recurrence of order in chaos - Semantic Scholar
October 3, 2005
13:55
01371
International Journal of Bifurcation and Chaos, Vol. 15, No. 9 (2005) 2865–2882 c World Scientific Publishing Company
RECURRENCE OF ORDER IN CHAOS G. CONTOPOULOS and M. HARSOULA Academy of Athens, Research Center of Astronomy, 4 Soranou Efessiou Str. 11527 Athens, Greece R. DVORAK and F. FREISTETTER Universit¨ at Wien, Institut f¨ ur Astronomie, T¨ urkenschanzstrasse 17, A-1180 Wien, Austria Received July 6, 2004; Revised August 4, 2004 The standard map x = x + y , y = y + (K/2π)sin(2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ∆K, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results. Keywords: Chaos; order; standard map; periodic orbits; recurrence; stability.
1. Introduction A problem of theoretical interest in highly perturbed dynamical systems is whether chaos is complete. For example, the standard map x = x + y (mod 1) (1) K sin(2πx) y =y+ 2π (where (mod 1) refers to both formulae), seems to become completely chaotic for K larger than about K = 7.6. Nevertheless, there are small islands of stability for arbitrarily large K. Such islands appear close to the points where the asymptotic curves of the unstable periodic orbits have tangencies [Newhouse, 1983]. The most general result for the standard map was derived by Duarte [1994] who has shown that the values of K when there are islands of stability
form a residual set everywhere dense. That is between any two values of K that differ by ∆K there are intervals of K for which the standard map has small islands of stability. The total measure of these intervals of K within a given ∆K may be small. Giorgilli and Lazutkin [2000] estimated that this measure is of order 1/K. But, although there may be many values of K without any islands at all, the standard map does not become an Anosov system for large K, because Anosov systems are structurally stable, i.e. a small perturbation gives again an Anosov system. The main question is now whether such islands are appreciable, i.e. whether they cover a part of phase space that is not very small. The numerical evidence is that for large values of K most islands are extremely small. For example, for K ≤ 6 there are appreciable islands of stability [Fig. l(a)], but these islands disappear for larger K
2865
October 3, 2005
2866
13:55
01371
G. Contopoulos et al.
of values of K, at least up to K = 200. These islands appear in a recurrent (almost periodic) way. Similar, albeit smaller, islands appear for even larger K. We study this recurrence phenomenon in Sec. 2, and give an explanation of it in Sec. 3. In Secs. 4 and 5 we study the underlying periodic orbits, and in Sec. 6 we present our conclusions.
2. Recurrence of Islands
(a)
We explored systematically the phase space (0, l) × (0, l) in the standard map for values of K beyond K = 4.5 with a step dK equal to dK = 0.0001. The total area of the islands was derived as follows. We divided the phase space (x, y) into 106 cells of size 0.001 × 0.001 and found how many cells contained (or not) iterates of an orbit starting in the chaotic sea. We tested different initial conditions (x0 , y0 ) and different numbers of iterations N . We found that we needed at least N = 108 iterations in order to find a reliable value of the area A of the islands (number of empty cells divided by 106 ). In some cases we had to reach N = 5 × 109 . This method cannot give details smaller than the grid size (0.001 × 0.001). Thus in several cases we used a finer grid, namely [(l/3000) × (1/3000)], in order to find more accurate values of A. In Fig. 2, we give the total size (area) of the islands as a function of K. The first large islands in this figure appear around K = 4π (Fig. 3). These islands cover over 0.2% of the total phase space at their maximum and they belong to two groups.
(b) Fig. 1. The structure of the phase space of the standard map for (a) K = 5 and (b) K = 8.
[K = 8; Fig. 1(b)] and any remaining islands seem to be below the level of detectability. However, for larger values of K further islands were found [Chirikov, 1970; Lichtenberg & Lieberman, 1992; Rom-Kedar & Zaslavsky, 1999; Zaslavsky, 2002]. In the present paper we make a systematic study of such islands. We found appreciable islands of stability for certain intervals
Fig. 2. The area A of the islands of groups I–III as a function of the control parameter K for values between 10 and 70.
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
2867
i.e. there are two period-4 islands near each other (Fig. 6). The maximum islands of this group, for K = 15.6, occupy 0.15% of the total area of phase space. However, these islands extend over smaller intervals of values of K.
Fig. 3. The islands around K = 4π in detail. The solid line represents islands outside the x-axis (group I). The dashed line refers to the islands on the x-axis (group II).
Group I represents islands outside the x-axis, as in Fig. 1(a). These islands surround two points belonging to a stable periodic orbit of period-2. The first point is between (x = 0.5, y = 0) and (x = 0.75, y = 0.5) while the second is symmetric to the first with respect to the center of Fig. 1(a) (x = y = 0.5). The sizes of the islands of group I are given by a solid line in Fig. 3. This group contains also bifurcations of this period-2 family of periodic orbits (see Sec. 4). The second group (group II) represents islands close to the x-axis (y = 0) (Fig. 4). Such islands appear for K a little larger than 4π (dashed line of Fig. 3). Similar islands of groups I and II appear close to K = 2nπ
(n ≥ 1)
Fig. 4.
Islands of group II near the x-axis for K = 6.7.
(2)
The islands of group I exist mainly for K ≤ 2nπ, but also a little beyond K = 2nπ, while the islands of group II exist for K > 2nπ. There are also islands, forming a third group (group III), that appear close to K = (2n + 1)π
(n ≥ 1)
(3)
This group contains islands of period-4, two of them being near the x-axis (y = 0) and two outside this axis (Fig. 5). These islands appear in pairs,
Fig. 5. One of the two sets of four islands for K = 15.596 (group III islands).
October 3, 2005
2868
13:55
01371
G. Contopoulos et al.
Fig. 7. The maximum size of the islands as a function of K for groups I and II (dark dots) and for group III (white squares). The curves represent Eq. (4). (a)
(b) Fig. 6. Islands of the group III for K = 15.5796. (a) One pair of islands on the x-axis and (b) another pair of islands outside the x-axis. The large dots represent the stable and unstable orbits. The scattered points belong to a chaotic orbit in the unstable manifold of the periodic orbit (x = 0.5, y = 0).
The maximum sizes of the islands decrease with K according to power laws A = aK b
(4)
where a = 0.72 and b = −2.30 for groups I and II and a = 1.22 and b = −2.45 for group III (Fig. 7). We notice that b is not very far from the theoretical
value b = −2 [Chirikov, 1979]. The islands become smaller than A = 0.00001 for K larger than about K ≈ 130. The intervals of ∆K where we have islands of the groups I, II and III are approximately inversely proportional to K (Secs. 4 and 5). Between the islands of groups I, II and III, there are smaller islands for very small intervals of values of K. Such islands appear in every interval ∆K, as predicted by the theorem of Duarte [1994]. The sizes of the islands around their maximum increase and decrease in a typical way. In Fig. 3 we see that as K increases the islands of the group I appear for the first time for K = 11.87 and their size increases abruptly. The curve A in Fig. 3 is fractal with an infinite number of intervals where it increases and decreases. The area of these islands (and their bifurcations) reaches and remains zero for K ≥ 12.8. The second set of islands (group II) starts at K = 4π. Their size increases and decreases in the same way as group I islands. The increase and decrease of the size of the islands is well understood [Contopoulos et al., 1999]. Let us consider the island around the stable invariant point O1 [Fig. 1(a)]. In fact there are two such invariant points, O1 and O1 , symmetric with respect to the center (x = y = 0.5), that are generated by bifurcation from the periodic orbit O (x = 0.5, y = 0) for K = 4 (see Sec. 4). Here we consider only the island O1 , around a point with x > 0.5. As K increases beyond K = 4 the orbit O becomes unstable and some chaos is generated around it. For K < 4.35 this chaotic domain does
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
not communicate with the large chaotic sea that fills most of the phase space. In fact there are closed invariant curves (KAM curves) surrounding O, and also O1 and O1 . However, for K > 4.35 these KAM curves are destroyed and the islands O1 and O1 are floating in the large chaotic sea [Fig. 1(a)]. The size of the island O1 is defined by the last KAM curve, that surrounds the stable periodic orbit O1 and separates the island from the large chaotic sea outside it. As K increases, the last KAM curve expands until it is destroyed at a critical value K = Kn/m and becomes a cantorus. Then the size of the island decreases abruptly, and is limited by a new last KAM curve. This transition is related to a resonance n/m. Namely for K < Kn/m there are m secondary islands inside and close to the original last KAM curve. Between these islands there are m unstable invariant points. Near these points there is chaos, that surrounds also the m islands, and is separated from the outer chaotic sea by the last KAM curve. When K > Kn/m the last KAM curve surrounding the m islands becomes a cantorus. Then the chaotic region between the m islands communicates with the large chaotic sea through the cantorus and the m islands are floating in the chaotic sea. Then the original island is surrounded only by KAM curves closer to its center than the m islands, and the new last KAM curve is inside the m islands. As K increases further this new KAM curve expands, until it is also destroyed. Some of the main resonances responsible for the abrupt decrease of the size (area) of the islands are marked in Fig. 3. The same resonances, like 3/5, 1/2, 3/7, 1/3, 1/5 appear in the interval K = 4.5 − 7.6 (Fig. 8). Figure 8 is similar to Fig. 2 of [Contopoulos et al., 1999]. There are only small differences due to the fact that in the previous paper we measured the distance of the last KAM curve from the central invariant point O1 along the y-direction, while here we give the area of the island. The point O1 becomes unstable for K = 2π ≈ 6.283 and for a little larger K (K = 6.335) the last KAM curve around O1 is destroyed and the islands that surround O1 vanish. However, if we measure the area of the islands we do not reach zero for K = 6.335. In fact, there are two new islands O2 and O2 on the left and on the right of O1 , that grow larger as K increases beyond K = 2π. The changes of the area A of the secondorder islands beyond K = 6.335 are similar to the changes of A along the original islands. The
Fig. 8.
2869
The sizes of the islands for K between 4.5 and 8.
second-order periodic orbits O2 and O2 are stable up to K = 6.594 and they become unstable for larger K. Then we have the formation of four stable points until K = 6.630, and so on (see Sec. 4). The successive intervals of stability decrease by a factor 8.72 [Benettin et al., 1980]. Finally, for K > 6.635 there are infinite bifurcating families, which are all unstable. In Figs. 3 and 8 a second set of islands, represented by dashed lines appears, for larger values of K (islands of group II). These islands are formed around stable periodic orbits on the x-axis (see Sec. 4). The increases and decreases of these islands follow the same pattern as in the main islands around O1 . The form of Fig. 3 seems to be universal. The islands near every value of K = 2nπ of groups I and II are similar to Fig. 3. Two examples are given in Fig. 9 for K around K = 6π and K = 50π. All these islands with n ≥ 2 surround irregular periodic orbits that are generated from zero at tangent bifurcations. The only exception are the islands with n = 1 and in particular for K ≤ 4.8 (Fig. 8). In this case the average size of the islands increases to large values as K decreases below K = 4.8. In fact, the islands around O1 and O1 are generated from the larger island around O (x = 0.5, y = 0) that exists for K < 4.35. The area of the islands at the resonance 1/3 in Figs. 3 and 8 vanishes for a particular value of K.
October 3, 2005
2870
13:55
01371
G. Contopoulos et al.
(a)
Fig. 10. The sizes of the islands of groups I and II for K near 6π (solid line). In this figure we superimposed the sizes of the islands near K = 4π (dashed line) after appropriate scalings in K and A.
same for K near 4π (dashed line). Then the quantity chi-square is given by the standard formula [Press et al., 1992]. 2
χ =
N (Ai1 − Ai2 )2 i=1
(b) Fig. 9. Islands of groups I and II for K around (a) K = 6π ≈ 18.5, and (b) K = 50π ≈ 157. Grid size (1/3000) × (1/3000).
