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Global periodic structure of integrable Duffing’s maps
Chaos, Solitons and Fractals 16 (2003) 233–244 www.elsevier.com/locate/chaos
Global periodic structure of integrable DuffingÕs maps Chieko Murakami a, Wakako Murakami a, Kei-ichi Hirose b, Yoshi. H. Ichikawa a,* a
College of Engineering, Chubu University, Kasugai-shi 487-8501, Japan b National Institute for Fusion Science, Toki-shi 509-5292, Japan Accepted 5 July 2002
Abstract Integrable DuffingÕs mappings have been constructed on the basis of a scheme presented by Suris. The Poincare– Birkhoff resonance condition determines the periodic behavior of the orbit close to the fixed points of the integrable mappings. In the region far apart from the fixed points, the nonlinear effect modifies the property of the periodic orbits. Here, the global behavior of periodic orbits of the integrable DuffingÕs maps is investigated by applying the Fourier analysis on the individual orbits. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Studies of nonlinear mappings are providing key informations on chaos in nonlinear dynamical systems [1,2]. Rapid development of computational physics has induced explosive expansion of our understanding of the nonlinear phenomena both in conservative and dissipative systems in physics and engineering. Though nonlinear mappings were commonly used as tools for the study of chaos, new interests are provoking the study of integrability on the basis of nonlinear mappings. Grammaticos and his collaborators [3,4] have been studying the Painleve property of integrable mappings, while Suris has presented a general scheme of integrable mappings [5] in addition to the works of Quispel and his collaborators [6,7]. In this connection, we have undertaken some exploration of the properties of integrable hyperbolic mappings [8], in which we have analyzed the phase space structure around the fixed point of the Sinh map, and have shown interesting feature of the global spectrum of the periodic orbits in the integrable Sinh map. In the present paper, we will investigate the global spectrum of the periodic orbits in the integrable DuffingÕs maps. These mappings attract our special interest because they are pertinent to describe the stationary envelope solitary waves in the nonlinear cubic Schrodinger equation. Discussing the basic properties of the first kind of DuffingÕs maps in the second section, we present a systematic analysis of the global spectrum of the periodic orbits in the Section 3. Analysis of the second kind of DuffingÕs map will be followed in the Section 4 and the Section 5. The last section is devoted for concluding discussions.
2. The basic properties of the integrable Duffing’s maps Referring to works of Ross and Thompson [9] and Suris [5], we have introduced the integrable DuffingÕs maps and have discussed the solutions of the Duffing equation [10]. Depending on the signs of the cubic nonlinear term of the Duffing equation, we have two types of the integrable DuffingÕs map
*
Corresponding author.
0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 1 9 4 - 7
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pnþ1 ¼ pn þ F ðÞ ðqn Þ qnþ1 ¼ qn þ pnþ1
ð1Þ
with the abbreviations of F ðþÞ ðqÞ ¼ 2
A q2 q 2 þ q2
ð2Þ
for the first kind, and F ðÞ ðqÞ ¼ 2
A þ q2 q 2 q2
ð3Þ
for the second kind of DuffingÕs map, respectively. As for the first kind of DuffingÕs map with Eq. (2), the fixed point at ðp ¼ 0; q ¼ 0Þ is stable for the values of A in the range of 4 < A < 0
ð4Þ
and is hyperbolic with reflection for A < 4. Hence, it undergoes the period doubling bifurcation yielding the period-2 orbit with the coordinate pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ qð2Þ ¼ ðA þ 4Þ; pð2Þ ¼ 2qð2Þ ; which remains to be stable for 1 < A < 4. The fixed point ðp ¼ 0;pffiffiffi q ¼ 0Þ turns to be unstable without reflection at 0 < A. For the positive value of A, the fixed point at ðp ¼ 0; q ¼ AÞ remains to be stable for A < 1. We can determine exactly the position of the period-4 orbit along the p ¼ 0 axis as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qð4Þ ¼ A ðA2 4Þ1=2 ; pð4Þ ¼ 0 ð6Þ We notice that the period-4 orbit exists for þ2 < A. Constructing the tangent map around the stable fixed points, we can obtain the Poincare–Birkhoff resonance condition as follows; around the fixed point at the origin, m m A ¼ 4 sin2 p ð7Þ n n which gives Table 1 for the period-n resonance with m ¼ 1. Here, we notice that at the value of A ¼ 2:00, the mapping function F þ is reduced to be linear as )2q, and we have a linear map 0 1 2 p p ð8Þ ¼ q0 1 1 q Since we have 4 1 2 1 ¼ 1 1 0
0 1
ð9Þ
we conclude that at the value of A ¼ 2:00 the first kind of integrable DuffingÕs map posses the intrinsic period-4 structure, namely the every orbit has the period-4. As it has been discussed in the analysis of the integrable Sinh map [8], the integrable mappings do not give rise to the Poincare–Birkhoff necklace of periodic islands around the stable fixed point, and hence the chaos never appears. As an illustration, let us examine the phase portrait of the first kind of DuffingÕs map at the value of A ¼ 3:20 in Fig. 1(a). We can identify three small spots at ðp ¼ 1:1; q ¼ 0:7Þ, ðp ¼ þ1:05; q ¼ þ0:35Þ and ðp ¼ þ0:0; q ¼ þ0:38Þ. This period-3 structure is best confirmed by constructing the momentum inversion symmetry lines as shown in Fig. 1(b).
