- Email: [email protected]
Bifurcations of Fibonacci generating functions
Chaos, Solitons and Fractals 33 (2007) 1240–1247 www.elsevier.com/locate/chaos
Bifurcations of Fibonacci generating functions ˇ enys b, Yasar Polatoglu a, Gu¨rsel Hacibekiroglu a, ¨ zer a,b,*, Antanas C Mehmet O c Ercument Akat , A. Valaristos d, A.N. Anagnostopoulos d b
a Istanbul Kultur University, E5 Karayolu Uzeri Sirinevler, 34191 Istanbul, Turkey Semiconductor Physics Institute, LT-01108 and Vilnius Gediminas Technical University, Sauletekio 11, LT-10223, Lithuania c Yeditepe University, 26 Agustos Campus Kayisdagi Street, Kayisßdagi 81120, Istanbul, Turkey d Aristotle University of Thessaloniki, GR-54124, Thessaloniki, Greece
Accepted 23 January 2006
Abstract In this work the dynamic behaviour of the one-dimensional family of maps Fp,q(x) = 1/(1 px qx2) is examined, for specific values of the control parameters p and q. Lyapunov exponents and bifurcation diagrams are numerically calculated. Consequently, a transition from periodic to chaotic regions is observed at values of p and q, where the related maps correspond to Fibonacci generating functions associated with the golden-, the silver- and the bronze mean. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Discrete time dynamical systems generated by the iteration of nonlinear maps provide simple and interesting examples of chaotic systems evolution. Even one-dimensional chaotic maps can describe very complicated dynamic behaviour, usually induced by their rich bifurcation structure. A typical feature of such maps is the existence of one or more parameters that control the nonlinearity. The quadratic family of maps, has drawn much attention and has been studied extensively due to its simplicity and clear exposition to period doubling route to chaos [1,2]. It is well known that iteration is one of the most powerful sources of self-similarity. Fibonacci sequences, the hyperbolic map xn+1 = 1/hxni, (where h i denotes remainder modulo 1), with its famous fixed points, the golden mean [1, 1, 1, . . .], the silver mean [2, 2, 2, . . .], and the bronze mean [3, 3, 3, . . .], all represent self-similar objects leading to chaos [3]. These three means belong to the family of metallic means [4–7] and appear astonishingly often in nature, sciences, even paintings, sculpture and music. On the other hand, many interesting results have been published recently, using members of the metallic means family to elucidate the transition from periodicity to quasi-periodicity.
* Corresponding author. Address: Istanbul Kultur University, E5 Karayolu Uzeri Sirinevler, 34191 Istanbul, Turkey. Tel.: +90 212 4514090; fax: +90 212 4512676. ¨ zer). E-mail address: [email protected] (M. O
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.095
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
1241
The recent progress on the technology of nano-layers of various materials (both semiconductors and magnetic ones) is of increasing interest in contemporary material science. Thickness and composition of multilayers can be controlled with high precision. When quantum structures of nano-layers are forming superlattices, the properties of these layers can deviate significantly from those of the original bulk materials. As a result of experimental developments, onedimensional quasi-periodic structures have been studied theoretically by several groups recently [8–10]. Most of the theoretical and numerical calculations have been performed for the Fibonacci sequences providing a kind of prototype for further studying quasi-periodic systems. On the other hand, using Fibonacci sequences, aperiodic systems have been studied based on substitution rules by means of matrix approach [11–20]. 1 In this work, we examine the behaviour of the family of maps F p;q ðxÞ ¼ 1pxqx 2 , for different values of p and q, which serve as the control parameters. This is the generating function of the generalized Fibonacci sequence described in the next section. It is observed that, as the parameters change, the behaviour of the maps progresses from periodicity through bifurcations to a state of chaos. Period doubling bifurcations and periodic windows are visualised, in a manner similar to the logistic map. The vicinities of the parameter-values where the above phenomena occur give rise to Fibonacci sequences associated with the golden mean, the silver mean and the bronze mean.
