- Email: [email protected]
Global periodic structure of integrable hyperbolic map
Chaos, Solitons and Fractals 15 (2003) 47–56 www.elsevier.com/locate/chaos
Global periodic structure of integrable hyperbolic map Kei-ichi Hirose b
a,*
, Chieko Murakami b, Wakako Murakami
b
a National Institute of Fusion Science, Toki 509-5292, Japan College of Engineering Chubu University, Kasugai 487-8501, Japan
Accepted 23 April 2002
Abstract Integrable hyperbolic mappings are constructed within a scheme presented by Suris. The Cosh map is a singular map, of which fixed point is unstable. The global behavior of periodic orbits of the Sinh map is investigated referring to the Poincare–Birkhoff resonance condition. Close to the fixed point, the periodicity is indeed determined from the Poincare–Birkhoff resonance condition. Increasing the distance from the fixed point, the orbit is affected by the nonlinear effect and the average periodicity varies globally. The Fourier transformation of the individual orbits determines overall spectrum of global variation of the periodicity. 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 work of Quispel and his collaborators [6,7]. Here, referring to the work of Suris, we undertake some exploration of the properties of integrable hyperbolic mappings, although we do not have suitable example to apply these mappings to any physical systems. We illustrate explicitly the characteristic feature of the integrable mappings, where all of the orbits prevail on the closed torus and no chaos appears in the whole range of the nonlinear parameter. In Section 2, we introduce the Cosh map and Sinh map, and investigate their basic properties. As for the Cosh map, there is no fixed point in the finite range of the phase space, even though it is an integrable map. Hence, we focus our interest to study the Sinh map. We present our analysis on the phase space structure around fixed point of the Sinh map in Section 3. On the contrary to the nonintegrable mappings, we do not observe the Poincare–Birkhoff bifurcation around the stable fixed point. This is one of the characteristic feature of the integrable mappings. We can confirm, however, that around the fixed point there appear periodic structure determined from the Poincare–Birkhoff resonance condition. In order to analyze this periodic structure, we examine the time sequence of the individual orbits with the help of the fast Fourier transform in Section 4. We present concluding remarks by summarizing the global spectrum of the periodic orbits in the integrable mappings in Section 5.
*
Corresponding author. E-mail address: [email protected] (K.-i. Hirose).
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 0 8 6 - 3
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K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
2. Cosh map and Sinh map Suris [5] has presented a theorem that the second order nonlinear difference equation xnþ1 2xn þ xn1 ¼ F ðxn ; Þ
ð1Þ
with the following three types of function F ðx; Þ: ðaÞ F ðx; Þ ¼
A þ Bx þ Cx2 þ Dx3 1 ðE þ 1=3Cx þ 1=2Dx2 Þ
ð2Þ
ðbÞ
2 ðx=2ÞðA sinðxxÞ þ B cosðxxÞ þ C sinð2xxÞ þ D cosð2xxÞÞ tan1 x 1 ðx=2ÞðA sinðxxÞ B cosðxxÞ þ C sinð2xxÞ D cosð2xxÞ þ EÞ
ð3Þ
1 1 þ aðB expðaxÞ þ D expð2axÞ EÞ ln a 1 aðA expðaxÞ þ C expð2axÞ þ EÞ
ð4Þ
F ðx; Þ ¼
ðcÞ F ðx; Þ ¼
are integrable in the sense that Eq. (1) posses a nontrivial symmetrical integral Uðxnþ1 ; xn Þ ¼ Uðxn ; xn1 Þ
ð5Þ
Suris conjectured that the proved theorem means the possibility of the construction of integrable difference schemes for the second order differential equations €x ¼ f0 ðxÞ with a right hand side of the forms, ðaÞ f0 ðxÞ ¼ A þ Bx þ Cx2 þ Dx3 ðbÞ
f0 ðxÞ ¼ A sinðxxÞ þ B cosðxxÞ þ C sinð2xxÞ þ D cosð2xxÞ
ðcÞ f0 ðxÞ ¼ A expðaxÞ þ B expðaxÞ þ C expð2axÞ þ D expð2axÞ
ð6Þ ð7Þ ð8Þ
To be specific, we restrict our interest to the case (c) with a special choice of B ¼ þA and B ¼ A, C ¼ D ¼ 0. Namely, we have f0 ðxÞ ¼ 2A coshðaxÞ and f0 ðxÞ ¼ 2A sinhðaxÞ, respectively. We introduce a normalized space variable qn as qn ¼ axn
ð9Þ
and the momentum variable pn in terms of pnþ1 ¼ qnþ1 qn
ð10Þ
Defining a new parameter A as aA !A 1 aE
ð11Þ
we can reduce Eq. (1) with Eq. (8) to the two dimensional area preserving map, pnþ1 ¼ pn þ F ðqn Þ
ð12Þ
qnþ1 ¼ qn þ Pnþ1
ð13Þ
with the abbreviation F ¼ ln
1 A expðqn Þ 1 A expðqn Þ
ð14Þ
The upper sign þ (and the lower sign ) stands for f0 ðxÞ ¼ 2A coshðqÞ (and 2A sinhðqÞÞ, defining the Cosh map (and the Sinh map), respectively. We find immediately that the Cosh map does not posses any fixed points, while the Sinh map has a fixed point at the origin (p ¼ 0, q ¼ 0) in the phase space. The stability of the fixed point (p ¼ 0, q ¼ 0) of the Sinh map is analyzed by constructing a tangent map at the point (p ¼ 0, q ¼ 0). Since the residue R at the point (p ¼ 0, q ¼ 0) is given as R¼
1 A 2 1A
ð15Þ
K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
49
the fixed point (p ¼ 0, q ¼ 0) is stable in the region of 1 < A < 0 and 2 < A < 1. In the region of 0 < A < 1, the fixed point is hyperbolic without reflection, while it becomes hyperbolic with reflection in the region of 1 < A < 2.
