A lot of strange attractors: Chaotic or not?R. Badard Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 023127 (2008); doi: 10.1063/1.2937016 View online: http://dx.doi.org/10.1063/1.2937016 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/18/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency Chaos 23, 023110 (2013); 10.1063/1.4804181 Detecting variation in chaotic attractors Chaos 21, 023128 (2011); 10.1063/1.3602221 Periodic solution and chaotic strange attractor for shunting inhibitory cellular neural networks with impulses Chaos 16, 033116 (2006); 10.1063/1.2225418 Chaotic pendulum: The complete attractor Am. J. Phys. 71, 250 (2003); 10.1119/1.1526465 Chaotic behavior and strange attractor in time-dependent solutions of the magnetohydrodynamic equationsfor the Faraday disc Phys. Plasmas 4, 3173 (1997); 10.1063/1.872554

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A lot of strange attractors: Chaotic or not?R. Badarda�

Institut National des Sciences Appliquées de Lyon, Dépt. Informatique, Bât. B. Pascal, INSA-Lyon,69621 Villeurbanne Cedex, France

�Received 5 December 2007; accepted 6 May 2008; published online 27 June 2008�

Iterations on R given by quasiperiodic displacement are closely linked with the quasiperiodicforcing of an oscillator. We begin by recalling how these problems are related. It enables us topredict the possibility of appearance of strange nonchaotic attractors �SNAs� for simple increasingmaps of the real line with quasiperiodic displacement. Chaos is not possible in this case �Lyapounovexponents cannot be positive�. Studying this model of iterations on R for larger variations, beyondcritical values where it is no longer invertible, we can get chaotic motions. In this situation we canget a lot of strange attractors because we are able to smoothly adjust the value of the Lyapounovexponent. The SNAs obtained can be viewed as the result of pasting pieces of trajectories, some ofwhich having positive local Lyapounov exponents and others having negative ones. This leads us tothink that the distinction between these SNAs and chaotic attractors is rather weak. © 2008 Ameri-can Institute of Physics. �DOI: 10.1063/1.2937016�

The quasiperiodic forcing of an oscillator is known toproduce strange nonchaotic attractors (SNAs). These at-tractors resemble chaotic ones but they do not have sen-sitivity to initial conditions, at least when characterizedby positive Lyapounov exponents. A typical route fortheir appearance is via an unsmooth saddle node bifurca-tion. Using a simple iteration formula it is possible toconstruct a system that stays for rather long time periodsin domains where it has a simple mode locked dynamicsand others where chaos is developing through a classicalperiod doubling sequence. Clearly, the nontrivial finitetime Lyapounov exponent can stay positive over longtime periods. The global attractor has a nontrivialLyapounov exponent that can be smoothly adjusted by aparameter. It can be made either positive or negative.Hence, continuously varying a parameter, it is possible toget a continuum of strange attractors some of which arechaotic (the exponent is positive) and others are not. Inthis situation the exponent does not seem to give a cleardistinction between these attractors. This leads to the fol-lowing question: Is sensitivity to initial conditions mea-sured from Lyapounov exponents a pertinent way tocharacterize chaos, and what kinds of properties are re-ally discriminating?

I. INTRODUCTION

Iterations on R given by a quasiperiodic displacementhave been soundly analyzed by Kwapisz.1 The starting pointis the study of iterations given by ti+1= ti+��ti�, where � iscontinuous and almost periodic. For instance, we will con-sider iterations where

��t� = K +H

�2��sin�2��t� +

Q

�2��sin�2��t + �� ,

where we generally assume � /� to be irrational.The possibility of appearance of different kinds of

strange attractors, chaotic or not, seems clear in this context.Many papers about SNAs have been published since the firstproof of their existence in Grebogi et al.2 The precedingmodel is adapted to show the difficulty we encounter whentrying to classify an attractor as chaotic or not. What prop-erties are typical of chaos?

