Physics Letters A 377 (2013) 1269–1273

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Physics Letters A

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Wada basins of strange nonchaotic attractors in a quasiperiodicallyforced system ✩

Yongxiang Zhang ∗

College of Science, Shenyang Agricultural University, Shenyang 110866, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 November 2012Received in revised form 9 March 2013Accepted 20 March 2013Available online 23 March 2013Communicated by A.R. Bishop

Keywords:Wada basinBasin of attractionStrange nonchaotic attractorsBasin cell

Whether Wada basins of strange nonchaotic attractors (SNAs) can exist has been an open problem. Herewe verify the existence of Wada basin for SNAs in a quasiperiodically forced Duffing map. We show thatthe SNAs’ basins are full Wada for a set of parameters of positive measure. We identify two types ofSNAs’ Wada basins by the basin cell method. It suggests that SNAs cannot be predicted reliably for thespecific initial conditions.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Nonlinear dynamical systems can have multiple coexisting at-tractors. The set of initial points which eventually approach eachparticular attractor is called its basin of attraction. The boundariesthat separate different basins of attraction are basin boundaries,which can be fractal (i.e., it contains a homoclinic point) [1,2]. Anunderstanding of basin boundaries is of great physical significance.For example, the fractal property of basin boundaries can make itdifficult to predict the behavior of dynamical systems. One of thegoals of nonlinear dynamics is to determine the global structureof a system such as basin boundaries. A particular type of basinboundary is called Wada basin boundary. A basin boundary satis-fies the Wada property if every open neighborhood of any pointon the boundary has a nonempty intersection with at least threedifferent basins. The Wada basin boundary was first introduced inthe physics literature by Kennedy and Yorke [3]. A beautiful ex-periment showing the Wada basin boundaries was reported in anoptical system published in Nature [4]. These dynamical systemshave three or more basins sharing the same boundary. Note thata Wada basin boundary is a fractal basin boundary but a fractalbasin boundary need not be a Wada basin boundary [5]. Althoughtypical fractal basin boundaries have been shown to be very com-mon in nonlinear dynamics, the number of examples displayingthe Wada property is still quite limited. So far, to our knowledge,

✩ Supported by the National Natural Science Foundation of China (No. 11002092).

* Tel.: +86 024 88493201.E-mail addresses: zyx9701@126.com, zyx9701@syau.edu.cn.

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2013.03.026

the study of Wada basin has been restricted to periodically drivensystems [5]. A quite important class of dynamical systems is thequasiperiodically forced systems. These systems are of interest be-cause they can present a special type of dynamics — strange non-chaotic attractors (SNAs) [6–9, and references therein]. SNAs areobjects that have chaotic attractor features like fractal dimensionand nondifferentiability (strangeness) but no exponential sensitiv-ity to initial conditions, i.e., its largest Lyapunov exponent is eitherzero or negative. Although this type of system has Wada basinboundaries for quasiperiodic attractors [10] and it has the inter-twined basin boundaries of SNAs [11], it is still not entirely clearwhether the SNAs have full Wada basins. Thus, it is of interest toinvestigate the nature of basins for SNAs in the quasiperiodicallyforced systems.

On the other hand, a thorough understanding of SNAs’ Wadabasin structure is important, since it provides an analysis for an-swering the question, “will there be a larger unpredictability ofsensitive dependence on initial conditions for SNAs in some sys-tems?” The knowledge of the SNAs on basin structures is also im-portant for addressing the question, “what routes and mechanismsfor the creation of SNAs with Wada basin can occur?” Of course,this interesting fact can be expected for other systems except forthe quasiperiodically forced two-dimensional dissipative map. Ourgoal in this Letter is twofold. First, we characterize and confirmthe creation of the SNA in a quasiperiodically forced system in-troduced by Holmes [12,13]. Some typical routes are identified,such as Heagy–Hammel route [14] and the gradual fractalizationroute [15–18]. Second, we describe some numerical experimentsand verify the existence of SNAs’ Wada basins. We use basin cellapproach to distinguish two types of Wada basins structure. Basin

1270 Y. Zhang / Physics Letters A 377 (2013) 1269–1273

cell notion was introduced by Nusse and Yorke [19] and it wasused to describe the Wada basin boundaries [20–22]. The basincells also play an important role to describe the bifurcations onbasin boundaries [20,21] and the geometry of mixing chaotic flows[22]. Basin cells provide much information about the associatedbasins. The corresponding basin can be seen as a basin cell (a cen-tral body of the basin) plus several arms that connect to it. Ourresults verify the abundance of SNAs with Wada basin. That is, fora positive measure set in the parameter space, SNAs’ basins arefull Wada. We confirm the existence of SNAs using the largest Lya-punov exponent and phase sensitivity function [23]. In the quasi-periodically forced system, we focus on the truncation of torusdoubling and the mechanisms for the creation of SNAs. At last,we observe the Wada property by the sections of the basins andsections of the invariant manifolds.