3. Poincare´–Birkhoff resonance of the Sinh map In the case of nonintegrable area preserving map, the tangent map at the stable fixed point determines the condition of Poincare–Birkhoff bifurcation around the stable fixed point. As for the integrable area preserving map, we observe that no chaos arises and every orbits form closed torus. In this situation, for the rotation number q ¼ ðm=nÞ2p, the eigenvalue of the tangent map determines the Poincare–Birkhoff resonance condition cos q ¼ 1 2R with Eq. (15), which in turn gives the value of nonlinear parameter Aðm=nÞ as Aðm=nÞ ¼ 1
1 cosð2pm=nÞ
ð16Þ
As for A < 0, taking m ¼ 1, we obtain the following values as shown in Table 1, while for A > 0, taking m ¼ 2, we have Table 2, respectively. Here, we illustrate the phase space portraits of the Sinh map in Fig. 1 (A ¼ 2:30), Fig. 2 (A ¼ 1:25), Fig. 3 (A ¼ 1:05), Fig. 4 (A ¼ 0:62) and Fig. 5 (A ¼ 0:42), respectively. For the positive value of A, we have Fig. 6 (A ¼ 2:236) and Fig. 7 (A ¼ 3:00). Appearance of the cycle-5 and the cycle-3 structure in Figs. 6 and 7 confirm the prediction of Table 2. Unlike the case of nonintegrable area preserving mappings, we do not have the
Table 1 Period number for Poincare–Birkhoff resonance (A < 0) A n
1 4
2.236 5
1.000 6
0.6039 7
0.4142 8
0.3054 9
Table 2 Period number for Poincare–Birkhoff resonance (A > 0) A n
2.000 4
2.236 5
3.000 6
5.494 7
þ1 8
Fig. 1. Phase space portrait at A ¼ 2:30. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:1) indicates slow modulation of the cycle-5 cyclic motion.
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K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
Fig. 2. Phase space portrait at A ¼ 1:25. The orbits in the outer region exhibit clear cycle-17 structure. The orbit with initial position (p ¼ 0:0, q ¼ 0:1) returns nearly to the starting position after the 17th iteration.
Fig. 3. Phase space portrait at A ¼ 1:05. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:1) shows slow modulation of the period-6 motion.
formation of the Poincare–Birkhoff neck lace of the periodic islands. We may call these cycle-n structure as the Poincare–Birkhoff resonance. Careful observation of Figs. 1 and 3 reveals that there appear the cycle-5 and the cycle-6 structure around the fixed point ðp; qÞ ¼ ð0; 0Þ. Similarly, Fig. 4 indicates occurrence of the cycle-7 structure. These observations are consistent with the analysis of Table 1. We want to notice, however, that the most striking feature of the phase portraits is the structure observed in Fig. 2, where we can identify the cycle-17 structure in the wide region centering the fixed point at the origin. These observations lead us to examine global periodic structure in the integrable Sinh map.
K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
51
Fig. 4. Phase space portrait at A ¼ 0:62. The orbit started at the initial position at (p ¼ 0:0, q ¼ 0:1) indicates slow modulation of the cycle-7.
Fig. 5. Phase space portrait at A ¼ 0:42. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:1) shows slow modulation of the cycle-8.