2. Generalized Fibonacci sequences and the associated generating functions A Fibonacci sequence is a sequence of integers, such that every integer is the sum of the two previous ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . The terms of the sequence satisfy the recurrence relation F nþ1 ¼ F n þ F n1
ðn ¼ 1; 2; . . .Þ
ð1Þ
with F0 = F1 = 1. The generating function for the Fibonacci numbers is f ðxÞ ¼
1 1 x x2
ð2Þ
since 1 X 1 2 3 4 5 ¼ 1 þ x þ 2x þ 3x þ 5x þ 8x þ ¼ F n xn 1 x x2 n¼0
ð3Þ
pffiffi of successive Fibonacci numbers, the famous golden mean / ¼ 1þ2 5 ¼ limn!1 FFnþ1 is obtained, with Taking ratios FFnþ1 n n continued fraction representation / = [1, 1, 1, . . .]. It can be shown that all Fibonacci numbers can be obtained by F n ¼ p1ffiffi5 f/nþ1 ð1=/Þnþ1 g.
The Fibonacci sequence defined above, may be generalized by considering the recurrence relation Gnþ1 ¼ pGn þ qGn1
ðn ¼ 1; 2; . . .Þ
ð4Þ
with G0 = a, G1 = b (a, b 2 Z) where p and q are natural numbers. In this case, the terms of the sequence are a, b, pb + qa, p(pb + qa) + qb, . . . Letting a = 1 and b = 1, the Fibonacci sequence is obtained for p = q = 1. In this paper, we consider (4), with G0 = 1 and G1 = p. The terms of the sequence become 1, p, p2 + q, p3 + 2pq, p4 + 3p2q + q2, . . . The corresponding generating function is F p;q ðxÞ ¼
1 1 px qx2
ð5Þ
since 1 ¼ 1 þ px þ ðp2 þ qÞx2 þ ðp3 þ 2pqÞx3 þ 1 px qx2
ð6Þ
In the next section, we shall examine the dynamics of (5) in the intervals 0.9 6 p, q 6 3.1, including the three special cases:
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
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1 (i) For p = 1 and q = 1, f ðxÞ ¼ 1xx 2 , is associated with the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . and the pffiffi 1þ 5 golden mean / ¼ 2 ¼ ½1. 1 (ii) For p = 2 and q = 1, f ðxÞ ¼ 12xx 2 , is associated with pffiffiffi the generalized Fibonacci sequence 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, . . . and the silver mean rAg ¼ 1 þ 2 ¼ ½ 2. 1 (iii) For p = 3 and q = 1, f ðxÞ ¼ 13xx 2 , is associated with the generalized Fibonacci sequence 1, 3, 10, 33, 109, pffiffiffiffi 360, 1189, 3927, 12 970, 42 837, . . . and the bronze mean rBr ¼ 3þ2 13 ¼ ½ 3.
3. Chaotic behaviour of the map We consider the recurrent form of (5) xn ¼
1 1 pxn1 qx2n1
ð7Þ
For any point x0 in the domain of the map, we study the orbit fxn g1 n¼0 , of it.
1
2
3
2 (a)
X
0
-2
-4
1
Lyapunov Exponent
(b)
0
-1 1
2
3
Parameter (p) Fig. 1. (a) The bifurcation diagram of function (7) in the (p, x) plane for q = 1 and 0.9 6 p 6 3.1. (b) The corresponding Lyapunov exponents.
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
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The two parameters p and q enter the map in an essential way. As these parameters are passing through some critical values, the sequence of iterates generated from the map also alters experiencing a transition from periodic to chaotic behaviour of the parameter. To study these transitions in detail we have calculated numerically the bifurcation diagrams and the corresponding Lyapunov exponents for different values of p and q. For this purpose we have used Mathcad [21–23] to calculate both the bifurcation diagrams and the Lyapunov exponents. Doing so, we took into account that the Lyapunov exponent can be found using the formula [24]
Fig. 2. A chaotic band appears by changing q from 1 to 1.001.
1.0010
Parameter (q)
1.0008 1.0006 1.0004 1.0002 1.0000 0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
Parameter (p) Fig. 3. The width of the chaotic band as a function of q. The line represents the best fitting obtained by Eq. (10).