The vanishing in Fig. 8 happens near K = 5.65 (compare with Figs. 6 and 7 of [Contopoulos et al., 1999]). A similar 1/3 resonance appears when the central period-2 orbit becomes unstable for K = 12.28 in Fig. 3. The variation of the size of the islands is similar for all the islands of groups I and II. A comparison of the size of the islands of groups I and II for K near 6π and near 4π after appropriate scaling is given in Fig. 10. A scaling is defined by the requirement that one maximum A and two main dips should coincide. Then a measure of the agreement between the two curves is given by the chi-square test. Let Ai1 be the value of the size of the islands for K near 6π (solid line) and Ai2 the
Ai1 + Ai2
(5)
where i extends from 1 to N , and N is of the order of 500. If χ2 is smaller than 10−2 the agreement is very good, while if χ2 is larger than 10−1 the agreement is bad. In Fig. 10 we compare the variations of the sizes of the islands near K = 6π and K = 4π. The value of chi-square is χ2 = 1.6 × 10−2 for orbits of group I and χ2 = 4.4 × 10−3 for group II. If we compare a curve of group I with a curve of group II we find again a very good agreement, namely χ2 = 8.5 × 10−3 . On the other hand the islands of group III increase and decrease in a somewhat different way [Figs. 11(a) and 11(b)]. These islands appear in pairs, thus they are different from the islands of groups I and II, but they are similar to each other. A comparison of the islands of group III after an appropriate scaling is given in Fig. 12. The corresponding chi-square is χ2 = 6.1 × 10−3 . But if we compare a curve of group III with a curve of group II, then chi-square is χ2 = 2.4 × 10−1 , i.e.
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
(a)
2871
Fig. 12. The sizes of the islands of group III for K = 5π (solid line) on which we have superimposed the sizes of the islands near K = 3π (dashed line) after appropriate scalings in K and A.
3. Explanation
(b) Fig. 11. The sizes of group III islands close to (a) K = 3π = 9.4, and (b) K = 9π = 28.3. Grid size (1/3000) × (1/3000).
much larger. Thus although the resonances in groups I, II and III are the same, the increase and decrease of the islands of group III is different from the other two cases. This is due to the fact that the islands of group III are close to each other and they affect the growth of both of them. Similar figures appear in maps similar to the standard map [Karney et al., 1982], and in Hamiltonian systems. e.g. systems of two coupled oscillators [Contopoulos et al., 1999] or the restricted three-body problem [H´enon, 1966, Fig. 19]. In fact as the perturbation increases the islands first increase in size, but later, when chaos becomes dominant, they decrease and disappear. However, a detailed comparison of the sizes of the islands in different systems is provided here for the first time.
The islands in the intervals of K near K = 12, K = 18, etc. are not only similar in size with the islands close to K = 6 but they are also located near the same regions of phase space (near (x = 0.7, y = 0.4)). These regions are around the stable periodic orbits whose existence is proven in Sec. 4. The area of the islands is limited by the asymptotic curves emanating from the periodic orbit O (x = 0.5, y = 0), which is unstable for K > 4. These asymptotic curves intersect at an infinity of homoclinic points (Fig. 13 for K = 4.7) and make oscillations between these homoclinic points, forming the so-called “lobes”. The main homoclinic point H is at about equal distances from O along the unstable and stable asymptotic curves U and S. The oscillations are relatively small in the region surrounding the stable periodic orbit O1 , for K not much larger than K = 4 and leave room for the invariant curves surrounding O1 . When K becomes larger (K = 6) the lobes formed by the asymptotic curves become longer. However, in Fig. 14 we see that the asymptotic curves still leave space for the appearance of islands around O1 . For even larger K the lobes of the asymptotic curves become even longer and their oscillations become larger and invade the region around O1 (Fig. 15 for K = 8). Furthermore, the orbit O1 and its higher order bifurcations become all unstable and their asymptotic curves intersect the asymptotic curves of the orbit O at an infinity of heteroclinic points,
October 3, 2005
2872
13:55
01371
G. Contopoulos et al.
Fig. 13. The asymptotic curves, unstable (thick line) and stable (thin line), from the orbit O (x = 0.5, y = 0) for K = 4.7.
Fig. 15. The asymptotic curves for K = 8, calculated over long times.
further, and they move to larger x and y. However, new stable periodic orbits are formed in the same regions for K ≈ 12.4 (Fig. 16). The reason is the following. As K increases the eigenvalue λ of the basic orbit O (x = 1/2, y = 0) increases almost linearly
Fig. 14. Parts of the asymptotic curves from O (x = 0.5, y = 0) for K = 6. Solid line for unstable and dashed line for stable. When the asymptotic curves go out to the square [(0, 1)×(0, 1)] they continue inside it because of the modulo 1.
producing a large degree of chaos over the whole phase space. The periodic orbit O1 and its bifurcations become more and more unstable as K increases
Fig. 16. The asymptotic curves for K = 12.4 leave again an empty region for an island near (x = 0.7, y = 0.4). The dot near (x = 0.9, y = 0.7) is the unstable periodic orbit O1 .
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
Fig. 17. The asymptotic curves, without modulo, extend to large distances as K increases. The tips of the unstable asymptotic curves U for K = 6, 12.4 and 18.8 are in the squares [(0, 1), (0, 1)], [(1, 2), (1, 2)], [(2, 3), (2, 3)] respectively. The tips of the stable asymptotic curves S are in the squares [(0, 1), (0, 1)], [(0, 1), (1, 2)], [(0, 1), (2, 3)] respectively.
with K. In fact a simple analysis gives 4 1/2 K 1+ 1− |λ| = 1 − 2 K =K −2−
2873
the regions close to the transposed tips U and S (Fig. 16) as in the case K = 6 (Fig. 14). As the structure of the transposed asymptotic curves for K = 12.4 in the region around (x = 0.7, y = 0.4) is similar to the structure of these curves for K = 6, the transposed curves of the case K = 12.4 leave an empty region containing an island (Fig. 16) in the same way as in Fig. 14 for K = 6. The only difference is that the lobes U and S near the tips of Fig. 16 are thinner than the corresponding lobes of Fig. 14, by a factor about 2, because the asymptotic curves in Fig. 17 for K = 12.4 are longer by a factor 2 than for K = 6. Therefore, the area of the islands in the case K = 12.4 is smaller than in the case K = 6 by a factor about 4. This is seen clearly if we superimpose the lobes of Figs. 14 and 16 (Fig. 18). Similar phenomena appear for K = 18.8 (Figs. 17 and 19). The tips of U and S are in the squares [(2, 3), (2, 3)] and [(0, 1), (2, 3)] respectively and roughly in the same position with respect to these squares as the tips of U and S for the case K = 6 with respect to the square [(0, 1), (0, 1)]. In this case the lobes of U and S near their tips are even thinner than in the case K = 12.4, therefore the islands are smaller. This phenomenon is repeated periodically in K with a period ∆K = 2π. As the lengths of U and S increase linearly with K, when K is near K = 2nπ
2 1 − 2 − ··· K K
Therefore the sizes of the asymptotic curves increase also almost linearly (Fig. 17). The lengths of U and S up to their respective tips (the maximum distances from O, where these curves turn again inwards), are of the same order for K = 6, thus they are of the same order for any value of K. When K = 12.4 the tip of the unstable asymptotic curve (U ) is in the square [(1, 2), (1, 2)] and roughly in the same position with respect to this square as the tip of U in the original square [(0, 1), (0, 1)] for K = 6. On the other hand, the tip of the curve S for K = 12.4 is in the square [(0, 1), (1, 2)] and roughly in the same position with respect to this square as the tip of S in the original square [(0, 1), (0, 1)] for K = 6. The forms of the curves U and S near their tips are similar. Therefore, when we bring back the parts of the curves U and S close to their tips to the original square [(0, 1), (0, 1)], because of the modulo 1, the lobes of U and S form a similar structure in
Fig. 18. Superposition of the tips of the lobes for K = 6 (solid line) and K = 12.4 (dashed line).
October 3, 2005
2874
13:55
01371
G. Contopoulos et al.
Fig. 19.
As in Fig. 14, for K = 18.8.
the tip of U is in the square [(n, n + 1), (n, n + 1)] and the tip of S is in the square [(0, 1), (n, n + 1)]. These tips and the lobes U and S near them are brought back to the square [(0, 1), (0, 1)], because of the modulo 1, and form a similar structure containing small islands of stability. This explains the periodic appearance of the islands of group I in the standard map for increasing K. The islands become insignificant for large K. Namely, A becomes smaller than 0.001% for K > 130.
4. Periodic Orbits of Periods 1 and 2 The periodic orbits in the standard map and their stability were studied by Chirikov [1979]. In the present paper we study in more detail the most important families of periodic orbits and we find the connections between them. The stable orbits are especially important because they are surrounded by islands of stability. The simplest period-1 family O has fixed x and y (x = 1/2, y = 0). This is stable for K < 4 and becomes unstable for K > 4. The H´enon [1965] stability parameter a is between −1 and 1 in the stable case, while |a| > 1 in the unstable case (Fig. 20). At K = 4 there is a period-doubling bifurcation. More families of period-1 appear on the x-axis whenever K goes beyond K = 2nπ (n ≥ 1). These are families
Fig. 20. The stability curves of the orbits of group I (solid line for stable and dashed line for unstable).
of group II and appear for y = 0 and K sin(2πx) = n(≡ 1) (mod 1) (6) 2π For example, if n = 1 and K = 2π we have x = 1/4. If K > 2π we have two values of x, one in the interval 0 < x < 1/4 and the other in the interval 1/4 < x < 1/2. Two more families have x symmetric to the above with respect to x = 1/2, namely in the intervals 3/4 < x < 1 and 1/2 < x < 3/4. These orbits have n negative. Similar families appear whenever K ≥ 2π|n|. The stability of these orbits was considered by Chirikov [1979]. The solutions in the intervals 0 < x < 1/4 and 3/4 < x < 1 are unstable, while the solutions in the intervals 1/4 < x < 1/2 and 1/2 < x < 3/4 are stable whenever K is in the intervals 2nπ < K < [(2nπ)2 + 16]1/2
(7)
The stable period-1 orbits are surrounded by islands of stability. The H´enon stability parameters [H´enon, 1965] of these families for n = 1 are given in Fig. 21. The intervals of stability ∆K = [(2nπ)2 + 16]1/2 − 2nπ are approximately equal to ∆K = 8/K. The period-2 family O1 generated at K = 4 is stable in the interval 4 < K < 2π (Fig. 20). An orbit starting at O1 has another point O1 symmetric with respect to the center (0.5, 0.5) [Fig. 1(a)].
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
2875
therefore y = 2x
(mod 1)
(14)
If we set 1 +δ 2 with 0 < δ < 1/2 we find x=
(15)
y = 2δ
(16)
and 2π(4δ + n − 1) (17) sin 2πδ The minimum value of K is given by 4 (18) K0 = cos 2πδ0 where π(n − 1) (19) tan 2πδ0 − 2πδ0 = 2 The values of δ0 for various integer values of n and the corresponding values of K0 are given in Table 1. The values of K0 are smaller than 2nπ, but they approach 2nπ for large n. Thus the differences ∆K0 approach 2π. The values of δ0 are smaller than 1/4 and they approach 1/4 for large n. The corresponding values of x0 at the minimum K approach 3/4. The family O1 is called “regular” because it is connected to the original family O (x = 1/2) that exists all the way from the unperturbed problem (K = 0). The families O1∗ , O1∗∗ , O1∗∗∗ , are “irregular” families, because they are not connected to the families of the unperturbed problem. (The terminology of “regular” and “irregular” families was introduced by Contopoulos [1970].) Every new stable irregular family is generated at a tangent bifurcation, (i.e. at a minimum K, where the tangent of the characteristic is vertical) together with an unstable periodic orbit of the same multiplicity (Fig. 22). K=
Fig. 21. The stability curves for the orbits of group II (solid line for stable and dashed line for unstable).
After two iterations the standard map gives K 2K sin 2πx + sin 2πx x = x + 2y + 2π 2π (mod 1) (8) K K sin 2πx + sin 2πx y = y + 2π 2π If x = x (mod 1), y = y (mod 1) we find K(sin 2πx + sin 2πx ) = 2π
(9)
where x is given by Eq. (1) and is an integer. Equation (9) was given by Chirikov [1979], who introduced the notion of “accelerator modes” if the images of y differ from y by an integer (see also [Rom-Kedar & Zaslavsky, 1999]). The case = 0 is the simplest one. Then Eq. (9) gives either x = n − x or
x =
1 n + 2
(first case)
(10)
+x
(second case)
(11)
First Case. This case refers to orbits of group I. Then Eqs. (1), (8) and (10) give K sin 2πx = 1 − n ≡ 0 (mod 1) (12) 2y + 2π hence K sin 2πx = x − y + 1 − n = n − x x = x + y + 2π (13)
Table 1. n
n, δ0 , K0 , ∆K0 for various families.
