis we have focused on tongues where Ω at the tip (vertex) takes the values 1 (or 0), 1/2 and 1/3. We searched for periodic orbits such that one of their points is a fixed point of one of the involutions. As the boundaries of the tongue are crossed from inside, periodic orbits are annihilated as a result of coalescence of a stable orbit with an unstable one. Near the tip of the tongue j/ p the

n X

2

sin[θ0 + (J0 − πΩ )n 0 + πΩ n 0 ] + 2πΩ n.

n 0 =1

(2.2) Substitution of this equation in the second equation of (1.1) yields θn = θ0 + J0 n + πΩ n(n − 1) + k˜

n X

(n − n 0 ) sin[θ0

n 0 =1 02

+ (J0 − πΩ )n 0 + πΩ n ].

(2.3)

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For a periodic orbit of period p the following equalities hold:

which can be summed ([22–25], see Eq. (36) of [4]) to give

θ p − θ0 = 2π s J p − J0 = 2π j,

G( p, j) = (2.4)

where the integers s and j are the winding numbers of θ and J ˜ around the torus. At the 0th order in k, j Ω= p

(2.5)

must hold, and so periodic orbits can be found only for rational ˜ periodic orbits can be found values of Ω . For nonvanishing k, in an interval around each rational value of Ω . In what follows ˜ assuming that (Ω − j ) we solve for periodic orbits for small k, p is also small (an assumption that will be verified a posteriori). Therefore we replace Ω by pj in the terms that are proportional to k˜ in (2.2) and (2.3). In order to find a periodic orbit of period p, we seek for a point (θ0 , J0 ) which is at once a point in this orbit and a fixed point of one of the involutions (1.4). The fixed points of I A satisfy J0 = 0

(2.6)

while the fixed points of I B satisfy J0 =

1˜ k sin θ0 + π Ω 2

1˜ k sin θ0 + π Ω + π. (2.8) 2 Since the periodic orbits of the map M of (1.1) are identical to those of its inverse, by time-reversal (1.6) we find that if in a periodic orbit of M there is a point with momentum Ji , then in a periodic orbit of this map, not necessarily the same one, there is a point with momentum −Ji . If there is only one periodic orbit which is stable (or unstable), then each point with momentum Ji has a partner with momentum −Ji in the same orbit, and so if p is odd then J0 = 0 is in this periodic orbit. Therefore J0 of (2.6) is in a periodic orbit of odd period p. On the other hand, for even p, (2.7) or (2.8) is typically in the periodic orbit. From (2.2) and (2.4) we find that in the leading order in k˜ a point on a periodic orbit of period p satisfies   j −Ω 2π p ! ) ( p X 1 k˜ iπ pj [(χ p −1)n 0 +n 0 2 ] iθ0 = e − c.c. (2.9) e 2i p n 0 =1 where χ p = 0 for odd p, where (2.6) holds, while χ p = 1 for even p where (2.7) holds and χ p = 1 + p/j where (2.8) holds. We will solve Eq. (2.9) for θ0 to the first order in k˜ and then show that it is consistent with (2.3) in this order. For this purpose it is instructive to use properties of Gauss sums [4,22–25]. We consider

m=1

e

iπ pj [(χ p −1)m+m 2 ]

where ξ( j, p) is leads to   j 2π −Ω = p

(2.10)

(2.11)

real (see Appendix A). Substitution in (2.9) k˜ √ sin ϑ, p

(2.12)

where ϑ = θ0 + ξ( j, p).

(2.13)

A θ0 that belongs to a periodic orbit must satisfy this equality. The Gauss sum (2.10) also satisfies (this is a corollary of Eqs. (34) and (35) of [4])  p  p + 1 G( p, j) X j p odd iπ p [(χ p −1)m+m 2 ] me = p2 (2.14)  (G( p, j) + 1) p even. m=1 2 Substitution in (2.3) for n = p, where p is odd, and using (2.4) results in:   k˜ p − 1 s Ω − ( p − 1) = √ sin ϑ. (2.15) 2π p 2 p 2 It is easy to see that it is consistent with (2.12) for s = For even p one finds in a similar way that