4. Global spectrum of periodic orbits in the Sinh map In order to examine the cycle-17 structure in Fig. 2, we follow a single orbit starting at the initial point (p ¼ 0:0, q ¼ þ0:1) for 200 iteration. Since Fig. 2 illustrates that the orbit started at the point (p ¼ 0:0, q ¼ 0:1) exhibits the dominant 17 cycle at least for short time, we follow the temporal sequence of the same orbit in Fig. 8, which reveals that this 17-cycle is composed of the sequences of 6-6-5 periodic repetitions. Referring back to the Poincare–Birkhoff
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K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
Fig. 6. Phase space portrait at A ¼ 2:236, where the cycle-5 structure is visible close at the center. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:05) shows slow modulation of the cycle-5. Orbits with the initial position at q > 0:10 along the p ¼ 0:0 axis diverges after several iterations.
Fig. 7. Phase space portrait at A ¼ 3:00. The structure at the central region indicates the m=n ¼ 2=6-cyclic structure. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:05) confirms the period-3 motion. Here, orbits with the initial position (p ¼ 0:0, q > 0:41) diverges within a few iteration.
resonance gives as Eq. (16), we find that at this value of A the occurrence of period-5 motion is in contradiction with the result of the linear map analysis. Inspired by such observation of the periodic structure of the integrable Sinh map, we develop systematic analysis of global spectrum of the periodic orbits presented in the Section 2, applying the method of the fast Fourier transfor-
K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
53
Fig. 8. The temporal sequence of the orbit with the initial position (p ¼ 0:0, q ¼ 0:1). Successive count of each steps suggests that the sequence consists of repetition of 6-6-5 cycles.
mation (FFT in short). Firstly, we notice in the limit of A ! 1, the Sinh map, Eq. (14) with the lower sign, is reduced to a linear map 0 p þ1 2 p ð17Þ ¼ q0 þ1 1 q Since we have 4 þ1 0 þ1 2 ¼ 0 þ1 þ1 1
ð18Þ
we conclude that in this limit the Sinh map posses the intrinsic period-4 structure. We apply the FFT spectrum analysis to the orbits along the p ¼ 0 axis toward the outside from the stable fixed point at the origin. Choosing 211 ¼ 2048 for the size of time domain, we observe positions of peak of the FFT spectrum. We present the FFT spectrum analysis of the orbits along the p ¼ 0 axis for various values of nonlinear parameter A in Figs. 9 and 10. Starting with the nonlinear parameter A ¼ 2:30, we find that the FFT spectrum analysis of the orbits along the p ¼ 0 axis gives rise to the variation of peak positions with the periodicity of 4.971 at the center to the larger value of 4.995 at q ¼ 0:5 and to the much larger value of 5.35 at larger value of q. Referring to Table 1, we should have the birth of the period-5 orbit at the value of A ¼ 2:236, which is confirmed as indicated by the curve (2) in Fig. 9. At the value of A ¼ 1:25, the curve (3) begins at the periodicity of 5.72 and increases over 5.85. Now, for the value of A ¼ 1:00, we notice the Sinh map is reduced exactly to a linear map. 0 p þ1 1 p ¼ ð19Þ q0 þ1 0 q Since we have 6 þ1 1 þ1 0 ¼ þ1 0 0 þ1
ð20Þ
we conclude that here the Sinh map has the period-6 structure for the entire region of the phase space at A ¼ 1:00, not restricted by the validity of linearized tangent map. This is indeed confirmed by our analysis of the orbits along the p ¼ 0 axis shown by the period-6 straight line in Fig. 9. Approaching to the next resonance at the value of A ¼ 0:6039, however, we find the opposite behavior of variation of the periodicity of the orbits. Namely, as shown in Fig. 10, variation of the observed FFT peak at the value of A ¼ 0:62 indicates that the periodicity of orbits decreases from 6.94 at the center to the smaller value of 6.87 at the
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K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
Fig. 9. Global variation of the periodicity of the orbits starting at the initial positions (p ¼ 0, 0 < q < 3:5) for the value of A ¼ 2:300, 2.236, 1.250 and 1.000.