δ0
Family
1
0
2
0.19538
O1∗
3
0.21516
4
0.22423
5
0.22951
O1
O1∗∗ O1∗∗∗
K0
∆K0
4 11.887
7.887
18.413
6.526
24.810
6.397
31.159
6.349
October 3, 2005
2876
13:55
01371
G. Contopoulos et al.
0 < x < 1/4 and 3/4 < x < 1. This is seen in Fig. 22 where all the families O1 , O1∗ , O1∗∗ , O1∗∗∗ . . . change from stable to unstable at x = 3/4 (i.e. δ = 1/4). The corresponding y are all equal to y = 1/2 at bifurcation, and the values of K are equal to K = 2nπ
Fig. 22. The characteristic diagram for the orbits O1 , O1∗ , O1∗∗ , O1∗∗∗ and some of their bifurcations.
The stability of the double periodic orbit is found by differentiating Eq. (8) with respect to x and y, namely dx
= A2 dx + B2 dy dy = C2 dx + D2 dy
(20)
The eigenvalues λ are given by λ2 − 2aλ + 1 = 0
(21)
where a = 1/2(A2 + D2 ) = 1 + K cos 2πx + K cos 2πx K2 cos 2πx cos 2πx (22) 2 is the H´enon stability parameter [H´enon, 1965]. From Eq. (10) we find +
cos 2πx = cos 2πx
(23)
and K2 cos2 2πx (24) 2 The orbits are stable if −1 < a < 1, hence a = 1 + 2K cos 2πx +
−4 < K cos 2πx < 0
(25)
0 < (2 + K cos 2πx)2
(26)
and
Therefore the orbits are stable if cos 2πx < 0, i.e. 1/4 < x < 3/4 and unstable if cos 2πx > 0, i.e.
(27)
Therefore the orbits O1 , O1∗ , O1∗∗ , O1∗∗∗ . . . are stable in the intervals K0 < K < 2nπ. The interval of stability ∆K = 2nπ − K0 is found approximately as follows. If we set π (28) 2πδ0 = − 2 in Eq. (19) we find πn (29) cot + = 2 hence 8 2 1+ 2 2 (30) ≈ πn 3π n Then from Eq. (18) we find 2 4 ≈ 2πn 1 − 2 2 K0 = sin π n
(31)
and 8 4 ≈ (32) πn K The inequality (26) is always satisfied, except if K cos 2πx = −K cos 2πδ = −2, when we have an equality. For example, if n = 0 we have δ = 0.1855 and K = 5.07. The stability curve of this family is tangent to the axis a = −1 at this point (Fig. 20). There we have the bifurcation of two stable and two unstable periodic orbits of period-4. These stable orbits produce resonant islands inside the islands surrounding O1 and O1 , as seen in Figs. 4 and 5 of [Contopoulos et al., 1999]. When these secondary islands escape in the chaotic sea, they generate a large dip (n/m = 1/2) in the curve giving A versus K (Fig. 8). Similar tangencies appear in the stability curves of the period-2 families for n ≥ 1. The characteristic of the period-2 family continues as unstable (Fig. 20) beyond the bifurcation point (x = 3/4, y = 1/2). As K increases both x and y increase and tend to the point x = y = 1 as K → ∞. The positions of this unstable periodic orbit for K = 12.4 and K = 18.8 are marked by dots in Figs. 16 and 19. ∆K ≈
Second Case. At the bifurcation point of group I family O1 at (x = 3/4, y = 1/2, K = 2π) we have generation of two families of equal period O2
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
(period-2, Fig. 20). These families belong to the second case of period-2 orbits given by Eq. (11). Then from Eqs. (1), (8), (9) and (11) we find 2y +
K sin 2πx = 0 (mod 1) 2π
x =
1 + x = x + y 2
(33)
and
=x+y+
K sin 2πx 2π
(mod 1)
(34)
hence y = y =
1 2
(mod 1)
(35)
and K=
2πn | sin 2πx|
(36)
Therefore the new families have y = 1/2 (constant), while two values of x are found for each K from Eq. (36). Similar period-2 families (equal period bifurcations) appear for n ≥ 2 at the same point (x = 3/4, y = 1/2, for K = 2πn). Their stability can be studied in the same way as above. They are stable in the intervals 2πn < K < [(2πn)2 + 4]1/2
(37)
The interval of stability ∆K is equal to ∆K ≈ 2/K. These families were discussed by Chirikov [1970], who gave the formulae (7) and (37). But we stress here that these families are bifurcations of the families O1 , O1∗ , O1∗∗ . . . for n = 1, 2, 3, respectively. An example of the stability curves of such families is given in Fig. 20. When such a family becomes unstable at a = −1 for K = [(2πn)2 + 4]1/2 it generates a double period family (period-4). This family becomes also unstable at a = −1 and generates a period-8 family, and so on. All the families beyond O2 are generated by period-doubling bifurcations. The stability intervals of every successive family decrease by a factor 8.72 [Benettin et al., 1980]. The islands around these successive bifurcating orbits are small. Their size is given in Fig. 8 beyond K = 2π = 6.283 as a continuation of the solid curve of Fig. 8 after the last KAM curve around O is destroyed at K = 6.335. The intervals of stability are: O1 : 2π − 4 = 2.283, O2 : [(2π)2 + 4]1/2 − 2π = 6.5938 − 6.2832 = 0.3106, O3 : 6.630184 − 6.593817 = 0.036367, O4 : 6.634364 − 6.630184 = 0.004180, O5 : 6.634844 − 6.634364 = 0.000480.
2877
Higher order intervals decrease by a factor 8.72 each time. Thus this sequence leads to an infinity of unstable periodic orbits beyond K = 6.6349 (Fig. 8). The total interval of stability before and beyond K = 2πn (solid line of Fig. 8) is equal to ∆K(group I) ≈ 10/K. Further Families. The sizes of the islands of group II (islands near the x-axis, Fig. 4) are given by dashed lines in Figs. 3 and 8. The periodic orbits that are surrounded by the islands with n = 1 start at K = 2π = 6.2832 and are stable up to K = [(2π)2 +16]1/2 = 7.4484 (Fig. 8). Then they generate a double period family that is stable in the interval 7.4484 − 7.7134, followed by a stable period-4 family in the interval 7.71341 − 7.74486, a stable period-8 family in the interval 7.74486 − 7.74848, and so on (Fig. 21). The intervals decrease by a factor that tends to 8.72. Thus beyond K = 7.74895 there is an infinity of unstable periodic orbits and all the islands are destroyed. Similar results appear for group II islands with n = 2 (Fig. 3). They start at K = 4π and are stable up to K = [(4π)2 + 16]1/2 = 13.1876. Then they generate a cascade of period-doubling bifurcations that lead to a destruction of all the islands beyond K = 13.3589. All the above families are generated with = 0 in Eq. (9). If is an integer different from 0 more period-2 families are generated. All of them
Fig. 23. The parameter K as a function of 2πδ for different (solid lines = −2, dashed lines = −1, dotted lines = 1).
October 3, 2005
2878
13:55
01371
G. Contopoulos et al.
are “irregular”, i.e. they are not connected to any families of the unperturbed problem. They are produced at various minimum values of K as a couple of one stable and one unstable family. Using Eqs. (8) and (9) we find K sin 2πδ = 2π −K sin 2πδ + sin 2π δ − 2 (38) where x = 1/2 + δ. For a fixed , Eq. (38) gives K as a function of δ. These functions have a minimum K. In fact, the factor of K in Eq. (38) is less than 2 absolutely, therefore K cannot be smaller than π||. Various families K = K(δ) for different are shown in Fig. 23. Near the minimum K one of the two branches is stable. However the corresponding islands of stability are extremely small.
Group III islands surround a set of period-4 orbits, and they are unexpectedly large. These orbits appear in pairs, forming two sets of four points each. Two points are on the x-axis and two outside this axis (Figs. 6(a) and 6(b)). Thus each set of four islands has two islands on the x-axis and two islands outside this axis. Here we discuss the period-4 orbits at the centers of these islands. We use the values of x , y as given by Eq. (1) and the values of x , y as given by Eq. (8). Then
+
x = x + 2y + 2z +
K sin 2πx 2π
x = x + 3y + 3z +
2K K sin 2πx + sin 2πx 2π 2π
(mod 1)
4y + 4z + + z+
2K 3K sin 2πx + sin 2πx 2π 2π
K sin 2πx = 0 2π
(mod 1)
(43)
K K K sin 2πx + sin 2πx + sin 2πx = 0 2π 2π 2π K sin 2π(x + z) = −n ≡ 0 2π
(mod 1) (44)
We can check that this solution satisfies both equations (43). From Eqs. (41) and (44) we find K[sin 2πx + sin 2π(x + z)] = 2π
(45)
as in the case of double periodic orbits [Eq. (9)]. From now on we consider only the case = 0. We have two solutions of this equation. Either (46)
or (39)
z=
1 2
(47)
The first solution gives 3K 4K sin 2πx + sin 2πx = x + 4y + 2π 2π +
y IV
(42)
One solution of these equations has y = 0 and
and xIV
(mod 1)
We have a periodic orbit of period-4 if (xIV = x, y IV = y) (mod 1). Then
x+z =1−x
2K 3K sin 2πx + sin 2πx 2π 2π
K sin 2πx 2π
x = x + y + z
y = z +
5. Period-4 Orbits
x = x + 3y +
(n = integer, 0 < z < 1) we find
2K K sin 2πx + sin 2πx 2π 2π
(48)
n+z K = 2π sin πz
(49)
and (mod 1) (40)
K K sin 2πx + sin 2πx =y+ 2π 2π
K K sin 2πx + sin 2πx + 2π 2π If we write K sin 2πx = n + z 2π
(1 − z) 2
x=
The corresponding value of x, as a function of K is given in Fig. 24 (for n = 1). The solution (49) has a minimum K = K0 if tan πz0 = π(n + z0 ), therefore (41)
K0 =
2 cos πz0
(50)
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
2879
We have stable orbits if −1 < a < 1. From Eq. (48) we find c = c = − cos πz
(54)
Kc = −2π(n + z) cot πz
(55)
and
Then 1 a = 1 + 8Kc + 10K 2 c2 + 4K 3 c3 + K 4 c4 2 1 = 1 + Kc(Kc + 2)2 (Kc + 4) 2 and we have stability if
Fig. 24. The characteristic diagram for periodic orbits of group III.
Table 2. n
πz0
K0
∆K0
1 2 3 4
1.3518168 1.4420665 1.4793437 1.4998232
9.2067 15.5794 21.8996 28.2034
6.3727 6.3202 6.3038
(58)
At the minimum K0 we have K0 c0 = −2. As z increases, x decreases from x0 = (1 − z0 )/2 (curve (1) of Fig. 24) and Kc increases until it becomes positive when z > 1/2. At the transition to instability z = 1/2 the value of K is K = (2n + 1)π
The values of πz0 and K0 at the minima of K for various values of n are given in Table 2. The values of πz0 are smaller than π/2, i.e. z0 < 1/2. The values of K0 are a little smaller than (2n + 1)π, but they tend to (2n + 1)π for large n. The differences ∆K0 are larger than 2π, but they tend to 2π as n increases. Therefore the corresponding islands are close to 2π apart. Differentiating Eqs. (40) we find dxIV = A4 dx + B4 dy dy IV = C4 dx + D4 dy
1 (57) −1 < 1 + Kc(Kc + 2)2 (Kc + 4) < 1 2 The stability curves (a versus K) for this family and its bifurcations are given in Fig. 25 for n = 1. The right inequality (57) gives −4 < Kc < 0
πz0 , K0 , ∆K0 for various n.
(56)
(59)
and x = 1/4. The interval of stability ∆K = (2n + 1)π − K0 is approximately equal to ∆K ≈
(51)
and if we set dxIV = λdx, dy IV = λdy, we find the eigenvalues λ from an equation of the form of Eq. (21), where the H´enon stability parameter is 3 a = 1 + 4K(c + c ) + K 2 (c2 + c2 ) 2 1 + 7K 2 cc + 2K 3 cc (c + c ) + K 4 c2 c2 2
(52)
with c = cos 2πx,
c = cos 2π(x + z)
(53)
Fig. 25. The stability curves for the periodic orbits of group III.
