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Generation of Fractals As Duffing Equation Orbits Generation of fractals as Duffing equation orbits Cite as: Chaos 29, 053113 (2019); https://doi.org/10.1063/1.5087760 Submitted: 04 January 2019 . Accepted: 29 April 2019 . Published Online: 14 May 2019 Marat Akhmet , Mehmet Onur Fen , and Ejaily Milad Alejaily ARTICLES YOU MAY BE INTERESTED IN Emergent dynamics of coordinated cells with time delays in a tissue Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 031101 (2019); https:// doi.org/10.1063/1.5092644 Unpredictability and robustness of chaotic dynamics for physical random number generation Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 033133 (2019); https:// doi.org/10.1063/1.5090177 Quantitative assessment of cerebral connectivity deficiency and cognitive impairment in children with prenatal alcohol exposure Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 041101 (2019); https:// doi.org/10.1063/1.5089527 Chaos 29, 053113 (2019); https://doi.org/10.1063/1.5087760 29, 053113 © 2019 Author(s). Chaos ARTICLE scitation.org/journal/cha Generation of fractals as Duffing equation orbits Cite as: Chaos 29, 053113 (2019); doi: 10.1063/1.5087760 Submitted: 4 January 2019 · Accepted: 29 April 2019 · Published Online: 14 May 2019 View Online Export Citation CrossMark Marat Akhmet,1,a) Mehmet Onur Fen,2 and Ejaily Milad Alejaily1 AFFILIATIONS 1Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey 2Department of Mathematics, TED University, 06420 Ankara, Turkey a)Author to whom correspondence should be addressed: [email protected]. Tel.: +90 312 210 5355. Fax: +90 312 210 2972. ABSTRACT Dynamics are constructed for fractals utilizing the motion associated with Dung equation. Using the paradigm of Fatou-Julia iteration, we develop iterations to map fractals accompanied with a criterion to ensure that the image is again a fractal. Because of the close link between mappings, dierential equations and dynamical systems, one can introduce dynamics for fractals through dierential equations such that they become points of the solution trajectory. There is no doubt that the dierential equations have a distinct role for studying chaos. Therefore, char- acterization of fractals as trajectory points is an important step toward a better understanding of the link between chaos and fractal geometry. Moreover, it would be helpful to enhance and widen the scope of their applications in physics and engineering. Published under license by AIP Publishing. https://doi.org/10.1063/1.5087760 Fractal and chaos are interesting features that characterize many parts from an initial square. The process starts with subdividing a natural phenomena, and they have a signicant impact on practi- solid square into nine identical subsquares and then removing the cal applications. Even though many studies have investigated the central one. In the next iterations, the same procedure is repeated relationship between chaos and fractals, still there is no formal to each of the remaining squares from the preceding iteration. In an theory to interpret this link. The link is more clear when frac- analogous way to the carpet, Sierpinski developed a triangular fractal tal dimension is used to measure the extent to which a trajectory known as the Sierpinski gasket. lls its phase space.1 In this research, we propose to connect the Involvement of the dynamics of iterative maps in fractal con- dynamics of dierential equations with fractals. For this purpose, struction was a critical step made by the French mathematicians a fractal mapping iteration is developed such that the motion asso- Pierre Fatou and Gaston Julia around 1917–1918, during their inde- ciated with Dung equation is considered as a map with a fractal pendent studies on the iteration of rational functions in the complex as an initial set. Thus, the orbit corresponding to the solution of plane.5,6 They described what we call today the Fatou-Julia iteration the dierential equation can be seen as a continuous sequence of (FJI).7 The iteration is dened over a domain D ⊆ C by fractals. zn+1 = F(zn), (1) I. INTRODUCTION AND PRELIMINARIES where F : D → D is a given function for the construction of the frac- 2 The term “fractal” was coined by Benoit Mandelbrot in 1975. tal set F. The points z0 ∈ D are included in the set F depending on He dened fractal as a set for which the Hausdor dimension strictly the boundedness of the sequence zn, n = 0, 1, 2, ..., and we say that exceeds the topological dimension. Dealing with fractals goes back the set F is constructed by FJI. to the 17th century when Gottfried Leibniz introduced the notions In