The average lifetime ($\tau(H)$) it takes for a randomly started trajectory
to land in a small region ($H$) on a chaotic attractor is studied. $\tau(H)$ is
an important issue for controlling chaos. We point out that if the region $H$
is visited by a short periodic orbit, the lifetime $\tau(H)$ strongly deviates
from the inverse of the naturally invariant measure contained within that
region ($\mu_N(H)^{-1}$). We introduce the formula that relates
$\tau(H)/\mu_N(H)^{-1}$ to the expanding eigenvalue of the short periodic orbit
visiting $H$.Comment: Accepted for publication in Phys. Rev. E, 3 PS figure

C. Grebogi

D. Auerbach

E. Bollt

E. Ott

G. Pianigiani

H. Buljan

J. Jacobs

J. Singer

J.P. Eckmann

K. Zyczkowski

M. Hénon

P. Cvitanović

P. Szepfalusy

T. Shinbrot

T. Tel

U. Dressler

V. Paar

W.L. Ditto

English

arXiv

The average lifetime [τ(H)] it takes for a randomly started trajectory to land in a small region (H) on a chaotic attractor is studied. τ(H) is an important issue for controlling chaos. We point out that if the region H is visited by a short periodic orbit, the lifetime τ(H) strongly deviates from the inverse of the naturally invariant measure contained within that region [μN(H)-1]. We introduce the formula that relates τ(H)/μN(H)-1 to the expanding eigenvalue of the short periodic orbit visiting H

