VOLUME 77, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 16 DECEMBER 1996

as 66045

Blowout Bifurcation Route to Strange Nonchaotic Attractors

Tolga Yalcınkaya* and Ying-Cheng Lai†

Department of Physics and Astronomy and Department of Mathematics, The University of Kansas, Lawrence, Kans(Received 2 May 1996; revised manuscript received 4 September 1996)

Strange nonchaotic attractors are attractors that are geometrically strange, but have nonpositive Lya-punov exponents. We show that for dynamical systems with an invariant subspace in which there isa quasiperiodic torus, the loss of the transverse stability of the torus can lead to the birth of a strangenonchaotic attractor. A physical phenomenon accompanying this route to strange nonchaotic attractorsis an extreme type of intermittency. [S0031-9007(96)01861-3]

PACS numbers: 05.45.+b

a

t

sora4

uun

i

us

r

nv

p

on-lly

tic-

.

-se

orusen

elyseugh

e-

tticis

ical

ffatnin-

are

Strange nonchaotic attractors are attractors thatgeometrically complicated, but typical trajectories othese attractors exhibit no sensitive dependence on iniconditionsasymptotically[1–11]. Here the wordstrangerefers to the complicated geometry of the attractor:strange attractor is not a finite set of points, and itnot piecewise differentiable. The wordchaotic refers toa sensitive dependence on initial conditions: trajectorioriginating from nearby initial conditions diverge exponentially in time. Strange nonchaotic attractors occurdissipative dynamical systems driven by severalincom-mensuratefrequencies (quasiperiodically driven system[1–11]. For example, it was demonstrated that in twfrequency quasiperiodically driven systems, there existgions of finite Lebesgue measure in the parameter spfor which there are strange nonchaotic attractors [3,More recent work demonstrates that a typical trajectoon a strange nonchaotic attractor actually possesses ptive Lyapunov exponents in finite time intervals, althougasymptotically the exponent is negative [7]. Strange nochaotic attractors can arise in physically relevant sitations such as quasiperiodically forced damped pendand quantum particles in quasiperiodic potentials [2], ain biological oscillators [4]. These exotic attractors havbeen observed in physical experiments [8,9].

While the existence of strange nonchaotic attractowas firmly established, a question that remains interestis how these attractors are created as a system paramchanges through a critical value, i.e., what the possibroutes to strange nonchaotic attractors are. One rowas investigated by Heagy and Hammel [5] who dicovered that, in quasiperiodically driven maps, thtransition from two-frequency quaisperiodicity to strangnonchaotic attractors occurs when a period-doubled tocollides with its unstable parent torus [5]. Near thcollision, the period-doubled torus becomes extremewrinkled and develops into a fractal set at the collisioalthough the Lyapunov exponent remains negatithroughout the collision process. Recently, Feudet al. found that the collision between a stable torus anan unstable one at a dense set of points leads to a stranonchaotic attractor [10]. A renormalization-grou

0031-9007y96y77(25)y5039(4)$10.00

renial

Ais

es-in

)-e-ce].

ryosi-hn--lad

e

rsngeterlete-eeusely,e

eldnge

analysis was also devised for the transition to strange nchaotic attractors in a particular class of quasiperiodicadriven maps [11].

In this paper, we present at route to strange nonchaoattractors in dynamical systems with a symmetric lowdimensional invariant subspaceS in the phase spaceSinceS is invariant, initial conditions inS result in tra-jectories which remain inS forever. We consider the casewherethere is a quasiperiodic torus inS [12]. Whetherthe torus attracts or repels initial conditions in the vicinity of S is determined by the sign of the largest transverLyapunov exponentLT computed for trajectories inS withrespect to perturbations in the subspaceT which is trans-verseto S. WhenLT is negative,S attracts trajectoriestransversely in the phase space, and the quasiperiodic tin S is also an attractor of the full phase space. WhLT is positive, trajectories in the vicinity ofS are repelledaway from it, and, consequently, the torus is transversunstable, and it is hence not an attractor of the full phaspace. Assume that a system parameter changes throa critical value ac; LT passes through zero from thnegative side. This bifurcation is referred to as the “blowout bifurcation” [13–15]. Our main result is that blowoubifurcation can lead to the birth of a strange nonchaoattractor. A physical phenomenon associated with throute to strange nonchaotic attractors is that the dynamvariables in the transverse subspaceT exhibit an extremetype of temporally intermittent bursting behavior: on-ointermittency [16]. Thus, our work also demonstrates thon-off intermittency can occur in quasiperiodically drivedynamical systems, whereas, to our knowledge, thesetermittencies have been reported only for systems thatdriven either randomly or chaotically.

