PHYSICAL REVIEW E VOLUME 55, NUMBER 4 APRIL 1997
Differentiable generalized synchronization of chaos
Brian R. Hunt,* Edward Ott,† and James A. Yorke*,‡ University of Maryland, College Park, Maryland 20742 ͑Received 13 September 1996͒ We consider simple Lyapunov-exponent-based conditions under which the response of a system to a chaotic drive is a smooth function of the drive state. We call this differentiable generalized synchronization ͑DGS͒. When DGS does not hold, we quantify the degree of nondifferentiability using the Ho¨lder exponent. We also discuss the consequences of DGS and give an illustrative numerical example. ͓S1063-651X͑97͒02704-9͔
PACS number͑s͒: 05.45.ϩb
I. INTRODUCTION stable if there is a region of y space B such that, for any two ͑1͒ ͑2͒ ͑1͒ initial y vectors y0 ,y0 B, we have limt ϱʈy͑t,x0 ,y0 ͒ ͑2͒ ͑1͒ →͑2͒ Recently, the concept of generalized synchronization has Ϫy͑t,x0 ,y0 ͒ʈϭ0, and y͑t,x0 ,y ͒ and y͑t,x0 ,y ͒ are in the been introduced ͓1͔ to characterize the dynamics of a re- interior of B for t sufficiently large. In the remainder of this sponse system that is driven by the output of a chaotic driv- paper we assume the response system is asymptotically ing system. Generalized synchronization ͑GS͒ is said to oc- stable. cur if, ignoring transients, the response y is uniquely There are various situations motivating consideration of determined by the current drive state x. That is, yϭ͑x͒, GS. The most natural such situation occurs in the one-way where is a function of x for x on the chaotic attractor of synchronization of two oscillators. An important special case the drive system ͑the attractor is assumed to be bounded͒.If of the dynamics ͑1b͒ is that of linear coupling, GS applies, the x dynamics can typically be topologically ˜ reconstructed from the y dynamics. ͑Depending on the di- g„y,h͑x͒…ϭf͑y͒ϩC•͑xϪy͒, ͑2͒ mension of the vector y and on the fractal dimension of the attractor in x, reconstruction may require formation of a de- between two nearly identical oscillatory systems, where C is lay coordinate vector from y ͓2͔.͒ a coupling matrix, and ˜f and f are close. In cases where the x In the case of continuous time ͑flows͒ we can write the dynamics is chaotic and C is properly chosen, exact stable combined drive-response system as synchronism, y(t)ϭx(t), occurs provided fϭ˜f ͑see ͓1,3,4͔ and references therein͒. Experimentally, one cannot expect dx/dtϭf͑x͒, ͑1a͒ exact equality of f and ˜f, and hence one cannot expect exact synchronism. Even so, GS might apply, with essentially the dy/dtϭg y,h x , 1b „ ͑ ͒… ͑ ͒ same useful practical consequences as exact synchronism. where xRk, yRl, and f:Rk Rk, h:Rk Rm, and g:Rl Another situation occurs when we cannot observe the sys- ϫRm Rl are continuously differentiable→ functions.→ In the tem state x directly, and Eq. ͑1b͒ models the response of the case of→ a discrete time drive-response system, we write measurement apparatus to the system state. Still another ex- ample is where the response y is a linearly filtered version of