J0 =

p X

peiξ( j, p) ,

(2.7)

or

G( p, j) =

√

 2π

 √ k˜ p Ω s − p = sin ϑ p 2 2

j ( p−1) . 2

(2.16)

and consistency with (2.12) is possible for s = pj 2 . Eq. (2.12) ˜ One finds solutions in determines the periodic orbits in small k. the range ˜ j − Ω ≤ k√ . (2.17) p 2π p This region has the form of a wedge and for small k˜ its boundaries are the lines Ω=

k˜ j ± √ , p 2π p

(2.18)

and some are presented in Fig. 1. Each tongue is characterized by p and j. The inequality ˜ (2.17) justifies our assumption that | pj −Ω | is at most of order k. We see that indeed for small k˜ the boundaries are well described by (2.18). The point ϑ on the orbit is given by (2.12). Special points are the two at the center of the tongue ϑ = 0, π and the two on its boundaries ϑ = π2 , 32 π . Since ξ( j, p) is known, one can determine θ0 from ϑ, as will be shown in Appendix A (see Table 1). The specific values of θ0 are of no importance for discussion in the rest of this section, since all the results may be ˜ for each value expressed in terms of ϑ. In the regime of small k, ˜ of k and Ω there are two periodic orbits, one is stable and the other is unstable. The two get closer and closer as the boundary

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˜ Ω ) phase diagram. For small k˜ the numerical results (dots) coincide with the analytically determined tongue Fig. 1. Various tongues, marked by ( p, j), in the (k, boundaries (2.18) (solid lines). Table 1 Numerically computed values of 1θ = ϑ −θ0 , compared to ξ( p, j) (see (2.13))

T p (θ0 ) =

p−1 Y

T (θi ).

Tongue ( p, j)

1θ = ϑ − θ0

ξ( p, j)

(2, 1) (3, 1) (3, 2) (4, 1) (4, 3) (5, 1) (5, 2) (5, 3) (5, 4) (6, 1) (6, 5) (7, 1) (7, 2) (7, 3) (7, 4) (7, 5) (7, 6)

0.7855 0.5235 −0.5235 0.7855 2.3567 0.6284 1.2573 −1.2573 −0.6284 0.7857 −2.3558 0.6733 −0.2251 2.0209 −2.0209 0.2251 −0.6733

π/4 ≈ 0.7853 π/6 ≈ 0.5235 −π/6 ≈ −0.5235 π/4 ≈ 0.7853 3π/4 ≈ 2.3561 π/5 ≈ 0.6283 2π/5 ≈ 1.2566 −2π/5 ≈ −1.2566 −π/5 ≈ −0.6283 π/4 ≈ 0.7853 −3π/4 ≈ −2.3561 3π/14 ≈ 0.6731 −π/14 ≈ −0.2243 9π/14 ≈ 2.0195 −9π/14 ≈ −2.0195 π/14 ≈ 0.2243 −3π/14 ≈ −0.6731

A periodic orbit is stable if |Tr[T p (θ0 )]| < 2

˜ 1 (θi ) T p (θi ) = T 0 + kT

The stability of an orbit can be determined as usual from the tangent map. The tangent map of (1.1) is 1 + k˜ cos θ T (θ) = 1

k˜ cos θ 1

(2.22)

where 

1 T0 = 1

0 1

 (2.23)

and cos θi T 1 (θi ) = 0

is approached and coalesce at the tongue boundaries. We tested ˜ numerically the validity of (2.12) for small k.

(2.21)

and unstable if the opposite inequality holds. In order to calculate the product (2.20) to the first order in k˜ we note that in this order (2.19) can be decomposed as





(2.20)

i=0

 cos θi . 0

(2.24)

With the help of the identities Tr(A + B) = Tr A + Tr B and Tr(AB) = Tr(B A) we find that, up to first order in k˜ p Tr(T p (θ0 )) = Tr T 0 + k˜

p X

( p−1)

Tr(T 0

T 1 (θi )).

(2.25)

i=1

 (2.19)

and the tangent map over p iterates along a period- p orbit (also termed Monodromy matrix) is given by

It is easy to see that   1 0 p T0 = p 1

(2.26)

R. Hihinashvili et al. / Physica D 226 (2007) 1–10

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and ( p−1) T0 T 1 (θi )

p cos θi = 0 

 cos θi . 0

(2.27)

Use of the Gauss sum (2.10) along with the result (2.11), and using the fact that the angles along a periodic orbit are given to the zeroth order in k˜ by the 2nd equation in (2.1), one finds that Tr(T p (θ0 )) = 2 + k˜ p 3/2 cos ϑ