Table 1 A n
)4.00 2
)3.00 3
)2.00 4
)1.38 5
)1.00 6
)0.7532 7
)0.5858 8
C. Murakami et al. / Chaos, Solitons and Fractals 16 (2003) 233–244
235
Fig. 1. (a) Phase space portrait of the first kind of integrable DuffingÕs map at A ¼ 3:20. (b) The momentum inversion symmetry lines cj ð0 < j < 3Þ for the first kind of DuffingÕs map at A ¼ 3:20.
Fig. 2. Phase space portrait at A ¼ 1:20.
At the value of A ¼ 1:20, we can observe clearly the period-5 streaks in Fig. 2. It is interesting to observe that at the value of A ¼ 0:90 as shown in Fig. 3, the period-6 streaks born in the center while the period-5 streaks remain in the outer fringe. In the following section, we will analyze the global change of the periodicity of orbits in the phase space.
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Fig. 3. Phase space portrait at A ¼ 0:90.
Table 2 A n
0.3431 8
0.4638 7
0.6667 6
1.056 5
2.00 4
6.00 3
1 2
Now, turning pffiffiffi to the positive values of A, we obtain the Poincare–Birkhoff resonance around the fixed point ðp ¼ 0; q ¼ AÞ as m m A ¼ 2 tan2 p n n
ð10Þ
which gives Table 2 for the period-n resonance with m ¼ 1. At the value of A ¼ þ0:70, we can count the period-6 streaks at the central region and the period-7 streaks in the fringe as shown in Fig. 4, while at the value of A ¼ þ2:10 we see the period-4 streaks are stretching over the entire circumference as shown in Fig. 5. Eq. (6) determines the positions of the period-4 orbits along the p ¼ 0 axis as q ¼ þ1:208 and q ¼ þ1:655. Referring to Table 2, we will examine the resonance at the value pffiffiffi of A ¼ þ6:00. In Fig. 6(a), we can observe the birth of the period-3 structure around the fixed point ðp ¼ 0; q ¼ 6 ¼ 2:45Þ. We should notice, however, that Eq. (6) suggests the existence of the period-4 orbits at ðp ¼ 0; q ¼ 0:585Þ and ðp ¼ 0; q ¼ 3:414Þ. We can identify this period-4 structure as thin streaks in the outermost fringe of the island. Fig. 6(b) illustrates the momentum inversion symmetry lines up to the order of 4. In the central region, the intersections of c3 with c0 (p ¼ 0 axis) identify the birth of the period-3 orbit, while at the outside fringe the intersections of c4 with c0 determine the period-4 coordinates. Such observation induces our interest to investigate the overall manifestation of the periodic orbits in the phase space. In the following section, we analyze the global spectrum of periodicity of the orbits with the help of the Fourier spectrum analysis.
C. Murakami et al. / Chaos, Solitons and Fractals 16 (2003) 233–244
Fig. 4. Phase space portrait of the first kind of DuffingÕs map around the fixed point of ðp ¼ 0; q ¼ cycle-6 streaks structures are clearly visible in the center.
Fig. 5. Phase space portrait around the fixed point of ðp ¼ 0; q ¼ visible.
237
pffiffiffiffiffiffiffiffiffi 0:70 ¼ 0:8366Þ at A ¼ 0:70. The
pffiffiffiffiffiffiffi 2:1 ¼ 1:425Þ at A ¼ 2:1. The cycle-4 streak structures are easily
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pffiffiffiffiffiffiffi Fig. 6. (a) Phase space portrait around the fixed point of ðp ¼ 0; q ¼ 6:0 ¼ 2:45Þ at A ¼ 6:0. The Poincare–Birkhoff resonance condition Eq. (7) determines birth of the period-3 structure at the center, while Eq. (6) determines the starting positions of the period-4 streaks on the p ¼ 0 axis. (b) The momentum inversion symmetry lines cj ð0 < j < 4Þ at A ¼ 6:0.
3. The global spectrum of periodic orbits in the first kind of Duffing’s map In order to understand overall behavior of the periodic orbits in the first kind of the integrable DuffingÕs map, we apply the fast Fourier transformation (FFT in short) to the orbits started from the initial positions along the p ¼ 0-axis. For the negative values of A, 4 < A < 0, we summarize our analysis in Fig. 7. Firstly, as discussed in the Section 2, we find the map is reduced to a linear map, Eq. (8), at A ¼ 2:00. Thus, we have the period-4 for the entire group of the orbits. As for other values of A, near the origin, the periodicity is determined by the resonance condition given by Eq. (7). The curb for the value of A ¼ 3:20 confirms the observation of Fig. 1. Eq. (7) for A ¼ 3:20 gives rise to a value of n ¼ 2:84 at the center. The curb passes through the line of the period-3 at the value of q ¼ 0:38, which is in accord with the observation of Fig. 1. Here, upon increasing the distance from the origin, the map approaches asymptotically to the linear map Eq. (8) in the limit of large q, and thus the average periodicity of the orbits approaches to the limiting value of 4. Now, as for A ¼ 0:9, Eq. (7) determines the average periodicity n ¼ 6:36 at q ¼ 0. As the result of FFT analysis of the orbits, we can confirm indeed that there occurs the period-6 orbit near the position of q ¼ 0:4 and the period-5 orbit around q ¼ 1:1. This observation is consistent with the phase space portrait of Fig. 3. It would be worth to notice that Eq. (7) can be regarded as a continuous function of n, not restricted n to be an integer. Secondly, for the positive value of A, referring to Table 2, we summarize the overall variation of the periodicity of the orbits obtained by the FFT analysis in Fig. 8. For the value of A ¼p0:3431, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi the orbit started at ðp ¼ 0; q ¼ 0:1Þ has a large periodicity of 12. Approaching to the fixed point ðp ¼ 0; q ¼ 0:3431 ¼ 0:586Þ, we confirm that the periodicity of the orbit is reduced down to 8, which is the value calculated by Eq. (10). Moving toward outside, we notice the curb goes up again to the larger periodicity and further out the FFT spectrum gives rise to complicated structure. Thepboundary of this transition is determined to be ffiffiffiffiffiffi the position of the homoclinic orbit with p ¼ 0, that is given by q ¼ 2A. In other words, the curves shown in Fig. 8 represent the global variation of the periodicitypof ffiffiffi orbits in the region bounded by the homoclinic orbit. The minimum of curbs occurs at the fixed point ðp ¼ 0; q ¼ AÞ, where the linearized tangent map determines the periodicity of the orbit in accord with Eq. (10). Lastly, the case of A ¼ þ6:00 calls our special interest. In Fig. 6(a), we have shown that near the fixed point pffiffiffiffiffiffiffiffiffi ðp ¼ 0; q ¼ 6:00 ¼ 2:45Þ there occurs the period-3 cycles as predicted by Eq. (10) and at the outer fringe there appear the period-4 structure. This behavior is in accord with Fig. 8. Now, referring to the phase space portrait shown in Fig. 9,
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239
Fig. 7. Global variation of the periodicity of the orbits for A < 0. The orbits start at ðp ¼ 0; 0 < q < 3Þ. The curbs give the values of periodicity determined from Eq. (7) at q < 0:4, and approach to the asymptotic value of the period-4 at larger values of q. The curbs for A ¼ 3:20 and )0.9 explain the observed structure of Figs. 1(a) and 3.