We first recall the relationship linking this problem to thestudy of the behavior of an oscillator with a quasiperiodicforcing. It is well known that the quasiperiodic forcing of anoscillator is suitable to analyze the birth of SNAs. Thus, wepredict that the preceding iterations can produce strange at-tractors, particularly when the map is increasing. In this casechaos is not possible, due to the lack of dimensions. A typicalroute leading to the creation of SNAs lies in an unsmoothcollision between an attractor and its dual repeller. This sce-nario has been much studied; for instance, one can see Pik-ovsky, Feudel, and Kurths.3,4

It leads us to the question, What happens when the mapis no longer increasing? By comparison with the classic non-increasing sinus map, it is certainly possible to have chaoticbehaviors. The rotation number �or mean displacement� is nolonger necessarily unique; it can take different values de-pending on the initial conditions. In this case chaos mayappear by a classic period doubling sequence. Similar stud-ies, for a quasiperiodically driven logistic map, have beendone by Negi et al.5

As we will see, in the quasiperiodic case, we can have ananalogous torus doubling sequence. We will show, in a suit-able example, that the state space can be partitioned in dif-ferent regions. In some regions the dynamics is clearly cha-otic and in other ones it is mode locked. In transition regionswe clearly see a torus doubling sequence leading to chaosa�Electronic mail: robert.badard@insa-lyon.fr.

CHAOS 18, 023127 �2008�

1054-1500/2008/18�2�/023127/8/$23.00 © 2008 American Institute of Physics18, 023127-1

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and then an inverse torus merging sequence leading back to amode locked situation. A trajectory going through such dif-ferent regions, some being expanding and other being con-tracting can be “more or less chaotic.” We are able, bychanging a parameter value, to adjust the relative weight ofthese regions, and so to adjust the value of a characteristicLyapounov exponent.

Generally Lyapounov exponents are used to classifystrange attractors as chaotic or not. A positive exponentmeans an exponential divergence between solutions startingfrom close initial conditions and this property is often asso-ciated with the sensitivity to initial conditions �SIC�. How-ever, there are other definitions of SIC that do not take ac-count of the speed of divergence; in particular, the mostpopular one was proposed by Devanney.6 Glendinning et al.7

studied how chaotic SNAs are and showed that generallythey verify SIC in the sense of Devanney �SICd�. Demir andKoçak8 compared SIC in these preceding senses, and theyexhibited examples showing that there are situations whereSICd holds but with a negative exponent, and on the con-trary, there are cases with a positive exponent without SICd.

Thus, setting apart the invertible case, it is rather diffi-cult to decide whether a strange attractor is chaotic or not. Aswe will see there is a smooth and fuzzy transition from SNAsto chaotic attractors.

Our approach is of the same kind as those used by Shuaiand Wong:9 We can act on the time a trajectory spends indifferent regions where contraction or expansion takes place.

II. PROBLEMS RELATED WITH THE QUASIPERIODICFORCING OF AN OSCILLATOR

In a preceding paper10 we explored how the study of thequasiperiodic forcing of an oscillator is linked to the study ofsimple increasing iterations on R. The motivation was toshow how simple iterations on R can produce complex,strange, but not chaotic attractors, in brief SNAs. The corre-spondence was obtained by taking a Poincaré section, reduc-ing the state space dimension by 1; then “unwrapping thetrajectories from a torus” gave us a family of simple itera-tions formulas on R.

A. The differential model

The model we started from was defined by the followingkind of differential equations on Rn:

�x

�t= 1 + aF�x,y,z�;

�y

�t= �; �1�

�z

�t= � .

We assumed that F was 1-periodic in x, y, z; i.e., F�x ,y ,z�=F�x+n ,y+m ,z+q�, where n, m, q�Z. Taking the trajecto-ries modulo 1 gives us a forced oscillator on the 3-torus. Thesystem �1� in R3 is easier to deal with; it can be viewed asequations on a lift of the torus.

The parameter a is the strength of the nonlinearity. Wesuppose that �F� is bounded by 1 and generally that �a� is lessthan 1 �in order to have x always increasing with time�.

In order to have uniqueness of the solutions, we gener-ally suppose that F satisfies the Lipchitz condition. Beinginterested by quasiperiodic forcing, we suppose that � ,�have no Diophantine relation �no resonance�. More preciselythe equation p�+q�=0, where p and q are taken in Z, hasonly the null solution: p=q=0.

B. H-maps: Unwrapping the trajectories

It is common to transform a differential problem to adiscrete iteration one by considering a Poincaré section. Itenables us to reduce by one the state dimension. The classicapproach considers the instants at which a variable, say, y,takes integer values. This leads to a classic model whoseparadigm is the forced sinus map.

Let us consider the instants at which the variable x takesinteger values. For any trajectory starting from x�0�=k, y�0�,z�0�, we consider the first point where x�t�=k+1, it corre-sponds to C3 to the next passage of the trajectory in the2-torus defined by x=0.

