Applied Mathematics and Computation 222 (2013) 559–563

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Applied Mathematics and Computation

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From Mandelbrot-like sets to Arnold tongues

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.08.001

⇑ Corresponding author.E-mail addresses: aendler@if.ufrgs.br (A. Endler), dfi2pcr@joinville.udesc.br (P.C. Rech).

Antonio Endler a, Paulo C. Rech b,⇑a Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Brazilb Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, Brazil

a r t i c l e i n f o

Keywords:Mandelbrot-like setsArnold tonguesParameter-spaceLyapunov exponentsCoupled maps

a b s t r a c t

A transition from Mandelbrot-like sets to Arnold tongues is characterized via a coupling oftwo non-identical quadratic maps proposed by us. A two-dimensional parameter-spaceconsidering the parameters of the individual quadratic maps was used to demonstratenumerically the event. The location of the parameter sets where Naimark–Sacker bifurca-tions occur, which is exactly the place where Arnold tongues of arbitrary periods are born,was computed analytically.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

We propose a two-dimensional nonlinear dynamical system, which consists in a coupling of two non-identical quadraticmaps, and is governed by the difference equations

xtþ1 ¼ a� x2t þ c1ðxt � ytÞ þ c2ðxt � ytÞ

2;

ytþ1 ¼ b� y2t þ c1ðyt � xtÞ þ c2ðyt � xtÞ2;

ð1Þ

where xt and yt represent dynamical variables, a and b are controlling parameters of the individual quadratic maps, c1 and c2

stand for coupling parameters, and t ¼ 0;1;2; . . . is the discrete time. Note that if we consider c2 ¼ 0, remains only the linearcoupling, and the system (1) can exhibits quasiperiodicity [1,2]. Otherwise, if we fix c1 ¼ 0, remains only the quadratic cou-pling in system (1). In this last case, the system can be simulated experimentally through an electronic circuit [3], and theparameter-space (b; a) corresponds to a Mandelbrot set, a typical fractal structure that can be described in terms of iterates ofcomplex functions [4,5]. In other words, when the coupling parameter c1 is equal to zero, system (1) is equivalent to thecomplex quadratic map ztþ1 ¼ A� z2

t with z ¼ xþ iy;A ¼ aþ ib, and i ¼ffiffiffiffiffiffiffi�1p

(for details, see Ref. [3]).Reports considering two-dimensional parameter-spaces of discrete-time nonlinear dynamical systems are frequent in the

literature in recent years [6–15]. Parameter-space plots allow us to see periodic or chaotic orbits on continuous sets ofparameters. Windows of periodicity may be seen embedded in chaotic or quasiperiodic regions. Arnold tongues embeddedin quasiperiodic regions are examples of these periodic windows. As is known, there is a change of the period-doubling routeto chaos in the single quadratic map, by a quasiperiodic route to chaos in the coupled system of two identical quadratic maps(for details, see Ref. [2] and references therein). The reason is that at some stage of the period-doubling bifurcation cascade, aNaimark–Sacker bifurcation occurs in the place of the period-doubling bifurcation. This change in the type of route to chaosmay be observed, for instance, in a dripping faucet experiment [16]. The single faucet has a period-doubling route to chaos.Two dripping faucets interacting in any way, may present a quasiperiodic route to chaos.

Our motivation in the investigation of the polynomial coupling (1) lies in the fact that it provides rich parameter-spaceplots, showing a transition from Mandelbrot-like sets to Arnold tongues, by the variation of only one of the two coupling

560 A. Endler, P.C. Rech / Applied Mathematics and Computation 222 (2013) 559–563

parameters. As a result, before the transition system (1) presents period-doubling bifurcations and periodic structures typ-ical of the Mandelbrot-like sets. Then, after the transition, we get Naimark–Sacker bifurcations and Arnold tongues. In ourcoupling (1) of two quadratic maps, the replacement of the period-doubling bifurcation by the Naimark–Sacker bifurcationoccurs when the linear coupling term becomes more important that the nonlinear coupling term. As we shall see in Section 2,this change of bifurcations in coupling (1) occurs for c2 ¼ 0:5 and c1 between 0 and 0.5. As far as we know, this is the firsttime that the transition period-doubling bifurcation/Naimark–Sacker bifurcations, obtained in a same coupling by variationof only one parameter, is reported.

2. Results and discussion

Typical parameter-space plots for the map (1) are shown in Fig. 1, and were numerically generated by using the methoddescribed in [17]. Each one of the diagrams show the complexity of the alternation of the stability domains in parameter-space, when the parameter interval is discretized in a mesh of 350 � 350 points equally spaced. The behavior of each(b; a) mesh point was always determined considering the initial condition (x0; y0) ¼ (0,0), and after a transient of 5 � 104

iterates. Different stability domains are distinguished by different tonalities, with the numbers signifying the periods ofthe more visible attractors. The grey region contiguous to the period-1 region, which appears more clearly in Fig. 1(c) and(d), indicates parameters which lead the system to quasiperiodic motion near the boundary NS, and to chaotic motion faraway from the same boundary. The basin of unbounded attractors is left in blank.