October 3, 2005
2880
13:55
01371
G. Contopoulos et al.
2/K (found in a similar way as the interval of stability of group I orbits). If z decreases below z0 , K increases, x increases beyond x0 = (1 − z0 )/2 (curve (2) of Fig. 24) and Kc decreases until it becomes smaller than −4. At the transition to instability K1 c1 = −4. For example, for n = 1 we find z0 = 0.4303, x0 = 0.2849, z1 = 0.36071, x1 = 0.31965 and K1 = 9.4392. The transition value K1 along the upper branch of x is slightly larger than the transition value K = 3π ≈ 9.4248 for instability along the lower branch of period-4 orbits with x < x0 . The left inequality (57) gives 0 < 4 + Kc(Kc + 2)2 (Kc + 4) = e
(60)
This is satisfied for −4 √< Kc < 0, except at the points√Kc = −2 + 2 = −0.5858, and Kc = −2 − 2 = −3.4142, where e = 0. As a consequence the stability curves of the families (1) and (2) are tangent to the axis a = −1 at one point each (both points are near K ≈ 9.3 for n = 1) (Fig. 25). Similar stability curves appear for all values of n. The fact that for a given K we have two values of x (Fig. 24) implies that we have the appearance of two stable orbits for the same values of K. These orbits are surrounded by pairs of islands (group III islands, Fig. 6). Similar pairs of islands were found in the web map by Zaslavsky et al. [1997]. At K = (2n + 1)π two period-4 families bifurcate from the previous family (equal period bifurcations along the lower period-4 family of Fig. 25 (family (1)), with z = 1/2. The first has x > 1/4 and the second x < 1/4. The corresponding y along the new families is y = z = 1/2 (constant) and K=
π(2n + 1) sin 2πx
(61)
The values of c and c [Eq. (53)] are c = −c = cos 2πx
(62)
K 2 c2 = K 2 − [π(2n + 1)]2
(63)
and
Then from Eq. (52) we find 1 a = 1 − 4K 2 c2 + K 4 c4 2
(64)
This orbit is stable if −1 < a < 1, therefore we have the inequalities K 2 c2 (K 2 c2 − 8) < 0 and
0 < 4 − 8K 2 c2 + K 4 c4 . These are written as
and
or
K 2 < [π(2n + 1)]2 + 8
(65)
√ K 2 c2 = K 2 − [π(2n + 1)]2 < 4 − 2 3
(66)
√ K 2 c2 = K 2 − [π(2n + 1)]2 > 4 + 2 3
(67)
For example, for n = 1 the new period-4 families are stable in the intervals 3π = 9.425 < K < 9.453 and 9.813 < K < 9.840. At K = 9.453 a period-8 family is generated followed by a cascade of period-doubling bifurcations (Figs. 24 and 25). Beyond K = 9.840 these families are unstable. At K = 9.840 each of them generates by bifurcation two equal period families that are stable in a small interval beyond this value of K. However, the corresponding islands of stability near K = 9.840 are very small. A similar bifurcation occurs along the upper branch (family (2) of Fig. 24) at K = K1 . The orbits have another intersection point with y = 0 and x = (n + z)/2. Therefore for n = 1 x =
(1 + z) 2
(68)
while K and Kc are given by Eqs. (49) and (55). Thus the same stability criteria apply to them. At the transition K = K1 start two bifurcating families of equal period, that generate a cascade of bifurcations. The stability curves of these orbits are shown as dashed lines in Fig. 25. We notice that the two islands of Fig. 6 are well separated for large K and they disappear at different values of K. It is remarkable that the intervals of stability for the periodic orbits beyond K = 2nπ and K = (2n + 1)π are given by similar, yet different, formulae. Namely the simple periodic orbits on the x-axis (group II) are stable in an interval ∆K = [(2nπ)2 + 16]1/2 − 2nπ ≈ 8/K [Eq. (7)], the double period orbits with constant y = 1/2 (also in group I) are stable in an interval ∆K = [(2nπ)2 + 4]1/2 − 2nπ ≈ 2/K [Eq. (37)], while the orbits of the group III with constant y = 1/2 are √ stable in 2 two intervals ∆K√= {[(2n + 1)π] + 4 − 2 3}1/2 − ∆K = {[(2n + √ 1)π]2 + (2n + 1)π ≈ (2 − 3)/K and √ 1/2 2 1/2 ≈ (2 − 3)/K 8} − {[(2n + 1)π] + 4 + 2 3} (Eq. 66). Therefore these stability intervals of the three groups of islands have similar forms.
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
6. Conclusions The most important results of the present study are the following: (i) Important islands of stability of the standard map appear in a recurrent way in three groups, for values of the nonlinearity parameter K near K = 2nπ (groups I and II) and K = (2n + 1)π (group III) with n ≥ 1. (ii) The maximum areas A of these islands follow similar power laws A = aK b , with a near 0.5 and b near −2 for all three groups. (iii) The islands exist in small intervals around the above values of K. The intervals of the groups I and II slightly overlap each other near K = 2nπ. (iv) The sizes of the islands within each of the above intervals ∆K increase and decrease in a characteristic, universal way. In particular A decreases abruptly near particular resonances. The curves giving A versus K are practically exactly the same after an appropriate scaling in the cases of groups I and II. The curves of group III agree with each other (after scaling) but they are quantitatively different from the curves of groups I and II, although they have similar dips at the same resonances. Similar increases and decreases of islands were found in several other problems. (v) The main difference of the islands of group III is that they appear in pairs (around two stable periodic orbits of period-4 each, belonging to two families (1) and (2)). These periodic orbits are generated, together with two unstable orbits, at the same tangent bifurcation. On the contrary, the islands of groups I and II are around regular periodic orbits, or around irregular orbits generated at a tangent bifurcation of the usual type (one stable and one unstable orbit). (vi) The islands of type I are surrounded by arcs of the asymptotic curves (unstable and stable) of the main periodic orbit O (x = 0.5, y = 0), near a point (x = 0.7, y = 0.4), and a symmetric point (x = 0.3, y = 0.6). These islands appear for the first time when K > 4. When K increases to K > 8 these islands are destroyed, but when K increases further to K ≈ 12 similar islands appear near the same point. This is due to the almost linear increase of the lengths of the asymptotic curves (up to their outermost tips) with K. In fact the
(vii)
(viii)
(ix)
(x)
2881
eigenvalue |λ| of the periodic orbit O is an almost linear function of K. When K increases from K ≈ 2π to K ≈ 4π the tips of the asymptotic curves move to the squares [(1, 2), (1, 2)] (unstable) and [(0, 1), (1, 2)] (stable), and when these squares are superimposed on the original square [(0, 1), (0, 1)] the structure of the arcs of the asymptotic curves near their tips is similar around the (same) points (x = 0.7, y = 0.4) and (x = 0.3, y = 0.6). Thus the asymptotic curves leave free space for the formation of similar islands of stability. Similar results appear whenever K increases by 2nπ. The islands surround stable periodic orbits that become unstable at particular critical values K = Kcr , and they are followed by a cascade of period doubling bifurcations at intervals decreasing by a factor near δ = 8.72. Every time K increases by 2π a new family appears that follows the same pattern of period-doubling bifurcations. These families are irregular, i.e. they are not connected to the families that exist for small K. The islands of the group II appear on the xaxis. These islands surround irregular stable periodic orbits on the x-axis. The first irregular periodic orbits appear when K ≥ 2π. Higher order islands appear whenever K ≥ 2nπ. Their stability and bifurcations can be found analytically. The periodic orbits of group II are stable in an interval ∆K beyond K = 2nπ. The main periodic orbits of group I are stable in an interval before K = 2nπ up to K = 2nπ. At K = 2nπ they become unstable, but they generate two equal period families of orbits that have (both) y = 1/2 (constant) and they are stable for an interval ∆K beyond 2nπ. These orbits are followed by a cascade of perioddoubling bifurcations. The periodic orbits of group III (family (1)) are stable in an interval ∆K before K = (2n + 1)π. Two islands of group III are on the x-axis and two more outside this axis. The corresponding periodic orbits are stable in an interval ∆K before K = (2n + 1)π, and become unstable at K = (2n + 1)π. There they generate two equal period orbits (period4) each, that have two points with y = 0 and two more points with y = 1/2 (constant) and they are stable in an interval ∆K beyond K = (2n + 1)π.
October 3, 2005
2882
13:55
01371
G. Contopoulos et al.
(xi) The minima K of the new irregular families of groups I and III are somewhat smaller than 2nπ and (2n+1)π respectively, while the minimum K of group II families is 2nπ. The stable periodic orbits of groups I and III (family (1)) become unstable at exactly 2nπ and (2n + 1)π and the intervals of stability are ∆K ≈ 8/K and ∆K ≈ 2/K, respectively. The orbits of groups I (in particular the equal period bifurcating families) and II, starting at K = n π = 2nπ and of group III (the equal period bifurcating families from family (1)) starting at K = n π = (2n+1)π are stable in intervals from K = n π up to√K = [(n π)2 + B]1/2 , with B = 4, 16 and 4− 2 3 in groups I, II, III, respectively. The corresponding intervals of stability are ∆K ≈ 2/K, ∆K ≈ 8/K and ∆K ≈ 0.27/K respectively. In group III (family(1)) the stable fam√ ily that becomes unstable for B = 4 − 2 3 becomes again stable √ when B is in an interval between B = 4 + 2 3 and B = 8 and this interval is again equal to ∆K ≈ 0.27/K. Similar results apply to the family (2) of group III. Thus the intervals of K that contain the main islands of the system (around the main periodic orbits of periods 1, 2 and 4) can be found theoretically. The decrease of the size of the islands can be also estimated theoretically. What was found empirically is the relatively large size of these islands. (xii) It seems probable that recurrences of order in chaos appear for most maps with a certain modulo. In fact, by bringing back the asymptotic curves in the original square, we may find new homoclinic tangencies, hence, according to the Newhouse theorem [Newhouse, 1983], new stable periodic orbits. In particular, we did find similar recurrencies in further maps with trigonometric terms. However, this problem requires a more detailed study.
Acknowledgments This work was partly supported by the 200/557 program of the Academy of Athens.
We thank Mr. Manolis Zoulias for his technical support.
References Benettin, G., Cercignani, C., Galgani, L. & Giorgilli, A. [1980] “Universal properties in conservative dynamical systems,” Lett. Nuovo Cim. 28, 1–4. Chirikov, B. V. [1979] “A universal instability of manydimensional oscillator systems,” Phys. Rep. 52, 263– 379. Contopoulos, G. [1970] “Orbits in highly perturbed dynamical systems. I,” Astron. J. 75, 96–107. Contopoulos, G., Harsoula, M., Voglis, N. & Dvorak, R. [1999] “Destruction of islands of stability,” J. Phys. A32, 5213–5232. Duarte, P. [1994] “Plenty of elliptic islands for the standard family of area preserving maps,” Ann. Inst. Henri Poincar´e 11, 359–407. Giorgilli, A. & Lazutkin, V. F. [2000] “Some remarks on the problem of ergodicity of the standard map,” Phys. Lett. A259, 359–367. H´enon, M. [1965] “Exploration num´erique du probl`eme restreint II,” Ann. Astrophys. 28, 992–1007. H´enon, M. [1966] “Exploration num´erique du probl`eme restreint III,” Bull Astron. 1, 57–79. Karney, C. F. F., Rechester, A. B. & White, R. B. [1982] “Effect of noise on the standard map,” Physica D4, 425–438. Lichtenberg, A. J. & Lieberman, M. A. [1992] Regular and Chaotic Dynamics, 2nd edition (Springer Verlag, NY). Newhouse, S. E. [1983] “Generic properties of conservative systems,” in Chaotic Behaviour of Deterministic Systems, eds. Looss, G., Helleman, R. H. G. & Stora, R. (North Holland, Amsterdam), pp. 443–451. Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. [1992] Numerical Recipes in Fortran (Cambridge Univ. Press). Rom-Kedar, V. & Zaslavsky, G. [1999] “Islands of accelerator modes and homoclinic tangles,” Chaos 9, 697– 705. Zaslavsky, G. M., Edelman, M. & Niyazov, B. A. [1997] “Self-similarity, renormalization, and phase space nonuniformity of the Hamiltonian chaotic dynamics,” Chaos 7, 159–181. Zaslavsky, G. M. [2002] “Chaos, fractional kinetics and anomalous transport,” Phys. Rep. 371, 461–580.