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
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k ¼ lim
N !1
N dxnþ1 1 X ln N n¼1 dxn
ð8Þ
which in the case of (7) becomes ! N 1 X p þ 2qxn k ln 2 ð1 pxn qx2n Þ N n¼1
ð9Þ
For the calculations of the Lyapunov exponent we have used instead of the general formula (8) the more specific (9). To avoid initial fluctuations we performed the averaging over the last 200 values of 10 000 iterations. In Fig. 1(a) the bifurcation diagram of (7) is shown, for q = 1 and 0.9 6 p 6 3.1. To guarantee that the steady state condition is always achieved, we have plotted only the last 200 values of 10 000 iterations. As it is evident from Fig. 1(a), there are two chaotic regimes around p = 1.7 and p = 2.3 occasionally interrupted by periodic orbits. To gain a better view to these infinitesimal regions we have numerically calculated the corresponding Lyapunov exponent for the same values of p. The results of this calculation are shown in Fig. 1(b). The two chaotic regimes of the previous figure correspond exactly to the two regimes of Fig. 1(b), where the Lyapunov exponent has positive values, occasionally
4
1
2
3
X
(a)
0
-3 1
(b)
Lyapunov Exponent
0
-2 1
2
3
Parameter (q) Fig. 4. (a) The bifurcation diagram of function (7) in the (q, x) plane for p = 1 and 0.9 6 q 6 3.1. (b) The corresponding Lyapunov exponents.
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
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interrupted by sharp and short transitions to negative values (periodic orbits). There are also three periodic windows around p = 1, p = 2 and p = 3. These values of p correspond exactly to the golden mean, silver mean and bronze mean provided that q = 1. In the first and second periodic windows we observe that bifurcations occur at p 1, p 1.3 and p 2. We have to remark that even small deviations of q from the value q = 1 are changing this picture dramatically. To show the influence of small q-deviations on the bifurcation diagram we have numerically calculated the bifurcation diagram for the following values of q = 1.00000, q = 1.00001, q = 1.00010 and q = 1.00100. The results of these calculations are shown in Fig. 2. As it can be deduced from these diagrams for q = 1.00000 and p = 1 only an orbit of period 3 exists. However, with slightly increasing q (q = 1.00001, q = 1.00010 and q = 1.00100) successively widening chaotic regimes coexist with the previous mentioned periodic one. To check the law between the width of the previous chaotic regimes and the deviation of q from the value 1, we have calculated diagrams similar to those of Fig. 2 for several values of q. The results are shown in Fig. 3 (full circles), where the deviation of q from value 1 is plotted vs. the widening of the chaotic regime. It is obvious that the curve of this figure follows a quadratic law of the form q ¼ A þ Bp þ Cp2
ð10Þ
The best fitting (full line) was obtained for the following values: A = 1.24663, B = 0.49294 and C = 0.24632.
4
0.96
1.00
1.04
1.00 Parameter (q)
1.04
X
(a)
0
-3 1
Lyapunov Exponent
(b)
0
-2
0.96
Fig. 5. (a) Enlargement of the previous diagram around p = 1. (b) The corresponding Lyapunov exponents.
1246
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
In Fig. 4(a) the bifurcation diagram of (7) is shown for p = 1 and 0.9 6 q 6 3.1. To guarantee that the steady state condition is always achieved, we have plotted again only the last 200 values of 10 000 iterations. As it is evident from Fig. 4(a), there are two chaotic regimes just before and after q = 1, the second one occasionally interrupted by periodic orbits. The corresponding behaviour of the Lyapunov exponent is shown in Fig. 4(b). The two chaotic regimes of Fig. 4(a) correspond exactly to the two regimes of Fig. 4(b), where the Lyapunov exponent has positive values. There are also two periodic windows, a narrow one to the left of q = 1 and a second wide one after q = 1.5. There are also clear nullifications of the Lyapunov exponents in these periodic windows corresponding to period doublings. At q = 3 (nickel mean), there exists an orbit of period 2. At q = 2 (copper mean), an orbit of period 4 appears and finally at q = 1 (golden mean), there is an orbit of period 3. The last case is displayed in detail in Fig. 5(a) and (b).