Fig. 8. Global variation of the periodicity of the orbits for 0 < A. Theporbits start at ðp ¼ 0; q ¼ 0:1Þ for A p <ffiffiffiffiffi 1:1 ffi and at ffiffiffi ðp ¼ 0; q ¼ 0:2Þ for A ¼ 2 and 6. The curbs go down to the minimum at q ¼ A, and go up sharply at the edge of q ¼ 2A, which is the coordinate of the homoclinic orbit on the p ¼ 0 axis.
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Fig. 9. Phase space portrait at A ¼ 6:00. The black circles represent the positions of the period-4 orbit determined from Eq. (6).
we investigate how the FFT spectrum appears in the outside across the homoclinic orbit. Fig. 10 illustrates our observation of the peak of FFT analysis for the orbits starting at ðp ¼ 0; 1 < q < 5Þ. At q ¼ 2:45, the FFT peak appears at the maximum value of 682, so that the period is given as 211 =682 ¼ 3:00. For the exact period-4 orbit at ðp ¼ 0; q ¼ 3:414Þ, the FFT peak will be at 512. The sharp drop of the curve at occurs at the homoclinic orbit with q ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 6 ¼ 3:464. For the orbit with the initial position at ðp ¼ 0; 5 < qÞ, we obtain several peaks of FFT spectrum. At the present stage of investigation, we can not give interpretation for this transition. We reserve this for our future study.
4. The basic properties of the second kind of Duffing’s map Now, turning to the second kind of DuffingÕs map, defined by the mapping function Eq. (3), we find that the fixed point at ðp ¼ 0; q ¼ 0Þ is stable for the value of A in the range of 4 < A < 0;
ð11Þ
and is hyperbolic pffiffiffiffiffiffiffiwith reflection for A < 4, and becomes unstable without reflection. The other fixed point at ðp ¼ 0; q ¼ AÞ exists only for the negative value of A, and always unstable. The coordinates of the period-2 orbit are determined as pffiffiffiffiffiffiffiffiffiffiffiffi qð2Þ ¼ A þ 4; pð2Þ ¼ 2qð2Þ ð12Þ which exists for 4 < A. We can obtain a formal expression for the period-4 orbit as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qð4Þ ¼ A ðA2 4Þ1=2 ; pð4Þ ¼ 0
ð13Þ
which exists for A < 4. The Poncare–Birkhoff resonance condition around the fixed point ðp ¼ 0; q ¼ 0Þ is given by the same expression Eq. (7) for the first kind of integrable DuffingÕs map. At the value of A ¼ 2:00, as in the case of the first kind of integrable DuffingÕs map, we have a linear map, Eq. (8), and hence here the every orbit has the period-4.
C. Murakami et al. / Chaos, Solitons and Fractals 16 (2003) 233–244
Fig. 10. The FFT peaks observed for A ¼ 6:00. The curb drops sharply at the edge of the homoclinic orbit at ðp ¼ 0; q ¼ 3:464Þ. Beyond the value of q ¼ 5, there occurs several peaks in the FFT spectrum.
241
pffiffiffiffiffi 12 ¼
The characteristic feature of the second kind of integrable DuffingÕs map is illustrated in Fig. 11 for the value of A ¼ 3:00, and in Fig. 12 for the value of A ¼ 0:9409. Fig. 11 indicates that the stable fixed point at the acpffiffiorigin ffi companies unstable period-2 orbit with q ¼ 1 to p ¼ 2, while the other fixed points at ðp ¼ 0; q ¼ 3Þ are hyperbolic with reflection. Here, we notice that the heteroclinic orbit joining the period-2 orbits intersect with the p ¼ 0 axis at q ¼ 0:4. As for pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the value of A ¼ 0:9409, we have the stable fixed point at the origin, and the fixed points at ðp ¼ 0; q ¼ 0:9409 ¼ 0:97Þ are hyperbolic without reflection. The period-2 orbit at ðpð2Þ ¼ 2qð2Þ ; qð2Þ ¼ 1:749Þ are unstable. We have carried out the FFT analysis for the periodic orbits around the stable fixed point at the origin. In Fig. 13, the every curbs start with the periodicity given in Table 1 as it was in Fig. 7. For the value of A ¼ 2:00, the map is reduced to a linear map with the intrinsic periodicity of 4. Moving outside to the edge of the heteroclinic pffiffiffiffiffiffiffi orbit, for the value of 2:00 < A, we observe that the curbs go up to large periodicity at the coordinate of q ¼ A. On the other hand, for the value of 4 < A < 2, we have opposite behavior. Namely for the value of A ¼ 3:00, the curb goes down from the period-3 at the center. As mentioned in the above, the heteroclinic orbit here gives the boundary at q ¼ 0:4. Beyond this edge, the FFT spectrum yields several peaks, suggesting transition to irregular motions. Basing on Fig. 13, we can reproduce accurate phase space portrait for the closed periodic orbit. For example, for the value of A ¼ 1:00, we will be able to observe the birth of period-6 structure in the center, and around p ¼ 0, q ¼ 0:7 we could observe the period-7 structure.