2. Quasiperiodically forced Duffing map

The study of iterated maps is becoming an important topic innonlinear dynamics. Such maps have been of interest to populationbiologists, mathematicians, engineers, physicists, and chemists [24,and references therein]. Several studies have been conducted oncubic maps, for example [25,26]. Such maps could be used to goodadvantage for understanding the dynamics of other types of mapsas well. Here, we shall devote ourselves to Wada basins of SNAsin the two-dimensional Duffing (cubic) map introduced by Holmes[12,13]. For this Duffing map, the coexistence of two attractors andfractal boundaries have been investigated [27]. Two types of Wadabifurcations and partially Wada basin boundaries (for the periodicattractors) have been investigated [28]. Its mathematical expres-sion is the following:

(xn+1, yn+1) = F (xn, yn):{ xn+1 = yn

yn+1 = μyn − y3n − J xn

(1)

where J is Jacobian (0 � J � 1) and μ means the bifurcationparameter. According to the symmetry of the map F (x, y) =F (−x,−y), the attractors and their basins have the same sym-metry.

In the present work, we investigate the dynamics of (2) withthe additional quasiperiodic forcing in R2 × T1,

(xn+1, yn+1, θn+1) = f (xn, yn, θn):⎧⎨⎩

xn+1 = yn

yn+1 = μ(1 + υ cos(2πθn)

)yn − y3

n − J xn

θn+1 = θn + ω (mod 1)

(2)

where ω and υ represent the frequency and amplitude of thequasiperiodic forcing, respectively. The determinant of the Jacobianmatrix (both two- and three-dimensional) is constant and equalto J , which controls the dissipativity of map. The variables of themap are denoted by x and y for clarity. The θ dynamics is a rigidirrational rotation, unaffected by the variation of the other vari-ables. The driving frequency is an irrational number, the inverseof the golden mean ratio, ω = (

√5 − 1)/2. The phase space of the

three-dimensional (3D) system is R2 × T1. Since the dynamics ofthe θ variable is uniform and ergodic, it is convenient to considerthe reduced phase space spanned by the variables x and y. Withquasiperiodic driving the periodic attractors become quasiperiodicattractors: period-n orbits of the system (1) are converted into toriwith n branches.

3. SNAs and Wada basins

In order to describe strange nonchaotic dynamics in system (2),we have found it useful to characterize SNAs through both the Lya-punov exponent λx in the x-direction, which is given by

λx = limN→∞

1

N

N∑i=1

ln

∣∣∣∣ ∂ f

∂xi

∣∣∣∣ (3)

and the phase sensitivity exponent which can be obtained fromphase sensitivity function ΓN [23]

ΓN(μ,υ, J) = minx0,y0,θ0

(max

0�n�N

∣∣∣∣∂xn

∂θ

∣∣∣∣)

(4)

∂xn∂θ

can be obtained by differentiating (2),

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂xn+1

∂θ= ∂ yn

∂θ

∂ yn+1

∂θ= −2πμυ sin(2πθn)yn

+ μ(1 + υ cos(2πθn)

)∂ yn

∂θ

− 3y2n∂ yn

∂θ− J

∂xn

∂θ

θn+1 = θn + ω (mod 1)

(5)

On an SNA, the function ΓN grows with the length of the or-bit N , as a power, i.e., ΓN ∼ Nγ , where γ is the phase sensitivityexponent. The exponent γ measures the sensitivity with respectto the phase of the quasiperiodic forcing and characterizes thestrangeness of an attractor in a quasiperiodically driven system.A smooth torus has a negative (or zero) Lyapunov exponent and nophase sensitivity (γ = 0). On the other hand, SNAs have negativeLyapunov exponents and high phase sensitivity (γ > 0). Chaotic at-tractors have positive Lyapunov exponents.