4. Comments It has to be mentioned here that the existence of period doublings is allowed and not guaranteed by the possession of a negative Schwarzian derivative Sf. Using expression (5) and after some elementary calculations we obtain the explicit expression for Sf 2 f 000 3 f 00 6q2 Sf ¼ 0 ¼ <0 ð11Þ 0 f 2 f ðp þ 2qxÞ2
Acknowledgements The authors thank NATO-project ICS.EAP.CLG 981947. One of the authors, M.O., acknowledges financial support from the Semiconductor Physics Institute, Vilnius, Lithuania in the frame of the EU project PRAMA (contract Nr. G5MA-CT-2002-04014).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Gulick D. Encounters with chaos. McGraw-Hill; 1992. Devaney RL. An introduction to chaotic dynamical systems. Addison-Wesley; 1989. Schroeder M. Fractals, chaos and power laws. W.H. Freeman; 1991. de Spinadel VW. La Familia de Numeros Metalicos. Cuadernos del CIMBAGE 2003;6:17–44. Available from: http:// www.econ.uba.ar/www/institutos/matematica/cimbage/cuarderno06/Numeros%20Metalicos.pdf. de Spinadel VW. Nonlinearity and the metallic means family. Nonlinear Anal 2001;47:4345–53. de Spinadel VW. The metallic means family and multifractal spectra. Nonlinear Anal 1999;36:721–45. de Spinadel VW. On characterization of the onset to chaos. Chaos, Solitons & Fractals 1997;8:1631–43. Mauriz PW, Vasconcelos MS, Albuquerque EL. Specific heat properties of electrons in generalized Fibonacci quasicrystals. Physica A 2003;329:101–13. Wang X, Grimm U, Schreiber M. Trace and antitrace maps for aperiodic sequences: extensions and applications. Phys Rev B 2000;62:14020–31. Domı´nguez-Adame F, Macia´ E, Me´ndez B, Roy CL, Khan A. Fibonacci superlattices of narrow-gap III–V semiconductors. Semicond Sci Technol 1995;10:797–802. Sil S, Karmakar SN, Moitra RK, Chakrabarti A. Extended states in one-dimensional lattices: application to the quasiperiodic copper-mean chain. Phys Rev B 1993;48:4192–5. Zhong JX, Xie T, You JQ, Yan JR. State transition in a family of generalized Fibonacci lattices. Zeitschrift fur Physik B 1992;87:223–7. Holzer M. Multifractal wave functions on a class of one-dimensional quasicrystals: exact f(alpha) curves and the limit of dilute quasiperiodic impurities. Phys Rev B 1991;44:2085–91. Severin M, Riklund R. Coexistence of localized, critical, and extended states in a one-dimensional substitutional non-Fibonaccian quasicrystal: a multifractal analysis. Phys Rev B 1989;39:10362–5. Severin M, Dulea M, Riklund R. Periodic and quasiperiodic wavefunctions in a class of one-dimensional quasicrystals: an analytical treatment. J Phys Condens Matter 1989;1:8851–8. Kolar M, Ali MK. Trace maps associated with general two-letter substitution rules. Phys Rev A 1990;42:7112–24. Benza VG, Kolar M, Ali MK. Phase transition in the generalized Fibonacci quantum Ising models. Phys Rev B 1990;41:9578–80. Kolar M, Ali MK. One-dimensional generalized Fibonacci tilings. Phys Rev B 1990;41:7108–12.
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[19] Ali MK, Gumbs G. Quasiperiodic dynamics for a generalized third-order Fibonacci series. Phys Rev B 1988;38:7091–3. [20] Gumbs G, Ali MK. Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices. Phys Rev Lett 1988;60:1081–4. [21] Hayward J. Chaotic iteration with MathCad, 1993. Available from: http://www.bham.ac.uk/ctimath/reviews/aug93/mathcad.pdf. [22] DiFranco D. Discovering chaotic iterations, 2002. Available from: http://www.mathcad.co.uk/mcadlib/apps/chaotic.mcd. [23] Kaufman M. The butterfly effect, 2004. Available from: http://www.csuohio.edu/physics/kaufman/Envphylab8.pdf. [24] Elert G. The chaos hypertextbook, 2004. Available from: http://hypertextbook.com/chaos/43.shtml.