It defines a map H : R2→R2, at least when x is strictlyincreasing with t and increases to infinity. Taking the valuesmodulo 1, it also defines a map H�1� : C2→C2. In the case ofmodel �1� these maps are well defined when we assume�a � �1.

From the evolution y=�t+y�0� and z=�t+z�0�, the mapH is completely described by a single function ��y ,z�, whichgives the time necessary for x to increase by 1, and thus Hcan be written as

�y�

z�� = �H1�y,z�

H2�y,z� � = �y

z� + ��y,z���

�� , �2�

where the prime denotes variables at the next step.Proposition 2-1 (Ref. 10): Let us suppose that in system

�1� we take �a � �1, we can then assert that the function��� , � � is continuous, periodic in each coordinate and strictlypositive and bounded. More precisely, for every y and z wehave 0�m���y ,z��M. The corresponding map H is a ho-meomorphism of R2 and H�1� is a homeomorphism of C2.

However, we will speak of an H map for a map like Eq.�2� when � is positive continuous and periodic but not nec-essarily bijective.

Starting from y0, z0 at t=0, the state will reach the pointy1=y0+�t1, z1=z0+�t1 at t1=0+��y0 ,z0�, then y2=y1

+���y1 ,z1�=y0+�t2, z2=z1+���y1 ,z1�=z2+�t2 at t2= t1

+��y1 ,z1�. Thus, the instants ti can be computed as ti+1= ti

+��y0+�ti ,z0+�ti�. When an H-map is bijective then all themaps �depending on y0 and z0� t+��y0+�t ,z0+�t� areinvertible.

From the iteration formulas

ti+1 = ti + ��y0 + �ti,z0 + �ti� = ti + ����ti� , �3�

we can draw back the trajectories on the torus C2, that is, theH�1� iterations, by plotting the points:

023127-2 R. Badard Chaos 18, 023127 �2008�

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�yi

zi� = �y0 + �ti�1�

z0 + �ti�1� � .

This drawing is an approximation of the attractor:

�m0cl���yi

zi�i, i � m ¯ � � ,

where “cl” stands for the closure operator. Another way torepresent a trace of the attractor consists in drawing couplesof time interval �Di ,Di+1 , Di= ti− ti−1 and to look for theformation of fractal sets. These points stay in the boundedregion �m ,M�2, so the limit set

�m0cl��� Di

Di + 1�i,i � m . . . � �

is necessarily compact. Such a set can have a more or lesscomplex structure.

C. Winding vectors and mean displacements

The mean displacement for iteration formula �3� is sim-ply defined as �=limn→�tn− t0� /n, when the limit exists. Aslong as the iteration formula defines a nondecreasing mapand � is positive, the limit exists and does not depend on t0,y0, or z0. In more general situations we can use some con-cepts of the mean displacement set. We use either the point-wise set, i.e., �P= s�, where s is the mean displacement fordifferent initial conditions, or the concept of set �M, intro-duced by Misiurewicz and Ziemian.11 The set �M is used forits interesting topological properties. It consists in taking thecollection of all the limits, i.e., limn→�tn− t0� /n, when t0 isvaried and may be taken dependent on n. More formally, itcan be written as �M =�m0cl��nm�tn− t0� /n , t0�R�.From this definition, �M is a compact interval of R.

It is clear that the concepts of rotation vectors for anH-map can be defined in the same way, working with thevariables y and z instead of t. From the equations yi=y0

+�ti, zi=z0+�ti, and the nondependence on y0, z0, it is clearthat we have

�HM = ��

���M .

Let us now recall a result about the existence and the non-dependence on initial conditions of the winding quantities.

Proposition 2-2: We consider an H-map and assume that� /� is irrational, the winding sets �H

M and �HP are equal and

not dependent on initial conditions. �HP, is just

��

���P,

where �P is the collection of mean displacements we wouldget in studying the iteration ti+1= ti+���ti ,�ti� for differentinitial conditions on t. In fact, all the formulas ti+1= ti+��y0

+�ti ,z0+�ti� give the same �M =�P. As long as we haveinvertibility, �P is reduced to a single point.