In Fig. 1(a), c1 ¼ 0, and one sees that the parameter-space exhibits a Mandelbrot set form [4]. According to the parameterc1 is increased from zero, this set is deformed as we can see in Fig. 1(b)–(d), where c1 ¼ 0:05; c1 ¼ 0:1, and c1 ¼ 0:5, respec-tively. Indeed, all the periodic structures are deformed, while the chaotic attractor in grey color which surrounds theperiod-1 region, not visible in Fig. 1(a) and (b), begins to grow. From some value of 0 6 c1 6 0:5 the linear coupling term

-1 0 1 2b

-1

0

1

2

a

1

23

4

56 78 910

(a)

c1 = 0.0

-1 0 1 2b

-1

0

1

2

a

c1 = 0.05

(b)

-1 0 1 2b

-1

0

1

2

a

c1 = 0.1

(c)

-1 0 1 2b

-1

0

1

2

a

NS

NS

c1 = 0.5

(d)

Fig. 1. Stability domains in parameter-space of the coupling (1), for c2 ¼ 0:5 and four values of c1. Numbers indicate periods of different attractors of morelow periodicity. (a) Mandelbrot set, for c1 ¼ 0:0. (b) Mandelbrot set lightly modified, for c1 ¼ 0:05. (c) c1 ¼ 0:1 and (d) c1 ¼ 0:5 showing, respectively, thechaotic or quasiperiodic grey region (see the text).

A. Endler, P.C. Rech / Applied Mathematics and Computation 222 (2013) 559–563 561

in (1) begins to have predominance over the quadratic coupling term, and quasiperiodic motion appears. Quasiperiodic mo-tion is a result of a Naimark–Sacker bifurcation [18] that, in this case, occurs for points located along the period-1 regionborder line NS. As a result, on the NS line the period-1 orbit disappears to give rise a one invariant closed curve, which isthe characteristic of the quasiperiodic motion. In Fig. 1(d) can be seen clearly, immersed in quasiperiodic region, resonancetongues which are periodic regions similar to the Arnold tongues observed in the circle map [19], mainly those associatedwith the rotation numbers 1/5 and 1/7. Note that Naimark–Sacker bifurcations of more high periodic orbits also occur insystem (1), as we can check in Fig. 1(d) for period-2 and period-4 orbits.

As is well known [18,20], a necessary condition to occurs a Naimark–Sacker bifurcation of a fixed point p0 of an n-dimen-sional map f, is that the Jacobian matrix, Df ðp0Þ, has a pair of complex eigenvalues, say m and m, with jmj ¼ 1. Under thiscondition, the coupling (1) gives us the following curve

Fig. 2.before,article.)

NS ¼ 256c22a4 þ 128 �4 2c2 � 1ð Þc2bþ 5c2 � 2ð Þc2

1

� �a3 � 16½�16 6c2

2 � 4c2 þ 1� �

b2 þ 8 5c2 þ 1ð Þc21b

þ 2 2c41 � 2c2

1 � 8c1 þ 3� �

c2 � 13c31 � 4c1 � 8

� �c1�a2 � 8 ð64 2c2 � 1ð Þc2b3 þ 16 5c2 þ 1ð Þc2

1b2

� 4 2 2c41 � 2c2

1 � 8c1 þ 3� �

c2 þ 11c41 � 2c2

1 � 3� �

bþ 4c21 � 2c1 � 1

� �4c4

1 þ 2c31 � c2

1 � 4c1 � 2� �

Þa

þ 256c22b4 þ 128 5c2 � 2ð Þc2

1b3 � 16 2 2c41 � 2c2

1 � 8c1 þ 3� �

c2 � 13c41 þ 4c2

1 þ 8c1� �

b2

� 8 4c21 � 2c1 � 1

� �4c4

1 þ 2c31 � c2

1 � 4c1 � 2� �

b� 4c1 þ 3ð Þ 4c21 � 2c1 � 1

� �2 ¼ 0; ð2Þ

which is the algebraic version to the border NS that surrounds the period-1 region in Fig. 1(d), if c1 ¼ c2 ¼ 0:5.Fig. 2 shows a magnification of the boxed region in Fig. 1(d), where now the color code is as follows: blue indicates peri-

odic orbits, while grey and green indicate quasiperiodic and chaotic orbits, respectively. The periodic tongues showed inFig. 2 are born from the period-1 region, at the points ða; bÞ ¼ ðD�;DþÞ where