xtϩ1ϭF͑xt͒, ͑1aЈ͒ the input ͑e.g., see ͓5͔͒. More generally, we can expand this viewpoint to regard Eq. ͑1b͒ as a nonlinear filter. While knowledge of the existence of a relation of the ytϩ1ϭG„yt ,H͑xt͒…, ͑1bЈ͒ form yϭ͑x͒ is useful, it is often important to also consider where we assume F is invertible, G͑y,H͒ is invertible in y, the continuity and smoothness of the function . For ex- and F, G, and H are continuously differentiable. ample, it is known in the context of filtering ͓5͔ that the Kocarev and Parlitz ͓3͔ formulated a condition for the relationship between the filtered signal and the system state, occurrence of GS for the system ͑1͒, which, after a slight although expressible ͓6͔ in the form yϭ͑x͒, can be such reformulation, can be stated as follows: GS occurs if, for all that the attractor reconstructed from y may have a larger initial x0 in a neighborhood of the chaotic attractor of the information dimension than the original attractor in x space. drive system, the response system is asymptotically stable. That is, the function may be ‘‘wild’’ enough that it Recall that the response system is said to be asymptotically changes the attractor’s information dimension. If we are in- terested, for example, in deducing the attractor dimension of the drive system from observations of the response y, this is *Also at Institute for Physical Science and Technology, Univer- undesirable. One can also cite other examples where suffi- sity of Maryland, College Park, MD 20742. ciently smooth is desirable ͑e.g., obtaining eigenvalues of †Also at Departments of Electrical Engineering and of Physics, unstable periodic orbits or Lyapunov exponents of the drive Institute for Plasma Research, and Institute for Systems Research, system from observations of the response͒. Thus we wish to University of Maryland, College Park, MD 20742. consider a stronger version of GS that we denote differen- ‡Also at Department of Mathematics, University of Maryland, tiable generalized synchronization ͑DGS͒. By this we simply College Park, MD 20742. mean that there is GS and the function ͑x͒ is continuously
1063-651X/97/55͑4͒/4029͑6͒/$10.0055 4029 © 1997 The American Physical Society 4030 BRIAN R. HUNT, EDWARD OTT, AND JAMES A. YORKE 55 differentiable for x on the chaotic attractor of the drive sys- history exponents will be those of the periodic orbit, which tem. In addition, if DGS does not hold, we wish to quantify in general are different from those of typical orbits on the the degree of nondifferentiability using the Ho¨lder exponent. attractor. Remark. Note that the attractor of the drive system may To begin our quantitative discussion, we first consider the be smooth in some directions and fractal in others, and that case of a map where the drive system attractor has a set of we are primarily interested in the function ͑x͒ evaluated for past-history Lyapunov exponents that at each point x on the x on the attractor of the drive system. In this context we attractor consists of kϪ1 positive exponents and one nega- define differentiability of as follows. If there exists a ma- tive exponent, denoted by Ϫhd͑x͒. In this case we expect the trix “x, such that for small ␦ with xϩ␦ on the attractor, attractor structure at each point x to be smooth in the kϪ1 expanding directions and to be fractal in the contracting di- ͑xϩ␦͒ϭ͑x͒ϩ␦•“xϩo͑ʈ␦ʈ͒, rection corresponding to the Lyapunov exponent Ϫhd͑x͒. This situation applies for many examples encountered in then we say is differentiable at the point x on the attractor. practice ͑e.g., for chaotic attractors of invertible two- Note that this defines differentiability requiring the evalua- dimensional maps such as the He´non map, the Ikeda map, tion of only at points on the drive attractor and that the etc.