Proof: Let y0, z0 be some initial conditions for system�2�, so we get the iteration formula ti+1= ti+��y0+�ti ,z0

+�ti�= fy0,z0�ti�. We know that ��y0+�t ,z0+�t� is quasi-periodic and t+��y0+�t ,z0+�t� is Lipchitz with a constant

K �it has a bounded derivative�. Now, starting from y0=z0

=0, we get ti+1= f0,0�ti�. From the quasiperiodicity, wecan find for every 0 an u such that�����u+ t� ,��u+ t��−��y0+�t ,z0+�t� � � for all t0. Fromuniform Lipchitz continuity, we deduce that for any 0,there is u such that supt0�����u+ t� ,��u+ t��−��y0+�t ,z0+�t� � � , and the composition satisfiessupt0��fy0,z0

2�t�− t�− �f0,02�u+ t�− �u+ t�� � � �1+K�.

Continuing the iteration n times, weget supt0��fy0,z0

n�t�− t��f0,0n�u+ t�− �u+ t�� � � �1+K��n−1�.

Thus, it is possible for any �0 to choose such that �1+K��n−1���, and thus to have supt0��fy0,z0

n�t�− t�− �f0,0

n�u+ t�− �u+ t�� � ��. This proves that it is possible forall initial conditions y0 and z0 to find u such that�1 /n��f0,0

n�u+ t�− �u+ t�� is as close as we want of�1 /n��fy0,z0

n�t�− t�. This clearly proves that the winding set�H

M is the same as the one we would get when taking initialconditions of the form y0=�u, z0=�u. Finally, ��� , � � beingstrictly positive, we can use the result �Theorem 2� fromKwapisz1 to conclude that �H

M =�HP. Thus, �H

M is merely thecollection of vectors

����

���, � � �P� ,

where �P is the set of the mean displacements we get fordifferent initial conditions t0 in the formula ti+1= ti

+���ti ,�ti�.In fact, �M is the same for the whole family of formulas

we obtained when we change the initial conditions on y0 andz0. This is also true for �P when � is positive, because in thiscase �P=�M, and so for any r in �M, there is t0 �dependingon r� for which �tn− t0� /n→r when n goes to infinity.

III. A SIMPLE MODEL GIVING A LOTOF STRANGE ATTRACTORS

Now we turn our attention to particular maps of thetorus. We will consider H-maps �Eqs. �2�� when ��y ,z� iscontinuous, periodic in y and z, and strictly positive. In orderto do numerical experimentations, we take ��y ,z� as trigono-metric polynomials. For instance, we will use the simplifiedformula

��y,z� = K +H

�2��sin�2�y� +

Q

�2��sin�2�z� , �4�

in which we suppose that K0 and K− �H � / �2��− �Q � / �2��0, in order to insure that � is positive.

In Ref. 10, we principally studied the case where invert-ibility is assumed and showed that SNAs can appear at anunsmooth saddle node bifurcation. Here we consider nonin-vertible situations. Thus, generally, we will not have a welldefined value for the mean displacement and we will have touse, for instance, �M. In fact, chaotic behavior �more pre-cisely, annular chaos� becomes possible.

023127-3 Strange attractors: Chaotic or not? Chaos 18, 023127 �2008�

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A. Computation of a nontrivial Lyapounov exponent

For such a model it is easy to characterize the SIC by thecomputation of Lyapounov exponents. From yi=y0+�ti andzi=z0+�ti, it is clear that in order to characterize the sensi-tivity to initial conditions �nontrivial Lyapounov exponent�we can merely compute the sensitivity of ti relative to t0.

From

� �tn

�t0� = � �tn

�tn − 1�� �tn−1

�tn − 2�¯ � �t1

�t0� ,

we can take the harmonic mean

limn→�� �tn

�tn−1�� �tn−1

�tn−2�¯ � �t1

�t0��1/n

.

Hence, taking the logarithm, we get as nontrivial Lyapounovexponent the quantity

� = limn→

1

n�i=0

n−1

log10��1 + ���

�y�yi,zi� + �

��

�z�yi,zi��� ,

where yi=y0+�ti and zi=z0+�ti.In particular, for the simplified example �4�, we easily

find �=limn→�1 /n��i=0n−1log10��1+�H cos�2��y0+�ti��

+�Q cos�2��z0+�ti�� � �, where ti are elements of the se-quence t0 , t1 , . . . , tn. To characterize SNAs, in particular theroute leading to their birth, Prasad et al.12–14 suggest to lookat the distribution of local Lyapounov exponents. They aremerely computed from the preceding formula by taking theaverage on finite time sequences.