5. Concluding discussions In the preceding sections, we have carried out detailed analysis of the global structure of the periodic orbits in the integrable DuffingÕs maps. Referring to the knowledge of the basic properties of the maps, such as the stability of the fixed points and the Poincare–Birkhoff resonance, we have observed the phase space portrait of the integrable DuffingÕs maps. Because of the integrability of the maps, any orbits never break up into fragments and no chaos occurs. We have observed, however, that among the closed orbits there appear periodical structure corresponding to the Poincare– Birkhoff resonance condition. Confirming its appearance by the symmetry analysis, we have applied the Fourier transformation to the orbits, and reduced the global variation of the periodicity of the orbits. As far as the periodic orbits bounded by the homoclinic and heteroclinic orbits are concerned, the Fourier transformation reproduces
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Fig. 11. Phase space portrait of the second kind of DuffingÕs map at A ¼ 3:00. The stable fixed point at the origin is encircled by the heteroclinic orbits joining the unstable period-2 orbits.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 12. Phase space portrait of the second kind of DuffingÕs map at A ¼ 0:9409. The fixed points at ðp ¼ 0; q ¼ 0:9409 ¼ 0:97Þ are hyperbolic without reflection.
C. Murakami et al. / Chaos, Solitons and Fractals 16 (2003) 233–244
243
Fig. 13. Global variation of the periodicity of the orbits for the second kindpofffiffiffiffiffiffiffi the DuffingÕs map. The curbs start at period determined by the linearized tangent map, and then go up sharply at the edge of q ¼ A for 2 < A < 0.
correctly the overall spectrum. However, we have found that in the outside of the homoclinic orbit there occurs many FFT peaks. At the present stage of studies, we do not have appropriate interpretation for such multipeaks spectrum. We reserve this for future investigation.
Acknowledgements We wish to express our thanks to Professor T. Kuroda for his generous support and constant encouragement extended to the present works. We are much obliged to Professor Y. Kimura of Nagoya University for his stimulating suggestion and critical discussion during the course of the present studies. They are also indebted to Dr. S. Saito of Nagoya University for his helpful discussions.
Appendix A. Momentum inversion symmetry of Duffing’s maps In order to make the present paper to be self-contained, we discuss briefly the momentum inversion symmetry of DuffingÕs maps. The two dimensional area preserving map 0 p ¼ p þ F ðqÞ T : ðA:1Þ q0 ¼ q þ p0 can be decomposed into the involution J0 and J1 defined as T ¼ J1 J0
ðA:2Þ
where denotes functional composition and J0 J0 ¼ J1 J1 ¼ 1. The inverse mapping T 1 expressed as T 1 ¼ J0 J1
ðA:3Þ
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C. Murakami et al. / Chaos, Solitons and Fractals 16 (2003) 233–244
and the jth involution Jj is defined as Jj ¼ T j J0 The momentum inversion symmetry is defined by ( p0 ¼ p J0 : q0 ¼ q p ( p0 ¼ p þ F ðq pÞ J1 : q0 ¼ q 2p þ F ðq pÞ
ðA:4Þ
ðA:5Þ ðA:6Þ
N-periodic orbit is determined by TNR ¼ R
ðA:7Þ
and a symmetry line cj for any integer j is formed by the set of fixed points of the involution Jj , cj : fRjJj R ¼ Rg
ðA:8Þ
The intersection of cj and ck is a periodic point. Its period N divides jj kj. The higher order symmetry lines are constructed by using the relations c2nþk ¼ T n ck
ðA:9Þ
c2nk ¼ Jn ck
ðA:10Þ
Although we can write down explicitly the expressions of the lower order symmetry lines as c0 : p ¼ 0
ðA:11Þ
c1 : 2p ¼ F ðqÞ
ðA:12Þ
c2 : 2p ¼ 2F ðqÞ
ðA:13Þ
c3 : 2p ¼ 2F ðqÞ F ðq þ p þ F ðqÞÞ
ðA:14Þ
cþ1 : 2p ¼ F ðq pÞ
ðA:15Þ
cþ2 : 2p ¼ 2F ðq pÞ
ðA:16Þ
the symmetry lines of Figs. 1(b) and 6(b) have been constructed by the use of Eqs. (A.9) and (A.10).
