Generation of 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

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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 . 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 .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 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 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 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. This is why we call these types of sets as “perforation sets.” points of the solution orbits. If the map t is bi-Lipschitzian [this is Let us introduce the map true, for instance, if the function g in (7) is Lipschitzian] then the set AtF for each xed t is a fractal. ψn(x) = B sin(Anx), (4) As our dierential equation, we shall consider the Dung equation where A = πan−1, B = π , and a, b are parameters. The recursive n b 00 0 3 formula is dened as follows: u + δu + βu + αu = γ cos ωt, (8) δ β α γ ω = ψ (x ) := x , where , , , , and are real parameters. Using the variables x u 0 0 0 and y = u0, one can show that Eq. (8) is equivalent to the nonau- xn = ψn(x0), n = 1, 2, ... . tonomous system To construct the perforation set, we start with the interval x0 = y, I = [0, 1] and include in the kth approximation of the set each point (9) y0 = −δy − βx − αx3 + γ cos ωt. x0 ∈ I that satises |xk| ≤ 1. Thus, for the Sierpinski carpet, we use a two-dimensional version of the map (4) which can be dened in the Let us denote by x(t, x0), y(t, y0) the solution of (9) with form  x(0, x0) = x0, y(0, y0) = y0. System (9) can be numerically solved ψ (x, y) = B sin(A x), B sin(A y) . (5) to construct a dynamical system with the motion A (x , y ) = n n n  t 0 0 (Atx0, Bty0), where Atx0 = x(t, x0) and Bty0 = y(t, y0). The procedure here is to determine the image sequence (xn, yn) of Applying this dynamics for the Julia set, the FMI (3) becomes each point (x0, y0) ∈ D, i.e., (xn, yn) = ψn(x0, y0). If we choose D = [0, 1] × [0, 1], the point (x , y ) is excluded from the set if the condi- A A 2 0 0 zn+1 = t −t(zn) + c, tion, |xn| > 1, |yn| > 1, is satised for some n ∈ N. For the values  of the parameters a = 3 and b = 3, the scheme gives the classical where if we let z = x + iy, then At(z) = Atx + iBty. For numeri- Sierpinski carpet. cal simulation, we consider an approximation of the Julia set, with

Chaos 29, 053113 (2019); doi: 10.1063/1.5087760 29, 053113-2 Published under license by AIP Publishing. Chaos ARTICLE scitation.org/journal/cha

FIG. 1. The trajectory of the Duffing dynamics for the Julia set and its sections.

FIG. 2. The trajectory of the Duffing dynamics for the Sierpinski carpet and its sections.

Chaos 29, 053113 (2019); doi: 10.1063/1.5087760 29, 053113-3 Published under license by AIP Publishing. Chaos ARTICLE scitation.org/journal/cha

c = −0.175 − 0.655i, as an initial set, and use the parameter val- Hausdor dimension is calculated through considering the self- ues δ = 0.04, β = 0, α = 1.5, γ = 0.2, and ω = 1 in system (9). similarity of the structure at dierent hierarchical levels. The self- The fractal’s trajectory for 0 ≤ t ≤ 1 is shown in Fig. 1(a). Figures similar fractals, like the Sierpinski carpet, are considered as eective 1(b)–1(e) exhibit the sections of the trajectory at the moments tools for studying the hierarchical structures.13,18 Thus, nding a t = 0.2, t = 0.6, t = 1.2, and t = 1.8. way to map this type of structures allows to create a new hierar- In the case of the Sierpinski carpet, we iteratively apply a motion chical structure with the same Hausdor dimension but dierent At to the scheme (5) to obtain the FMI mechanical properties if one considers bi-Lipschitz maps.

(ξ , η ) = A ψ A− (ξ, η) , n n t n t  ACKNOWLEDGMENTS

where At(x, y) = (Atx, Bty). The iteration is applied for an approx- The authors wish to express their sincere gratitude to the refer- imation of the Sierpinski carpet as an initial set. The fractal tra- ees for the helpful criticism and valuable comments, which helped to jectory for 0 ≤ t ≤ 2 is shown in Fig. 2(a), whereas Figs. 2(b)–2(e) improve the paper signicantly. M. Akhmet and M. O. Fen have been display the sections of the trajectory at the specic times t = 0.5, supported by a grant (Project No. 118F161) from TÜBİTAK, the Sci- t = 1.0, t = 1.5, and t = 2.0. The values δ = 0.08, β = 0, entic and Technological Research Council of Turkey. E. M. Alejaily α = 1, γ = 0.2, and ω = 1 are used in the simulation. is supported by a scholarship from the Ministry of Education, Libya.