B. A kind of torus doubling sequence

From the � expression �4�, we get a family of iterationformulas �depending on the initial conditions y0 and z0�:

ti+1 = ti + K +H

�2��sin�2��y0 + �ti��

+Q

�2��sin�2��z0 + �ti�� . �5�

In order to have � positive, we assume K− �H � / �2��− �Q � / �2��0.

When the derivative can take negative values, i.e.,��H � + ��Q � 1, then this map is no longer increasing. How-ever, there can be long time periods where it is still increas-ing and others where it is not. In the first case the localLyapounov exponent is negative or null, but in the secondcase it can be positive. The mixing of these cases can cer-tainly give SNAs, in the sense that the global Lyapounovexponent can be maintained negative.

More particularly, taking y0=z0=0, �=1, and �=1+ ��and � have to be incommensurate, so we assume that is asmall irrational� we would have a modulation of the nonlin-ear term. Taking Q=H, the iteration can be rewritten ti+1

= ti+K+ �H /��cos�� ti�sin�2��1+ 12 �ti�. For a small

value, the factor �H /��cos�� ti� plays the role of a slowmodulation. It stays for long time periods close to �H /�and in other ones close to 0. When this factor is close to 0 thedynamics cannot be chaotic, the local Lyapounov exponent

remains nonpositive and we have a single local windingvalue. As the modulation factor increases in absolute valuewe get a kind of sinus map with a larger and larger nonlinearterm. When this nonlinear term is sufficiently large, say,�H � �1 /2� / �1+ �, invertibility is lost. When �H� is still in-creased, chaos appears by the classic period doubling se-quence. When this occurs the local Lyapounov exponent be-comes positive in some regions, but the global one, whichcan be viewed as an average on different regions, remainsnegative. Thus, as we take H larger and larger, the globalLyapounov exponent increases. In the same way, in regionswhere annular chaos is developing the mean displacement isdependent on initial conditions.

The preceding discussion relies on considering rationalapproximations of � /� and the irrationality is taken into ac-count as a slow phase shift. In the preceding model we con-sider �=1+ , and take =0, and discuss about the solutionswhen the phase is varied. However, while this discussionseems correct, we are in a rather different situation. For in-stance, when we said that chaos appears by a classic perioddoubling sequence, this is not really the case. The quasiperi-odicity disables the possibility to have cycles and thus toobserve any period doubling bifurcation. Now, in this con-text, cycles are replaced by links made from a one-dimensional torus. The period doubling sequence merely be-comes a stable torus, corresponding to a mode lockedsituation, which in certain regions loses its stability and bi-furcates, giving two pieces of torus. In the preceding ex-ample, the doubling sequence leading to chaos is followedby an inverse merging sequence bringing back the system ina region where it is mode locked. This merely comes fromthe particular model we have taken. When the phase betweeny and z is close to 0�1�, we have the most important nonlin-ear term and the more chance to get chaotic trajectories; onthe contrary, when the phase is close to 1

2 �1�, the nonlinearterm vanishes and chaos is impossible. Thus, when chaos�annular chaos� is developing in a region it is decaying inanother one.

When chaotic regions appear the mean displacement isnot necessarily unique. As soon as the region with annularchaos �we can speak of local chaos in this case� takes apositive measure the mean displacement can be a full inter-val. Thus, this kind of SNA is rather troublesome; it verifiesa property typical of chaos but there is no SIC in the sense ofpositive Lyapounov exponent; however, we will see thatSICd certainly holds.

In order to test the existence of strange attractors Pik-ovsky and Feudel4 proposed either to measure the phase sen-sitivity �PS� or to show that for every rational approximationof � /� there are bifurcations when we vary the initial phase.In fact, these two approaches have similar motivations; i.e.,showing the fractal nature of the attractor.

In our case the phase sensitivity can be tested by observ-ing the distance between trajectories starting, respectively,from �y0 ,z0� and �y0� ,z0��, with y0�=y0−e� and z0�=z0+e�.The direction of the initial error is taken orthogonal to thedirection of the iterations. Due to the quasiperiodicity it isunlikely that the error vanishes. But as long as the maximumof this distance stays in the order of the initial error

023127-4 R. Badard Chaos 18, 023127 �2008�

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erI= �e � ��2+�2�1/2, there is no strangeness. On the contrary,when the distance increases and reaches values much largerthan erI �even if we decrease e the trajectories tend to sepa-rate with a same error�, then there is certainly a strange at-tractor. Pikovsky4 studied the speed of separation with thenumber n of iterates and roughly found nm, with m�0.97.