13:55
01371
International Journal of Bifurcation and Chaos, Vol. 15, No. 9 (2005) 2865–2882 c World Scientific Publishing Company
RECURRENCE OF ORDER IN CHAOS G. CONTOPOULOS and M. HARSOULA Academy of Athens, Research Center of Astronomy, 4 Soranou Efessiou Str. 11527 Athens, Greece R. DVORAK and F. FREISTETTER Universit¨ at Wien, Institut f¨ ur Astronomie, T¨ urkenschanzstrasse 17, A-1180 Wien, Austria Received July 6, 2004; Revised August 4, 2004 The standard map x = x + y , y = y + (K/2π)sin(2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ∆K, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results. Keywords: Chaos; order; standard map; periodic orbits; recurrence; stability.
1. Introduction A problem of theoretical interest in highly perturbed dynamical systems is whether chaos is complete. For example, the standard map x = x + y (mod 1) (1) K sin(2πx) y =y+ 2π (where (mod 1) refers to both formulae), seems to become completely chaotic for K larger than about K = 7.6. Nevertheless, there are small islands of stability for arbitrarily large K. Such islands appear close to the points where the asymptotic curves of the unstable periodic orbits have tangencies [Newhouse, 1983]. The most general result for the standard map was derived by Duarte [1994] who has shown that the values of K when there are islands of stability
form a residual set everywhere dense. That is between any two values of K that differ by ∆K there are intervals of K for which the standard map has small islands of stability. The total measure of these intervals of K within a given ∆K may be small. Giorgilli and Lazutkin [2000] estimated that this measure is of order 1/K. But, although there may be many values of K without any islands at all, the standard map does not become an Anosov system for large K, because Anosov systems are structurally stable, i.e. a small perturbation gives again an Anosov system. The main question is now whether such islands are appreciable, i.e. whether they cover a part of phase space that is not very small. The numerical evidence is that for large values of K most islands are extremely small. For example, for K ≤ 6 there are appreciable islands of stability [Fig. l(a)], but these islands disappear for larger K
2865
October 3, 2005
2866
13:55
01371
G. Contopoulos et al.
of values of K, at least up to K = 200. These islands appear in a recurrent (almost periodic) way. Similar, albeit smaller, islands appear for even larger K. We study this recurrence phenomenon in Sec. 2, and give an explanation of it in Sec. 3. In Secs. 4 and 5 we study the underlying periodic orbits, and in Sec. 6 we present our conclusions.
2. Recurrence of Islands
(a)
We explored systematically the phase space (0, l) × (0, l) in the standard map for values of K beyond K = 4.5 with a step dK equal to dK = 0.0001. The total area of the islands was derived as follows. We divided the phase space (x, y) into 106 cells of size 0.001 × 0.001 and found how many cells contained (or not) iterates of an orbit starting in the chaotic sea. We tested different initial conditions (x0 , y0 ) and different numbers of iterations N . We found that we needed at least N = 108 iterations in order to find a reliable value of the area A of the islands (number of empty cells divided by 106 ). In some cases we had to reach N = 5 × 109 . This method cannot give details smaller than the grid size (0.001 × 0.001). Thus in several cases we used a finer grid, namely [(l/3000) × (1/3000)], in order to find more accurate values of A. In Fig. 2, we give the total size (area) of the islands as a function of K. The first large islands in this figure appear around K = 4π (Fig. 3). These islands cover over 0.2% of the total phase space at their maximum and they belong to two groups.
(b) Fig. 1. The structure of the phase space of the standard map for (a) K = 5 and (b) K = 8.
[K = 8; Fig. 1(b)] and any remaining islands seem to be below the level of detectability. However, for larger values of K further islands were found [Chirikov, 1970; Lichtenberg & Lieberman, 1992; Rom-Kedar & Zaslavsky, 1999; Zaslavsky, 2002]. In the present paper we make a systematic study of such islands. We found appreciable islands of stability for certain intervals
Fig. 2. The area A of the islands of groups I–III as a function of the control parameter K for values between 10 and 70.
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
2867
i.e. there are two period-4 islands near each other (Fig. 6). The maximum islands of this group, for K = 15.6, occupy 0.15% of the total area of phase space. However, these islands extend over smaller intervals of values of K.
Fig. 3. The islands around K = 4π in detail. The solid line represents islands outside the x-axis (group I). The dashed line refers to the islands on the x-axis (group II).
Group I represents islands outside the x-axis, as in Fig. 1(a). These islands surround two points belonging to a stable periodic orbit of period-2. The first point is between (x = 0.5, y = 0) and (x = 0.75, y = 0.5) while the second is symmetric to the first with respect to the center of Fig. 1(a) (x = y = 0.5). The sizes of the islands of group I are given by a solid line in Fig. 3. This group contains also bifurcations of this period-2 family of periodic orbits (see Sec. 4). The second group (group II) represents islands close to the x-axis (y = 0) (Fig. 4). Such islands appear for K a little larger than 4π (dashed line of Fig. 3). Similar islands of groups I and II appear close to K = 2nπ
(n ≥ 1)
Fig. 4.
Islands of group II near the x-axis for K = 6.7.
(2)
The islands of group I exist mainly for K ≤ 2nπ, but also a little beyond K = 2nπ, while the islands of group II exist for K > 2nπ. There are also islands, forming a third group (group III), that appear close to K = (2n + 1)π
(n ≥ 1)
(3)
This group contains islands of period-4, two of them being near the x-axis (y = 0) and two outside this axis (Fig. 5). These islands appear in pairs,
Fig. 5. One of the two sets of four islands for K = 15.596 (group III islands).
October 3, 2005
2868
13:55
01371
G. Contopoulos et al.
Fig. 7. The maximum size of the islands as a function of K for groups I and II (dark dots) and for group III (white squares). The curves represent Eq. (4). (a)
(b) Fig. 6. Islands of the group III for K = 15.5796. (a) One pair of islands on the x-axis and (b) another pair of islands outside the x-axis. The large dots represent the stable and unstable orbits. The scattered points belong to a chaotic orbit in the unstable manifold of the periodic orbit (x = 0.5, y = 0).
The maximum sizes of the islands decrease with K according to power laws A = aK b
(4)
where a = 0.72 and b = −2.30 for groups I and II and a = 1.22 and b = −2.45 for group III (Fig. 7). We notice that b is not very far from the theoretical
value b = −2 [Chirikov, 1979]. The islands become smaller than A = 0.00001 for K larger than about K ≈ 130. The intervals of ∆K where we have islands of the groups I, II and III are approximately inversely proportional to K (Secs. 4 and 5). Between the islands of groups I, II and III, there are smaller islands for very small intervals of values of K. Such islands appear in every interval ∆K, as predicted by the theorem of Duarte [1994]. The sizes of the islands around their maximum increase and decrease in a typical way. In Fig. 3 we see that as K increases the islands of the group I appear for the first time for K = 11.87 and their size increases abruptly. The curve A in Fig. 3 is fractal with an infinite number of intervals where it increases and decreases. The area of these islands (and their bifurcations) reaches and remains zero for K ≥ 12.8. The second set of islands (group II) starts at K = 4π. Their size increases and decreases in the same way as group I islands. The increase and decrease of the size of the islands is well understood [Contopoulos et al., 1999]. Let us consider the island around the stable invariant point O1 [Fig. 1(a)]. In fact there are two such invariant points, O1 and O1 , symmetric with respect to the center (x = y = 0.5), that are generated by bifurcation from the periodic orbit O (x = 0.5, y = 0) for K = 4 (see Sec. 4). Here we consider only the island O1 , around a point with x > 0.5. As K increases beyond K = 4 the orbit O becomes unstable and some chaos is generated around it. For K < 4.35 this chaotic domain does
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
not communicate with the large chaotic sea that fills most of the phase space. In fact there are closed invariant curves (KAM curves) surrounding O, and also O1 and O1 . However, for K > 4.35 these KAM curves are destroyed and the islands O1 and O1 are floating in the large chaotic sea [Fig. 1(a)]. The size of the island O1 is defined by the last KAM curve, that surrounds the stable periodic orbit O1 and separates the island from the large chaotic sea outside it. As K increases, the last KAM curve expands until it is destroyed at a critical value K = Kn/m and becomes a cantorus. Then the size of the island decreases abruptly, and is limited by a new last KAM curve. This transition is related to a resonance n/m. Namely for K < Kn/m there are m secondary islands inside and close to the original last KAM curve. Between these islands there are m unstable invariant points. Near these points there is chaos, that surrounds also the m islands, and is separated from the outer chaotic sea by the last KAM curve. When K > Kn/m the last KAM curve surrounding the m islands becomes a cantorus. Then the chaotic region between the m islands communicates with the large chaotic sea through the cantorus and the m islands are floating in the chaotic sea. Then the original island is surrounded only by KAM curves closer to its center than the m islands, and the new last KAM curve is inside the m islands. As K increases further this new KAM curve expands, until it is also destroyed. Some of the main resonances responsible for the abrupt decrease of the size (area) of the islands are marked in Fig. 3. The same resonances, like 3/5, 1/2, 3/7, 1/3, 1/5 appear in the interval K = 4.5 − 7.6 (Fig. 8). Figure 8 is similar to Fig. 2 of [Contopoulos et al., 1999]. There are only small differences due to the fact that in the previous paper we measured the distance of the last KAM curve from the central invariant point O1 along the y-direction, while here we give the area of the island. The point O1 becomes unstable for K = 2π ≈ 6.283 and for a little larger K (K = 6.335) the last KAM curve around O1 is destroyed and the islands that surround O1 vanish. However, if we measure the area of the islands we do not reach zero for K = 6.335. In fact, there are two new islands O2 and O2 on the left and on the right of O1 , that grow larger as K increases beyond K = 2π. The changes of the area A of the secondorder islands beyond K = 6.335 are similar to the changes of A along the original islands. The
Fig. 8.
2869
The sizes of the islands for K between 4.5 and 8.
second-order periodic orbits O2 and O2 are stable up to K = 6.594 and they become unstable for larger K. Then we have the formation of four stable points until K = 6.630, and so on (see Sec. 4). The successive intervals of stability decrease by a factor 8.72 [Benettin et al., 1980]. Finally, for K > 6.635 there are infinite bifurcating families, which are all unstable. In Figs. 3 and 8 a second set of islands, represented by dashed lines appears, for larger values of K (islands of group II). These islands are formed around stable periodic orbits on the x-axis (see Sec. 4). The increases and decreases of these islands follow the same pattern as in the main islands around O1 . The form of Fig. 3 seems to be universal. The islands near every value of K = 2nπ of groups I and II are similar to Fig. 3. Two examples are given in Fig. 9 for K around K = 6π and K = 50π. All these islands with n ≥ 2 surround irregular periodic orbits that are generated from zero at tangent bifurcations. The only exception are the islands with n = 1 and in particular for K ≤ 4.8 (Fig. 8). In this case the average size of the islands increases to large values as K decreases below K = 4.8. In fact, the islands around O1 and O1 are generated from the larger island around O (x = 0.5, y = 0) that exists for K < 4.35. The area of the islands at the resonance 1/3 in Figs. 3 and 8 vanishes for a particular value of K.
October 3, 2005
2870
13:55
01371
G. Contopoulos et al.
(a)
Fig. 10. The sizes of the islands of groups I and II for K near 6π (solid line). In this figure we superimposed the sizes of the islands near K = 4π (dashed line) after appropriate scalings in K and A.
same for K near 4π (dashed line). Then the quantity chi-square is given by the standard formula [Press et al., 1992]. 2
χ =
N (Ai1 − Ai2 )2 i=1
(b) Fig. 9. Islands of groups I and II for K around (a) K = 6π ≈ 18.5, and (b) K = 50π ≈ 157. Grid size (1/3000) × (1/3000).