We consider the following class ofN-dimensional dy-namical systems,

dxdt

­ Fsx, z, pd ,

(1)dzdt

­ v ,

where x is Nx dimensional,z is Nz dimensional,Nx 1

Nz ­ N , v ; sv1, v2, . . . , vNzd is a frequency vector,

© 1996 The American Physical Society 5039

VOLUME 77, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 16 DECEMBER 1996

-

iceb

-

an

h

2-

taa

tux-

,a

fb

hct1

iiu

iscta

f

icrge.ries

th

thcef ann-

lsi-

ness

usin-ndas

sky

m:

andp is a bifurcation parameter. The functionF satisfiesFs0, z, pd ­ 0 so that x ­ 0 defines the invariant subspaceS in which the dynamics is described by [17]dzydt ­ v. The frequenciessv1, v2, . . . , vNz d are incom-mensurate so that thez dynamics gives a quasiperiodtorus. The largest transverse and nontrivial ovall Lyapunov exponents of the systems are givenLT ­ limt !`

1t lnfjdxstdjyjdxs0djg and L ­ limt !`

1t

3 lnfjdXstdjyjdXs0djg, respectively, where the infinitesimal vectorsdxstd and dXstd are evolved accordingto ddxstdydt ­ s≠Fy≠xdjx­0. dx and ddXstdydt ­s≠Fy≠xd ? dX for random initial vectorsdxs0d anddXs0d[13]. To illustrate our findings, we consider a physicexample, mathematically described by the followiversion of Eq. (1),

dxdt

­ y ,

dydt

­ 2 ky 2 gx3 1 sb 1 f1 sinz1 1 f2 sinz2d

3 sins2pxd ,(2)

dz1

dt­ v1,

dz2

dt­ v2 ,

where the invariant subspaceS is given bysx, yd ­ s0, 0d,v1 and v2 are two incommensurate frequencies so tthere is a two-frequency quasiperiodic torus inS sz1, z2d forwhich the largest Lyapunov exponent is zero. In Eq. (k (dissipation),g, b, f1, andf2, are parameters. Equation (2) is a slightly modified version of the experimenmodel used by Zhou, Moss, and Bulsara, which is relevto the radio-frequency-driven superconducting quaninterference device (SQUID) [9]. In our numerical eperiments, we arbitrarily choosek as the bifurcation parameter and fix other parameters atg ­ 2.0, b ­ 21.1,v1 ­ 2.25, v1yv2 ­

12 s

p5 1 1d (the golden mean)

f1 ­ 3.5, andf2 ­ 5.0. Numerically we observe thatblowout bifurcation occurs atkc ø 4.17, whereLT . 0s, 0d for k , kc s.kcd.

Figure 1(a) shows thesx, yd projection of a trajectory o50 000 iterations (after 10 000 preiterations) on the stroscopic surface of section defined byz1stnd ­ 2np sn ­1, 2, . . .d for k ­ 4.1 sLT ø 0.019d. The largest nontriv-ial Lyapunov exponent of the system isL ø 20.134.Thus, there is no positive Lyapunov exponent for tparameter setting. The geometric shape of the attrahowever, appears strange, as can be seen from Fig.Qualitatively, the strangeness can be understood aslows. Fork & kc, the transverse Lyapunov exponentslightly positive. Thus, a typical trajectory on the torusS is transversely unstable. There are time intervals ding which the trajectory, when it is in the vicinity ofS, isrepelled fromS. As can be verified numerically, at thparameter setting there are apparently no other attrain the phase space [18]. Thus, the trajectory comes bto the neighborhood ofS intermittently in the course o

5040

r-y

lg

at

),

lntm-

o-

isor,(a).