(2.28)

where ϑ is related to θ0 by (2.13). At the boundaries of the tongue given by (2.18) the stable and unstable orbits coalesce and Tr(T p (θ0 )) = 2. Inside the tongue the orbits with π2 < ϑ < 32 π are stable and the ones in the other part are unstable. This was numerically tested. 3. Numerical analysis of periodic orbits and stability borders ˜ Ω ) phase diagram was In the previous section the (k, analyzed in the framework of perturbation theory for small ˜ This approach is valid near the tip of a tongue. In this k. section the phase diagram is studied in the non-perturbative region. In this region, the edges of a tongue deviate from the straight lines (2.18) (except in the case p = 1, where (2.18) is exact). Furthermore, this region is characterized by instabilities and bifurcations, which are observed as k˜ is increased. We numerically explore this behavior for some low-period tongues. As periodic orbits coalesce at the edges of a tongue, their precise numerical determination is difficult near the edges; and this, in turn, makes the precise determination of the edges themselves somewhat delicate. An algorithm we have developed to locate the boundary is presented in Appendix B. From (2.12) one finds that if ϑs and ϑu belong to the stable and unstable orbits respectively, then, in the vicinity of the ˜ boundary, and for small k, (ϑu − ϑs )2 = 16π

√ |δΩ | p , k˜

(3.1)

where δΩ is the distance from the boundary in Ω . This equation can be used to check consistency of results. The region of existence of a stable ( j/ p) orbit is bounded on the left and on the right by the edges of the tongue, and from above by a stability border, which is observed in the interior of the tongue (see, e.g. Fig. 2). Whereas at the edges of the ( j/ p) tongue, period p orbits, whether stable or unstable, disappear, at the stability border stable periodic orbits become unstable, and, in the cases considered in this work, a period-doubling cascade follows. Loss of stability is foreshadowed by the perturbative equation (2.28), because the second term in it is negative for the stable orbit, and so it would lead to violation of (2.21), and hence to instability, for |k˜ p 3/2 cos(ϑ)| > 4. According to this argument, the location of the stability boundary in k˜ decreases with the period as p −3/2 (in agreement with earlier work [4]). However, this instability occurs for relatively large ˜ where perturbation theory is no longer valid, and values of k,

Fig. 2. Phase diagram of the j = 1, p = 1 tongue. Full lines show the analytical boundaries of the tongue (straight lines emanating from the point marked by 1 on the horizontal axis) and the analytical stability border of period 1 orbits. Superimposed dots show numerical results. The meaning of symbols (circle, rhombi) is explained in the text. The dash-dotted line marks the stability border of the p = 2 orbit which emerges at the period doubling of the p = 1 orbit. The region delimited by the dotted-line rectangle is described in Fig. 3.

therefore (2.28) may just provide hints there. In particular, it suggests that stable regions are more likely to be found close to the margins of a tongue, where cos(ϑ) is small. This is indeed observed in numerical results: as k˜ increases, the stability border asymptotically approaches the edges of the tongue. All periodic orbits which are calculated in this section are found by the involution method. As explained in Appendix B, they are numerically found from the requirement that one point in the orbit is a fixed point of an involution. In the case of odd period p, this is involution I A of (1.4); for even p, it is typically I B of (1.4). 3.1. The tongue j = 1, p = 1 For this tongue many results are known analytically, and were presented in [5,6,10]. In particular the boundary of the tongue, and the stability border satisfy ([6], Eq. (32)): k˜b = 2π|Ω − 1|, p k˜s = 16 + 4π 2 (Ω − 1)2 .

(3.2) (3.3)

These are shown in Fig. 2 along with numerically obtained data (the latter are shown in order to check consistency of our numerical method of finding the boundary and the stability border). As k˜ increases beyond k˜s the typical scenario of a tripling bifurcation is observed as shown in Fig. 4: a period doubling bifurcation takes place, such that the period 1 orbit turns unstable and a period 2 stable periodic orbit appears. Around the p = 2 orbit, a resonant orbit of period 2 × 3 is formed, as is shown in Fig. 4 for the case marked by a circle in Fig. 2. For Ω = 1, k˜ = 2π a pitchfork bifurcation of the p = 2 orbit into two p = 2 orbits takes place. The regions where such orbits exist are bound by dashed lines in Fig. 2. They are magnified in Fig. 3 and described in detail in the corresponding caption. The bifurcation observed on moving upwards across the stability border is actually the 1st in a period-doubling cascade

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R. Hihinashvili et al. / Physica D 226 (2007) 1–10

Fig. 3. A magnification of the region delimited by the rectangle in Fig. 2. Stable period 2 orbits are found below the line JHG and inside the loop BHB, as well as between the lines EF and BF’, and between BD’ and CD. In the kite shaped region AEBC, two stable orbits of period 2 are found. A very complicated behaviour is found in the vicinity of H where period 2 and period 4 stable orbits coexist.