Formally, we could formulate the PS �imitating the defi-nition of the SICd� as: there is an h0 such that for every�y0 ,z0� and e0, there is n0 such that the nth iterates�yn ,zn�, starting from y0 ,z0, and �yn� ,zn�� starting from y0�=y0−e� and z0�=z0+e�, are such that d��yn ,zn� , �yn� ,zn���h.

With such a definition, it is obvious that PS impliesSICd.

Experimentally, we test the PS by evaluating the maxi-mum of the error between trajectories from �y0 ,z0� and �y0

−e� ,z0+e�� and show that this quantity is fairly indepen-dent of e �but also of y0 ,z0�.

Let us clarify these points by numerical computations.

C. Numerical experiments

In order to illustrate this phenomenon we makesimulations with the preceding model. Taking arbitrarilyclose to 0, we feel that chaos has more chances to appear inregions where the nonlinear term is large. In fact, for =0,we have to study the iterations: ti+1= ti+K+ �H /��cos����sin�2�ti+��� for different � values. In par-ticular, for �=k, we have the situation with the larger non-linear term. We show in Fig. 1�a� the bifurcation diagramwhen K=0.35 and H is varied from 0.69 to 0.72 and inparallel, in Fig. 1�b�, we draw the corresponding Lyapounovexponent. We find, as is well known, the appearance of astable three-cycle �around H=0.359�, which becomes un-stable and gives birth to a stable six-cycle �around H=0.639 24�, then this six-cycle becomes unstable and givesbirth to a 12-cycle �around H=0.691 45�, and so on. As wesee, in these figures, chaos appears when H is close to 0.703.

We use the iteration formula �5� and take �=1, �=1+�5 /1000, K=0.35, and H=Q. In order to have � positive,we still assume K− �H � /�0. We draw the attractor bywrapping it back to the torus in Fig. 2 and by drawing thesequence of time intervals �Di ,Di+1 in Fig. 3. Figures 2 and3 are drawn, respectively, in the same conditions.

We increase the parameter value H and draw the corre-sponding attractors. As we predicted, we have a region nearthe lines z= �� /��y+z0, when z0 is close to −1,0 ,1, whereannular chaos is developing �we clearly see the torus dou-bling mechanism in Figs. 2�a� and 3�a��. In the contrary,

when z0 is close to − 12 , 1

2, we are in a region where we have

partial mode locking and the attractor is locally a smoothcurve. When we increase H the attractor becomes “more andmore chaotic,” as we see the Lyapounov exponent increaseswith H. Taking H=0.73 �Figs. 2�a� and 3�a��, we find asexponent �=−0.0826; however, it is obvious that in a regionclose to the diagonal the local Lyapounov exponents arepositive. Taking H=0.9 �Figs. 2�b� and 3�b��, we find �=0.036, it is positive. Clearly, as Q is increased from 0.73 to0.9 the Lyapounov exponent increases continuously fromnegative to positive values, however the attractor still have

regions where it seems mode locked �strong contractiontakes place� and regions where annular chaos is developingalong a kind of torus doubling sequence. The exponent be-comes non-negative when H is close to 0.874.

As we said in Sec. III B, we can study the PS by drawingthe maximum of the distance between trajectories startingfrom close initial conditions. We take e=10−9 and considersequences of length 5�105. We draw in Figs. 4�a� and 4�c�the maximum of the distance and the corresponding expo-nent when H=Q is varied from 0.66 to 0.76. Figures 4�b�and 4�d� are merely a zoom on the preceding graphs when His in �0.67,0.74�.

These figures are rather independent from the startingpoint �y0 ,z0� and the initial error e. They show that PS andthus the SICd are satisfied as soon as H=Q0.678. It cer-

FIG. 1. Birth of chaos after the classic period doubling cascade. We drawthe bifurcation diagram �a� and the corresponding evolution of theLyapounov exponent �b� for the iteration formula �5� when �=�=1, y0

=z0=0, K=0.35, and H=Q is varied from 0.69 to 0.72. A stable lengththree-cycle appears at H=0.359; then it becomes unstable and gives birth toa stable length six-cycle at H=0.639 24, then another doubling appears atH=0.691 45, . . ..