In general, the invariant torus of saddle type on the basinboundary is computed by numerical approximations. Also, sincetheir stable and unstable manifolds are two-dimensional, it ishard to approximate these [29]. The special form of the quasi-periodically forced map allows for a technique to find the invarianttorus [30] and this technique can apply to similar Hénon map [31].Since the dynamics on the torus is simply the rigid rotation by ωand since the torus is parameterized in θ , we can easily expressthe invariance of the torus in term of the parameterization [32].The resulting equation can then be solved with Newton’s method.For the computation of the two-dimensional stable and unstablemanifolds of the circle, we use the method presented in [30]. Thisidea is to consider the two-dimensional manifold as a collection ofone-dimensional torus in planes defined by θ = constant. In sys-tem (2), basin boundaries are two-dimensional surfaces, in otherwords, the stable manifolds of the torus are two-dimensional sur-faces or segments. Such surfaces can be fractal or Wada. Withquasiperiodic driving the saddles (p = 0, in the absence of quasi-periodic driving) become some invariant curves of saddle type.The closure of the two-dimensional stable manifold of some in-variant curves forms the basin boundary of the attractor. Since itis difficult to study basin structures and critical surfaces in threedimensions, we fix the variable θ and study the reduced basinstructure and critical curves [the invariant sections of manifoldsfor θ = constant] in two-dimensional (x, y) surfaces in order to un-derstand the Wada property from an intuitive point of view (theseare in fact cross-sections of the critical surfaces at the chosen valueof θ ).

We focus on a representative region in the parameter space(2.8 � μ � 2.92,0.018 � υ � 0.032 for the fixed J = 0.6), wherethe SNAs’ basins are full Wada. The mechanism of the creation ofSNAs is due to the truncation of torus-doubling bifurcation (grad-ual fractalization of a torus and Heagy–Hammel route). Here, ouraim is twofold: one is to investigate the nature of SNAs and theirtransitions. The other one is to verify the Wada property and an-alyze the Wada basin structure. Note that multi-stability of the

Y. Zhang / Physics Letters A 377 (2013) 1269–1273 1271

coexisting attractors can be observed. We examine the bifurcationstructure of two symmetric attractors with separate symmetricbasins of attraction, where the phase diagram is quite identical.Fig. 1 shows a phase diagram (in the μ–υ planes) with single

Fig. 1. Parts of the parameter plane of the map (2) in the μ–υ parameter plane atthe fixed value J = 0.6. Regular, chaotic, SNAs (with Wada basin) and divergence(infinity) regimes are shown in white, green, yellow and black, respectively. Forthe case of regular attractor, a torus and some doubled torus exist in the whiteregions denoted by 1T , 2T and 4T , respectively. The doubling bifurcation curvesrepresented by the black solid line. Two types of Wada basins structure for SNAsare denoted by I and II (blue dashed line), respectively. Typical dynamical transi-tions such fractalization (routes A, C and D), Heagy–Hammel (routes B) can occurthrough the interruption of torus doubling bifurcation. (For interpretation of the ref-erences to color in this figure legend, the reader is referred to the web version ofthis Letter.)

attractors for reasons of simplicity and clarity (here we take the at-tractors with negative values as an example). Regular, chaotic, SNAs(with Wada basin) and divergence (infinity) regimes are shown inwhite, green, yellow and black, respectively. For the case of regu-lar attractor, a torus, a doubled torus and secondary doubled torusexist in the white regions denoted by 1T , 2T and 4T , respec-tively. The doubling bifurcation curves represented by the blacksolid line. Two types of Wada basins structure can be observed inregions I and II (dashed line in blue), respectively. We observe thatthe basin sections possess the Wada property for the different θ .The 6-sided basin cells (in the basin sections shown in Fig. 2a) andthe 8-sided basin cells (in the basin sections shown in Fig. 2b) canexist in regions I and II, respectively. The transitions (between Iand II denoted by the red arrow) can be induced by the basin cellbifurcation in the system (1). Typical dynamical transitions suchas the fractalization (the routes A, C and D) and Heagy–Hammelroute (the route B) can be investigated. The route A (or D) is the2T torus gradual fractalization and the route C is a transition from4T torus to chaos via the SNA. The SNAs (by the route A) haveWada basin sections formed by the 6-sided basin cell. The SNAs(by the route C and route D) have Wada basin sections formed bythe 8-sided basin cell.