Fig. 4. Phase portrait for k˜ = 5.5825 > k˜s , and Ω = 1.

(marked in Fig. 2 at Ω = 1.3692 by rhombi). Let us denote by δθn the difference in θ between the points separated by half a period in the orbit of period 2n p, when it becomes unstable; p is the original period at the beginning of the period-doubling cascade. We follow the cascade up to n = 4 and find that the numerical results as n increases are consistent with: 1 k˜n − k˜∞ ∼ n δ 1 δθn ∼ n α

(3.4) (3.5)

where k˜n are values of k˜ where the (n + 1)-th bifurcation is observed. The scaling factors δ and α are universal numbers for the period-doubling route to chaos, in area preserving systems with 2-dimensional phase space. Our estimates agree within 5% with theoretical predictions of the renormalization group treatment, δ ≈ 8.721, α ≈ −4.018 [17]. Similar agreement was found for other tongues discussed below.

Fig. 5. Phase diagram for the p = 2, j = 1 tongue. The region of coexistence of two stable periodic orbits of period 2 is framed by the dotted-line rectangle, and magnified in Fig. 6. The thick dashed lines mark the tongue boundaries (found numerically), and the dash-dotted line is the stability border (found numerically). The solid lines indicate the perturbative analytical result of Eq. (2.18) for the boundary of the tongue. The small rhombi and triangles indicate the period doubling cascade for Ω = 0.6376 and Ω = 0.3624, respectively.

Fig. 6. A magnification of the region delimited by the dashed-line rectangle of Fig. 5. At the dash-dotted lines period-2 orbits lose stability, and stable period4 orbits are formed. On the dashed lines coalescence of one of the period-2 orbits with an unstable periodic orbit occurs. The stability loss at the dashdotted lines is followed by a period doubling cascade. The small rhombi and triangles indicate this cascade for Ω = 0.5122 and Ω = 0.4878, respectively.

3.2. The tongue j = 1, p = 2 The structure of this tongue is presented in Figs. 5 and 6. First we study the phase portrait at the center of the tongue ˜ For k˜ = 3.0 (marked by “o” (Ω = 1/2) for various values of k. in Fig. 5), which is below the stability line, the phase portrait is presented in Fig. 7 (2nd and 3rd slice from top). In the upper slice, the function Fθ (θ0 ) is plotted, the zeros of which are used to locate θ0 of periodic orbits according to the involution method, as explained in Appendix B. A similar analysis was ˜ but keep it performed for all periodic orbits. As we increase k, ˜ lower than the stability border ks , a new stable orbit of period two appears at k˜ ≥ π. Further bifurcations are found for k˜ > π, and these are described in Fig. 6.

R. Hihinashvili et al. / Physica D 226 (2007) 1–10

Fig. 7. Phase portrait for p = 2, j = 1, k˜ = 3.0 and Ω = 1/2. The upper slice presents the function Fθ (θ0 ). The middle slice shows the period two stable orbit, and the lower slice is a magnification of one of its islands.

The bifurcation leading from one stable period-2 orbit to two distinct stable orbits with period 2, which is numerically observed at k˜ = π , is easily obtained analytically. It is a standard pitchfork bifurcation. To see this, we iterate twice the map (1.1). The condition for a period two orbit (modulo 2π) is ! ˜ π k = 0, (3.6) k˜ sin θ0 + k˜ sin θ0 + sin θ0 + 2 2 where Ω = 1/2 and (2.7) were used. When k˜ ≤ π there is one stable orbit. At k˜ = π, this trajectory is {(π/2, π ), (3π/2, π)} and is marginally stable (the trace of ˜ two the tangent map Tr[T2 (θ0 )] = 2). When k˜ = π + δ k, new stable trajectories emerge. To find these zeros of Eq. (3.6) we replace θ0 = π/2 + δθ0 and neglect terms of order of ˜ 0 2 ) ∼ O(δθ0 4 ) and higher. This yields the following O(δ kδθ cubic equation for δθ0 :  s  s ˜ ˜ 2δ k 2δ k + δθ0  ·  − δθ0  = 0. (3.7) δθ0 ·  π π The same equation is obtained from the substitution θ0 = 3π/2 + δθ0 using (2.8). Eq. (3.7) yields three periodic orbits with period 2. One of them has δθ0 = 0. It continues the unique orbit at k˜ = π and, like that orbit, it is marginally stable at this order; numerically, it was found that it is actually ˜ The other two orbits have unstable, q and that 0 6= δθ0 = o(δ k). δθ0 = ±

2δ k˜ π .