The vanishing in Fig. 8 happens near K = 5.65 (compare with Figs. 6 and 7 of [Contopoulos et al., 1999]). A similar 1/3 resonance appears when the central period-2 orbit becomes unstable for K = 12.28 in Fig. 3. The variation of the size of the islands is similar for all the islands of groups I and II. A comparison of the size of the islands of groups I and II for K near 6π and near 4π after appropriate scaling is given in Fig. 10. A scaling is defined by the requirement that one maximum A and two main dips should coincide. Then a measure of the agreement between the two curves is given by the chi-square test. Let Ai1 be the value of the size of the islands for K near 6π (solid line) and Ai2 the
Ai1 + Ai2
(5)
where i extends from 1 to N , and N is of the order of 500. If χ2 is smaller than 10−2 the agreement is very good, while if χ2 is larger than 10−1 the agreement is bad. In Fig. 10 we compare the variations of the sizes of the islands near K = 6π and K = 4π. The value of chi-square is χ2 = 1.6 × 10−2 for orbits of group I and χ2 = 4.4 × 10−3 for group II. If we compare a curve of group I with a curve of group II we find again a very good agreement, namely χ2 = 8.5 × 10−3 . On the other hand the islands of group III increase and decrease in a somewhat different way [Figs. 11(a) and 11(b)]. These islands appear in pairs, thus they are different from the islands of groups I and II, but they are similar to each other. A comparison of the islands of group III after an appropriate scaling is given in Fig. 12. The corresponding chi-square is χ2 = 6.1 × 10−3 . But if we compare a curve of group III with a curve of group II, then chi-square is χ2 = 2.4 × 10−1 , i.e.
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
(a)
2871
Fig. 12. The sizes of the islands of group III for K = 5π (solid line) on which we have superimposed the sizes of the islands near K = 3π (dashed line) after appropriate scalings in K and A.
3. Explanation
(b) Fig. 11. The sizes of group III islands close to (a) K = 3π = 9.4, and (b) K = 9π = 28.3. Grid size (1/3000) × (1/3000).
much larger. Thus although the resonances in groups I, II and III are the same, the increase and decrease of the islands of group III is different from the other two cases. This is due to the fact that the islands of group III are close to each other and they affect the growth of both of them. Similar figures appear in maps similar to the standard map [Karney et al., 1982], and in Hamiltonian systems. e.g. systems of two coupled oscillators [Contopoulos et al., 1999] or the restricted three-body problem [H´enon, 1966, Fig. 19]. In fact as the perturbation increases the islands first increase in size, but later, when chaos becomes dominant, they decrease and disappear. However, a detailed comparison of the sizes of the islands in different systems is provided here for the first time.
The islands in the intervals of K near K = 12, K = 18, etc. are not only similar in size with the islands close to K = 6 but they are also located near the same regions of phase space (near (x = 0.7, y = 0.4)). These regions are around the stable periodic orbits whose existence is proven in Sec. 4. The area of the islands is limited by the asymptotic curves emanating from the periodic orbit O (x = 0.5, y = 0), which is unstable for K > 4. These asymptotic curves intersect at an infinity of homoclinic points (Fig. 13 for K = 4.7) and make oscillations between these homoclinic points, forming the so-called “lobes”. The main homoclinic point H is at about equal distances from O along the unstable and stable asymptotic curves U and S. The oscillations are relatively small in the region surrounding the stable periodic orbit O1 , for K not much larger than K = 4 and leave room for the invariant curves surrounding O1 . When K becomes larger (K = 6) the lobes formed by the asymptotic curves become longer. However, in Fig. 14 we see that the asymptotic curves still leave space for the appearance of islands around O1 . For even larger K the lobes of the asymptotic curves become even longer and their oscillations become larger and invade the region around O1 (Fig. 15 for K = 8). Furthermore, the orbit O1 and its higher order bifurcations become all unstable and their asymptotic curves intersect the asymptotic curves of the orbit O at an infinity of heteroclinic points,
October 3, 2005
2872
13:55
01371
G. Contopoulos et al.
Fig. 13. The asymptotic curves, unstable (thick line) and stable (thin line), from the orbit O (x = 0.5, y = 0) for K = 4.7.
Fig. 15. The asymptotic curves for K = 8, calculated over long times.
further, and they move to larger x and y. However, new stable periodic orbits are formed in the same regions for K ≈ 12.4 (Fig. 16). The reason is the following. As K increases the eigenvalue λ of the basic orbit O (x = 1/2, y = 0) increases almost linearly
Fig. 14. Parts of the asymptotic curves from O (x = 0.5, y = 0) for K = 6. Solid line for unstable and dashed line for stable. When the asymptotic curves go out to the square [(0, 1)×(0, 1)] they continue inside it because of the modulo 1.
producing a large degree of chaos over the whole phase space. The periodic orbit O1 and its bifurcations become more and more unstable as K increases
Fig. 16. The asymptotic curves for K = 12.4 leave again an empty region for an island near (x = 0.7, y = 0.4). The dot near (x = 0.9, y = 0.7) is the unstable periodic orbit O1 .
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
Fig. 17. The asymptotic curves, without modulo, extend to large distances as K increases. The tips of the unstable asymptotic curves U for K = 6, 12.4 and 18.8 are in the squares [(0, 1), (0, 1)], [(1, 2), (1, 2)], [(2, 3), (2, 3)] respectively. The tips of the stable asymptotic curves S are in the squares [(0, 1), (0, 1)], [(0, 1), (1, 2)], [(0, 1), (2, 3)] respectively.
with K. In fact a simple analysis gives 4 1/2 K 1+ 1− |λ| = 1 − 2 K =K −2−
2873
the regions close to the transposed tips U and S (Fig. 16) as in the case K = 6 (Fig. 14). As the structure of the transposed asymptotic curves for K = 12.4 in the region around (x = 0.7, y = 0.4) is similar to the structure of these curves for K = 6, the transposed curves of the case K = 12.4 leave an empty region containing an island (Fig. 16) in the same way as in Fig. 14 for K = 6. The only difference is that the lobes U and S near the tips of Fig. 16 are thinner than the corresponding lobes of Fig. 14, by a factor about 2, because the asymptotic curves in Fig. 17 for K = 12.4 are longer by a factor 2 than for K = 6. Therefore, the area of the islands in the case K = 12.4 is smaller than in the case K = 6 by a factor about 4. This is seen clearly if we superimpose the lobes of Figs. 14 and 16 (Fig. 18). Similar phenomena appear for K = 18.8 (Figs. 17 and 19). The tips of U and S are in the squares [(2, 3), (2, 3)] and [(0, 1), (2, 3)] respectively and roughly in the same position with respect to these squares as the tips of U and S for the case K = 6 with respect to the square [(0, 1), (0, 1)]. In this case the lobes of U and S near their tips are even thinner than in the case K = 12.4, therefore the islands are smaller. This phenomenon is repeated periodically in K with a period ∆K = 2π. As the lengths of U and S increase linearly with K, when K is near K = 2nπ
2 1 − 2 − ··· K K
Therefore the sizes of the asymptotic curves increase also almost linearly (Fig. 17). The lengths of U and S up to their respective tips (the maximum distances from O, where these curves turn again inwards), are of the same order for K = 6, thus they are of the same order for any value of K. When K = 12.4 the tip of the unstable asymptotic curve (U ) is in the square [(1, 2), (1, 2)] and roughly in the same position with respect to this square as the tip of U in the original square [(0, 1), (0, 1)] for K = 6. On the other hand, the tip of the curve S for K = 12.4 is in the square [(0, 1), (1, 2)] and roughly in the same position with respect to this square as the tip of S in the original square [(0, 1), (0, 1)] for K = 6. The forms of the curves U and S near their tips are similar. Therefore, when we bring back the parts of the curves U and S close to their tips to the original square [(0, 1), (0, 1)], because of the modulo 1, the lobes of U and S form a similar structure in
Fig. 18. Superposition of the tips of the lobes for K = 6 (solid line) and K = 12.4 (dashed line).
October 3, 2005
2874
13:55
01371
G. Contopoulos et al.
Fig. 19.
As in Fig. 14, for K = 18.8.
the tip of U is in the square [(n, n + 1), (n, n + 1)] and the tip of S is in the square [(0, 1), (n, n + 1)]. These tips and the lobes U and S near them are brought back to the square [(0, 1), (0, 1)], because of the modulo 1, and form a similar structure containing small islands of stability. This explains the periodic appearance of the islands of group I in the standard map for increasing K. The islands become insignificant for large K. Namely, A becomes smaller than 0.001% for K > 130.
4. Periodic Orbits of Periods 1 and 2 The periodic orbits in the standard map and their stability were studied by Chirikov [1979]. In the present paper we study in more detail the most important families of periodic orbits and we find the connections between them. The stable orbits are especially important because they are surrounded by islands of stability. The simplest period-1 family O has fixed x and y (x = 1/2, y = 0). This is stable for K < 4 and becomes unstable for K > 4. The H´enon [1965] stability parameter a is between −1 and 1 in the stable case, while |a| > 1 in the unstable case (Fig. 20). At K = 4 there is a period-doubling bifurcation. More families of period-1 appear on the x-axis whenever K goes beyond K = 2nπ (n ≥ 1). These are families
Fig. 20. The stability curves of the orbits of group I (solid line for stable and dashed line for unstable).
of group II and appear for y = 0 and K sin(2πx) = n(≡ 1) (mod 1) (6) 2π For example, if n = 1 and K = 2π we have x = 1/4. If K > 2π we have two values of x, one in the interval 0 < x < 1/4 and the other in the interval 1/4 < x < 1/2. Two more families have x symmetric to the above with respect to x = 1/2, namely in the intervals 3/4 < x < 1 and 1/2 < x < 3/4. These orbits have n negative. Similar families appear whenever K ≥ 2π|n|. The stability of these orbits was considered by Chirikov [1979]. The solutions in the intervals 0 < x < 1/4 and 3/4 < x < 1 are unstable, while the solutions in the intervals 1/4 < x < 1/2 and 1/2 < x < 3/4 are stable whenever K is in the intervals 2nπ < K < [(2nπ)2 + 16]1/2
(7)
The stable period-1 orbits are surrounded by islands of stability. The H´enon stability parameters [H´enon, 1965] of these families for n = 1 are given in Fig. 21. The intervals of stability ∆K = [(2nπ)2 + 16]1/2 − 2nπ are approximately equal to ∆K = 8/K. The period-2 family O1 generated at K = 4 is stable in the interval 4 < K < 2π (Fig. 20). An orbit starting at O1 has another point O1 symmetric with respect to the center (0.5, 0.5) [Fig. 1(a)].
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
2875
therefore y = 2x
(mod 1)
(14)
If we set 1 +δ 2 with 0 < δ < 1/2 we find x=
(15)
y = 2δ
(16)
and 2π(4δ + n − 1) (17) sin 2πδ The minimum value of K is given by 4 (18) K0 = cos 2πδ0 where π(n − 1) (19) tan 2πδ0 − 2πδ0 = 2 The values of δ0 for various integer values of n and the corresponding values of K0 are given in Table 1. The values of K0 are smaller than 2nπ, but they approach 2nπ for large n. Thus the differences ∆K0 approach 2π. The values of δ0 are smaller than 1/4 and they approach 1/4 for large n. The corresponding values of x0 at the minimum K approach 3/4. The family O1 is called “regular” because it is connected to the original family O (x = 1/2) that exists all the way from the unperturbed problem (K = 0). The families O1∗ , O1∗∗ , O1∗∗∗ , are “irregular” families, because they are not connected to the families of the unperturbed problem. (The terminology of “regular” and “irregular” families was introduced by Contopoulos [1970].) Every new stable irregular family is generated at a tangent bifurcation, (i.e. at a minimum K, where the tangent of the characteristic is vertical) together with an unstable periodic orbit of the same multiplicity (Fig. 22). K=
Fig. 21. The stability curves for the orbits of group II (solid line for stable and dashed line for unstable).
After two iterations the standard map gives K 2K sin 2πx + sin 2πx x = x + 2y + 2π 2π (mod 1) (8) K K sin 2πx + sin 2πx y = y + 2π 2π If x = x (mod 1), y = y (mod 1) we find K(sin 2πx + sin 2πx ) = 2π
(9)
where x is given by Eq. (1) and is an integer. Equation (9) was given by Chirikov [1979], who introduced the notion of “accelerator modes” if the images of y differ from y by an integer (see also [Rom-Kedar & Zaslavsky, 1999]). The case = 0 is the simplest one. Then Eq. (9) gives either x = n − x or
x =
1 n + 2
(first case)
(10)
+x
(second case)
(11)
First Case. This case refers to orbits of group I. Then Eqs. (1), (8) and (10) give K sin 2πx = 1 − n ≡ 0 (mod 1) (12) 2y + 2π hence K sin 2πx = x − y + 1 − n = n − x x = x + y + 2π (13)
Table 1. n
n, δ0 , K0 , ∆K0 for various families.