Let us first give an introduction to Wada basin (for periodic at-tractors) in the unforced map (1) in order to understand the basinstructure (for SNAs) in the quasiperiodically forced system (2). InRef. [28], we concentrate on three types of basin cells (the 6-sidedbasin cell, the 8-sided basin cell and the 10-sided basin cell) andtheir transitions. These basin cells have been used to verify theexistence of Wada basins for periodic attractors rigorously. The di-versity of basin cells suggests the abundance of Wada basins inthe system. For example, Fig. 2c and Fig. 2d are two typical Wadabasins formed by the 6-sided basin cell and the 8-sided basin

Fig. 2. Schematic drawings of the basin cells: (a) A 6-sided basin cell generated by a period-3 orbit; (b) An 8-sided basin cell generated by a period-4 orbit; (c) A 6-sidedbasin cell (as shown in light gray) is topologically equivalent to the cell (a). It is verified that the system (1) (μ = 3.0, J = 0.6) has Wada basin. The red area is the basinof a positive period-1 attractor, the black area is the basin of another negative period-1 attractor and the white area is the basin of the attractor at infinity. The hyperbolicsaddles are shown in green ‘∗’. Their boundaries consist of the stable and unstable manifolds [marked by yellow curves]. (d) An 8-sided basin cell is topologically equivalentto the cell (b). It is verified that the system (1) (μ = 2.8, J = 0.6) has Wada basin. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this Letter.)

1272 Y. Zhang / Physics Letters A 377 (2013) 1269–1273

Fig. 3. The negative SNAs and their basin section plot (θ = 0.5). (a) Projection of the negative SNAs (blue) for μ = 2.88, υ = 0.01985, J = 0.6 in the (θ, x) plane. Box (1)denotes the plot of phase sensitivity function ΓN vs N . Box (2) denotes the largest Lyapunov exponent diagram and the largest Lyapunov exponent of SNAs is approximately−0.006. (b) A 6-sided basin cell (gray) and three channels determine the topological structure of the basins. It is verified that the system (2) has Wada basin section (DB:The red area is the basin of a positive SNA, the blue area is the basin of another negative SNA and the white area is the basin of the attractor at infinity). (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this Letter.)

cell in the unforced map (1). With quasiperiodic driving the basinboundaries are two-dimensional Wada surfaces. Since it is difficultto observe the Wada basin structures in three dimensions, we willfix the variable θ and study the reduced Wada basin structure inthe following sections.

For SNAs in I regimes (Fig. 1 in the μ–υ parameter plane),we will describe the gradual fractalization route where a torusgets increasingly wrinkled and then transits to an SNA as wechange the system parameter. In this route, a period-2n torus be-comes wrinkled and then the wrinkled attractor gradually losesits smoothness and forms a 2n-band SNA as the system param-eter is varied. Such a phenomenon is a gradual fractalization ofthe doubled torus. For example (a point a marked by red ‘◦’ inFig. 3a), the nature of the SNA is shown in Fig. 3a. A negative SNAis shown in blue for μ = 2.88, J = 0.6, υ = 0.01985 (in the θ–xplane). Box (1) denotes the plot of phase sensitivity function ΓN

vs N and the ΓN grows unboundedly with the power-law relationwith γ ≈ 0.89. The largest Lyapunov exponent of the attractor isapproximately −0.006 [see the Box (2)]. We verify that the attrac-tor is indeed an SNA. We can observe that the coexisting attractor(the positive attractor) is also an SNA. We also investigate the basinstructure by a cross-section with θ = 0.5 (see Fig. 3b, the sim-ilarly topological property for other θ = constant). The red areais the basin of a positive SNA, the blue area is the basin of an-other negative SNA and the white area is the basin of the attractorat infinity. It is easy to visualize from an intuitive point of viewthat it satisfies the Wada property. Fig. 3c shows the section withthe plane θ = 0.5 of invariant 2D manifolds associated to invariantcurves. A saddle-hyperbolic period-3 orbit on the basin boundaryis marked by yellow ‘◦’. The basin cell (the gray region) has beenchosen so that the sum of the lengths of the stable and unstablesegments of the six manifolds is minimal. One branch of the un-stable manifold (marked by green curves) of each of the points ofthe saddles intersects all three basins. A 6-sided basin cell (gray)and three channels determine the topological structure of the basinsection. We have found that the boundaries of all basins coincide.