D� ¼12

cos að Þ � 1ð Þ cos að Þ � c1ð Þ � 14� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic1

2 þ 1� cos2 að Þð Þ cos að Þ � 1þ c1ð Þ2q

; ð3Þ

a ¼ 2pn=k, and n=k is the rotation number. The labeled tongues in Fig. 2 have rotation numbers 2=5;5=12;8=19;4=7 and 4=9.For a most general case where c2 – 0:5, the tongues are born at ða; bÞ ¼ ðD�;DþÞ, where now

D� ¼12

cos að Þ � 1ð Þ cosðaÞ � c1ð Þ � 14� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic1

2 þ 1� cos2 að Þð Þ cos að Þ � 1þ c1ð Þ2

4c2 � 1

s: ð4Þ

Trajectories in the phase-space can be used to describe the dynamics in quasiperiodic region of the parameter-space. Forinstance, Fig. 3 shows the behavior of the system (1) for four points roughly aligned along the solid linea ¼ �0:7664bþ 1:7524 shown in the parameter-space of Fig. 2. One sees, therefore, four trajectories for different valuesof the (b; a) pair. For (b; a) ¼ (1.5476, 0.5663), signifying the point A located close to the boundary NS in grey region ofthe Fig. 2, an invariant closed curve (inner curve in Fig. 3) is observed, and the motion is said quasiperiodic. As we move awayfrom the NS boundary, always walking along the solid line, this closed curve is deformed (outer curve in Fig. 3), but the

1.5 1.6 1.7b

0.4

0.5

0.6

a

.

...

1

12

19

2

5

7

9NS

A

B

C

D

Magnification of the box in Fig. 1(d). Blue indicates periodicity regions, grey indicates quasiperiodic motion, and green indicates chaotic motion. Asnumbers indicate periods. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this

-0.2 0.0 0.2 0.4 0.6 0.8x

0.0

0.5

1.0

1.5

2.0

y

c1 = c

2 = 0.5

Fig. 3. Trajectories in phase-space for the map (1), with c1 ¼ c2 ¼ 0:5. (b; a) ¼ (1.5476, 0.5663) (inner curve), (b; a) ¼ (1.6482, 0.4895) (outer curve),(b; a) ¼ (1.6940, 0.4541) (circles), and (b; a) ¼ (1.7449, 0.4151) (complicated structure).

562 A. Endler, P.C. Rech / Applied Mathematics and Computation 222 (2013) 559–563

motion yet is quasiperiodic. Now (b; a) ¼ (1.6482, 0.4895), signifying the point B located close to the period-7 resonance ton-gue. A mode-locked solution characterized by a period-7 resonance tongue is obtained, being represented by the seven cir-cles in Fig. 3, for (b; a) ¼ (1.6940, 0.4541), which is the point C located inside the period-7 blue region. Finally, a chaoticattractor is obtained, and it is represented by the complicated structure in Fig. 3, for which (b; a) ¼ (1.7449, 0.4151) signifythe point D along the solid line in Fig. 2, now in green region after the period-7 resonance tongue, but before the divergenceregion in white, not much visible in Fig. 2. Note that this route to chaos via quasiperiodicity is not a exclusive occurrence tomoving along the solid line in Fig. 2. No matter which the line, since it goes from a point close to the border NS with theperiod-1 region, to a point in the divergence region, crossing the grey region, chaotic region will be reached. All trajectoriesin plots of Fig. 3 were initialized at (x0; y0) ¼ (0,0), and plotted with 5 � 103 points, after a transient of 5 � 104 iterates.

Alternatively, Lyapunov exponents and bifurcation diagrams can be used to study the dynamics of the system (1). A bifur-cation diagram for the variable x as a function of the parameter b is shown in Fig. 4(a), while the dependence of the largestLyapunov exponent k on same parameter is shown in Fig. 4(b), both plots for points along the same solid linea ¼ �0:7664bþ 1:7524 in Fig. 2, and for c1 ¼ c2 ¼ 0:5 as before. For all plots the initial conditions and transient are the sameused before, and 103 values of the parameter b were considered. In bifurcation diagram 50 points were plotted for each b

-0.2

0.2

0.6

1.0

x

1.55 1.6 5 1.75

b

-0.2

0.0

0.2

λ

(a)

(b)

c1=c

2=0.5

Fig. 4. (a) Bifurcation diagram showing the variable x. (b) The largest Lyapunov exponent plot. Both graphs as a function of the parameter b, fora ¼ �0:7664bþ 1:7524 and c1 ¼ c2 ¼ 0:5. In the Lyapunov exponent plot, the dashed line locates k ¼ 0.

A. Endler, P.C. Rech / Applied Mathematics and Computation 222 (2013) 559–563 563

value, while the average involved in Lyapunov exponent calculations was made over 106 iterates, after to eliminate the tran-sient behavior.