Bifurcations of Fibonacci generating functions ˇ enys b, Yasar Polatoglu a, Gu¨rsel Hacibekiroglu a, ¨ zer a,b,*, Antanas C Mehmet O c Ercument Akat , A. Valaristos d, A.N. Anagnostopoulos d b
a Istanbul Kultur University, E5 Karayolu Uzeri Sirinevler, 34191 Istanbul, Turkey Semiconductor Physics Institute, LT-01108 and Vilnius Gediminas Technical University, Sauletekio 11, LT-10223, Lithuania c Yeditepe University, 26 Agustos Campus Kayisdagi Street, Kayisßdagi 81120, Istanbul, Turkey d Aristotle University of Thessaloniki, GR-54124, Thessaloniki, Greece
Accepted 23 January 2006
Abstract In this work the dynamic behaviour of the one-dimensional family of maps Fp,q(x) = 1/(1 px qx2) is examined, for specific values of the control parameters p and q. Lyapunov exponents and bifurcation diagrams are numerically calculated. Consequently, a transition from periodic to chaotic regions is observed at values of p and q, where the related maps correspond to Fibonacci generating functions associated with the golden-, the silver- and the bronze mean. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Discrete time dynamical systems generated by the iteration of nonlinear maps provide simple and interesting examples of chaotic systems evolution. Even one-dimensional chaotic maps can describe very complicated dynamic behaviour, usually induced by their rich bifurcation structure. A typical feature of such maps is the existence of one or more parameters that control the nonlinearity. The quadratic family of maps, has drawn much attention and has been studied extensively due to its simplicity and clear exposition to period doubling route to chaos [1,2]. It is well known that iteration is one of the most powerful sources of self-similarity. Fibonacci sequences, the hyperbolic map xn+1 = 1/hxni, (where h i denotes remainder modulo 1), with its famous fixed points, the golden mean [1, 1, 1, . . .], the silver mean [2, 2, 2, . . .], and the bronze mean [3, 3, 3, . . .], all represent self-similar objects leading to chaos [3]. These three means belong to the family of metallic means [4–7] and appear astonishingly often in nature, sciences, even paintings, sculpture and music. On the other hand, many interesting results have been published recently, using members of the metallic means family to elucidate the transition from periodicity to quasi-periodicity.
* Corresponding author. Address: Istanbul Kultur University, E5 Karayolu Uzeri Sirinevler, 34191 Istanbul, Turkey. Tel.: +90 212 4514090; fax: +90 212 4512676. ¨ zer). E-mail address: [email protected] (M. O
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.095
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
1241
The recent progress on the technology of nano-layers of various materials (both semiconductors and magnetic ones) is of increasing interest in contemporary material science. Thickness and composition of multilayers can be controlled with high precision. When quantum structures of nano-layers are forming superlattices, the properties of these layers can deviate significantly from those of the original bulk materials. As a result of experimental developments, onedimensional quasi-periodic structures have been studied theoretically by several groups recently [8–10]. Most of the theoretical and numerical calculations have been performed for the Fibonacci sequences providing a kind of prototype for further studying quasi-periodic systems. On the other hand, using Fibonacci sequences, aperiodic systems have been studied based on substitution rules by means of matrix approach [11–20]. 1 In this work, we examine the behaviour of the family of maps F p;q ðxÞ ¼ 1pxqx 2 , for different values of p and q, which serve as the control parameters. This is the generating function of the generalized Fibonacci sequence described in the next section. It is observed that, as the parameters change, the behaviour of the maps progresses from periodicity through bifurcations to a state of chaos. Period doubling bifurcations and periodic windows are visualised, in a manner similar to the logistic map. The vicinities of the parameter-values where the above phenomena occur give rise to Fibonacci sequences associated with the golden mean, the silver mean and the bronze mean.
2. Generalized Fibonacci sequences and the associated generating functions A Fibonacci sequence is a sequence of integers, such that every integer is the sum of the two previous ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . The terms of the sequence satisfy the recurrence relation F nþ1 ¼ F n þ F n1
ðn ¼ 1; 2; . . .Þ
ð1Þ
with F0 = F1 = 1. The generating function for the Fibonacci numbers is f ðxÞ ¼
1 1 x x2
ð2Þ
since 1 X 1 2 3 4 5 ¼ 1 þ x þ 2x þ 3x þ 5x þ 8x þ ¼ F n xn 1 x x2 n¼0
ð3Þ
pffiffi of successive Fibonacci numbers, the famous golden mean / ¼ 1þ2 5 ¼ limn!1 FFnþ1 is obtained, with Taking ratios FFnþ1 n n continued fraction representation / = [1, 1, 1, . . .]. It can be shown that all Fibonacci numbers can be obtained by F n ¼ p1ffiffi5 f/nþ1 ð1=/Þnþ1 g.