practice, one cannot verify the boundedness for innitely of recursive self-similarity.3 A considerable leap in the construction long iterations. This is why in simulation we x an integer k and a of fractals was performed in 1883 by Georg Cantor, as he discovered bounded subset M ⊂ C, then the obtained set is the collection of the most essential and inuential fractal known as the Cantor set. all points z0 ∈ D such that the points zn, n = 1, 2, ... , k, belong to Waclaw Sierpinski was one of the mathematicians who made M. In what follows, we call such a set a kth “approximation” of the signicant contributions in the eld of fractals. He introduced the fractal F. famous square fractal in 1916,4 known as the Sierpinski carpet. The The most popular fractals, Julia and Mandelbrot sets, are gener- 2 fractal is generated by a recursive process of removing symmetrical ated using the iteration of the quadratic map F(zn) = zn + c, where c Chaos 29, 053113 (2019); doi: 10.1063/1.5087760 29, 053113-1 Published under license by AIP Publishing. Chaos ARTICLE scitation.org/journal/cha 0 is a complex number. Julia sets contain the points z0 ∈ C correspond- Now, let Φ : D → D be an invertible function. Then, the fractal ing to the bounded sequence zn, whereas the Mandelbrot set is the set mapping scheme can be dened by of parameter values c ∈ C such that {z }, z = 0, remains bounded. n 0 Φ−1(ξ , η ) = ψ Φ−1(ξ , η ) . (6) To map fractals, a new method attained by FJI is suggested. The n n n 0 0 method is based on involving a map Φ in the iteration (1) to dene In order to obtain the mapped Sierpinski carpet, FΦ , the domain of the Fractal Mapping Iteration (FMI) (6) is restricted only to the points (ξ0, η0) that belong to the mapped 0 −1 −1 −1 domain D . Thus, if we let (un, vn) = Φ (ξn, ηn), then (ξ0, η0) is Φ (xn+1) = F Φ (xn) , (2) included in FΦ if at least one of |un| and |vn| is less than or equal 2 N where x0 is a number in C or in R . In this recursive equation, the to 1 for all n ∈ . FJI is applied to the preimage of the mapped set. To determine a One can show that, for the FMIs (3) and (6), the set FΦ is merely F kth approximation of the mapped fractal FΦ , we consider a bounded the image of under the map Φ. However, the following question subset M ⊂ D, where D is the domain of the original FJI. The bound- arises here: Is the mapped set a fractal? The answer is “yes” if the −1 edness of the sequence Φ (xn), n = 1, 2, ... , k, in M, is examined map Φ satises a bi-Lipschitz condition. This result is stated in the for all points x0 ∈ D. Then, the kth approximation of FΦ is the following lemma. 8 Rn Rm set containing only the points x0 corresponding to the bounded Lemma 1. Let E ⊆ . If f : E → is a bi-Lipschitz sequences. The FMI (2) is described in a general form, i.e., it is valid function, i.e., there exist real numbers l1, l2 > 0 such that l1|u − v| for any function F. ≤ |f (u) − f (v)| ≤ l2|u − v|, for all u, v ∈ E, then In the present paper, we shall apply the FMI to fractals and con- dim f (E) = dim (E), struct continuous dynamics using dierential equations, namely, the H H classical Dung equation. As examples, the Julia set and the Sier- where dimH denotes the Hausdor dimension. pinski carpet are considered. To map the Julia set, the FMI takes the form II. MAIN RESULT −1 −1 2 Φ (z + ) = Φ (z ) + c. (3) n 1 n We connect dierential equations with fractals by involving their dynamics as maps in FMI. The general idea of constructing a The mapped Julia set FΦ is the set of parameter values z0 ∈ C such −1 dynamics for fractal is to use the motion of a dynamical system with a that the sequence Φ (zn) remains bounded. For the Sierpinski carpet, the idea of FJI is adopted to develop a fractal as an initial set. The motion of dynamical system is dened by A = ϕ( ) ϕ scheme for constructing the set. The technique of the FJI is based on tx0 t, x0 , where is the solution of a two-dimensional system detecting the points of a fractal set through the boundedness of their of ordinary dierential equations, iterations under a specic map. Here, we shall extend the technique to x0 = g(t, x), (7) include any possible criterion for grouping points in a given domain. A F We use a map that constructs a set that is similar to the Cantor set with ϕ(0, x0) = x0. Thus, we construct dynamics of sets t , where F in the generation way but dierent in structure. The purpose of such the fractal is the initial value. Through this procedure, the dier- sets is to cut out successively smaller parts (holes) in the Sierpinski ential equations are involved in fractals such that the latter become A carpet.
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