By increasing b from 1.51 along the line a ¼ �0:7664bþ 1:7524, the system is periodic (period-1 orbit), characterized by alargest Lyapunov exponent less than zero, until b ¼ 1:5277 where a quasiperiodic region is reached (see also Fig. 2). Whilequasiperiodic motion takes place, the exponent remains at zero value, and the orbit appears to fill out the entire interval ofthe variable x. This quasiperiodic motion is, therefore, originated from a period-1 orbit, and a consequence of a supercriticalNaimark–Sacker bifurcation [18] that occurs at b ¼ 1:5277. As a consequence of this bifurcation, one invariant closed curve iscreated, whose examples can be seen in Fig. 3 (inner and outer curves). There are mode-locked states (resonance tongues)immersed in the quasiperiodic region, for which the exponent is also less than zero. It is important to note the agreementbetween Figs. 2–4.

3. Summary

We propose a coupling of two non-identical quadratic maps with two coupling terms, one linear and the other quadratic(see Eqs. (1)). We have shown explicitly through numerical simulations that, depending on coupling parameter values, thissystem can generates either a parameter-space typical of a Mandelbrot set (see Fig. 1(a)), or a parameter-space characterizedby regions where periodic structures, the Arnold tongues, appear immersed in quasiperiodic regions (see Figs. 1(d) and 2).We also have determined, this time analytically, the location in parameter-space where Naimark–Sacker bifurcations occur(see Eq. (2)), that is, the location where Arnold tongues of arbitrary period are born. Therefore, we conclude that, as the cou-pling parameters are varied in the considered discrete-time system, a transition from a Mandelbrot-like set to Arnold ton-gues takes place.

Acknowledgement

PCR thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq, Brazil, for the financial support.

References

[1] P.C. Rech, M.W. Beims, J.A.C. Gallas, Naimark–Sacker bifurcations in linearly coupled quadratic maps, Physica A 342 (2004) 351–355.[2] P.C. Rech, M.W. Beims, J.A.C. Gallas, Generation of quasiperiodic oscillations in pairs of coupled maps, Chaos Solitons Fract. 33 (2007) 1394–1410.[3] O.B. Isaeva, S.P. Kusnetsov, V.I. Ponomarenko, Mandelbrot set in coupled logistic maps and in an electronic experiment, Phys. Rev. E 64 (2001)

055201(R).[4] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982.[5] A. Endler, J.A.C. Gallas, Mandelbrot-like sets in dynamical systems with no critical points, C. R. Acad. Sci. Paris Ser. I 342 (2006) 681–684.[6] E.N. Lorenz, Compound windows of the Hénon map, Physica D 237 (2008) 1689–1704.[7] P.C. Rech, Chaos control in a discrete time system through asymmetric coupling, Phys. Lett. A 372 (2008) 4434–4437.[8] G.A. Casas, P.C. Rech, Dynamics of asymmetric couplings of two Hénon maps, Int. J. Bifurcation Chaos 20 (2010) 153–160.[9] G.A. Casas, P.C. Rech, Numerical study of a three-dimensional Hénon map, Chin. Phys. Lett. 28 (2011) 010203.

[10] E.L. Brugnago, P.C. Rech, Chaos suppression in a sine square map through a nonlinear coupling, Chin. Phys. Lett. 28 (2011) 110506.[11] A. Celestino, C. Manchein, H.A. Albuquerque, M.W. Beims, Ratchet Transport and Periodic Structures in Parameter Space, Phys. Rev. Lett. 106 (2011)

234101.[12] D.F.M. Oliveira, E.D. Leonel, Parameter space for a dissipative Fermi–Ulam model, New J. Phys. 13 (2011) 123012.[13] S.L.T. de Souza, A.A. Lima, I.L. Caldas, R.O. Medrano-T, Z.O. Guimarães-Filho, Self-similarities of periodic structures for a discrete model of a two-gene

system, Phys. Lett. A 376 (2012) 1290–1294.[14] X.-F. Li, Y.-D. Chu, H. Zhang, Fractal structures in a generalized square map with exponential terms, Chin. Phys. B 21 (2012) 030203.[15] P.C. Rech, The dynamics of a symmetric coupling of three modified quadratic maps, Chin. Phys. B 22 (2013) 080202.[16] M.B. Reyes, R.D. Pinto, A. Tufaile, J.C. Sartorelli, Heteroclinic behavior in a dripping faucet experiment, Phys. Lett. A 300 (2002) 192–198.[17] J.A.C. Gallas, Structure of the parameter space of the Hénon map, Phys. Rev. Lett. 70 (1993) 2714–2717.[18] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 2003.[19] H.G. Schuster, Deterministic Chaos, An Introduction, VCH, Weinheim, 2005.[20] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 2002.