Fig. 10. Global variation of the periodicity of the orbits starting at the initial positions (p ¼ 0, 0 < q < 3:5) for the value of A ¼ 0:62, 0.6039, 0.550, 0.42 and 0.4142.
outer region of q ¼ 1:00 and further down to 6.50 at the far outside. It is interesting to observe again at the resonate value of A ¼ 0:6039, the FFT peaks keeps the constant value of the mode number 293 for the wide range of q up to 0.5. It should be noticed that the periodicity of 2048=293 ¼ 6:98976 confirms here the resonance takes place with the period-7. Moving to out side form the origin, we observe that the periodicity decreases down to 6.6. Next, observing the global variation of the periodicity of the orbits at the value of A ¼ 0:55, as shown by the curve (7), and at value of A ¼ 0:42, shown by the curve (8), we examine the variation of the periodicity at the next resonance with A ¼ 0:4142. Here, we observe that the FFT peak keeps the constant value 256 ¼ 28 up to q ¼ 0:25, so that we have the exact value of 23 ¼ 8. Beyond this region, the nonlinear effect modifies the Poincare–Birkhoff resonance and the periodicity decreases very rapidly down to 7. We explain this asymptotic behavior in the last section. Turning to positive value of A, we carry out the similar observation for the values of A at and between the Poincare– Birkhoff resonance. The results shown in Fig. 11 confirms that close to the fixed point at the origin the resonance
K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
55
Fig. 11. Global variation of the periodicity of the orbits starting at the initial positions (p ¼ 0, 0 < q < 0:3) for the value of A ¼ 2:15, 2.236, 2.50, 3.00, 4.00, 5.494, 10.00 and 30.00.
condition given as Eq. (16) determines the average periodicity for continuous variation of n, not restricted to the integer. Crossing the resonance at A ¼ 2:236 for n=m ¼ 5=2 ¼ 2:5, at A ¼ 3:000 for n=m ¼ 6=2 ¼ 3:0 and at A ¼ 5:494 for n=m ¼ 7=2 ¼ 3:5, we observe that, upon increasing the value of A to the larger values 10 and 30, the curve approaches to the asymptotic limit of n=m ¼ 8=2 ¼ 4. Here, we notice that the mapping function equation (14) becomes singular at value of q log A, so the curves (1) and (2) diverges at relatively smaller values of q.
5. Summary and concluding discussions Here, we have examined the global variation of the periodicity of the integrable orbits in the Sinh map. We present results of observation of global variation of the periodicity of the orbits along the p ¼ 0 axis in Figs. 9–11. For the values of nonlinear parameter A given by Tables 1 and 2, the periodicity of the orbit close to the origin takes the value determined by the Poincare–Birkhoff resonance condition, Eq. (16). We notice that the Sinh map is reduced to a linear map equation (19) at the value of A ¼ 1:00, where the periodicity of all of the orbits is given as 6. Furthermore in the limit of large q, since the mapping function F ðqÞ is approximated as F ðqÞ ¼ ln jAj q
ð21Þ
we get p0 ¼ p ln jAj q, and q0 ¼ p ln jAj, thus, defining a shifted coordinate r ¼ q þ ln jAj, we can reduce the Sinh map, Eqs. (12) and (13) with (14), as 0 þ1 1 p p ¼ ð22Þ r0 þ1 0 r which is the same linear map as Eq. (19). Therefore, we can conclude that in the large q limit, the Sinh map with A < 0 is reduced asymptotically to the linear map Eq. (22), and thus the average periodicity asymptotically decreases down to the period-6, as can be seen in Fig. 10. As for the case of A > 0, Fig. 11 illustrates that the global periodicity of the orbits with A > 0 is in accord with the linearized resonance condition summarized in Table 2. In conclusion, we give heuristic explanation for the appearance of the cycle-17 structure at A ¼ 1:25. As observed in Fig. 8, the temporal sequence of the orbit started at (p ¼ 0:0, q ¼ 0:0) is composed of the sequence of 6-6-5, so that the average period turns out to be 5.667 which is indeed confirmed by observation shown in Fig. 9. Going toward outside from the fixed point at the origin, we see that the occurrence of the cycle-6 motion becomes more predominant and the orbit appears to be a sequence of 6-6-6-5 cyclic repetitions. Thus, the average period exceeds 5.8. Our
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K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
preliminary investigation on other integrable mappings such as the Duffing map and the K–dV map suggests similar variation of the global periodic orbits. We will report on these cases in the forthcoming paper.
Acknowledgements We wish to express our thanks to Professor Y.H. Ichikawa for his stimulating suggestion to the present work. They are also obliged to Dr. S. Sait^ o for his discussions.
References [1] [2] [3] [4] [5] [6] [7]
Lichtenberg AJ, Lieberman MA. Regular and chaotic dynamics. New York: Springer; 1992. Reichl LE. The transition to chaos. New York: Springer; 1992. Grammaticos B, Ramani A, Papageorgiou V. Do integrable mappings have the Painleve property? Phys Rev Lett 1991;67:1825–8. Ramani A, Grammaticos 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.