References [1] [2] [3] [4] [5] [6] [7] [8]
Lichtenberg A, Lieberman MA. Regular and Stochastic Motion. New York: Springer; 1983. Atlee Jackson E. In: Perspectives of Nonlinear Dynamics, vols.1 and 2. New York: Cambridge University Press; 1989. Grammaticos B, Ramani A, Papageorgiou V. Do integrable mappings have the Painleve property? Phys Rev Lett 1991;67:1825–8. Ramani A, Gramamticos B, Hietarinta J. Discrete versions of Painleve equations. Phys Rev Lett 1991;67:1829–32. Suris YuB. Integrable mappings of the standard type. Funk Anal Ego Priloz 1989;23:84–5. Quispel GRW, Roberts JAG, Thompson CJ. Integrable mappings and Soliton equations. Phys Lett 1988;126:419–21. Quispel GRW, Roberts JAG, Thompson CJ. Integrable mappings and Soliton equations II. Physica D 1989;34:183–92. Hirose K, Murakami C, Murakami W. Global periodic structure of integrable hyperbolic map. Chaos, Solitons and Fractals, in press. [9] Ross KA, Thompson C. Interation of some discretization of the nonlinear Schrodinger equation. Physica 1986;135 A:551–8. [10] Murakami W, Murakami C, Hirose K, Ichikawa YH. Integrable DuffingÕs maps and solutions of the Duffing equation. Chaos, Solitons and Fractals, in press.
Global periodic structure of integrable DuffingÕs maps Chieko Murakami a, Wakako Murakami a, Kei-ichi Hirose b, Yoshi. H. Ichikawa a,* a
College of Engineering, Chubu University, Kasugai-shi 487-8501, Japan b National Institute for Fusion Science, Toki-shi 509-5292, Japan Accepted 5 July 2002
Abstract Integrable DuffingÕs mappings have been constructed on the basis of a scheme presented by Suris. The Poincare– Birkhoff resonance condition determines the periodic behavior of the orbit close to the fixed points of the integrable mappings. In the region far apart from the fixed points, the nonlinear effect modifies the property of the periodic orbits. Here, the global behavior of periodic orbits of the integrable DuffingÕs maps is investigated by applying the Fourier analysis on the individual orbits. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Studies of nonlinear mappings are providing key informations on chaos in nonlinear dynamical systems [1,2]. Rapid development of computational physics has induced explosive expansion of our understanding of the nonlinear phenomena both in conservative and dissipative systems in physics and engineering. Though nonlinear mappings were commonly used as tools for the study of chaos, new interests are provoking the study of integrability on the basis of nonlinear mappings. Grammaticos and his collaborators [3,4] have been studying the Painleve property of integrable mappings, while Suris has presented a general scheme of integrable mappings [5] in addition to the works of Quispel and his collaborators [6,7]. In this connection, we have undertaken some exploration of the properties of integrable hyperbolic mappings [8], in which we have analyzed the phase space structure around the fixed point of the Sinh map, and have shown interesting feature of the global spectrum of the periodic orbits in the integrable Sinh map. In the present paper, we will investigate the global spectrum of the periodic orbits in the integrable DuffingÕs maps. These mappings attract our special interest because they are pertinent to describe the stationary envelope solitary waves in the nonlinear cubic Schrodinger equation. Discussing the basic properties of the first kind of DuffingÕs maps in the second section, we present a systematic analysis of the global spectrum of the periodic orbits in the Section 3. Analysis of the second kind of DuffingÕs map will be followed in the Section 4 and the Section 5. The last section is devoted for concluding discussions.
2. The basic properties of the integrable Duffing’s maps Referring to works of Ross and Thompson [9] and Suris [5], we have introduced the integrable DuffingÕs maps and have discussed the solutions of the Duffing equation [10]. Depending on the signs of the cubic nonlinear term of the Duffing equation, we have two types of the integrable DuffingÕs map
*
Corresponding author.
0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 1 9 4 - 7
234
C. Murakami et al. / Chaos, Solitons and Fractals 16 (2003) 233–244
pnþ1 ¼ pn þ F ðÞ ðqn Þ qnþ1 ¼ qn þ pnþ1
ð1Þ
with the abbreviations of F ðþÞ ðqÞ ¼ 2
A q2 q 2 þ q2
ð2Þ
for the first kind, and F ðÞ ðqÞ ¼ 2
A þ q2 q 2 q2
ð3Þ
for the second kind of DuffingÕs map, respectively. As for the first kind of DuffingÕs map with Eq. (2), the fixed point at ðp ¼ 0; q ¼ 0Þ is stable for the values of A in the range of 4 < A < 0
ð4Þ
and is hyperbolic with reflection for A < 4. Hence, it undergoes the period doubling bifurcation yielding the period-2 orbit with the coordinate pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ qð2Þ ¼ ðA þ 4Þ; pð2Þ ¼ 2qð2Þ ; which remains to be stable for 1 < A < 4. The fixed point ðp ¼ 0;pffiffiffi q ¼ 0Þ turns to be unstable without reflection at 0 < A. For the positive value of A, the fixed point at ðp ¼ 0; q ¼ AÞ remains to be stable for A < 1. We can determine exactly the position of the period-4 orbit along the p ¼ 0 axis as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qð4Þ ¼ A ðA2 4Þ1=2 ; pð4Þ ¼ 0 ð6Þ We notice that the period-4 orbit exists for þ2 < A. Constructing the tangent map around the stable fixed points, we can obtain the Poincare–Birkhoff resonance condition as follows; around the fixed point at the origin, m m A ¼ 4 sin2 p ð7Þ n n which gives Table 1 for the period-n resonance with m ¼ 1. Here, we notice that at the value of A ¼ 2:00, the mapping function F þ is reduced to be linear as )2q, and we have a linear map 0 1 2 p p ð8Þ ¼ q0 1 1 q Since we have 4 1 2 1 ¼ 1 1 0
0 1
ð9Þ
we conclude that at the value of A ¼ 2:00 the first kind of integrable DuffingÕs map posses the intrinsic period-4 structure, namely the every orbit has the period-4. As it has been discussed in the analysis of the integrable Sinh map [8], the integrable mappings do not give rise to the Poincare–Birkhoff necklace of periodic islands around the stable fixed point, and hence the chaos never appears. As an illustration, let us examine the phase portrait of the first kind of DuffingÕs map at the value of A ¼ 3:20 in Fig. 1(a). We can identify three small spots at ðp ¼ 1:1; q ¼ 0:7Þ, ðp ¼ þ1:05; q ¼ þ0:35Þ and ðp ¼ þ0:0; q ¼ þ0:38Þ. This period-3 structure is best confirmed by constructing the momentum inversion symmetry lines as shown in Fig. 1(b).
