Intermittency Route to Strange Nonchaotic Attractors

VOLUME 79, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 24 NOVEMBER 1997

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Intermittency Route to Strange Nonchaotic Attractors

Awadhesh Prasad, Vishal Mehra, and Ramakrishna RamaswamySchool of Physical Sciences, Jawaharlal Nehru University,

New Delhi 110067, India(Received 25 April 1997)

Strange nonchaotic attractors (SNA) arise in quasiperiodically driven systems in the neighborof a saddle-node bifurcation whereby a strange attractor is replaced by a periodic (torus) attraThis transition is accompanied by Type-I intermittency. The largest nontrivial Lyapunov exponL is a good order parameter for this route from chaos to SNA to periodic motion: the signaturdistinctive and unlike that for other routes to SNA. In particular,L changes sharply at the SNAto torus transition, as does the distribution of finite-time orN-step Lyapunov exponents,PsLN d.[S0031-9007(97)04588-2]

PACS numbers: 05.45.+b

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Strange nonchaotic attractors (SNA’s), which are commonly found in quasiperiodically forced systems, are gemetrically strange—they are properly described byfractal dimension—but the largest nontrivial LyapunoexponentL is negative, implying nonchaotic dynamics.

Since they were first described [1], and shown to bgeneric in quasiperiodically driven nonlinear systems [2a number of characteristics of SNA’s have been studieincluding the important question of how they are createThree main mechanisms or scenarios for the creationSNA’s have been advanced. Heagy and Hammelidentified the birth of a SNA with the collision betweena period-doubled torus and its unstable parent. Thmechanism is analogous to the attractor-merging crisis toccurs in chaotic systems [4]. Kaneko [5,6] has describthe “fractalization” of a torus, namely, the increasinwrinkling of tori which leads to the appearance of a SNwithout any interaction with a nearby unstable periodorbit. This route to SNA (and eventually to chaos) haalso been observed in several systems. Yal¸cinkaya andLai [7] have recently shown a third route to SNA’s viaa blowout bifurcation [8], namely, through the loss otransverse stability of a torus.

We describe a new mechanism for the creationSNA’s. As a function of driving parameter, a strangattractor disappears and is eventually replaced by a ofrequency torus through an analog of the saddle-nodefurcation. In the vicinity of this crisislike phenomenon[4] the attractor is strange and nonchaotic. We show ththe dynamics at this transition is intermittent, and the scing behavior is characteristic of Type-I intermittency [9Furthermore, the signature of the transition inL (or in thedistribution of finite-time Lyapunov exponents) is distinctive and very different from the routes to SNA that havbeen hitherto discussed. The present mechanism is geral and is likely to be operative in any quasiperiodicaldriven system.

For definiteness, we illustrate our results using thquasiperiodically forced logistic map [3]

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5 2 1dy2, the golden mean, and2 # a # 4. This system is convenient to study since thphenomenology is smoothly related to that of the logistmap in the limit of e ! 0. Since the driving term ismultiplicative, it is clear that the motion will remain boundif af1 1 e coss2pfndg [ f0, 4g, and thus for anya, thelargest value ofe allowed is4ya 2 1. It is preferable towork with the rescaled driving parameter,e0 eys4ya 2

1d since this scales the region of interest in parameter spato 0 # e0 # 1.

Figure 1 is a phase diagram of the system showing tdifferent possible dynamical behaviors—periodic, strangnonchaotic, and chaotic attractors, corresponding to tsymbolsP, S, andC. The different phases are characterizethrough the largest nonzero Lyapunov exponentL limN!`

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PNi1 ln js1 2 2xidaf1 1 e coss2pfidgj, which

we calculate typically from106 iterations of the map, aswell as the phase-sensitivity exponent [10]. The logistmap with additive driving [6] has identical behavior.

SNA’s occur along the boundary of the chaotic region[11], when L takes small negative values. In thea-e0

plane, the regions ofP, S, andC behavior can be interwo-ven. In particular, there are two distinct chaotic phasethe regions markedC1 and C2 which are separated by anarrow tongue where the motion is periodic. The behaior of L as a function ofa for e0 1, namely, along theupper edge of Fig. 1, is shown in Fig. 2(a). The nonmontonicity of L as a function ofa is typical, and severaldynamical transitions can be described:P ! S, S ! C,and C ! S (note that periodic motion here correspondto a n-frequency torus, withn 1, 2, or 4; highern suchas 3, 6, 8, etc. can also be observed in small windowsparameter space).