For SNAs in II regimes (Fig. 1 in the μ–υ parameter plane), wewill also find the gradual fractalization route (the route C and theroute D) and the Wada basins. Their Wada basins structure [theircentral bodies are some 8-sided basin cells] are different fromthose in I regimes. Here, we are interested in the Heagy–Hammelroute (route B) in which a period-2n torus gets wrinkled and be-comes an SNA. That is, the doubling bifurcation is truncated dueto the collision of the doubled torus with its unstable parent. For

example (a point b marked by red ‘◦’ in Fig. 1), the nature of theSNA and its transition are shown in Fig. 4a. Two coexisting SNAsare shown in red and blue for μ = 2.82, J = 0.6, υ = 0.0231 (inthe θ–x plane). It is shown that the transition from quasiperiodicattractors [e.g. Box (1) for υ = 0.0227] to chaotic attractors [e.g.Box (2) for υ = 0.0235] can occur through Heagy–Hammel mecha-nism. In the present case, the truncation of the torus-doubling be-gins when the four strands of the 4T attractor become extremelywrinkled. At this transition, the strands are seen to come closer tothe unstable period-2T orbit, lose their continuity when the new4T attractor collide with the unstable parent 2T attractor. A mag-nified part of the lower SNA and unstable parent torus are givenin Box (4). Box (3) denotes the plot of phase sensitivity functionΓN vs N and the ΓN grows unboundedly with the power-law re-lation with γ ≈ 0.78. The largest Lyapunov exponents of the twoattractors are approximately −0.002. We verify that the two co-existing attractors are indeed SNAs. We also investigate the basinstructure by a cross-section with θ = 0.5 (see Fig. 4b, the similarlytopological property for other θ = constant). The red area is thebasin of a positive SNA, the blue area is the basin of another nega-tive SNA and the white area is the basin of the attractor at infinity.It is easy to visualize from an intuitive point of view that it satis-fies the Wada property. The 8-sided basin cell can be observed andwe can see the basin consist of the central body and four channels.These channels are infinitely long and wind in a very complicatedpattern without crossing each other.

We remark that these SNAs with Wada basins are indeed abun-dant (Fig. 1) in the Duffing map (2). We have done much morenumerical experiments and we have shown that it could be possi-ble to exist in other parameter values.

4. Conclusion

In the Letter, we study SNAs’ Wada basins in quasiperiodicallyforced dissipative systems using the Duffing map as a characteristicexample. Numerical evidence shows some SNAs have Wada basinsfor a set of parameters of positive measure. We distinguish twotypes of basins structure and identify two routes to the creationof SNAs. The routes include transitions to chaos via SNAs from aperiod-doubled torus. We find that the two distinct routes (thefractalization route and the Heagy–Hammel route) for the trun-cation of the torus-doubling bifurcation and the creation of SNAsoccur in this model (2). We have verified that these SNAs’ basinsare full Wada. To distinguish between the basins structure, we use

Y. Zhang / Physics Letters A 377 (2013) 1269–1273 1273

Fig. 4. Coexisting SNAs and their basin section plot (θ = 0.5). (a) Projection of the two coexisting SNAs (red and blue) for μ = 2.82, υ = 0.0231, J = 0.6 in the (θ, x) planeindicating the transition from quasiperiodic attractors [e.g. Box (1)] to chaotic attractors [e.g. Box (2)] through Heagy–Hammel mechanism. Box (3) denotes the plot of phasesensitivity function ΓN vs N . The largest Lyapunov exponent of SNAs is approximately −0.002. A magnified part of the lower SNA and unstable parent torus (black line) aregiven in Box (4); (b) An 8-sided basin cell (gray) and four channels determine the topological structure of the basins. It is verified that the system (2) has Wada basin section(DB). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

the basin cells to describe them. The Wada property reflects acomplete robust structure of SNAs (that is, those structures thatpersist under small perturbations in the system). It suggests thatSNAs cannot be predicted reliably for the specific initial conditionson the basin boundaries. Although the SNAs with Wada basins canbe observed in such a Duffing system, we suggest that it will prob-ably turn out to be widespread in a variety of systems.

Acknowledgements

The authors thank the anonymous reviewers for their helpfulcomments and suggestions which led to an improvement of thisLetter. Some computations have been made using the softwareDYNAMICS [33]. This work was also partially supported by theNSFC (No. 11002052).

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