fol-snr-

orsck

FIG. 1. (a) A trajectory of 50 000 points on the stroboscopsection for Eq. (2) atk ­ 4.1 (see text for other parametevalues). Apparently, the attractor is geometrically stran(b) Singular-continuous spectrum analysis of the time sehxnj: shown is log10jXsV, Tdj2 versus log10 T . We havejXsV, T dj2 , T 1.26. The subplot shows the corresponding pasReX, Im Xd.

time evolution. Since the trajectory is bounded in box and y, the asymptotic attractor in the full phase spadevelops a strange shape: It apparently consists oinfinite number of points, and it is not piecewise differetiable. SinceL is negative, despiteLT * 0, the attractoris nonchaotic. Thus, ask decreases through the criticavaluekc, a strange nonchaotic attractor is born: The potiveness of the transverse Lyapunov exponentLT givesrise to strangeness of the attractor, and the negativeof the largest nontrivial Lyapunov exponentL guaranteesthe attractor is nonchaotic.

To qualitatively verify that the attractor fork & kc

is strange nonchaotic, we perform a singular-continuospectrum analysis that was first proposed in thevestigation of models of quasiperiodic lattices aquasiperiodically forced quantum systems [19], and wlater applied to strange nonchaotic attractors by Pikovand Feudel [6]. Take a time serieshxnj or h ynj on thesurface of section. Compute the following Fourier suXsV, T d ­

PTn­1 xnei2pnV, where V is proportional

VOLUME 77, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 16 DECEMBER 1996

,xeecgeg

gn

oc

-fb

into

r

a.

ni-

alv

t

y.

isy,hatnt

orus

o be

n.w-wotic,

shen

to the ratio of the two incommensurate frequenciesthe quasiperiodic driving. WhenT is regarded as timethe quantity XsV, T d defines a path in the compleplane sReX, Im Xd. For discrete spectrum, we havjXsV, T dj2 , T 2, while for continuous spectrum, thpath is described by Brownian motion and, henjXsV, T dj2 , T . The spectrum associated with strannonchaotic attractors falls somewhat in between thtwo categories. It was demonstrated [6] that for strannonchaotic attractors, the quantityXsV, T d has the fol-lowing two distinct features: (1)jXsV, Tdj2 , Ta , wherea fi 1, 2; and (2) the pathsReX, Im Xd is fractal. Fig-ure 1(b) shows, forV ­ s

p5 1 1dy2, log10 jXsV, Tdj2

versus log10 T and the pathsReX, Im Xd (subplot) com-puted from the time serieshxnj in Fig. 1(a). We havea ø 1.26 and, the pathsReX, Im Xd apparently exhibitsa fractal self-similar structure. These results stronsuggest that the attractor in Fig. 1(a) is indeed stranonchaotic.

A feature associated with the birth of the strange nchaotic attractor is the occurrence of on-off intermitten[16] when k & kc (LT being slightly positive). This isshown in Fig. 2, where the time seriesh ynj is plotted fork ­ 4.1. We see that there are time intervals whenyn

stays neary ­ 0 (the “off” state), but there are also intermittent bursts ofyn (the “on” state) away from the ofstate. This is a typical consequence of the blowoutfurcation [13–15]. Note that the on-off intermittencyFig. 2 is, in fact, produced by a quasiperiodic drivingthe transverse dynamics.

To understand whyL remains negative in parameteregimes after the blowout bifurcation whereLT * 0, weconsider the following analyzable two-dimensional mthat captures the essential feature of physical model Eqon the surface of section

xn11 ­1

2psa coszn 1 bd sins2pxnd ,

(3)zn11 ­ szn 1 2pvd mods2pd ,

FIG. 2. On-off intermittency atk ­ 4.1.

of

e,esee

lyge

n-y

i-

p(2)

where a and b are parameters, andv [ s0, 1d is anirrational number so that thez dynamics is a map onthe circle that generates a quasiperiodic torus with uform invariant densityrszd ­ 1ys2pd in z [ f0, 2pg(two-frequency quasiperiodicity). The one-dimensioninvariant subspace isx ­ 0. The transverse Lyapunoexponent isLT ­ limn !`