Global periodic structure of integrable hyperbolic map Kei-ichi Hirose b
a,*
, Chieko Murakami b, Wakako Murakami
b
a National Institute of Fusion Science, Toki 509-5292, Japan College of Engineering Chubu University, Kasugai 487-8501, Japan
Accepted 23 April 2002
Abstract Integrable hyperbolic mappings are constructed within a scheme presented by Suris. The Cosh map is a singular map, of which fixed point is unstable. The global behavior of periodic orbits of the Sinh map is investigated referring to the Poincare–Birkhoff resonance condition. Close to the fixed point, the periodicity is indeed determined from the Poincare–Birkhoff resonance condition. Increasing the distance from the fixed point, the orbit is affected by the nonlinear effect and the average periodicity varies globally. The Fourier transformation of the individual orbits determines overall spectrum of global variation of the periodicity. 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 work of Quispel and his collaborators [6,7]. Here, referring to the work of Suris, we undertake some exploration of the properties of integrable hyperbolic mappings, although we do not have suitable example to apply these mappings to any physical systems. We illustrate explicitly the characteristic feature of the integrable mappings, where all of the orbits prevail on the closed torus and no chaos appears in the whole range of the nonlinear parameter. In Section 2, we introduce the Cosh map and Sinh map, and investigate their basic properties. As for the Cosh map, there is no fixed point in the finite range of the phase space, even though it is an integrable map. Hence, we focus our interest to study the Sinh map. We present our analysis on the phase space structure around fixed point of the Sinh map in Section 3. On the contrary to the nonintegrable mappings, we do not observe the Poincare–Birkhoff bifurcation around the stable fixed point. This is one of the characteristic feature of the integrable mappings. We can confirm, however, that around the fixed point there appear periodic structure determined from the Poincare–Birkhoff resonance condition. In order to analyze this periodic structure, we examine the time sequence of the individual orbits with the help of the fast Fourier transform in Section 4. We present concluding remarks by summarizing the global spectrum of the periodic orbits in the integrable mappings in Section 5.
*
Corresponding author. E-mail address: [email protected] (K.-i. Hirose).
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 0 8 6 - 3
48
K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
2. Cosh map and Sinh map Suris [5] has presented a theorem that the second order nonlinear difference equation xnþ1 2xn þ xn1 ¼ F ðxn ; Þ
ð1Þ
with the following three types of function F ðx; Þ: ðaÞ F ðx; Þ ¼
A þ Bx þ Cx2 þ Dx3 1 ðE þ 1=3Cx þ 1=2Dx2 Þ
ð2Þ
ðbÞ
2 ðx=2ÞðA sinðxxÞ þ B cosðxxÞ þ C sinð2xxÞ þ D cosð2xxÞÞ tan1 x 1 ðx=2ÞðA sinðxxÞ B cosðxxÞ þ C sinð2xxÞ D cosð2xxÞ þ EÞ
ð3Þ
1 1 þ aðB expðaxÞ þ D expð2axÞ EÞ ln a 1 aðA expðaxÞ þ C expð2axÞ þ EÞ
ð4Þ
F ðx; Þ ¼
ðcÞ F ðx; Þ ¼
are integrable in the sense that Eq. (1) posses a nontrivial symmetrical integral Uðxnþ1 ; xn Þ ¼ Uðxn ; xn1 Þ
ð5Þ
Suris conjectured that the proved theorem means the possibility of the construction of integrable difference schemes for the second order differential equations €x ¼ f0 ðxÞ with a right hand side of the forms, ðaÞ f0 ðxÞ ¼ A þ Bx þ Cx2 þ Dx3 ðbÞ
f0 ðxÞ ¼ A sinðxxÞ þ B cosðxxÞ þ C sinð2xxÞ þ D cosð2xxÞ
ðcÞ f0 ðxÞ ¼ A expðaxÞ þ B expðaxÞ þ C expð2axÞ þ D expð2axÞ
ð6Þ ð7Þ ð8Þ
To be specific, we restrict our interest to the case (c) with a special choice of B ¼ þA and B ¼ A, C ¼ D ¼ 0. Namely, we have f0 ðxÞ ¼ 2A coshðaxÞ and f0 ðxÞ ¼ 2A sinhðaxÞ, respectively. We introduce a normalized space variable qn as qn ¼ axn
ð9Þ
and the momentum variable pn in terms of pnþ1 ¼ qnþ1 qn
ð10Þ
Defining a new parameter A as aA !A 1 aE
ð11Þ
we can reduce Eq. (1) with Eq. (8) to the two dimensional area preserving map, pnþ1 ¼ pn þ F ðqn Þ
ð12Þ
qnþ1 ¼ qn þ Pnþ1
ð13Þ
with the abbreviation F ¼ ln
1 A expðqn Þ 1 A expðqn Þ
ð14Þ
The upper sign þ (and the lower sign ) stands for f0 ðxÞ ¼ 2A coshðqÞ (and 2A sinhðqÞÞ, defining the Cosh map (and the Sinh map), respectively. We find immediately that the Cosh map does not posses any fixed points, while the Sinh map has a fixed point at the origin (p ¼ 0, q ¼ 0) in the phase space. The stability of the fixed point (p ¼ 0, q ¼ 0) of the Sinh map is analyzed by constructing a tangent map at the point (p ¼ 0, q ¼ 0). Since the residue R at the point (p ¼ 0, q ¼ 0) is given as R¼
1 A 2 1A
ð15Þ
K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
49
the fixed point (p ¼ 0, q ¼ 0) is stable in the region of 1 < A < 0 and 2 < A < 1. In the region of 0 < A < 1, the fixed point is hyperbolic without reflection, while it becomes hyperbolic with reflection in the region of 1 < A < 2.