Paar, Vladimir

Buljan, Hrvoje

University of Zagreb Repository

PHYSICAL REVIEW E OCTOBER 2000VOLUME 62, NUMBER 4Bursts in the chaotic trajectory lifetimes preceding controlled periodic motionV. Paar and H. BuljanDepartment of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia~Received 9 May 2000!The average lifetime @t~H!# it takes for a randomly started trajectory to land in a small region ~H! on achaotic attractor is studied. t(H) is an important issue for controlling chaos. We point out that if the region His visited by a short periodic orbit, the lifetime t(H) strongly deviates from the inverse of the naturallyinvariant measure contained within that region @mN(H)21#. We introduce the formula that relatest(H)/mN(H)21 to the expanding eigenvalue of the short periodic orbit visiting H.PACS number~s!: 05.45.Gg, 05.45.AcControlling chaos by stabilizing one of the many unstableperiodic orbits embedded within a given chaotic attractor isattainable with small, time-dependent changes in an acces-sible system parameter @1–3#. The idea is to observe a typi-cal trajectory of the uncontrolled system for some transienttime, until it falls sufficiently close to the desired periodicorbit, and then to activate the control mechanism. An impor-tant issue related to the utilization of this method is the av-erage lifetime of chaotic transients that precede the con-trolled periodic motion @1,3–5#.Suppose that the uncontrolled chaotic attractor A de-scribes the asymptotic behavior of the dynamical systemO:D→D , D#Rm, also referred to as the original system.Let jW5Ok(jW ) be a point on a particular unstable periodicorbit @6,7# that we wish to stabilize. Furthermore, let thevicinity of the orbit H[He(jW ) be an m-dimensional ball ofradius e!1 centered at jW . The probability that a randomlystarted trajectory does not reach H up to time t is ;e2t/t(H).The average lifetime t[t(H) is strongly correlated with thevisitation frequency of typical trajectories to the region H,which is described in terms of the naturally invariant mea-sure (mN) contained within H2mN(H) @1,4,8#. Obviously, ifa certain region on a given chaotic attractor is visited morefrequently by typical trajectories, the average lifetime it takesfor an orbit to land in that region will be smaller. In thepresent paper we address the following question: What is thedeviation of t from mN(H)21 as a function of jW and e?We will demonstrate the existence of bursts in the life-times, i.e., significant deviations of t from mN(H)21, whichappear when the H region encompasses a point on a shortperiodic orbit. In contrast to the overall t.mN(H)21 behav-ior, at these exceptional positions, the lifetime t is consider-ably prolonged as compared to mN(H)21. As the length ofthe shortest cycle visiting H increases, the parameter of thisdeviation, S(H)[t/mN(H)21, decreases rapidly towards 1.We will introduce a formula that relates the parameter S(H)to the repelling properties ~expanding eigenvalue! of theshortest cycle within H. Furthermore, we will demonstratethat S(H) is independent of e ~for e!1). This is consistentwith the previously reported scaling t;mN(H)21 ~see, e.g.,Ref. @1,4#!.The present paper is motivated by the previous investiga-tion of the logistic map with a hole @9#. In this paper wepresent a theoretical explanation for the phenomenologicalPRE 621063-651X/2000/62~4!/4869~4!/$15.00result reported in Ref. @9# and generalize it to one-dimensional ~1D! noninvertible and 2D invertible chaoticmaps. ~From our considerations a conjecture follows thatsimilar phenomena occur generally in chaotic systems.!It will be useful to define an auxiliary modified map@9,10#M ~jW !5H O~jW8!, jW8PD\Houtside of the basin of A , jW8PH .~1!A typical trajectory of the map O remains on the chaoticattractor forever, while the same trajectory in the map Meventually escapes through the region H, from now on alsoreferred to as the hole. The average lifetime of chaotic tran-sients created by the map M is equal to t(H), which we havedefined above. Similar maps with a forbidden gap regionarise in the context of communicating with chaos @11#, and incalculation of the topological entropy @12#.To illustrate the concept of bursts, we consider two cha-otic 1D maps: ~i! the asymmetric tent map O(x)5k1x , x,k1 ; O(x)5k2(12x), x.k1 , k1 ,k2.0, k1211k22151,and ~ii! the sinusoidal map O(x)5sin px. Figure 1 displays tas a function of the position of the hole j @H5(j2e ,j1e)#, for the two paradigmatic maps ~see also Ref. @9#!. Thewidth of the hole is kept constant (e50.005). Both graphsexhibit some common features: ~i! the overall behavior oflifetimes follows the mN(H)21 pattern; ~ii! strong local de-viations from the mN(H)21 behavior—the bursts, observedas leaps in the lifetimes, occur when the hole interval sweepsacross a short periodic orbit; ~iii! the bursts are more signifi-cant if the length of the short periodic orbit is smaller.The explanation of the burst phenomenon requires thecomparison of two concepts: ~i! the conditionally invariantmeasure @13–15# ~also referred to as the c measure!