The intermittency route to SNA that we describe in thiLetter occurs along the boundary of the chaotic regioC2 and the periodic regionT on its right, namely, at the

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FIG. 1. Phase diagram for the forced logistic map (schematobtained by calculating the Lyapunov exponent as a functiona and the rescaled parametere0 defined ase0 eys4ya 2 1din a 100 3 100 grid. P and C correspond to periodic (torus)and chaotic attractors. The shaded region along the boundof P and C corresponds to SNA (markedS). The boundariesseparating the different regions are convoluted, and regionsSNA and chaotic attractors are interwoven in a complicatmanner. The intermittent SNA is found on the edge of theC2region marked I.

point markedI in Figs. 1 and 2(a). An enlarged view isshown in Fig. 2(b) where the bars indicate the varianin the Lyapunov exponent estimated from several comptations. There is a transition from a chaotic attractora SNA ata ø 3.405 802 with L changing linearly [12]through zero. Theintermittenttransition from the SNA toa torus—the phenomenon described here—occurs ata ac ; 3.405 808 8 . . . . At this transition, the distinctivesignature is an abrupt change in the dependence ofL onthe parametera, as well as a concurrent marked reduction in the variance inL [see Fig. 2(b); the bars have beecomputed from 50 samples of105 steps]. The disappear-ance of the SNA or the chaotic attractor is accompaniedintermittent dynamics, which can be conveniently studieby coevolving two trajectories with identicalsx0, f0d ande0, with differenta: since the angular coordinate remainidentical, the distance between the trajectories is simpthe difference in thexn ’s. We find that the time betweenbursts shows the scalingt , sac 2 ad2u, with u ø 1y2(the numerical value we obtain is0.52 6 0.04). This tran-sition to an intermittent SNA [Fig. 2(c)] occurs all alongthe right boundary ofC2. In thea-e plane, this is a regionof nearly constante.

The abrupt death of the strange nonchaotic attractorthrough a quasiperiodic analog of the saddle-node bifurction where a period 1 orbit is born. Consider a sequencemaps that are rational approximations to Eq. (1) by settiv vk FkyFk11, whereFk is thekth Fibonacci num-ber. TheFk11th iterate of the map is a function ofx alone(sincefFk11 ; f0). This allows for the construction of abifurcation diagram as a function ofa for fixed e. Thecase ofk 2, namely,v 1y2 has been studied earlier

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FIG. 2. (a) The largest Lyapunov exponentL versusa atthe upper edge of the phase plane, namely, ate0 1 and2 # a # 4. C1, C2 correspond to the two chaotic phaseidentified in Fig. 1. T denotes periodic motion (tori); SNA’s arefound along the boundaries ofC1 andC2, in the neighborhoodof the points markedA, B, and I. (b) An enlarged view of(a) near the intermittent transition markedI. The bars indicatethe variance inL computed from 50 samples of length105.(c) The intermittent SNA ata 3.405 808 ande0 1.

by Sanju and Varma [13] as the period-2 modulatedgistic map. Between the first period doubling bifurcatioand the two-band merging crisis, the attractor of the (umodulated) logistic map consists of two branches. (Fthe logistic map, these bifurcations occur ata 3 anda ø 3.678 857 . . . , respectively.) The effect of period-2modulation is to split the two branches of the attractor,shifting one branch to higher values ofa [13]. The ex-tent to which the two branches are shifted relative to oanother depends on the value ofe. The single attracting or-bit of period 2 in the unmodulated map becomes two serate attracting orbits for small values of the modulatioand depending on how the branches are shifted, namon the strength of the modulation, two very differen


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attractors—for example, chaotic and periodic—can coeist. For sufficiently largee the two branches can in factbecome disjoint, namely, the periodic orbit of one brancof the attractor is shifted beyond the two-band merginpoint (see Fig. 2 of Ref. [13]). When this happens, thperiodic attractor appears as at a saddle-node bifurcatigiving rise to a single periodic attracting orbit of period 1

It can easily be verified that this mechanism operatwith small modifications for the higher rational approximationsvk; this is a case of period-Fk11 modulation [13].Thus, in the limit ofvk ! v s

p5 2 1dy2, we suggest

that the disappearance of the chaotic attractor and the Sis through this analog of a saddle-node bifurcation.

Viewed as a function of decreasinga, this transitionshares some of the features of a widening crisis in unforcsystems [4,14,15]. The discontinuous change ofL at asaddle-node bifurcation is softened by the quasiperiodforcing, and fora , ac, the Lyapunov exponent showsthe scaling (see Fig. 3)

L 2 Lc , sac 2 adm, (2)

with the exponentm 0.37 6 0.03 at e0 1. To withinthe quoted error bar, the same exponent is obtained whthe probability density in the burst phase is fit to a powelaw [14–16]. The remnants of the old torus which arnow embedded in the SNA constitute one of the weakcoupled components of the SNA. These exponents sapproximately constant along theC2 boundary.

There are at least two other mechanisms for the creatof SNA’s that are operative in this system. Along thboundary of the chaotic regionC1 in Fig. 1, which appearsat the end of the (truncated) period doubling cascadSNA’s that are formed can appear as a result of thcollision between a period-doubled torus and its unstabparent [3]. SNA’s are also created through the proceof fractalization [5] which occurs along the entire lefedge of regionC2 (for example, the cases studied bNishikawa and Kaneko [6]). Along theC1 boundary bothmechanisms operate and representative points are mar

FIG. 3. Scaling of the probability density (pB) in the burstphase (1) and L (h) at the intermittent transition atI,for ac 3.405 808 8 . . . and Lc 20.0283 at e0 1. Themeasured exponents areø0.37.