They are stable at this order, sharing the same

7

˜ and are numerically found to be value of Tr[T2 ] = 2 − 2π δ k, really stable. On further moving upwards along the axis, that marks the center of the tongue (that is, increasing k˜ while Ω = 1/2), the two period-2 stable orbits persist until point B is reached. There they turn unstable, and give rise to two stable period-4 orbits. Two stable period-2 orbits (which we term “a” and “b” for convenience) are actually observed inside the whole kiteshaped region AE BC in Fig. 6. Each of them is observed, alone, in a larger region. For instance, “a” is observed in between the lines AD, E D 0 , and “b” is observed in between AF and C F 0 . On the line AD, “a” coalesces with an unstable orbit and disappears, and “b” disappears in the same way along the line AF. Across the line E D 0 , “a” bifurcates to a period-4 orbit, and the same happens to “b” across C F 0 . Resonances of order 3 similar to those found for the p = 1 tongue and presented in Fig. 4 are found also here. As k˜ is increased above the stability border (dot-dashed line in Fig. 5), a period doubling cascade is started. A few steps are marked by rhombi and triangles in Figs. 5 and 6. So far, the periodic orbits in the tongue p = 2, j = 1 were found identifying one point of the orbit as a fixed point of I B . There is also at least one stable periodic orbit where none of its points are fixed points of I B , but some are fixed points of I A . For Ω = 1/2 one such orbit of period 4 is {X 0 = (0, 0), X 1 = (π, 0), X 2 = (0, π ), X 3 = (π, π )}, as is easily checked. The points X 0 and X 2 are fixed points of I A while X 1 and X 3 are interchanged by this involution. We checked (by calculating the trace of the tangent map) that the orbit is stable for 0 < k˜ ≤ 0.7320 and 2.7320 ≤ k˜ ≤ 2.8284. 3.3. The tongue j = 1, p = 3 Our numerical findings about the j = 1, p = 3 tongue are summarized by Figs. 8 and 9. The stable period-3 orbit loses its stability going upwards across the stability border, the details of which are shown in Fig. 9. However, such a stable orbit reappears on further ˜ it is observed in the region bound by the lines increasing k; 0 L R and L R 0 . Between A and B in Fig. 9 there exist two stable periodic orbits of period 3 which we call “a” and “b”. Both turn unstable at B. “a” coalesces with an unstable orbit and disappears on the line AD, and “b” disappears in the same way along AF. When crossing C F 0 , “b” loses stability, and the same happens to “a” along E D 0 . 4. Orbits that are not related to involutions In the previous sections periodic orbits that include fixed points either of I A or I B were found. Here we give examples of periodic orbits, that do not include such fixed points at all. Such orbits cannot be computed by the method of involutions, which was used in the previous sections, and we shall describe a method of computing them. First we find such orbits for k˜ = 0. For Ω = j/ p, a point (J0 , θ0 ) belongs to a periodic orbit of period p if J0 p + π j ( p − 1) = 2π s, with s integer, and θ0 arbitrary, as one can see from (1.1) and (2.3). It is easy to find

8

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Fig. 8. Phase diagram for the p = 3, j = 1 tongue, indicating the region of coexistence of two stable period-3 orbits, which is magnified in Fig. 9. The solid line marks the tongue boundaries found numerically, and the dash-dotted line is the stability border also found numerically. The dashed lines bound the region in which two period-3 orbits coexist. As each of these lines is crossed from the inside of this region, one periodic orbit coalesces with an unstable orbit, and is annihilated. At the stability border a period doubling cascade begins. The rhombi and triangles illustrate such a cascade for Ω = 0.3936 and Ω = 0.2285, respectively. Between lines L R and L 0 R 0 a stable period 3 orbit is again observed.