Table 1 A n
)4.00 2
)3.00 3
)2.00 4
)1.38 5
)1.00 6
)0.7532 7
)0.5858 8
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Fig. 1. (a) Phase space portrait of the first kind of integrable DuffingÕs map at A ¼ 3:20. (b) The momentum inversion symmetry lines cj ð0 < j < 3Þ for the first kind of DuffingÕs map at A ¼ 3:20.
Fig. 2. Phase space portrait at A ¼ 1:20.
At the value of A ¼ 1:20, we can observe clearly the period-5 streaks in Fig. 2. It is interesting to observe that at the value of A ¼ 0:90 as shown in Fig. 3, the period-6 streaks born in the center while the period-5 streaks remain in the outer fringe. In the following section, we will analyze the global change of the periodicity of orbits in the phase space.
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Fig. 3. Phase space portrait at A ¼ 0:90.
Table 2 A n
0.3431 8
0.4638 7
0.6667 6
1.056 5
2.00 4
6.00 3
1 2
Now, turning pffiffiffi to the positive values of A, we obtain the Poincare–Birkhoff resonance around the fixed point ðp ¼ 0; q ¼ AÞ as m m A ¼ 2 tan2 p n n
ð10Þ
which gives Table 2 for the period-n resonance with m ¼ 1. At the value of A ¼ þ0:70, we can count the period-6 streaks at the central region and the period-7 streaks in the fringe as shown in Fig. 4, while at the value of A ¼ þ2:10 we see the period-4 streaks are stretching over the entire circumference as shown in Fig. 5. Eq. (6) determines the positions of the period-4 orbits along the p ¼ 0 axis as q ¼ þ1:208 and q ¼ þ1:655. Referring to Table 2, we will examine the resonance at the value pffiffiffi of A ¼ þ6:00. In Fig. 6(a), we can observe the birth of the period-3 structure around the fixed point ðp ¼ 0; q ¼ 6 ¼ 2:45Þ. We should notice, however, that Eq. (6) suggests the existence of the period-4 orbits at ðp ¼ 0; q ¼ 0:585Þ and ðp ¼ 0; q ¼ 3:414Þ. We can identify this period-4 structure as thin streaks in the outermost fringe of the island. Fig. 6(b) illustrates the momentum inversion symmetry lines up to the order of 4. In the central region, the intersections of c3 with c0 (p ¼ 0 axis) identify the birth of the period-3 orbit, while at the outside fringe the intersections of c4 with c0 determine the period-4 coordinates. Such observation induces our interest to investigate the overall manifestation of the periodic orbits in the phase space. In the following section, we analyze the global spectrum of periodicity of the orbits with the help of the Fourier spectrum analysis.
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Fig. 4. Phase space portrait of the first kind of DuffingÕs map around the fixed point of ðp ¼ 0; q ¼ cycle-6 streaks structures are clearly visible in the center.
Fig. 5. Phase space portrait around the fixed point of ðp ¼ 0; q ¼ visible.
237
pffiffiffiffiffiffiffiffiffi 0:70 ¼ 0:8366Þ at A ¼ 0:70. The
pffiffiffiffiffiffiffi 2:1 ¼ 1:425Þ at A ¼ 2:1. The cycle-4 streak structures are easily
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pffiffiffiffiffiffiffi Fig. 6. (a) Phase space portrait around the fixed point of ðp ¼ 0; q ¼ 6:0 ¼ 2:45Þ at A ¼ 6:0. The Poincare–Birkhoff resonance condition Eq. (7) determines birth of the period-3 structure at the center, while Eq. (6) determines the starting positions of the period-4 streaks on the p ¼ 0 axis. (b) The momentum inversion symmetry lines cj ð0 < j < 4Þ at A ¼ 6:0.
