Fibonacci Bifurcation

Chaos, Solitons and Fractals 33 (2007) 1240–1247 www.elsevier.com/locate/chaos

Bifurcations of Fibonacci generating functions
ˇ ¨ Mehmet Ozer a,b,*, Antanas Cenys b, Yasar Polatoglu a, Gursel Hacibekiroglu a, ¨ c d Ercument Akat , A. Valaristos , 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, Kayis dagi 81120, Istanbul, Turkey ß 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 À px À qx2) isexamined, 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 themetallic 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. ¨ E-mail address: m.ozer@iku.edu.tr (M. Ozer).

0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.01.095

¨ M. Ozer et al. / Chaos, Solitons and Fractals 33 (2007) 1240–1247

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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 formingsuperlattices, 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 furtherstudying 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 1ÀpxÀqx2 , for different values of p and q, which serve as the control parameters. This is the generating function of the generalized Fibonacci...