Fig. 10. The two period-6 orbits found at Ω = 1/2 for k˜ = 0.345. The two lines indicate the invariant lines under the I B involution (as given by (2.7) and (2.8)), thus exhibiting that none of the points of these periodic orbits is a fixed point of the I B involution. It is also evident that none of the points is a fixed point of I A , since none of the points is located on the invariant line of I A which is given by (2.6). Points in one periodic orbit are marked by ai i = 1, 2, . . . , 6 and points in the other by bi . The involution I A transforms the former orbit into the latter, as follows: a1 −→ b1 , a2 −→ b6 , a3 −→ b5 , a4 −→ b4 , a5 −→ b3 , a6 −→ b2 , while I B transforms them as follows: a1 −→ b2 , a2 −→ b1 , a3 −→ b6 , a4 −→ b5 , a5 −→ b4 , a6 −→ b3 .

5. Summary and discussion

Fig. 9. A magnification of the region delimited by the rectangle in Fig. 8.

numerically values of j, p, s such that none of the points in such an orbit is fixed under I A or I B : for instance, with p = 6, j = 3, and s = 2, it is easy to check that none of the fixedpoint conditions (2.6), (2.7) or (2.8) is satisfied for any of the points in the periodic orbit. Such an orbit can then be used to find periodic orbits where none of the points is a fixed point of the involutions, also for k˜ > 0. For this purpose take k˜ > 0 but small. Iterate p times the line of the fixed points of the map with k˜ = 0, generating a new line M p (J0 , θ0 ). Let ( J˜0 , θ˜0 ) be a point of the intersection of these two lines. Although this is not necessarily a fixed point of M p for k˜ > 0, it is reasonable to look for one in its vicinity, and indeed we found such orbits of period p = 6. They are plotted in Fig. 10. These orbits are also related by time-reversal symmetry (1.5) and (1.6).

In this paper the tongues with j/ p = 1, 1/2, 1/3 were analyzed in the perturbative regime. A large variety of scenarios for the disappearance of orbits, the loss of their stability and bifurcations was found. We expect similar behavior for the higher order tongues, but the verification is left for further research. For higher values of k˜ some novel results may be found but such an exploration is out of the scope of the present paper. The research in the present work was confined to the classical regime. The fingerprints of the rich classical behavior that was found on the quantum dynamics in the semiclassical regime may be of great interest. In particular the effect of bifurcations on the eigenstates or the scattering states of the quantum system may reveal novel results. Effects similar to the ones found for quantum ratchets [26] and “flooding” [27] are expected. Also states on islands around periodic orbits related by symmetry may result in states of novel properties. These may lead to effects observable in atom optics experiments. Acknowledgments This work was partly supported by the Israel Science Foundation (ISF), by the US-Israel Binational Science Foundation (BSF), by the Minerva Center of Nonlinear Physics of Complex Systems by the fund for promotion of research at the Technion, by the Shlomo Kaplansky academic chair and by the Institute of Theoretical Physics at the Technion. Highly instructive discussions with Roberto

R. Hihinashvili et al. / Physica D 226 (2007) 1–10

Artuso, Andreas Buchleitner, Michael d’Arcy, Simon Gardiner, Shai Haran, Zhao-Yuan Ma, Zeev Rudnick, Gil Summy, are acknowledged. I.G. acknowledges partial support from the MIUR-PRIN project “Order and Chaos in extended nonlinear systems: coherent structures, weak stochasticity and anomalous transport”. Appendix A. Gauss sums and periodic orbits In this appendix some properties of Gauss sums that are relevant for the present paper will be reviewed. Specifically we focus on the sum (2.10) that is related to p X iπ [lm+ jm 2 ] ˜ j, p, l) = 1 G( ep , p m=1

(A.1)

where in our case, l = j (χ p − 1) vanishes for even p if (2.7) holds, and it is equal to p if (2.8) holds, while it takes the value − j for odd p. Using elementary calculations, one can show [4] that in our case (A.1) is just (2.11) divided by p, namely ˜ j, p, l). G( j, p) = p G(

(A.2)

The calculation of ξ( j, p) is more complicated and will be summarized in what follows. In the standard number theoretical literature ([24,25]), sums like (A.1) are calculated for l = 0, even j and odd p. Hannay and Berry [22] generalized to other cases of (A.1) (See also [23]). They show that this sum reduces to the average of the summands and is given, for j and p mutually prime, by Eq. (14) there, which reads: ! "  2 #  1 iπ iπ j l j 2   exp − ( p − 1) − ( j \ p) √   p 4 p p  p     p odd,! j even, l even        iπ i4π j 1 j  √ exp − ( p − 1) − (4 j \ p)2 l 2 ˜ j, p, l) = p p 4 p G(    p odd, j odd, l odd   ! "   2 #   1 iπ iπ j l  p 2  √ exp j− ( j \ p)   p 4 p p  j   p even, j odd, l even.