3. The global spectrum of periodic orbits in the first kind of Duffing’s map In order to understand overall behavior of the periodic orbits in the first kind of the integrable DuffingÕs map, we apply the fast Fourier transformation (FFT in short) to the orbits started from the initial positions along the p ¼ 0-axis. For the negative values of A, 4 < A < 0, we summarize our analysis in Fig. 7. Firstly, as discussed in the Section 2, we find the map is reduced to a linear map, Eq. (8), at A ¼ 2:00. Thus, we have the period-4 for the entire group of the orbits. As for other values of A, near the origin, the periodicity is determined by the resonance condition given by Eq. (7). The curb for the value of A ¼ 3:20 confirms the observation of Fig. 1. Eq. (7) for A ¼ 3:20 gives rise to a value of n ¼ 2:84 at the center. The curb passes through the line of the period-3 at the value of q ¼ 0:38, which is in accord with the observation of Fig. 1. Here, upon increasing the distance from the origin, the map approaches asymptotically to the linear map Eq. (8) in the limit of large q, and thus the average periodicity of the orbits approaches to the limiting value of 4. Now, as for A ¼ 0:9, Eq. (7) determines the average periodicity n ¼ 6:36 at q ¼ 0. As the result of FFT analysis of the orbits, we can confirm indeed that there occurs the period-6 orbit near the position of q ¼ 0:4 and the period-5 orbit around q ¼ 1:1. This observation is consistent with the phase space portrait of Fig. 3. It would be worth to notice that Eq. (7) can be regarded as a continuous function of n, not restricted n to be an integer. Secondly, for the positive value of A, referring to Table 2, we summarize the overall variation of the periodicity of the orbits obtained by the FFT analysis in Fig. 8. For the value of A ¼p0:3431, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi the orbit started at ðp ¼ 0; q ¼ 0:1Þ has a large periodicity of 12. Approaching to the fixed point ðp ¼ 0; q ¼ 0:3431 ¼ 0:586Þ, we confirm that the periodicity of the orbit is reduced down to 8, which is the value calculated by Eq. (10). Moving toward outside, we notice the curb goes up again to the larger periodicity and further out the FFT spectrum gives rise to complicated structure. Thepboundary of this transition is determined to be ffiffiffiffiffiffi the position of the homoclinic orbit with p ¼ 0, that is given by q ¼ 2A. In other words, the curves shown in Fig. 8 represent the global variation of the periodicitypof ffiffiffi orbits in the region bounded by the homoclinic orbit. The minimum of curbs occurs at the fixed point ðp ¼ 0; q ¼ AÞ, where the linearized tangent map determines the periodicity of the orbit in accord with Eq. (10). Lastly, the case of A ¼ þ6:00 calls our special interest. In Fig. 6(a), we have shown that near the fixed point pffiffiffiffiffiffiffiffiffi ðp ¼ 0; q ¼ 6:00 ¼ 2:45Þ there occurs the period-3 cycles as predicted by Eq. (10) and at the outer fringe there appear the period-4 structure. This behavior is in accord with Fig. 8. Now, referring to the phase space portrait shown in Fig. 9,
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Fig. 7. Global variation of the periodicity of the orbits for A < 0. The orbits start at ðp ¼ 0; 0 < q < 3Þ. The curbs give the values of periodicity determined from Eq. (7) at q < 0:4, and approach to the asymptotic value of the period-4 at larger values of q. The curbs for A ¼ 3:20 and )0.9 explain the observed structure of Figs. 1(a) and 3.
Fig. 8. Global variation of the periodicity of the orbits for 0 < A. Theporbits start at ðp ¼ 0; q ¼ 0:1Þ for A p <ffiffiffiffiffi 1:1 ffi and at ffiffiffi ðp ¼ 0; q ¼ 0:2Þ for A ¼ 2 and 6. The curbs go down to the minimum at q ¼ A, and go up sharply at the edge of q ¼ 2A, which is the coordinate of the homoclinic orbit on the p ¼ 0 axis.
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Fig. 9. Phase space portrait at A ¼ 6:00. The black circles represent the positions of the period-4 orbit determined from Eq. (6).
we investigate how the FFT spectrum appears in the outside across the homoclinic orbit. Fig. 10 illustrates our observation of the peak of FFT analysis for the orbits starting at ðp ¼ 0; 1 < q < 5Þ. At q ¼ 2:45, the FFT peak appears at the maximum value of 682, so that the period is given as 211 =682 ¼ 3:00. For the exact period-4 orbit at ðp ¼ 0; q ¼ 3:414Þ, the FFT peak will be at 512. The sharp drop of the curve at occurs at the homoclinic orbit with q ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 6 ¼ 3:464. For the orbit with the initial position at ðp ¼ 0; 5 < qÞ, we obtain several peaks of FFT spectrum. At the present stage of investigation, we can not give interpretation for this transition. We reserve this for our future study.
4. The basic properties of the second kind of Duffing’s map Now, turning to the second kind of DuffingÕs map, defined by the mapping function Eq. (3), we find that the fixed point at ðp ¼ 0; q ¼ 0Þ is stable for the value of A in the range of 4 < A < 0;
ð11Þ
and is hyperbolic pffiffiffiffiffiffiffiwith reflection for A < 4, and becomes unstable without reflection. The other fixed point at ðp ¼ 0; q ¼ AÞ exists only for the negative value of A, and always unstable. The coordinates of the period-2 orbit are determined as pffiffiffiffiffiffiffiffiffiffiffiffi qð2Þ ¼ A þ 4; pð2Þ ¼ 2qð2Þ ð12Þ which exists for 4 < A. We can obtain a formal expression for the period-4 orbit as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qð4Þ ¼ A ðA2 4Þ1=2 ; pð4Þ ¼ 0
ð13Þ
which exists for A < 4. The Poncare–Birkhoff resonance condition around the fixed point ðp ¼ 0; q ¼ 0Þ is given by the same expression Eq. (7) for the first kind of integrable DuffingÕs map. At the value of A ¼ 2:00, as in the case of the first kind of integrable DuffingÕs map, we have a linear map, Eq. (8), and hence here the every orbit has the period-4.
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Fig. 10. The FFT peaks observed for A ¼ 6:00. The curb drops sharply at the edge of the homoclinic orbit at ðp ¼ 0; q ¼ 3:464Þ. Beyond the value of q ¼ 5, there occurs several peaks in the FFT spectrum.