1n

Pnj­1 lnja coszj 1 bj ­

12p

3R2p

0 lnja cosz 1 bj dz. We obtain LT ­ ln jbj 2

lnsss2yh1 1 f1 2 saybd2gjddd if a # b, and LT ­ lnjbj 1

ln a2b if a . b. We have, for example,ac ­ 2 for

the casea . b . 0, where LT # 0 for a # ac andLT . 0 for a . ac. The nontrivial Lyapunov exponenis L ø LT 1 l, wherel ;

Rlnj coss2pxdjrsxd dx , 0

and rsxd is the invariant density ofx for a . ac. Notethat for a , ac we have l ­ 0 and henceL ­ LT

because in this casex approaches zero asymptoticallFor a * ac whereLT * 0, it is possible to haveL & 0becausel , 0. Thus a strange nonchaotic attractorborn at the blowout bifurcation with on-off intermittencas can also be verified numerically. We stress tit is essential to have quasiperiodicity in the invariasubspace. When frequencies are locked on the t(corresponding tov’s being rational), trajectories onlyhave a finite number of possiblez values. Numericalanalysis indicates that the attractor does not appear tstrange in the phase space.

We now describe two issues related to the bifurcatio(1) Variation of the Lyapunov exponent after the blo

out bifurcation.—We address the following question: hodoes the strange nonchaotic attractor become a chaattractor ask decreases fromkc? To answer this questionnotice thatL ø LT 1 l, wherel , 0. Thus,L becomespositive whenLT . jlj. As k decreases,LT increasesmonotonically nearkc, but l is not monotonic. Thisis shown in Fig. 3, whereLT , L, and l versusk areplotted for 2000 values ofk in [3.2, 4.8]. For each valueof k, LT , L, andl are computed with 50 000 iterationand 10 000 preiterations on the surface of section. WLT and jlj have comparable magnitudes,L can change

FIG. 3. LT , L, andl versus the parameterk.

5041

VOLUME 77, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 16 DECEMBER 1996

tlyrain-

sy[3pofo

iti-

eto

ngeenher

m-oif-cai-

odon

riva

erdtee

ys

hisced.Sutantesa.

ica

v.

r.,

n

F.

s-

ev.

ther,

s-

,

w-the

ible

otd,

orshe

,

s.

from negative to positive and vice versa. Consequenthere exist several parameter intervals for chaotic atttors sL . 0d which are interspersed with parametertervals for strange nonchaotic attractorssL , 0d. Thisbehavior has actually been observed in other physicaltems such as the quasiperiodically forced pendulumWe believe that the existence of two competing exnents, i.e.,LT andl, is responsible for the alternation ostrange nonchaotic and chaotic attractors. We haveserved that whenk is decreased further through some crcal value,LT is sufficiently large so thatLT , jlj doesnot occur, the system possesses a positive LyapunovponentL and, consequently, strange nonchaotic attracare no longer possible.

(2) Transient on-off intermittent behavior precedithe blowout bifurcation.—Before the birth of the strangnonchaotic attractor, a typical trajectory exhibits transion-off intermittent behavior before finally approaching tinvariant subspacex ­ y ­ 0. For a given parametevalue k, the average transient lifetime,tskd, depends onthe parameter differencejk 2 kcj. Numerically, we findthe following scaling law:tskd , jk 2 kcj

21. This canbe understood by noting thattskd , 1yLT , and fork nearkc, we haveLT , jk 2 kcj.

In summary, we have shown that two distinct dynaical phenomena, strange nonchaotic attractor and onintermittency, commonly thought as arising in very dferent contexts in the study of nonlinear systems,actually be closely related. The link is the blowout bfurcation that destabilizes, transversely, the quasiperitorus in the invariant subspace. Our study thus demstrates that blowout bifurcation can occur even if the ding is not chaotic or random but quasiperiodic. Asconsequence, on-off intermittency can arise in quasipodically driven dynamical systems. We have presentephysical example for which the blowout bifurcation routo strange nonchaotic attractor can be observed numcally, and we believe that this route can be tested in phcal experiments.

We thank U. Feudel for valuable discussions. Twork was partly sponsored by the Air Force Offiof Scientific Research, Air Force Materiel CommanUSAF, under Grant No. F49620-96-1-0066. The UGovernment is authorized to reproduce and distribreprints for Governmental purposes notwithstandingcopyright notation thereon. This work was also supporby the General Research Fund at the University of Kanand by the NSF/K*STAR EPSCoR Program in Kansas

*Electronic address: tolga@math.ukans.edu†Electronic address: lai@poincare.math.ukans.edu

[1] C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke, Phys(Amsterdam)13D, 261 (1984).