͒; the considerations also translate readily to flows with definition includes directions cutting across the fractal struc- one negative Lyapunov exponent. Let Ϫhr͑x͒ denote the ture of the attractor. least-negative response-system past-history Lyapunov expo- nent corresponding to the point „x,yϭ͑x͒… ͑recall that we ¨ assume the response to be asymptotically stable, implying II. HOLDER EXPONENT AND DGS that all response exponents are nonpositive; we consider the Assuming GS applies, we first consider the Ho¨lder expo- case where the Lyapunov exponents of the response system nent ␥͑x͒ of the function ͑x͒ evaluated at the point ͑x,y͒, are negative͒. where x is on the drive attractor and yϭ͑x͒. For points x Our principal results for the case where the drive has only and xϩ␦ on the drive attractor, we define the Ho¨lder expo- one negative exponent are the following. Their application to nent ␥͑x͒ of ͑x͒ at x as the general case, in which there may be more than one nega- tive exponent, is discussed later. ͑i͒ The Ho¨lder exponent of the function ͑x͒ at a point x ␥͑x͒ϭ liminf͕logʈ͑xϩ␦͒Ϫ͑x͒ʈ/logʈ␦ʈ͖, ͑3͒ ␦ 0 on the drive attractor is one if hr͑x͒уhd͑x͒. For typical sys- → tems, if hr͑x͒Ͻhd͑x͒ the Ho¨lder exponent is if the right-hand side is less than one, and x ϭ1ifthe ␥͑ ͒ ␥͑x͒ϭh ͑x͒/h ͑x͒. ͑4͒ right-hand side is greater than one. r d It can be easily shown that ␥͑x͒Ͼ0 implies that is con- We discuss the meaning of the phrase ‘‘typical system’’ sub- tinuous at x, and that is not differentiable at x for ␥͑x͒Ͻ1. sequently. ͓Including the atypical cases, we have that, in If ͑x͒ is differentiable at x, then ␥͑x͒ϭ1. If ͑x͒ is discon- general, if hr͑x͒Ͻhd͑x͒, then ␥͑x͒уhr͑x͒/hd͑x͒.͔ tinuous at x, then ␥͑x͒ϭ0. On the other hand, ␥͑x͒ϭ1 does ͑ii͒ The function ͑x͒ is differentiable for all x on the not necessarily imply that ͑x͒ is differentiable at x, nor drive attractor ͑i.e., DGS applies͒,if does ␥͑x͒ϭ0 necessarily imply that ͑x͒ is discontinuous at x. hr͑x͒Ͼhd͑x͒ ͑5͒ We proceed to determine ␥͑x͒ in terms of the dynamics of the drive and response systems. At each point x on the drive for all x on the drive attractor. attractor, and at each corresponding point yϭ͑x͒ of the Recall that the existence of a Ho¨lder exponent, ␥͑x͒Ͼ0, response, we imagine that we evaluate the past-history claimed in ͑i͒, means that is continuous for points x on the Lyapunov exponents. That is, for time TϾ0 we look at the T drive-system attractor. In some cases, results in the literature preimage of x, evaluate the finite-time Lyapunov exponents ͓7͔ on the existence of invariant manifolds and their persis- over the orbit segment traveling from the T preimage of x to tence under perturbation yield conclusions similar to ͑but x, and then let T ϱ. For almost every point x with respect generally weaker than͒͑i͒and ͑ii͒. to the attractor’s→ natural measure, the same set of numbers We now give a heuristic argument for ͑i͒ and ͑ii͒. For for the past-history Lyapunov exponents will be found, and simplicity, consider the case where x is a point on a period p these are also the same as the forward Lyapunov exponents unstable periodic orbit embedded in the drive attractor. Let at x. ͑We call such points typical.