241
pffiffiffiffiffi 12 ¼
The characteristic feature of the second kind of integrable DuffingÕs map is illustrated in Fig. 11 for the value of A ¼ 3:00, and in Fig. 12 for the value of A ¼ 0:9409. Fig. 11 indicates that the stable fixed point at the acpffiffiorigin ffi companies unstable period-2 orbit with q ¼ 1 to p ¼ 2, while the other fixed points at ðp ¼ 0; q ¼ 3Þ are hyperbolic with reflection. Here, we notice that the heteroclinic orbit joining the period-2 orbits intersect with the p ¼ 0 axis at q ¼ 0:4. As for pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the value of A ¼ 0:9409, we have the stable fixed point at the origin, and the fixed points at ðp ¼ 0; q ¼ 0:9409 ¼ 0:97Þ are hyperbolic without reflection. The period-2 orbit at ðpð2Þ ¼ 2qð2Þ ; qð2Þ ¼ 1:749Þ are unstable. We have carried out the FFT analysis for the periodic orbits around the stable fixed point at the origin. In Fig. 13, the every curbs start with the periodicity given in Table 1 as it was in Fig. 7. For the value of A ¼ 2:00, the map is reduced to a linear map with the intrinsic periodicity of 4. Moving outside to the edge of the heteroclinic pffiffiffiffiffiffiffi orbit, for the value of 2:00 < A, we observe that the curbs go up to large periodicity at the coordinate of q ¼ A. On the other hand, for the value of 4 < A < 2, we have opposite behavior. Namely for the value of A ¼ 3:00, the curb goes down from the period-3 at the center. As mentioned in the above, the heteroclinic orbit here gives the boundary at q ¼ 0:4. Beyond this edge, the FFT spectrum yields several peaks, suggesting transition to irregular motions. Basing on Fig. 13, we can reproduce accurate phase space portrait for the closed periodic orbit. For example, for the value of A ¼ 1:00, we will be able to observe the birth of period-6 structure in the center, and around p ¼ 0, q ¼ 0:7 we could observe the period-7 structure.
5. Concluding discussions In the preceding sections, we have carried out detailed analysis of the global structure of the periodic orbits in the integrable DuffingÕs maps. Referring to the knowledge of the basic properties of the maps, such as the stability of the fixed points and the Poincare–Birkhoff resonance, we have observed the phase space portrait of the integrable DuffingÕs maps. Because of the integrability of the maps, any orbits never break up into fragments and no chaos occurs. We have observed, however, that among the closed orbits there appear periodical structure corresponding to the Poincare– Birkhoff resonance condition. Confirming its appearance by the symmetry analysis, we have applied the Fourier transformation to the orbits, and reduced the global variation of the periodicity of the orbits. As far as the periodic orbits bounded by the homoclinic and heteroclinic orbits are concerned, the Fourier transformation reproduces
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Fig. 11. Phase space portrait of the second kind of DuffingÕs map at A ¼ 3:00. The stable fixed point at the origin is encircled by the heteroclinic orbits joining the unstable period-2 orbits.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 12. Phase space portrait of the second kind of DuffingÕs map at A ¼ 0:9409. The fixed points at ðp ¼ 0; q ¼ 0:9409 ¼ 0:97Þ are hyperbolic without reflection.
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Fig. 13. Global variation of the periodicity of the orbits for the second kindpofffiffiffiffiffiffiffi the DuffingÕs map. The curbs start at period determined by the linearized tangent map, and then go up sharply at the edge of q ¼ A for 2 < A < 0.
correctly the overall spectrum. However, we have found that in the outside of the homoclinic orbit there occurs many FFT peaks. At the present stage of studies, we do not have appropriate interpretation for such multipeaks spectrum. We reserve this for future investigation.
Acknowledgements We wish to express our thanks to Professor T. Kuroda for his generous support and constant encouragement extended to the present works. We are much obliged to Professor Y. Kimura of Nagoya University for his stimulating suggestion and critical discussion during the course of the present studies. They are also indebted to Dr. S. Saito of Nagoya University for his helpful discussions.
Appendix A. Momentum inversion symmetry of Duffing’s maps In order to make the present paper to be self-contained, we discuss briefly the momentum inversion symmetry of DuffingÕs maps. The two dimensional area preserving map 0 p ¼ p þ F ðqÞ T : ðA:1Þ q0 ¼ q þ p0 can be decomposed into the involution J0 and J1 defined as T ¼ J1 J0
ðA:2Þ
where denotes functional composition and J0 J0 ¼ J1 J1 ¼ 1. The inverse mapping T 1 expressed as T 1 ¼ J0 J1
ðA:3Þ
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and the jth involution Jj is defined as Jj ¼ T j J0 The momentum inversion symmetry is defined by ( p0 ¼ p J0 : q0 ¼ q p ( p0 ¼ p þ F ðq pÞ J1 : q0 ¼ q 2p þ F ðq pÞ
ðA:4Þ
ðA:5Þ ðA:6Þ
N-periodic orbit is determined by TNR ¼ R
ðA:7Þ
and a symmetry line cj for any integer j is formed by the set of fixed points of the involution Jj , cj : fRjJj R ¼ Rg
ðA:8Þ
The intersection of cj and ck is a periodic point. Its period N divides jj kj. The higher order symmetry lines are constructed by using the relations c2nþk ¼ T n ck
ðA:9Þ
c2nk ¼ Jn ck
ðA:10Þ
Although we can write down explicitly the expressions of the lower order symmetry lines as c0 : p ¼ 0
ðA:11Þ
c1 : 2p ¼ F ðqÞ
ðA:12Þ
c2 : 2p ¼ 2F ðqÞ
ðA:13Þ
c3 : 2p ¼ 2F ðqÞ F ðq þ p þ F ðqÞÞ
ðA:14Þ
cþ1 : 2p ¼ F ðq pÞ
ðA:15Þ
cþ2 : 2p ¼ 2F ðq pÞ
ðA:16Þ
the symmetry lines of Figs. 1(b) and 6(b) have been constructed by the use of Eqs. (A.9) and (A.10).
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