5042

,c-

s-].-

b-

x-rs

t

-ff

n

ic-

-

i-a

ri-i-

,.eyds

[2] A. Bondeson, E. Ott, and T. M. Antonsen, Jr., Phys. ReLett. 55, 2103 (1985).

[3] F. J. Romeiras and E. Ott, Phys. Rev. A35, 4404 (1987);F. J. Romeiras, A. Bondeson, E. Ott, T. M. Antonsen, Jand C. Grebogi, Physica (Amsterdam)26D, 277 (1987).

[4] M. Ding, C. Grebogi, and E. Ott, Phys. Rev. A39, 2593(1989); M. Ding and J. A. Scott Kelso, Int. J. BifurcatioChaos4, 553 (1994).

[5] J. F. Heagy and S. M. Hammel, Physica (Amsterdam)70D,140 (1994).

[6] A. S. Pikovsky and U. Feudel, J. Phys. A27, 5209 (1994).[7] A. S. Pikovsky and U. Feudel, Chaos5, 253 (1995); Y. C.

Lai, Phys. Rev. E53, 57 (1996).[8] W. L. Ditto, M. L. Spano, H. T. Savage, S. N. Rauseo, J.

Heagy, and E. Ott, Phys. Rev. Lett.65, 533 (1990).[9] T. Zhou, F. Moss, and A. Bulsara, Phys. Rev. A45, 5394

(1992).[10] U. Feudel, J. Kurths, and A. S. Pikovsky, Physica (Am

terdam)88D, 176 (1995).[11] S. P. Kuznetsov, A. S. Pikovsky, and U. Feudel, Phys. R

E 51, R1629 (1995).[12] For a definition of attractors for dynamical systems wi

invariant subspaces, see, for example, J. C. AlexandJ. A. Yorke, Z. You, and I. Kan, Int. J. Bifurcation Chao2, 795 (1992); E. Ott, J. C. Alexander, I. Kan, J. C. Sommerer, and J. A. Yorke, Physica (Amsterdam)76D, 384(1994).

[13] E. Ott and J. C. Sommerer, Phys. Lett. A188, 39(1994).

[14] P. Ashwin, J. Buescu, and I. Stewart, Phys. Lett. A193,126 (1994); Nonlinearity9, 703 (1996).

[15] Y. C. Lai and C. Grebogi, Phys. Rev. E52, R3313 (1995).[16] E. A. Spiegel, Ann. New York Acad. Sci.617, 305 (1981);

A. S. Pikovsky, Z. Phys. B55, 149 (1984); H. Fujisaka andT. Yamada, Prog. Theor. Phys.74, 919 (1985);75, 1087(1986); H. Fujisaka, H. Ishii, M. Inoue, and T. YamadaProg. Theor. Phys.76, 1198 (1986); A. S. Pikovsky andP. Grassberger, J. Phys. A24, 4587 (1991); A. S.Pikovsky, Phys. Lett. A165, 33 (1992); N. Platt, E. A.Spiegel, and C. Tresser, Phys. Rev. Lett.70, 279 (1993);J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E49,1140 (1994).

[17] In general, when the system does not have a skeproduct structure, one should also consider terms inz equations such asex, ejxjx, or even higher-order termsin x that vanish atx ­ 0. Near the invariant subspace (jxj

small), these terms are small. Thus, they have a negligeffect on our conclusions.

[18] The cubic term in Eq. (2) stipulates that trajectories do nwander far away from the invariant subspace. We finnumerically, that there are apparently no other attractin the phase space except the one in the vicinity of tinvariant subspace.

[19] S. Aubry, C. Godreche, and J. M. Luck, Europhys. Lett.4,639 (1987); J. Stat. Phys.51, 1033 (1988); C. GodrecheJ. M. Luck, and F. Vallet, J. Phys. A20, 4483 (1987);J. M. Luck, H. Orland, and U. Smilansky, J. Stat. Phy53, 551 (1988).