023127-5 Strange attractors: Chaotic or not? Chaos 18, 023127 �2008�

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tainly signifies that strange attractors are present before theappearance of local chaos by the torus doubling sequence atH�0.704.

We clearly see in Fig. 4�a� that when H0.7419, thedistances explode. This phenomenon appears at the last bandmerging of the local chaotic attractor. From this point theerror between the trajectories can make jumps resembling arandom walk. We visualize the errors along the two trajecto-ries in Figs. 5�a� and 5�b�. The jumps happen when the tra-jectories enter the region of local one band chaos; then in theregion of mode locked regime we see the error recalling themerging phase �the two trajectories can be on different links�and they merge in the center of the mode locked region whenit remains only one torus.

This example shows that the PS and so the SICd are notpertinent to decide when strange attractors are chaotic or not.The Lyapounov exponent is no longer pertinent with respectto this goal. Perhaps the birth of local chaos, in particular thepresence of a dense set of unstable torus, could be pertinent.However. in a more general situation this could be difficult todetect; perhaps the analyze of the distribution of local expo-nents, as proposed by Prasad,12,13 could give an efficient tool.

IV. CONCLUSION

We have shown that simple iterations on R, constructedfrom a quasiperiodic displacement, are linked to the quasip-eriodic forcing of an oscillator. We know that SNAs seem tobe reminiscent in situations where an oscillator is quasiperi-odically forced. It suggests to us that iterations on R can

FIG. 2. From SNAs to annular chaos. We use the iteration map �5� with�=1, �=1+�5 /1000, y0=z0=0, and K=0.35, H=Q. We draw the attractoron a 2-torus. For H=0.73 �a�, we find �=−0.0826; at H=0.9 �b�, we find�= +0.036.

FIG. 3. From SNAs to annular chaos. The conditions are those of Fig. 2. Wedraw the sequence of time intervals between iteration times. It is merely theset of points �ti+1− ti , ti+2− ti+1 . For H=0.73 �a�, �=−0.0826; at H=0.9 �b�,�= +0.036.

023127-6 R. Badard Chaos 18, 023127 �2008�

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exhibit SNAs. As long as the iteration formula is invertiblewe cannot have chaotic motion. When this formula is notinvertible then chaos becomes possible.

From a very simple model, we have shown how we canget a lot of strange attractors, chaotic or not. It simply relies

on a kind of pasting, end to end, pieces of trajectories, someof which having positive local Lyapounov exponent valuesand the others having negative ones.

Continuously varying a parameter enables us to adjustthe value of the Lyapounov exponent from negative to posi-tive values. Usually the attractor is said to be chaotic whenthere is SIC �in the sense of positive Lyapounov exponents�.At a first glance it does not seem to have a characteristicchange in the nature of the attractor as the sign of theLyapounov exponent changes. The presence of such attrac-tors is robust in the sense that they exist for a full set ofparameter values. It leads us to think that neither a distinc-tion from the sign of Lyapounov exponents nor from the PSis really pertinent.

ACKNOWLEDGMENTS

We would like to thank an anonymous referee for hisconstructive comments. They enable us to improve our argu-mentation. We would like to very thank Delphine Badard forher help in reading this article.

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FIG. 4. Phase sensitivity. In �a� and �b� we draw the evolution of the max of the distance between trajectories starting, respectively, from �y0 ,z0� and�y0� ,z0��= �y0−e� ,z0+e��. We still use formula �5� with �=1, �=1+�5 /1000, K=0.35, H=Q, e=10−9, the length of the sequence is 5�105. In �c� and �d� wedraw the corresponding evolution of the Lyapounov exponent. These graphics remain similar when we change the initial condition y0, z0, and the error e. Welocate the region of PS �H0.679�, before the appearance of local chaos at H�0.703. We clearly see an explosion after the last crisis, when H0.7419.

FIG. 5. Phase sensitivity: a random walk. The conditions are those of Fig. 4,with H=Q=0.742, and y0=0, z0=0.5 �in order to start in the mode lockedregion�. We draw the evolution of the differences between the commutationtimes ti− ti�. In the region of local chaos the error makes steps like a randomwalk. In �a� we draw a long time sequence and in �b� we only draw the veryfirst steps.

023127-7 Strange attractors: Chaotic or not? Chaos 18, 023127 �2008�

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023127-8 R. Badard Chaos 18, 023127 �2008�

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