͒ However, usually there is ␦0 denote an initial displacement from the point x, where also a dense set of points x on the attractor for which the xϩ␦0 is on the drive attractor. Using Eq. ͑1a͒ or Eq. ͑1aЈ͒ past-history Lyapunov exponents are different from those of take the displaced point forward in time by the amount n, typical points. The set of all these atypical points has zero ␦ ␦ , where nϭmp and m is an integer. Similarly, take 0→ n natural measure, but these points are nevertheless significant the point yϩ␦y0ϭ͑xϩ␦0͒ forward the same time n using in our considerations. Thus, in general, we must regard the Eq. ͑1b͒ or Eq. ͑1bЈ͒, ␦y0 ␦yn . According to the Hartman- past-history Lyapunov exponents as x dependent. The sim- Grobman theorem, there→ exists a change of variables that plest example illustrating x dependence of the past-history makes the dynamics linear in a finite region about the peri- Lyapunov exponents is the case where x lies precisely on the odic orbit ͑assuming generic eigenvalues͒. Thus it suffices to unstable manifold of an unstable periodic orbit in the chaotic consider the dynamics as linear. Assume ␦0 to be chosen to attractor of the drive system. In that case, as T ϱ, the T lie in the eigendirection of the linearized drive system corre- preimage of x approaches the periodic orbit. Thus→ the past- sponding to the eigenvalue yielding the contractive 55 DIFFERENTIABLE GENERALIZED SYNCHRONIZATION... 4031 x-Lyapunov exponent, Ϫhd͑x͒. We wish to examine the be- Which of these two behaviors dominates at large n is deter- havior of ␦y0ϭ͑xϩ␦0͒Ϫ͑x͒ as it is iterated forward in mined by whether hdϾhr or hdϽhr . We have after a large time. To do this we write the linearized drive-response sys- number of iterates tem as znϩ1ϭZzn where exp͓Ϫhr͑x͒n͔,ifhr͑x͒Ͻhd͑x͒ x ␦y0 ␦ynϳ , zϭ , → ͭ exp͓Ϫhd͑x͒n͔,ifhr͑x͒уhd͑x͒ ͫyͬ ͑13a͒
XO ␦ ␦ ϳexp͓Ϫh ͑x͒n͔, ͑13b͒ Zϭ . ͑6͒ 0→ n d ͫ WYͬ where ␦ynϭ͑xϩ␦n͒Ϫ͑x͒. For hr͑x͒Ͻhd͑x͒, we have Here x is a k vector, y is an l vector, z isa(kϩl) vector, O from Eq. ͑13͒, is the lϫk zero matrix, and the matrices X, Y, and W have dimensions kϫk, lϫl, and kϫl, respectively. Generically, X ␥͑x͒ ʈ͑xϩ␦n͒Ϫ͑x͒ʈϳʈ␦nʈ , ͑14͒ and Y have distinct eigenvalues,
where ␥͑x͒ is given by Eq. ͑4͒. Thus, since ␦n 0asn ϱ, Xexpϭxpexp , pϭ1,2,...,k, ͑7͒ → → Eq. ͑14͒ yields Eq. ͑4͒. Similarly, for the case hr͑x͒уhd͑x͒, Eqs. ͑13͒ again yield Eq. ͑14͒, but with ␥͑x͒ϭ1. This argu- Ye ϭ e , qϭ1,2,...,l. ͑8͒ yq yq yq ment applies for x being a periodic point. Another argument, not given here, shows that the same result applies to the case The matrix Z has eigenvalues xp , yq with corresponding (kϩl)-dimensional eigenvectors e and e , where x is any nonperiodic point on the attractor. ͑In the case zxp zyq of a periodic orbit the past-history and forward Lyapunov exponents are the same. When x is nonperiodic they are not Zezxpϭxpezxp , pϭ1,...,k, ͑9͒ necessarily the same, and it is the past-history exponents that are relevant.͒ Zezyqϭyqezyq , qϭ1,...,l. ͑10͒ In order to demonstrate ͑ii͒, we note that when The eigenvectors are hr͑x͒Ͼhd͑x͒, taking ␦0ϭ␦0exp , for any pϭ1,2,...,k, the lin- ear dynamics ͑7͒ always results in exp ezxpϭ , ͫ eˆp ͬ e n xp ␦znХxp␦0 , ͑15͒ ͫ eˆp ͬ O ezyqϭ , ͑11͒ ͫeyqͬ for sufficiently large n. Thus, since the exp generically span the x space, we have that the derivatives of x exist and Ϫ1 ͑ ͒ where eˆpϭ͑xp1lϪY͒ Wexp , O is the k-dimensional zero are given by vector, the lϫl unit matrix is denoted 1l and the indicated inverse exists assuming the generic condition xpÞyq for exp•“xϭeˆp , pϭ1,2,...,k. ͑16͒ all p and q. Our original question of what happens to ␦y0 as it is iterated can be addressed by considering the entire sys- ͓In the case of complex , we can take the real and imagi- tem with initial displacement xp nary parts of ͑16͒.