—theconcept associated with the modified system, and ~ii! thenaturally invariant measure @7,3# of the original system. Inorder to define these measures, imagine that we cover thechaotic attractor with cells ~I! from a very fine grid. Then werandomly distribute a large number ~N! of points on the grid,and evolve them under the dynamics O for a long time T.Suppose that all initial points are colored blue, and that apoint irretrievably changes color from blue to red immedi-ately after its first entrance into the region H. Thus, the pointxW T5OT(xW 0) at time T is blue if Ot(xW 0)„H for t4869 ©2000 The American Physical Society4870 PRE 62V. PAAR AND H. BULJANP$0,1, . . . ,T21%, and red otherwise. In the limit T→‘ , thefraction of points found in a given cell I, is just the naturalmeasure contained within that cell: mN(I)5NT(I)/N; N5( INT(I), where NT(I)5bT(I)1rT(I) denotes the totalnumber of points in a given cell I at time T @3#. The numberof blue ~red! points within a given cell I at time T is denotedas bT(I) rT(I). The points that change color from blue tored under the action of the map O, are those that wouldescape the attractor under the action of the map M. Hence, ifwe were to evolve exactly the same initial conditions usingthe map M for the same time T, the number of survivingpoints ~blue points! would be BT5( IbT(I);N exp(2T/t).In the limit N→‘ , T→‘ , the distribution of blue pointsconverges to the c measure of the modified system @13–15#.The fraction of blue points in a given cell I is simply the cmeasure contained within that cell: mC(I)5bT(I)/BT .The blue point at any time t.0, was certainly not in thehole H at time t21. Therefore, M 21(I)[O21(I)\Hand bT(I)5bT21M 21(I), which divided by BT5BT21exp21/t yields the well-known relation for the cmeasure @13–15#mC~I !5e1/tmCM 21~I !. ~2!By summing the equation above over all the cells I we obtain1t52ln@12mC~H !#.mC~H !. ~3!We emphasize the importance of this observation. The aver-age lifetime it takes for a typical trajectory to reach the smallregion H on the attractor is an inverse of the c measurecontained within that region mC(H)21, which may signifi-cantly differ from the inverse of the natural measuremN(H)21.FIG. 1. t(j) ~solid line! and mN(H)21 ~dashed line! vs j for ~a!the asymmetric tent map (k12150.39, k22150.61) and ~b! the sin pxmap. H5(j2e ,j1e), e50.005.As an illustration, in Fig. 2 we display the c measure forthe modified version of the map sin px, in comparison to thenatural measure of the original map. For the c measure inFig. 2~a!, the hole has been positioned at an arbitrary point,but not on the short periodic orbit. In this case, we observethat mC(H).mN(H), i.e., mN(H)21 is a good approximationfor the lifetime. In contrast, in Fig. 2~b! we display mC forthe modified map sin px, with the hole positioned on thefixed point. We notice that mC(H) strongly deviates frommN(H). This case corresponds to the burst labeled 1 in Fig.1~b!. The overall agreement of the two measures is evident inboth Figs. 2~a! and 2~b!. However, at locations above thefirst few images of the hole, mC takes the shape of a well,with values that are considerably lower than mN . When thehole lies on the fixed point @Fig. 2~b!#, the wells are justabove the hole itself. This results in a pronounced deviationof t5mC(H)21 from mN(H)21, which manifests as a burst.Now we compare the two measures globally. A chaoticrepeller is a set of points on the attractor that never visit thehole @14,15,11#. A trajectory that starts close to the repeller,does not escape the attractor for a long time. Therefore, theblue points at a large time T are located along the unstablemanifold of the repeller. Their distribution along this mani-fold defines the c measure @14#. Thus, the natural measure isconstructed from all the points ~at time T) on all the unstablemanifolds, whereas the c measure results only from points onparts of these manifolds, the parts that extend from the re-peller up to the hole. As we reduce the size of the hole e , theFIG. 