The Fibonacci sequence defined above, may be generalized by considering the recurrence relation Gnþ1 ¼ pGn þ qGn1
ðn ¼ 1; 2; . . .Þ
ð4Þ
with G0 = a, G1 = b (a, b 2 Z) where p and q are natural numbers. In this case, the terms of the sequence are a, b, pb + qa, p(pb + qa) + qb, . . . Letting a = 1 and b = 1, the Fibonacci sequence is obtained for p = q = 1. In this paper, we consider (4), with G0 = 1 and G1 = p. The terms of the sequence become 1, p, p2 + q, p3 + 2pq, p4 + 3p2q + q2, . . . The corresponding generating function is F p;q ðxÞ ¼
1 1 px qx2
ð5Þ
since 1 ¼ 1 þ px þ ðp2 þ qÞx2 þ ðp3 þ 2pqÞx3 þ 1 px qx2
ð6Þ
In the next section, we shall examine the dynamics of (5) in the intervals 0.9 6 p, q 6 3.1, including the three special cases:
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
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1 (i) For p = 1 and q = 1, f ðxÞ ¼ 1xx 2 , is associated with the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . and the pffiffi 1þ 5 golden mean / ¼ 2 ¼ ½1. 1 (ii) For p = 2 and q = 1, f ðxÞ ¼ 12xx 2 , is associated with pffiffiffi the generalized Fibonacci sequence 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, . . . and the silver mean rAg ¼ 1 þ 2 ¼ ½ 2. 1 (iii) For p = 3 and q = 1, f ðxÞ ¼ 13xx 2 , is associated with the generalized Fibonacci sequence 1, 3, 10, 33, 109, pffiffiffiffi 360, 1189, 3927, 12 970, 42 837, . . . and the bronze mean rBr ¼ 3þ2 13 ¼ ½ 3.
3. Chaotic behaviour of the map We consider the recurrent form of (5) xn ¼
1 1 pxn1 qx2n1
ð7Þ
For any point x0 in the domain of the map, we study the orbit fxn g1 n¼0 , of it.
1
2
3
2 (a)
X
0
-2
-4
1
Lyapunov Exponent
(b)
0
-1 1
2
3
Parameter (p) Fig. 1. (a) The bifurcation diagram of function (7) in the (p, x) plane for q = 1 and 0.9 6 p 6 3.1. (b) The corresponding Lyapunov exponents.
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
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The two parameters p and q enter the map in an essential way. As these parameters are passing through some critical values, the sequence of iterates generated from the map also alters experiencing a transition from periodic to chaotic behaviour of the parameter. To study these transitions in detail we have calculated numerically the bifurcation diagrams and the corresponding Lyapunov exponents for different values of p and q. For this purpose we have used Mathcad [21–23] to calculate both the bifurcation diagrams and the Lyapunov exponents. Doing so, we took into account that the Lyapunov exponent can be found using the formula [24]
Fig. 2. A chaotic band appears by changing q from 1 to 1.001.
1.0010
Parameter (q)
1.0008 1.0006 1.0004 1.0002 1.0000 0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
Parameter (p) Fig. 3. The width of the chaotic band as a function of q. The line represents the best fitting obtained by Eq. (10).