3. Poincare´–Birkhoff resonance of the Sinh map In the case of nonintegrable area preserving map, the tangent map at the stable fixed point determines the condition of Poincare–Birkhoff bifurcation around the stable fixed point. As for the integrable area preserving map, we observe that no chaos arises and every orbits form closed torus. In this situation, for the rotation number q ¼ ðm=nÞ2p, the eigenvalue of the tangent map determines the Poincare–Birkhoff resonance condition cos q ¼ 1 2R with Eq. (15), which in turn gives the value of nonlinear parameter Aðm=nÞ as Aðm=nÞ ¼ 1
1 cosð2pm=nÞ
ð16Þ
As for A < 0, taking m ¼ 1, we obtain the following values as shown in Table 1, while for A > 0, taking m ¼ 2, we have Table 2, respectively. Here, we illustrate the phase space portraits of the Sinh map in Fig. 1 (A ¼ 2:30), Fig. 2 (A ¼ 1:25), Fig. 3 (A ¼ 1:05), Fig. 4 (A ¼ 0:62) and Fig. 5 (A ¼ 0:42), respectively. For the positive value of A, we have Fig. 6 (A ¼ 2:236) and Fig. 7 (A ¼ 3:00). Appearance of the cycle-5 and the cycle-3 structure in Figs. 6 and 7 confirm the prediction of Table 2. Unlike the case of nonintegrable area preserving mappings, we do not have the
Table 1 Period number for Poincare–Birkhoff resonance (A < 0) A n
1 4
2.236 5
1.000 6
0.6039 7
0.4142 8
0.3054 9
Table 2 Period number for Poincare–Birkhoff resonance (A > 0) A n
2.000 4
2.236 5
3.000 6
5.494 7
þ1 8
Fig. 1. Phase space portrait at A ¼ 2:30. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:1) indicates slow modulation of the cycle-5 cyclic motion.
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K.-i. Hirose et al. / Chaos, Solitons and Fractals 15 (2003) 47–56
Fig. 2. Phase space portrait at A ¼ 1:25. The orbits in the outer region exhibit clear cycle-17 structure. The orbit with initial position (p ¼ 0:0, q ¼ 0:1) returns nearly to the starting position after the 17th iteration.
Fig. 3. Phase space portrait at A ¼ 1:05. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:1) shows slow modulation of the period-6 motion.
formation of the Poincare–Birkhoff neck lace of the periodic islands. We may call these cycle-n structure as the Poincare–Birkhoff resonance. Careful observation of Figs. 1 and 3 reveals that there appear the cycle-5 and the cycle-6 structure around the fixed point ðp; qÞ ¼ ð0; 0Þ. Similarly, Fig. 4 indicates occurrence of the cycle-7 structure. These observations are consistent with the analysis of Table 1. We want to notice, however, that the most striking feature of the phase portraits is the structure observed in Fig. 2, where we can identify the cycle-17 structure in the wide region centering the fixed point at the origin. These observations lead us to examine global periodic structure in the integrable Sinh map.
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51
Fig. 4. Phase space portrait at A ¼ 0:62. The orbit started at the initial position at (p ¼ 0:0, q ¼ 0:1) indicates slow modulation of the cycle-7.
Fig. 5. Phase space portrait at A ¼ 0:42. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:1) shows slow modulation of the cycle-8.