δ0
Family
1
0
2
0.19538
O1∗
3
0.21516
4
0.22423
5
0.22951
O1
O1∗∗ O1∗∗∗
K0
∆K0
4 11.887
7.887
18.413
6.526
24.810
6.397
31.159
6.349
October 3, 2005
2876
13:55
01371
G. Contopoulos et al.
0 < x < 1/4 and 3/4 < x < 1. This is seen in Fig. 22 where all the families O1 , O1∗ , O1∗∗ , O1∗∗∗ . . . change from stable to unstable at x = 3/4 (i.e. δ = 1/4). The corresponding y are all equal to y = 1/2 at bifurcation, and the values of K are equal to K = 2nπ
Fig. 22. The characteristic diagram for the orbits O1 , O1∗ , O1∗∗ , O1∗∗∗ and some of their bifurcations.
The stability of the double periodic orbit is found by differentiating Eq. (8) with respect to x and y, namely dx
= A2 dx + B2 dy dy = C2 dx + D2 dy
(20)
The eigenvalues λ are given by λ2 − 2aλ + 1 = 0
(21)
where a = 1/2(A2 + D2 ) = 1 + K cos 2πx + K cos 2πx K2 cos 2πx cos 2πx (22) 2 is the H´enon stability parameter [H´enon, 1965]. From Eq. (10) we find +
cos 2πx = cos 2πx
(23)
and K2 cos2 2πx (24) 2 The orbits are stable if −1 < a < 1, hence a = 1 + 2K cos 2πx +
−4 < K cos 2πx < 0
(25)
0 < (2 + K cos 2πx)2
(26)
and
Therefore the orbits are stable if cos 2πx < 0, i.e. 1/4 < x < 3/4 and unstable if cos 2πx > 0, i.e.
(27)
Therefore the orbits O1 , O1∗ , O1∗∗ , O1∗∗∗ . . . are stable in the intervals K0 < K < 2nπ. The interval of stability ∆K = 2nπ − K0 is found approximately as follows. If we set π (28) 2πδ0 = − 2 in Eq. (19) we find πn (29) cot + = 2 hence 8 2 1+ 2 2 (30) ≈ πn 3π n Then from Eq. (18) we find 2 4 ≈ 2πn 1 − 2 2 K0 = sin π n
(31)
and 8 4 ≈ (32) πn K The inequality (26) is always satisfied, except if K cos 2πx = −K cos 2πδ = −2, when we have an equality. For example, if n = 0 we have δ = 0.1855 and K = 5.07. The stability curve of this family is tangent to the axis a = −1 at this point (Fig. 20). There we have the bifurcation of two stable and two unstable periodic orbits of period-4. These stable orbits produce resonant islands inside the islands surrounding O1 and O1 , as seen in Figs. 4 and 5 of [Contopoulos et al., 1999]. When these secondary islands escape in the chaotic sea, they generate a large dip (n/m = 1/2) in the curve giving A versus K (Fig. 8). Similar tangencies appear in the stability curves of the period-2 families for n ≥ 1. The characteristic of the period-2 family continues as unstable (Fig. 20) beyond the bifurcation point (x = 3/4, y = 1/2). As K increases both x and y increase and tend to the point x = y = 1 as K → ∞. The positions of this unstable periodic orbit for K = 12.4 and K = 18.8 are marked by dots in Figs. 16 and 19. ∆K ≈
Second Case. At the bifurcation point of group I family O1 at (x = 3/4, y = 1/2, K = 2π) we have generation of two families of equal period O2
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
(period-2, Fig. 20). These families belong to the second case of period-2 orbits given by Eq. (11). Then from Eqs. (1), (8), (9) and (11) we find 2y +
K sin 2πx = 0 (mod 1) 2π
x =
1 + x = x + y 2
(33)
and
=x+y+
K sin 2πx 2π
(mod 1)
(34)
hence y = y =
1 2
(mod 1)
(35)
and K=
2πn | sin 2πx|
(36)
Therefore the new families have y = 1/2 (constant), while two values of x are found for each K from Eq. (36). Similar period-2 families (equal period bifurcations) appear for n ≥ 2 at the same point (x = 3/4, y = 1/2, for K = 2πn). Their stability can be studied in the same way as above. They are stable in the intervals 2πn < K < [(2πn)2 + 4]1/2
(37)
The interval of stability ∆K is equal to ∆K ≈ 2/K. These families were discussed by Chirikov [1970], who gave the formulae (7) and (37). But we stress here that these families are bifurcations of the families O1 , O1∗ , O1∗∗ . . . for n = 1, 2, 3, respectively. An example of the stability curves of such families is given in Fig. 20. When such a family becomes unstable at a = −1 for K = [(2πn)2 + 4]1/2 it generates a double period family (period-4). This family becomes also unstable at a = −1 and generates a period-8 family, and so on. All the families beyond O2 are generated by period-doubling bifurcations. The stability intervals of every successive family decrease by a factor 8.72 [Benettin et al., 1980]. The islands around these successive bifurcating orbits are small. Their size is given in Fig. 8 beyond K = 2π = 6.283 as a continuation of the solid curve of Fig. 8 after the last KAM curve around O is destroyed at K = 6.335. The intervals of stability are: O1 : 2π − 4 = 2.283, O2 : [(2π)2 + 4]1/2 − 2π = 6.5938 − 6.2832 = 0.3106, O3 : 6.630184 − 6.593817 = 0.036367, O4 : 6.634364 − 6.630184 = 0.004180, O5 : 6.634844 − 6.634364 = 0.000480.
2877
Higher order intervals decrease by a factor 8.72 each time. Thus this sequence leads to an infinity of unstable periodic orbits beyond K = 6.6349 (Fig. 8). The total interval of stability before and beyond K = 2πn (solid line of Fig. 8) is equal to ∆K(group I) ≈ 10/K. Further Families. The sizes of the islands of group II (islands near the x-axis, Fig. 4) are given by dashed lines in Figs. 3 and 8. The periodic orbits that are surrounded by the islands with n = 1 start at K = 2π = 6.2832 and are stable up to K = [(2π)2 +16]1/2 = 7.4484 (Fig. 8). Then they generate a double period family that is stable in the interval 7.4484 − 7.7134, followed by a stable period-4 family in the interval 7.71341 − 7.74486, a stable period-8 family in the interval 7.74486 − 7.74848, and so on (Fig. 21). The intervals decrease by a factor that tends to 8.72. Thus beyond K = 7.74895 there is an infinity of unstable periodic orbits and all the islands are destroyed. Similar results appear for group II islands with n = 2 (Fig. 3). They start at K = 4π and are stable up to K = [(4π)2 + 16]1/2 = 13.1876. Then they generate a cascade of period-doubling bifurcations that lead to a destruction of all the islands beyond K = 13.3589. All the above families are generated with = 0 in Eq. (9). If is an integer different from 0 more period-2 families are generated. All of them
Fig. 23. The parameter K as a function of 2πδ for different (solid lines = −2, dashed lines = −1, dotted lines = 1).
October 3, 2005
2878
13:55
01371
G. Contopoulos et al.
are “irregular”, i.e. they are not connected to any families of the unperturbed problem. They are produced at various minimum values of K as a couple of one stable and one unstable family. Using Eqs. (8) and (9) we find K sin 2πδ = 2π −K sin 2πδ + sin 2π δ − 2 (38) where x = 1/2 + δ. For a fixed , Eq. (38) gives K as a function of δ. These functions have a minimum K. In fact, the factor of K in Eq. (38) is less than 2 absolutely, therefore K cannot be smaller than π||. Various families K = K(δ) for different are shown in Fig. 23. Near the minimum K one of the two branches is stable. However the corresponding islands of stability are extremely small.
Group III islands surround a set of period-4 orbits, and they are unexpectedly large. These orbits appear in pairs, forming two sets of four points each. Two points are on the x-axis and two outside this axis (Figs. 6(a) and 6(b)). Thus each set of four islands has two islands on the x-axis and two islands outside this axis. Here we discuss the period-4 orbits at the centers of these islands. We use the values of x , y as given by Eq. (1) and the values of x , y as given by Eq. (8). Then
+
x = x + 2y + 2z +
K sin 2πx 2π
x = x + 3y + 3z +
2K K sin 2πx + sin 2πx 2π 2π
(mod 1)
4y + 4z + + z+
2K 3K sin 2πx + sin 2πx 2π 2π
K sin 2πx = 0 2π
(mod 1)
(43)
K K K sin 2πx + sin 2πx + sin 2πx = 0 2π 2π 2π K sin 2π(x + z) = −n ≡ 0 2π
(mod 1) (44)
We can check that this solution satisfies both equations (43). From Eqs. (41) and (44) we find K[sin 2πx + sin 2π(x + z)] = 2π
(45)
as in the case of double periodic orbits [Eq. (9)]. From now on we consider only the case = 0. We have two solutions of this equation. Either (46)
or (39)
z=
1 2
(47)
The first solution gives 3K 4K sin 2πx + sin 2πx = x + 4y + 2π 2π +
y IV
(42)
One solution of these equations has y = 0 and
and xIV
(mod 1)
We have a periodic orbit of period-4 if (xIV = x, y IV = y) (mod 1). Then
x+z =1−x
2K 3K sin 2πx + sin 2πx 2π 2π
K sin 2πx 2π
x = x + y + z
y = z +
5. Period-4 Orbits
x = x + 3y +
(n = integer, 0 < z < 1) we find
2K K sin 2πx + sin 2πx 2π 2π
(48)
n+z K = 2π sin πz
(49)
and (mod 1) (40)
K K sin 2πx + sin 2πx =y+ 2π 2π
K K sin 2πx + sin 2πx + 2π 2π If we write K sin 2πx = n + z 2π
(1 − z) 2
x=
The corresponding value of x, as a function of K is given in Fig. 24 (for n = 1). The solution (49) has a minimum K = K0 if tan πz0 = π(n + z0 ), therefore (41)
K0 =
2 cos πz0
(50)
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
2879
We have stable orbits if −1 < a < 1. From Eq. (48) we find c = c = − cos πz
(54)
Kc = −2π(n + z) cot πz
(55)
and
Then 1 a = 1 + 8Kc + 10K 2 c2 + 4K 3 c3 + K 4 c4 2 1 = 1 + Kc(Kc + 2)2 (Kc + 4) 2 and we have stability if
Fig. 24. The characteristic diagram for periodic orbits of group III.
Table 2. n
πz0
K0
∆K0
1 2 3 4
1.3518168 1.4420665 1.4793437 1.4998232
9.2067 15.5794 21.8996 28.2034
6.3727 6.3202 6.3038
(58)
At the minimum K0 we have K0 c0 = −2. As z increases, x decreases from x0 = (1 − z0 )/2 (curve (1) of Fig. 24) and Kc increases until it becomes positive when z > 1/2. At the transition to instability z = 1/2 the value of K is K = (2n + 1)π
The values of πz0 and K0 at the minima of K for various values of n are given in Table 2. The values of πz0 are smaller than π/2, i.e. z0 < 1/2. The values of K0 are a little smaller than (2n + 1)π, but they tend to (2n + 1)π for large n. The differences ∆K0 are larger than 2π, but they tend to 2π as n increases. Therefore the corresponding islands are close to 2π apart. Differentiating Eqs. (40) we find dxIV = A4 dx + B4 dy dy IV = C4 dx + D4 dy
1 (57) −1 < 1 + Kc(Kc + 2)2 (Kc + 4) < 1 2 The stability curves (a versus K) for this family and its bifurcations are given in Fig. 25 for n = 1. The right inequality (57) gives −4 < Kc < 0
πz0 , K0 , ∆K0 for various n.
(56)
(59)
and x = 1/4. The interval of stability ∆K = (2n + 1)π − K0 is approximately equal to ∆K ≈
(51)
and if we set dxIV = λdx, dy IV = λdy, we find the eigenvalues λ from an equation of the form of Eq. (21), where the H´enon stability parameter is 3 a = 1 + 4K(c + c ) + K 2 (c2 + c2 ) 2 1 + 7K 2 cc + 2K 3 cc (c + c ) + K 4 c2 c2 2
(52)
with c = cos 2πx,
c = cos 2π(x + z)
(53)
Fig. 25. The stability curves for the periodic orbits of group III.