¨ zer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247 M. O
1244
k ¼ lim
N !1
N dxnþ1 1 X ln N n¼1 dxn
ð8Þ
which in the case of (7) becomes ! N 1 X p þ 2qxn k ln 2 ð1 pxn qx2n Þ N n¼1
ð9Þ
For the calculations of the Lyapunov exponent we have used instead of the general formula (8) the more specific (9). To avoid initial fluctuations we performed the averaging over the last 200 values of 10 000 iterations. In Fig. 1(a) the bifurcation diagram of (7) is shown, for q = 1 and 0.9 6 p 6 3.1. To guarantee that the steady state condition is always achieved, we have plotted only the last 200 values of 10 000 iterations. As it is evident from Fig. 1(a), there are two chaotic regimes around p = 1.7 and p = 2.3 occasionally interrupted by periodic orbits. To gain a better view to these infinitesimal regions we have numerically calculated the corresponding Lyapunov exponent for the same values of p. The results of this calculation are shown in Fig. 1(b). The two chaotic regimes of the previous figure correspond exactly to the two regimes of Fig. 1(b), where the Lyapunov exponent has positive values, occasionally
4
1
2
3
X
(a)
0
-3 1
(b)
Lyapunov Exponent
0
-2 1
2
3
Parameter (q) Fig. 4. (a) The bifurcation diagram of function (7) in the (q, x) plane for p = 1 and 0.9 6 q 6 3.1. (b) The corresponding Lyapunov exponents.
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interrupted by sharp and short transitions to negative values (periodic orbits). There are also three periodic windows around p = 1, p = 2 and p = 3. These values of p correspond exactly to the golden mean, silver mean and bronze mean provided that q = 1. In the first and second periodic windows we observe that bifurcations occur at p 1, p 1.3 and p 2. We have to remark that even small deviations of q from the value q = 1 are changing this picture dramatically. To show the influence of small q-deviations on the bifurcation diagram we have numerically calculated the bifurcation diagram for the following values of q = 1.00000, q = 1.00001, q = 1.00010 and q = 1.00100. The results of these calculations are shown in Fig. 2. As it can be deduced from these diagrams for q = 1.00000 and p = 1 only an orbit of period 3 exists. However, with slightly increasing q (q = 1.00001, q = 1.00010 and q = 1.00100) successively widening chaotic regimes coexist with the previous mentioned periodic one. To check the law between the width of the previous chaotic regimes and the deviation of q from the value 1, we have calculated diagrams similar to those of Fig. 2 for several values of q. The results are shown in Fig. 3 (full circles), where the deviation of q from value 1 is plotted vs. the widening of the chaotic regime. It is obvious that the curve of this figure follows a quadratic law of the form q ¼ A þ Bp þ Cp2
ð10Þ
The best fitting (full line) was obtained for the following values: A = 1.24663, B = 0.49294 and C = 0.24632.
4
0.96
1.00
1.04
1.00 Parameter (q)
1.04
X
(a)
0
-3 1
Lyapunov Exponent
(b)
0
-2
0.96
Fig. 5. (a) Enlargement of the previous diagram around p = 1. (b) The corresponding Lyapunov exponents.
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In Fig. 4(a) the bifurcation diagram of (7) is shown for p = 1 and 0.9 6 q 6 3.1. To guarantee that the steady state condition is always achieved, we have plotted again only the last 200 values of 10 000 iterations. As it is evident from Fig. 4(a), there are two chaotic regimes just before and after q = 1, the second one occasionally interrupted by periodic orbits. The corresponding behaviour of the Lyapunov exponent is shown in Fig. 4(b). The two chaotic regimes of Fig. 4(a) correspond exactly to the two regimes of Fig. 4(b), where the Lyapunov exponent has positive values. There are also two periodic windows, a narrow one to the left of q = 1 and a second wide one after q = 1.5. There are also clear nullifications of the Lyapunov exponents in these periodic windows corresponding to period doublings. At q = 3 (nickel mean), there exists an orbit of period 2. At q = 2 (copper mean), an orbit of period 4 appears and finally at q = 1 (golden mean), there is an orbit of period 3. The last case is displayed in detail in Fig. 5(a) and (b).
4. Comments It has to be mentioned here that the existence of period doublings is allowed and not guaranteed by the possession of a negative Schwarzian derivative Sf. Using expression (5) and after some elementary calculations we obtain the explicit expression for Sf 2 f 000 3 f 00 6q2 Sf ¼ 0 ¼ <0 ð11Þ 0 f 2 f ðp þ 2qxÞ2
Acknowledgements The authors thank NATO-project ICS.EAP.CLG 981947. One of the authors, M.O., acknowledges financial support from the Semiconductor Physics Institute, Vilnius, Lithuania in the frame of the EU project PRAMA (contract Nr. G5MA-CT-2002-04014).
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