2. mC ~solid line! in comparison to mN ~dashed line! for thesin px map (e50.005). ~a! mC for the hole positioned on j50.66.The hole is not visited by a short periodic orbit. Note that mC(H).mN(H). ~b! mC for the hole located on the fixed point. Note thatmC(H),mN(H). In both ~a! and ~b! mC and mN are globally iden-tical, except at the first 3-4 images of the hole, which are plottedunderneath the graphs.PRE 62 4871BURSTS IN THE CHAOTIC TRAJECTORY LIFETIMES . . .repeller and its unstable manifold grow. Consequently, mCgradually approaches mN , and for sufficiently small e , thetwo measures are practically identical. ~For e50, mC be-comes mN @13–15#!.However, the deviation of the lifetime t5mC(H)21 frommN(H)21 depends only on the values of the two measureswithin the hole, and therefore is a local quantity. In order tomake a more accurate comparison of mC and mN , we intro-duce the following definitions. Consider a set P,D suchthat mN(P).0. We define the quantitya~P !5mC~P !/mN~P !, ~4!which describes the relation between mC and mN within P.We also define the influence i(P) of the hole on the set P asi~P !5mNO2l(P)~P !øHmNO2l(P)~P !. ~5!l(P) denotes the smallest integer for which the sectionO2l(P)(P)øH becomes nonempty. The natural invariantmeasure within O2l(P)(P) is mapped to P in l(P) iterates.The influence is just a fraction 0<i(P)<1 of mN(P) thatis mapped from the hole in the last l(P) time steps.Let Pe[Pe(xW ),D be an m-dimensional ball of radius e~the same radius as the hole! centered at xW . We ask the fol-lowing question: Given a chaotic attractor and choosing thehole region, what is the behavior of a(Pe)5mC(Pe)/mN(Pe) as the position of Pe on the attractor ischanged?By using Eq. ~2! and the identity mN(Pe)5mNO21(Pe), we can writea~Pe!5el/t12i~Pe!aM 2l~Pe!, ~6!where l[l(Pe) ~in what follows, l[l(Pe)).Concerning the first factor in Eq. ~6!, note that the averagelifetime typically scales like t;1/eDp(jW ) @Dp(jW ) denotes thepointwise dimension at j¢] @1,4,3#, whereas the minimal num-ber of iterates l for which O2l(Pe)øHÞB scales like l;ln(1/e) @4#. Therefore, exp(l/t).11l/t.1.If the influence i(Pe) is small, the second factor in Eq. ~6!is .1. We argue that i(Pe), the influence of the hole on theregion Pe decreases exponentially with l. For the 2D originalmaps, O2l(Pe) is a narrow region that is stretched along thestable direction and squeezed along the unstable one @4#. Theintersection of O2l(Pe) with the hole H is roughly a rect-angle of length e and width e exp(2l1l). For the 1D maps,O2l(Pe)øH is an interval of width ;e exp(2l1l). In bothcases, l1 denotes the positive Lyapunov exponent obtainedfor typical initial conditions on the attractor. Since the natu-ral measure is concentrated along the unstable manifolds@7,3#, we can relate mNO2l(Pe)øH;exp(2l1l). Thus,due to the chaoticity of the map O we obtain i(Pe);exp(2l1l).Concerning the third factor in Eq. ~6!, we consider the setM 2l(Pe) and the value aM 2l(Pe) in dependence of l. Forthe 2D maps, the set M 2l(Pe)[O2l(Pe)\H is stretched ex-ponentially fast with increasing l along the stable manifolds,and thus crosses many of the unstable manifolds that carryboth the natural and the c measure. For the 1D maps, thenumber of disjoint intervals that make the lth preimage ofPe , grows exponentially with l. Furthermore, they are scat-tered all over the attractor. Due to the chaoticity of the mapO, in both cases M 2l(Pe) becomes more democratic withlarger l, in the sense that the value aM 2l(Pe) reflects theglobal agreement between the two measures. Thus, insofar asl is not small, aM 2l(Pe).1.We conclude that for l[l(Pe), larger than some criticalvalue ~call it lc), all of the three factors in Eq. ~6! are .1and therefore mC(Pe).mN(Pe). This is consistent with theglobal agreement between the two measures. The criticalvalue lc depends on the chaoticity of the original map. Forexample, we may take lc to be the smallest integer for whiche2l1lc,0.1 ~e.g., for the sin(px) map this gives lc;324).When H maps to Pe in just a few iterates, so that l,lc , asignificant difference is observed between mC(Pe) andmN(Pe) ~this explains the wells in Fig. 2!.Coming back to the average lifetimes, if the hole does notmap back to itself in just a few iterates, i.e., if the shortestperiodic orbit within H has a period