4. Global spectrum of periodic orbits in the Sinh map In order to examine the cycle-17 structure in Fig. 2, we follow a single orbit starting at the initial point (p ¼ 0:0, q ¼ þ0:1) for 200 iteration. Since Fig. 2 illustrates that the orbit started at the point (p ¼ 0:0, q ¼ 0:1) exhibits the dominant 17 cycle at least for short time, we follow the temporal sequence of the same orbit in Fig. 8, which reveals that this 17-cycle is composed of the sequences of 6-6-5 periodic repetitions. Referring back to the Poincare–Birkhoff
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Fig. 6. Phase space portrait at A ¼ 2:236, where the cycle-5 structure is visible close at the center. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:05) shows slow modulation of the cycle-5. Orbits with the initial position at q > 0:10 along the p ¼ 0:0 axis diverges after several iterations.
Fig. 7. Phase space portrait at A ¼ 3:00. The structure at the central region indicates the m=n ¼ 2=6-cyclic structure. The orbit started at the initial position (p ¼ 0:0, q ¼ 0:05) confirms the period-3 motion. Here, orbits with the initial position (p ¼ 0:0, q > 0:41) diverges within a few iteration.
resonance gives as Eq. (16), we find that at this value of A the occurrence of period-5 motion is in contradiction with the result of the linear map analysis. Inspired by such observation of the periodic structure of the integrable Sinh map, we develop systematic analysis of global spectrum of the periodic orbits presented in the Section 2, applying the method of the fast Fourier transfor-
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Fig. 8. The temporal sequence of the orbit with the initial position (p ¼ 0:0, q ¼ 0:1). Successive count of each steps suggests that the sequence consists of repetition of 6-6-5 cycles.
mation (FFT in short). Firstly, we notice in the limit of A ! 1, the Sinh map, Eq. (14) with the lower sign, is reduced to a linear map 0 p þ1 2 p ð17Þ ¼ q0 þ1 1 q Since we have 4 þ1 0 þ1 2 ¼ 0 þ1 þ1 1
ð18Þ
we conclude that in this limit the Sinh map posses the intrinsic period-4 structure. We apply the FFT spectrum analysis to the orbits along the p ¼ 0 axis toward the outside from the stable fixed point at the origin. Choosing 211 ¼ 2048 for the size of time domain, we observe positions of peak of the FFT spectrum. We present the FFT spectrum analysis of the orbits along the p ¼ 0 axis for various values of nonlinear parameter A in Figs. 9 and 10. Starting with the nonlinear parameter A ¼ 2:30, we find that the FFT spectrum analysis of the orbits along the p ¼ 0 axis gives rise to the variation of peak positions with the periodicity of 4.971 at the center to the larger value of 4.995 at q ¼ 0:5 and to the much larger value of 5.35 at larger value of q. Referring to Table 1, we should have the birth of the period-5 orbit at the value of A ¼ 2:236, which is confirmed as indicated by the curve (2) in Fig. 9. At the value of A ¼ 1:25, the curve (3) begins at the periodicity of 5.72 and increases over 5.85. Now, for the value of A ¼ 1:00, we notice the Sinh map is reduced exactly to a linear map. 0 p þ1 1 p ¼ ð19Þ q0 þ1 0 q Since we have 6 þ1 1 þ1 0 ¼ þ1 0 0 þ1
ð20Þ
we conclude that here the Sinh map has the period-6 structure for the entire region of the phase space at A ¼ 1:00, not restricted by the validity of linearized tangent map. This is indeed confirmed by our analysis of the orbits along the p ¼ 0 axis shown by the period-6 straight line in Fig. 9. Approaching to the next resonance at the value of A ¼ 0:6039, however, we find the opposite behavior of variation of the periodicity of the orbits. Namely, as shown in Fig. 10, variation of the observed FFT peak at the value of A ¼ 0:62 indicates that the periodicity of orbits decreases from 6.94 at the center to the smaller value of 6.87 at the
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Fig. 9. Global variation of the periodicity of the orbits starting at the initial positions (p ¼ 0, 0 < q < 3:5) for the value of A ¼ 2:300, 2.236, 1.250 and 1.000.