The symbol ( j \ p) marks the inverse (mod p) of j, notably j ( j \ p) = 1

mod p

(A.4)

which can be calculated with the help of (whenever j is mutually prime with p): ( j \ p) = j ϕ( p)−1

mod p,

A more straightforward way of determining the value of a Legendre symbol is given by the next formula, which is based on Fermat’s little theorem   p−1 j = j 2 mod p (A.7) p keeping in mind that p is an odd prime and that j is not divisible by p. For our purposes it is convenient to write the Jacobi symbol in the form       iπ j j = exp 1− . (A.8) p p 2 It is useful to note that      −j j −1 = p p p and that   iπ −1 = e 2 ( p−1) p and also that   j =1 1

(A.5)

where ϕ( p) is Euler’s totient function, that is, the number  of integers less than p that have no common divisors with p. ab is the Jacobi symbol, where a and b are integers so that b is odd and does not divide a. The Jacobi symbol assumes the  values ±1 only, and is the product of the Legendre symbols qa for all the prime factors q of b. The Legendre symbol is defined by  1 if there is an integer      a m such that m 2 = a mod q ≡ (A.6) q −1 if there is no integer    m such that m 2 = a mod q.

(A.9)

(A.10)

(A.11)

for all integers j. Other useful formulae are given in Appendix B of [22] and in [24] and [25] (see also [23]). We note that in our case for an even p, l = 0, while for an odd p, l = − j. In this case (A.1) takes the form ˜ j, p, l) = √1 eiξ( p, j) G( p

(A.3)

9

where, for even p corresponding to (2.7),     π j p ξ( p, j) = 1− + , j 2 2 and for even p corresponding to (2.8)     π j p 2 ξ( p, j) = 1− − 2 ( j \ p) , j 2 p while for odd p   π  1 j  1 − − ( p − 1)   p  2 2     1 j  2  ( j \ p) l 2 j even  − 2 p   ξ( p, j) = π  1 j   1− − ( p − 1)   p  2  2   8 j  2  − (4 j \ p) l 2 j odd. p

(A.12)

(A.13)

(A.14)

(A.15)

The phase ξ( p, j) determines the relation between θ0 and ϑ via (2.13) and can be used to check the accuracy of the calculations. At the center of the tongue ϑ = 0 or ϑ = π while at the boundaries ϑ = π2 or ϑ = 3π 2 . In Table 1 we compare the values of the numerical results 1θ = ϑ − θ0 for k˜ = 0.001 and θ0 at the center of some tongues with ξ( p, j) calculated from (A.13) and (A.15).

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Appendix B. A numerical method for the calculation of periodic orbits In the numerical solution for periodic orbits we make use of (2.6) and (2.7) in order to eliminate J0 . Then we impose the periodic orbit conditions (2.4) and require that θ0 is a common zero of the functions FJ (θ0 ) = k˜

p X

sin θ j + 2π Ω p − 2π j,

(B.1)

j=1

where j is determined by the tongue ((2.4) and (2.5)), and Fθ (θ0 ) = p J0 + k˜

p X

( p − j) sin θ j + πΩ p( p − 1) − 2πs,

j=1

(B.2) that are obtained after iterating the map (1.1) p times. We solved for zeros of FJ (θ0 ) and sin[ 12 Fθ (θ0 )]. This is because j is known, while s is unknown, and since the function Fθ (θ0 ) should be taken mod 2π . Then only the common zeros of FJ (θ0 ) and sin[ 21 Fθ (θ0 )] are kept. The elimination of J0 by the involution method reduced the problem to one of finding zeroes of a function of a single variable, which is a relatively easy task. We used the MATLAB7 routine “fzero” for this purpose [28]. In order to locate the boundary, we note that the zeroes corresponding to the stable and unstable orbits are separated by an extremum of the function in question, say FJ (θ0 ). At the boundary these zeroes coalesce and at this extremum the relevant function vanishes (at the extremum FJ = 0). The extremum can be easily found by the MATLAB7 routine “fminbnd” [28]. The ˜ Ω ) space is the locus of points boundary of a tongue in the (k, of vanishing extremum. It was found by successive change of k˜ and Ω starting from the vertex k˜ = 0, Ω = j/ p. The boundary was found by the overshoot method, making use of the fact that the value of the function at the extremum changes sign when the boundary of a tongue is crossed. References [1] E. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge UK, 2002. [2] P. Cvitanovic, R. Artuso, P. Dahlqvist, R. Mainieri, G. Tanner, G. Vattay, N. Whelan, A. Wirzba, Classical and Quantum Chaos Part I: Deterministic Chaos, 2003. at: www.nbi.dk/ChaosBook version 10.01.06.