larger than lc , thenmC(H).mN(H), or simply t.mN(H)21. This explains theoverall behavior of lifetimes ~see Fig. 1!. On the other hand,if H encompasses a short periodic orbit ~period[l(H),lc),the two measures differ within the hole. Quantitatively, wesubstitute Pe→H in Eq. ~6! and approximate el(H)/t.1 andmC(M 2l(H)).mN(M 2l(H)) . This results int.12i~H !21mN~H !21. ~7!Although l(H) is small, the approximation a(M 2l(H)).1 isjustified if l(M 2l(H)).lc , or simply, if the period of thesecond shortest orbit within H exceeds lc . We have testedrelation ~7! and consequently the approximations that lead toit in a number of systems. We have compared t(H) withmN(H)21 by changing the position of H from ‘‘the mostexceptional’’ point, the fixed point, to longer cycles. In Fig.3 we display a test of Eq. ~7! for the generalized baker’s map~see Ref. @3#, p. 75, la50.35, lb50.40, a50.40, and b50.60), and for the He´non map ~see Ref. @16# a51.4,b50.3). Recalling that i(H) decreases exponentially withl(H), and considering Eq. ~7!, we see that the parameterS(H)512i(H)21 decreases rapidly towards 1 with theincrease of l(H) ~see Fig. 3!. Equation ~7! is robust and canbe applied for holes of different shapes, as long as t@1. If~for the 2D maps! we tailor the hole as a rectangle with sidesof length e parallel to the stable and unstable manifold seg-ments, and center it on a short periodic orbit, thent.~12Lu21!21mN~H !21. ~8!Lu denotes the magnitude of the expanding eigenvalue ofthat orbit. Equation ~8! also applies to 1D maps. Note thatthe approximation i(H).Lu21 assumes that the natural mea-sure is smooth along the unstable direction within H. Weobserve that t/mN(H)21 is independent of e . This is in ac-cordance with the statement that the lifetime t scales with ejust like mN(H)21 @1,4#.Let us consider an application of Eq. ~8!. Suppose that wewish to control a chaotic system around an unstable fixed4872 PRE 62V. PAAR AND H. BULJANpoint. In order to obtain the position of the fixed point, itsunstable eigenvalue, and other information required for thecontrol, an observation of the free running system is needed@1,2#. From this observation, we can also evaluate the visita-FIG. 3. Numerically evaluated parameter t/mN(H)21 for theHenon map ~diamonds! and the Baker map ~circles! in comparisonto 12i(H)21 ~horizontal bars!. The hole of radius e50.005 iscentered on the shortest periodic orbit @period5l(H)# visiting H.tion frequency to the e vicinity of the fixed point, i.e.,mNH[He(jW ). e is determined by the maximally alloweddeviation of the control parameter from its nominal value@1,2#. The question of interest is how many iterates areneeded on the average (t), before a chaotic trajectory entersthe region H, when the control becomes attainable @1#. Theprediction given by mN(H)21 is an underestimate, since weare on the fixed point. For example, if the underlying dynam-ics of the system is the asymmetric tent map ~with the sameparameters as in Fig. 1!, and if H5(0.621 118 01 . . .20.002, 0.621 118 0110.002), the estimate for the life-time mN(H)21 gives 250 iterates. On the other hand, thenumerically calculated lifetime is .627 iterates, which ismore than twice as long. The lifetime obtained from formula~8! is 641 iterates, which is very close to the numericallycalculated lifetime. Thus, Eq. ~8! can be utilized to easily andaccurately obtain t from an observation of the free runningsystem.In summary, we have studied the average lifetime (t) ittakes for a randomly started orbit to land in a small region~H! on a chaotic attractor. That problem was introduced inRef. @1# as an important issue for controlling chaos. Ourmain result is that if a low-period unstable periodic orbitvisits the region H, then the lifetime t significantly deviatesfrom the inverse of the natural measure contained within HmN(H)21. The parameter of this deviation, t/mN(H)21, isa function of the expanding eigenvalue of that low-periodorbit.@1# E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 64, 1196~1990!.@2# W.L. Ditto, S.N. Rauseo, and M.L. Spano, Phys. Rev. Lett. 65,3211 ~1990!; J. Singer, Y-Z. Wang, and H.H. Bau, ibid. 66,1123 ~1991!; U. Dressler and G. Nitsche, ibid. 68, 1 ~1992!; T.Shinbrot, C. Grebogi, E. Ott, and J.A. Yorke, Nature ~London!363, 411 ~1993!.@3# E. 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Bursts in the chaotic trajectory lifetimes preceding controlled periodic motion

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