October 3, 2005
2880
13:55
01371
G. Contopoulos et al.
2/K (found in a similar way as the interval of stability of group I orbits). If z decreases below z0 , K increases, x increases beyond x0 = (1 − z0 )/2 (curve (2) of Fig. 24) and Kc decreases until it becomes smaller than −4. At the transition to instability K1 c1 = −4. For example, for n = 1 we find z0 = 0.4303, x0 = 0.2849, z1 = 0.36071, x1 = 0.31965 and K1 = 9.4392. The transition value K1 along the upper branch of x is slightly larger than the transition value K = 3π ≈ 9.4248 for instability along the lower branch of period-4 orbits with x < x0 . The left inequality (57) gives 0 < 4 + Kc(Kc + 2)2 (Kc + 4) = e
(60)
This is satisfied for −4 √< Kc < 0, except at the points√Kc = −2 + 2 = −0.5858, and Kc = −2 − 2 = −3.4142, where e = 0. As a consequence the stability curves of the families (1) and (2) are tangent to the axis a = −1 at one point each (both points are near K ≈ 9.3 for n = 1) (Fig. 25). Similar stability curves appear for all values of n. The fact that for a given K we have two values of x (Fig. 24) implies that we have the appearance of two stable orbits for the same values of K. These orbits are surrounded by pairs of islands (group III islands, Fig. 6). Similar pairs of islands were found in the web map by Zaslavsky et al. [1997]. At K = (2n + 1)π two period-4 families bifurcate from the previous family (equal period bifurcations along the lower period-4 family of Fig. 25 (family (1)), with z = 1/2. The first has x > 1/4 and the second x < 1/4. The corresponding y along the new families is y = z = 1/2 (constant) and K=
π(2n + 1) sin 2πx
(61)
The values of c and c [Eq. (53)] are c = −c = cos 2πx
(62)
K 2 c2 = K 2 − [π(2n + 1)]2
(63)
and
Then from Eq. (52) we find 1 a = 1 − 4K 2 c2 + K 4 c4 2
(64)
This orbit is stable if −1 < a < 1, therefore we have the inequalities K 2 c2 (K 2 c2 − 8) < 0 and
0 < 4 − 8K 2 c2 + K 4 c4 . These are written as
and
or
K 2 < [π(2n + 1)]2 + 8
(65)
√ K 2 c2 = K 2 − [π(2n + 1)]2 < 4 − 2 3
(66)
√ K 2 c2 = K 2 − [π(2n + 1)]2 > 4 + 2 3
(67)
For example, for n = 1 the new period-4 families are stable in the intervals 3π = 9.425 < K < 9.453 and 9.813 < K < 9.840. At K = 9.453 a period-8 family is generated followed by a cascade of period-doubling bifurcations (Figs. 24 and 25). Beyond K = 9.840 these families are unstable. At K = 9.840 each of them generates by bifurcation two equal period families that are stable in a small interval beyond this value of K. However, the corresponding islands of stability near K = 9.840 are very small. A similar bifurcation occurs along the upper branch (family (2) of Fig. 24) at K = K1 . The orbits have another intersection point with y = 0 and x = (n + z)/2. Therefore for n = 1 x =
(1 + z) 2
(68)
while K and Kc are given by Eqs. (49) and (55). Thus the same stability criteria apply to them. At the transition K = K1 start two bifurcating families of equal period, that generate a cascade of bifurcations. The stability curves of these orbits are shown as dashed lines in Fig. 25. We notice that the two islands of Fig. 6 are well separated for large K and they disappear at different values of K. It is remarkable that the intervals of stability for the periodic orbits beyond K = 2nπ and K = (2n + 1)π are given by similar, yet different, formulae. Namely the simple periodic orbits on the x-axis (group II) are stable in an interval ∆K = [(2nπ)2 + 16]1/2 − 2nπ ≈ 8/K [Eq. (7)], the double period orbits with constant y = 1/2 (also in group I) are stable in an interval ∆K = [(2nπ)2 + 4]1/2 − 2nπ ≈ 2/K [Eq. (37)], while the orbits of the group III with constant y = 1/2 are √ stable in 2 two intervals ∆K√= {[(2n + 1)π] + 4 − 2 3}1/2 − ∆K = {[(2n + √ 1)π]2 + (2n + 1)π ≈ (2 − 3)/K and √ 1/2 2 1/2 ≈ (2 − 3)/K 8} − {[(2n + 1)π] + 4 + 2 3} (Eq. 66). Therefore these stability intervals of the three groups of islands have similar forms.
October 3, 2005
13:55
01371
Recurrence of Order in Chaos
6. Conclusions The most important results of the present study are the following: (i) Important islands of stability of the standard map appear in a recurrent way in three groups, for values of the nonlinearity parameter K near K = 2nπ (groups I and II) and K = (2n + 1)π (group III) with n ≥ 1. (ii) The maximum areas A of these islands follow similar power laws A = aK b , with a near 0.5 and b near −2 for all three groups. (iii) The islands exist in small intervals around the above values of K. The intervals of the groups I and II slightly overlap each other near K = 2nπ. (iv) The sizes of the islands within each of the above intervals ∆K increase and decrease in a characteristic, universal way. In particular A decreases abruptly near particular resonances. The curves giving A versus K are practically exactly the same after an appropriate scaling in the cases of groups I and II. The curves of group III agree with each other (after scaling) but they are quantitatively different from the curves of groups I and II, although they have similar dips at the same resonances. Similar increases and decreases of islands were found in several other problems. (v) The main difference of the islands of group III is that they appear in pairs (around two stable periodic orbits of period-4 each, belonging to two families (1) and (2)). These periodic orbits are generated, together with two unstable orbits, at the same tangent bifurcation. On the contrary, the islands of groups I and II are around regular periodic orbits, or around irregular orbits generated at a tangent bifurcation of the usual type (one stable and one unstable orbit). (vi) The islands of type I are surrounded by arcs of the asymptotic curves (unstable and stable) of the main periodic orbit O (x = 0.5, y = 0), near a point (x = 0.7, y = 0.4), and a symmetric point (x = 0.3, y = 0.6). These islands appear for the first time when K > 4. When K increases to K > 8 these islands are destroyed, but when K increases further to K ≈ 12 similar islands appear near the same point. This is due to the almost linear increase of the lengths of the asymptotic curves (up to their outermost tips) with K. In fact the
(vii)
(viii)
(ix)
(x)
2881
eigenvalue |λ| of the periodic orbit O is an almost linear function of K. When K increases from K ≈ 2π to K ≈ 4π the tips of the asymptotic curves move to the squares [(1, 2), (1, 2)] (unstable) and [(0, 1), (1, 2)] (stable), and when these squares are superimposed on the original square [(0, 1), (0, 1)] the structure of the arcs of the asymptotic curves near their tips is similar around the (same) points (x = 0.7, y = 0.4) and (x = 0.3, y = 0.6). Thus the asymptotic curves leave free space for the formation of similar islands of stability. Similar results appear whenever K increases by 2nπ. The islands surround stable periodic orbits that become unstable at particular critical values K = Kcr , and they are followed by a cascade of period doubling bifurcations at intervals decreasing by a factor near δ = 8.72. Every time K increases by 2π a new family appears that follows the same pattern of period-doubling bifurcations. These families are irregular, i.e. they are not connected to the families that exist for small K. The islands of the group II appear on the xaxis. These islands surround irregular stable periodic orbits on the x-axis. The first irregular periodic orbits appear when K ≥ 2π. Higher order islands appear whenever K ≥ 2nπ. Their stability and bifurcations can be found analytically. The periodic orbits of group II are stable in an interval ∆K beyond K = 2nπ. The main periodic orbits of group I are stable in an interval before K = 2nπ up to K = 2nπ. At K = 2nπ they become unstable, but they generate two equal period families of orbits that have (both) y = 1/2 (constant) and they are stable for an interval ∆K beyond 2nπ. These orbits are followed by a cascade of perioddoubling bifurcations. The periodic orbits of group III (family (1)) are stable in an interval ∆K before K = (2n + 1)π. Two islands of group III are on the x-axis and two more outside this axis. The corresponding periodic orbits are stable in an interval ∆K before K = (2n + 1)π, and become unstable at K = (2n + 1)π. There they generate two equal period orbits (period4) each, that have two points with y = 0 and two more points with y = 1/2 (constant) and they are stable in an interval ∆K beyond K = (2n + 1)π.
October 3, 2005
2882
13:55
01371
G. Contopoulos et al.
(xi) The minima K of the new irregular families of groups I and III are somewhat smaller than 2nπ and (2n+1)π respectively, while the minimum K of group II families is 2nπ. The stable periodic orbits of groups I and III (family (1)) become unstable at exactly 2nπ and (2n + 1)π and the intervals of stability are ∆K ≈ 8/K and ∆K ≈ 2/K, respectively. The orbits of groups I (in particular the equal period bifurcating families) and II, starting at K = n π = 2nπ and of group III (the equal period bifurcating families from family (1)) starting at K = n π = (2n+1)π are stable in intervals from K = n π up to√K = [(n π)2 + B]1/2 , with B = 4, 16 and 4− 2 3 in groups I, II, III, respectively. The corresponding intervals of stability are ∆K ≈ 2/K, ∆K ≈ 8/K and ∆K ≈ 0.27/K respectively. In group III (family(1)) the stable fam√ ily that becomes unstable for B = 4 − 2 3 becomes again stable √ when B is in an interval between B = 4 + 2 3 and B = 8 and this interval is again equal to ∆K ≈ 0.27/K. Similar results apply to the family (2) of group III. Thus the intervals of K that contain the main islands of the system (around the main periodic orbits of periods 1, 2 and 4) can be found theoretically. The decrease of the size of the islands can be also estimated theoretically. What was found empirically is the relatively large size of these islands. (xii) It seems probable that recurrences of order in chaos appear for most maps with a certain modulo. In fact, by bringing back the asymptotic curves in the original square, we may find new homoclinic tangencies, hence, according to the Newhouse theorem [Newhouse, 1983], new stable periodic orbits. In particular, we did find similar recurrencies in further maps with trigonometric terms. However, this problem requires a more detailed study.
Acknowledgments This work was partly supported by the 200/557 program of the Academy of Athens.
We thank Mr. Manolis Zoulias for his technical support.
References Benettin, G., Cercignani, C., Galgani, L. & Giorgilli, A. [1980] “Universal properties in conservative dynamical systems,” Lett. Nuovo Cim. 28, 1–4. Chirikov, B. V. [1979] “A universal instability of manydimensional oscillator systems,” Phys. Rep. 52, 263– 379. Contopoulos, G. [1970] “Orbits in highly perturbed dynamical systems. I,” Astron. J. 75, 96–107. Contopoulos, G., Harsoula, M., Voglis, N. & Dvorak, R. [1999] “Destruction of islands of stability,” J. Phys. A32, 5213–5232. Duarte, P. [1994] “Plenty of elliptic islands for the standard family of area preserving maps,” Ann. Inst. Henri Poincar´e 11, 359–407. Giorgilli, A. & Lazutkin, V. F. [2000] “Some remarks on the problem of ergodicity of the standard map,” Phys. Lett. A259, 359–367. H´enon, M. [1965] “Exploration num´erique du probl`eme restreint II,” Ann. Astrophys. 28, 992–1007. H´enon, M. [1966] “Exploration num´erique du probl`eme restreint III,” Bull Astron. 1, 57–79. Karney, C. F. F., Rechester, A. B. & White, R. B. [1982] “Effect of noise on the standard map,” Physica D4, 425–438. Lichtenberg, A. J. & Lieberman, M. A. [1992] Regular and Chaotic Dynamics, 2nd edition (Springer Verlag, NY). Newhouse, S. E. [1983] “Generic properties of conservative systems,” in Chaotic Behaviour of Deterministic Systems, eds. Looss, G., Helleman, R. H. G. & Stora, R. (North Holland, Amsterdam), pp. 443–451. Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. [1992] Numerical Recipes in Fortran (Cambridge Univ. Press). Rom-Kedar, V. & Zaslavsky, G. [1999] “Islands of accelerator modes and homoclinic tangles,” Chaos 9, 697– 705. Zaslavsky, G. M., Edelman, M. & Niyazov, B. A. [1997] “Self-similarity, renormalization, and phase space nonuniformity of the Hamiltonian chaotic dynamics,” Chaos 7, 159–181. Zaslavsky, G. M. [2002] “Chaos, fractional kinetics and anomalous transport,” Phys. Rep. 371, 461–580.