[3] A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Springer, New York, 1983. [4] I. Guarneri, L. Rebuzzini, S. Fishman, Arnold tongues and quantum accelerator modes, Nonlinearity 19 (2006) 1141–1164. [5] S. Fishman, I. Guarneri, L. Rebuzzini, Phys. Rev. Lett. 89 (2002) 084101. [6] S. Fishman, I. Guarneri, L. Rebuzzini, J. Stat. Phys. 110 (2003) 911. [7] M. Schell, S. Fraser, R. Kapral, Phys. Rev. A 28 (1983) 373. [8] H.G. Schuster, Deterministic Chaos, an Introduction, VCH, Weinheim, 1995. [9] M.H. Jensen, P. Bak, T. Bohr, Phys. Rev. A 30 (1984) 1960. [10] M.B. d’Arcy, G.S. Summy, S. Fishman, I. Guarneri, Phys. Scripta 69 (2004) C25. [11] M.K. Oberthaler, R.M. Godun, M.B. d’Arcy, G.S. Summy, K. Burnett, Phys. Rev. Lett. 83 (1999) 4447; R.M. Godun, M.B. d’Arcy, M.K. Oberthaler, G.S. Summy, K. Burnett, Phys. Rev. A 62 (2000) 013411. [12] S. Schlunk, M.B. d’Arcy, S.A. Gardiner, G.S. Summy, Phys. Rev. Lett. 90 (2003) 124102. [13] Z.Y. Ma, M.B. d’Arcy, S.A. Gardiner, Phys. Rev. Lett. 93 (2004) 164101. [14] J.D. Meiss, The standard map, in: A. Scott (Ed.), Encyclopaedia of Nonlinear Sciences, Routledge, NY, 2005. [15] J.S.W. Lamb, J.A.G. Roberts, Physica D 112 (1998) 1. [16] J.M. Greene, J. Math. Phys. 20 (1979) 1183. [17] J.M. Greene, R.S. Mackay, F. Vivaldi, M.J. Feigenbaum, Physica 3D (1981) 468. [18] T. Oliker, Characterization of periodic orbits by topological considerations for a kicked particle, M.Sc. Thesis, Technion, 2005; T. Oliker, B. Wajnryb (in press). [19] J. Farey, On a curious property of vulgar fractions, London, Edinburgh and Dublin Phil. Mag 47 (1816) 385. See, however, a historical note in Ref. [20], p. 36. [20] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon press, Oxford, England, 1979 (Ch. 3). [21] I. Niven, H.S. Zuckerman, An Introduction to the Theory of Numbers, Wiley, NY, 1960. [22] J.H. Hannay, M.V. Berry, Physica D 1 (1980) 267. [23] M.V. Berry, S. Klein, J. Modern Opt. 43 (1996) 2139; M.V. Berry, E. Bodenschatz, J. Modern Opt. 46 (1999) 349. [24] S. Lang, Algebraic Number Theory, Addison-Wesley, New York, 1970. [25] P.G.L. Dirichlet, with supplements by R. Dedekind, in: Lectures on Number Theory, in: History of Mathematics, vol. 16, American Mathematical Society, 1999. [26] T.S. Monteiro, P.A. Dando, N. Hutchings, M. Isherwood, Phys. Rev. Lett. 89 (2002) 194102; T. Jonckheere, M. Isherwood, T.S. Monteiro, Phys. Rev. Lett. 91 (2003) 253003; P.H. Jones, M. Goonasekera, H.E. Saunders-Singer, T.S. Monteiro, D.R. Meacher. preprint (physics/0504096). [27] R. Ketzmerick, A.G. Monastra, Phys. Rev. Lett. 94 (2005) 054102. [28] MATLAB7 User-guide.