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as B and A, respectively, in Fig. 2(a). A fractalizingtorus gets increasingly wrinkled as parameters are variand eventually goes chaotic, without any period-doublinbifurcation. These two routes can be differentiated frothe geometry of the SNA: in the torus collision mechanis[3], a period-2k torus gives rise to a2k21-band SNA, whilethrough fractalization [5,6], the result is a2k-band SNA.

The intermittent SNA is similar in many respects ta reentrant phase: the SNA so created is structuraldistinct from the SNA born through other mechanismThis distinction also extends to the corresponding toand examination of successive iterates of the lines1y2, fdunder the mapf of Eq. (1) gives an indication of how thisoccurs. By an extension of thee 0 case [17], this linewill be attracted to the 1-torus,

limk!`

fks1y2, fd ! T . (3)

[The attractor of the dynamics, whether periodic, strannonchaotic, or chaotic, is confined in a strip in configuration space, the upper boundary of which is the linfs1y2, fd, and the lower boundary is theinfimum offjs1y2, fd, j $ 2.] The convergence behavior of the lineto the attractor, quantified through the phase sensitivindex [10] shows that for smaller values of nonlinearitand driving, the convergence is slower than for highnonlinearity. Indeed, the torus stabilized at largea, e0

(in Fig. 1) is relatively smooth, as opposed to the highstructured and fractalizing torus at smaller values ofa.

To further characterize this distinction, we examine thdistribution of finite-time Lyapunov exponentsPsLN d inFig. 4, namely, the Lyapunov exponent computed fromN-step orbit on the attractor. AlthoughL is negative, forshort times the Lyapunov exponent on a SNA is positiv[10]. The distributionPsLN d has not been studied in detail, but it is known to have a significant positive tail whichdoes not vanish even for largeN. For the intermittentSNA, PsLN d decays slowly both as a function of the (local) LN as well asN. This distinction is a consequence othe fact that the variance inL increases drastically throughthis transition [in Fig. 2(b)]. In contrast, at the other transitions from a torus to a SNA (or vice versa), the varance does not change significantly and with increasingN,the PsLN d becomes essentially Gaussian. The differenin the distributions is not visible forN , 50 [Fig. 4(a)]while at N , 1000 the distinct character of the intermit-tent SNA is clearly seen [Fig. 4(b)]. These observatioare consistent with a coarser interweaving of the positiand negative local Lyapunov exponent regions on thetermittent SNA. Furthermore, this transition is robust tlow-amplitude noise [18], as has been verified by addingsmall random component to thex dynamics. The scalingbehavior inL and the probability density in the burst phaswere observed to hold [19] with slightly modified valueof the exponent.

Such quasiperiodically driven systems are interestifrom a variety of points of view. A potential novel use

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FIG. 4. (a) Distribution of N-step or local Lyapunov ex-ponents, computed fromN 50 step segments of a longtrajectory for the intermittent SNA at the point markedIcontrasted with SNA’s (born through other mechanisms) at tpoints markedA andB in Fig. 2(a). These correspond, respectively, to I (1), A (,), andB (h). (b) Same as (a), except thatN 1000.

of SNA’s that can be envisaged is for secure communictions [20]: two independent SNA’s can be easily synchronized since the Lyapunov exponents are negative, whthe signals from such systems provide effective maskisince they appear erratic owing to the underlying strangeometry.

In summary, in the present Letter we have describednew mechanism for the creation of strange nonchaotictractors through intermittency, whereby a chaotic attractis eventually replaced by a quasiperiodic one. In the intemediate region, there are SNA’s. The distinctive signatuof this transition is a sharp change in the Lyapunov eponent which shows large fluctuations and scaling behaior on the SNA side of the transition. This mechanismwhich is operative at high nonlinearity and large ampltude quasiperiodic forcing, proceeds via astabilizationofchaotic motion and is thus like a reentrant phase transitio

The route to intermittent SNA is a general one. Whave studied a variety of related systems [19] and find ththe phenomenology is very similar: the phase diagram this obtained for the driven logistic map is typical. Intermittent SNA’s always occur in analogous regions in paramter space, namely, where a tongue of periodic behavseparates two chaotic areas with differing characterist[21]. We expect that in situations where the amplitudequasiperiodic forcing is an conveniently varied paramete

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for example, this mechanism for the creation of SNAmay be experimentally observed.

This work was supported by Grant No. SPSyMO-5y92from the Department of Science and Technology, IndWe thank Sanju for discussions.

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uniform random variable in the unit interval. Ate0 1,for instance, the transition survives for additive noise oamplitude up tos 1026, while at lowere0 the transitionis robust to even largers ø 1024. In other systems withintermittent SNA’s the transition may survive to highenoise levels.

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[21] We have located intermittent SNA’s in a number osystems such as the forced sine map, as well as higdimensional examples such as the ring and Hénon ma[11] using this strategy [19].

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Intermittency Route to Strange Nonchaotic Attractors