Fig. 10. Global variation of the periodicity of the orbits starting at the initial positions (p ¼ 0, 0 < q < 3:5) for the value of A ¼ 0:62, 0.6039, 0.550, 0.42 and 0.4142.
outer region of q ¼ 1:00 and further down to 6.50 at the far outside. It is interesting to observe again at the resonate value of A ¼ 0:6039, the FFT peaks keeps the constant value of the mode number 293 for the wide range of q up to 0.5. It should be noticed that the periodicity of 2048=293 ¼ 6:98976 confirms here the resonance takes place with the period-7. Moving to out side form the origin, we observe that the periodicity decreases down to 6.6. Next, observing the global variation of the periodicity of the orbits at the value of A ¼ 0:55, as shown by the curve (7), and at value of A ¼ 0:42, shown by the curve (8), we examine the variation of the periodicity at the next resonance with A ¼ 0:4142. Here, we observe that the FFT peak keeps the constant value 256 ¼ 28 up to q ¼ 0:25, so that we have the exact value of 23 ¼ 8. Beyond this region, the nonlinear effect modifies the Poincare–Birkhoff resonance and the periodicity decreases very rapidly down to 7. We explain this asymptotic behavior in the last section. Turning to positive value of A, we carry out the similar observation for the values of A at and between the Poincare– Birkhoff resonance. The results shown in Fig. 11 confirms that close to the fixed point at the origin the resonance
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Fig. 11. Global variation of the periodicity of the orbits starting at the initial positions (p ¼ 0, 0 < q < 0:3) for the value of A ¼ 2:15, 2.236, 2.50, 3.00, 4.00, 5.494, 10.00 and 30.00.
condition given as Eq. (16) determines the average periodicity for continuous variation of n, not restricted to the integer. Crossing the resonance at A ¼ 2:236 for n=m ¼ 5=2 ¼ 2:5, at A ¼ 3:000 for n=m ¼ 6=2 ¼ 3:0 and at A ¼ 5:494 for n=m ¼ 7=2 ¼ 3:5, we observe that, upon increasing the value of A to the larger values 10 and 30, the curve approaches to the asymptotic limit of n=m ¼ 8=2 ¼ 4. Here, we notice that the mapping function equation (14) becomes singular at value of q log A, so the curves (1) and (2) diverges at relatively smaller values of q.
5. Summary and concluding discussions Here, we have examined the global variation of the periodicity of the integrable orbits in the Sinh map. We present results of observation of global variation of the periodicity of the orbits along the p ¼ 0 axis in Figs. 9–11. For the values of nonlinear parameter A given by Tables 1 and 2, the periodicity of the orbit close to the origin takes the value determined by the Poincare–Birkhoff resonance condition, Eq. (16). We notice that the Sinh map is reduced to a linear map equation (19) at the value of A ¼ 1:00, where the periodicity of all of the orbits is given as 6. Furthermore in the limit of large q, since the mapping function F ðqÞ is approximated as F ðqÞ ¼ ln jAj q
ð21Þ
we get p0 ¼ p ln jAj q, and q0 ¼ p ln jAj, thus, defining a shifted coordinate r ¼ q þ ln jAj, we can reduce the Sinh map, Eqs. (12) and (13) with (14), as 0 þ1 1 p p ¼ ð22Þ r0 þ1 0 r which is the same linear map as Eq. (19). Therefore, we can conclude that in the large q limit, the Sinh map with A < 0 is reduced asymptotically to the linear map Eq. (22), and thus the average periodicity asymptotically decreases down to the period-6, as can be seen in Fig. 10. As for the case of A > 0, Fig. 11 illustrates that the global periodicity of the orbits with A > 0 is in accord with the linearized resonance condition summarized in Table 2. In conclusion, we give heuristic explanation for the appearance of the cycle-17 structure at A ¼ 1:25. As observed in Fig. 8, the temporal sequence of the orbit started at (p ¼ 0:0, q ¼ 0:0) is composed of the sequence of 6-6-5, so that the average period turns out to be 5.667 which is indeed confirmed by observation shown in Fig. 9. Going toward outside from the fixed point at the origin, we see that the occurrence of the cycle-6 motion becomes more predominant and the orbit appears to be a sequence of 6-6-6-5 cyclic repetitions. Thus, the average period exceeds 5.8. Our
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preliminary investigation on other integrable mappings such as the Duffing map and the K–dV map suggests similar variation of the global periodic orbits. We will report on these cases in the forthcoming paper.
Acknowledgements We wish to express our thanks to Professor Y.H. Ichikawa for his stimulating suggestion to the present work. They are also obliged to Dr. S. Sait^ o for his discussions.
References [1] [2] [3] [4] [5] [6] [7]
Lichtenberg AJ, Lieberman MA. Regular and chaotic dynamics. New York: Springer; 1992. Reichl LE. The transition to chaos. New York: Springer; 1992. Grammaticos B, Ramani A, Papageorgiou V. Do integrable mappings have the Painleve property? Phys Rev Lett 1991;67:1825–8. Ramani A, Grammaticos 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.