III. CONCLUSION REFERENCES 1 In this paper, we propose to connect fractals with dierential F. C. Moon, Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers (Wiley, New York, 1992). equations by considering dynamics of the Dung equation as a tra- 2B. B. Mandelbrot, Les Objets Fractals: Forme, Hasard, et Dimension (Flammarion, jectory initiated at a fractal set. The dynamics are constructed using Paris, 1975). the fractal mapping iteration, which is dened on the basis of the 3O. Zmeskal, P. Dzik, and M. Vesely, “Entropy of fractal systems,” Comput. Math. Fatou-Julia iteration. We consider the Julia set and the Sierpinski Appl. 66, 135–146 (2013). carpet as two examples of the fractal sets. For the Sierpinski car- 4W. Sierpinski, “Sur une Corbe Cantorienue qui contient une image biunivoquet pet, an iteration scheme is constructed to generate the set, then a et continué detoute Corbe donée,” C. R. Acad. Paris 162, 629–632 (1916). 5G. Julia, “Mémoire sur l’itération des fonctions rationelles,” J. Math. Pures Appl. fractal mapping scheme is formulated. Since the Dung equation 8, 47–245 (1918). possesses chaotic solutions for specic values of the parameters in (8), 6P. Fatou, Sur les équations fonctionnelles, I, Bull. Soc. Math. France 47, 161–271 the results of the research can be useful for investigating the fractal (1919); Sur les équations fonctionnelles, II, Bull. Soc. Math. France 48, 33–94 nature of the chaotic . (1920); Sur les équations fonctionnelles, III, Bull. Soc. Math. France 48, 208–314 Important applications can be considered by taking into account (1920). 7 the relationship between the fractal theory of motion and quantum B. B. Mandelbrot, Fractals and Chaos: The Mandelbrot Set and Beyond (Springer, 9,10 New York, 2004). mechanics. In the scale relativity theory, fractals are considered as a 8K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd geometric framework of atomic scale motions such that the quantum ed. (Wiley, Chichester, 2003). behavior can be viewed as particles moving on fractal trajectories. 9L. Nottale, Fractal Space-Time and Microphysics Towards a Theory of Scale Rela- One can suppose that by composing the scale relativity theory with tivity (World Scientic, Singapore, 1993). dynamics of fractals developed in this paper, we will be able to under- 10L. Nottale, Scale Relativity and Fractal Space-Time: A New Approach in Unifying stand better the fractal nature of the world. A possible connection Relativity and Quantum Mechanics (Imperial College Press, London, 2011). 11Y. Dong, M. Dai, and D. Ye, “Non-homogeneous fractal hierarchical weighted between fractal mappings and quantum mechanics through the scale networks,” PLoS One, 10(4), e0121946 (2015). relativity theory can provide important applications for the former in 12Z. Zhang, Y. Li, S. Gao, S. Zhou, J. Guan, and M. Li, “Trapping in scale-free various elds such as biology, cosmology, and fractal geodesics (see networks with hierarchical organization of modularity,” Phys. Rev. E 80, 051120 Ref. 10 and the relevant references therein). (2009). Owing to the important roles of Sierpinski fractals in several 13J. J. Kozak and V.Balakrishnan, “Analytic expression for the mean time to absorp- applications like weighted networks, trapping problems, antenna tion for a random walker on the Sierpinski gasket,” Phys. Rev. E 65, 021105 11–15 (2002). engineering, city planning, and urban growth, we expect that the 14A. Kansal and J. Kaur, “Sierpinski gasket fractal array antenna,” Int. J. Comp. Sci. results of the present study will be helpful in the elds of applica- Commun. 1(2), 133–136 (2010). tions. One of the crucial applications of fractals involves optimization 15D. Triantakonstantis, “Urban growth prediction modelling using fractals and theory. Fractal geometry is used to solve some classes of optimiza- theory of chaos,” Open J. Civil Eng. 2(2), 81–86 (2012). tion problems such as supply chain management and hierarchical 16D. Rayneau-Kirkhope, Y.Mao, and R. Farr, “Optimization of fractal space frames design.16,17 In Ref. 16, for instance, the properties of a particu- under gentle compressive load,” Phys. Rev. E 87, 063204 (2013). 17S. Liu, H. Dong, and W. Zhao, “Optimization model based on the fractal theory lar hierarchical structure are established. The authors constructed in supply chain management,” Adv. Mater. Res. 694–697, 3554–3557 (2013). the relationship between the Hausdor dimension of the optimal 18J. A. Riera, “Relaxation of hierarchical models dened on Sierpinski gasket structure and loading for which the structure is optimized. The fractals,” J. Phys. A Math. Gen. Phys. 19, L869–L873 (1986).

Chaos 29, 053113 (2019); doi: 10.1063/1.5087760 29, 053113-4 Published under license by AIP Publishing.