͔ In the above discussion obtaining ͑i͒ it was assumed that ␦0 ␦z0ϭ , the drive-response system was typical in that the chosen ͫ␦y0ͬ direction for ␦0 ͑namely, ␦0ϭ␦0exd͒ yields ␦y0 , which results in a nonzero component of ␦y0Ϫ␦0eˆd along the response- where the initial x displacement is ␦0ϭ␦0exd and exd is the x eigenvector corresponding to the contractive eigendirection system eigendirection corresponding to the exponent Ϫhr͑x͒. with eigenvalue , ͉ ͉ϭexp͑Ϫh ͒. Writing ␦z as As an example where this is not the case, consider the xd xd d 0 situation where the response system is of the form ͑2͒ and the drive and response are exactly matched, ˜fϭf. In this case exd O ␦z0ϭ␦0 ϩ , ͑12͒ we can have exact synchronism, yϭx ͓i.e., ͑x͒ϭx͔. Thus, ͫ eˆd ͬ ͫ␦y0Ϫ␦0eˆdͬ even though Eq. ͑5͒ may be violated, the surface yϭ͑x͒ we see that the first vector on the right-hand side of Eq. ͑12͒ is still smooth ͑it is the hyperplane xϭy͒. However, a generic n ˜ evolves with time as ͉xd͉ ϭexp͑Ϫnhd͒. Typically we ex- perturbation of the function f away from f restores the pect that ␦y0Ϫ␦0eˆd has no special relationship to the eigen- validity of Eq. ͑4͒. On the other hand, we note that, if this directions of Y. In this case all of the eigendirections of Y perturbation is small, the resulting component of ␦y0 along will be ‘‘excited.’’ For large time n the response eigenvalue the response-system eigendirection corresponding to the n of largest magnitude will dominate, ͉yr͉ ϭexp͑Ϫnhr͒. exponent Ϫhr͑x͒ is expected to be small ͑it is zero when Thus, for large n, ␦yn consists of two components: one ex- the perturbation is zero͒. Thus in this case we might, cited by the first vector on the right-hand side of Eq. ͑12͒ for example, expect a ␦ dependence of the form ␥(x) decaying as exp͑Ϫnhd͒, and one excited by the second vec- ʈ(xϩ␦)Ϫ(x)ʈХKʈ␦ʈ to apply with a relatively small tor on the right-hand side of Eq. ͑12͒ decaying as exp͑Ϫnhr͒. value of K. A small enough value of K would have the effect 4032 BRIAN R. HUNT, EDWARD OTT, AND JAMES A. YORKE 55 that, in the presence of limited precision measurement and/or small noise, the consequences of nondifferentiability may be unobservable. Note that having only one negative drive exponent im- plies fractal structure of the drive attractor in the eigendirec- tion corresponding to Ϫhd͑x͒. Thus our stipulation that we can pick a small ␦0 aligned along the contracting direction such that xϩ␦0 is a point on the attractor can be satisfied. In the case where there are several contracting directions for the response system, the attractor may be ‘‘empty’’ in some of the contracting directions ͑i.e., for some of the contracting FIG. 1. The generalized baker’s map, Eqs. ͑17͒. directions, there may be no small ␦0 aligned along that con- ͑1͒ ͑2͒Ͻ tracting direction such that both x and xϩ␦0 are on the at- ͑1͒ axn , if xn ␣ xnϩ1ϭ ͑1͒ ͑2͒ , ͑17a͒ tractor͒. In that case, by slightly extending our consider- ͭ 1ϩbxn , if xn у␣ ations, our previous discussion can still be applied. To do this, at each point x on the attractor we consider those nega- 1 x͑2͒ ,ifx͑2͒Ͻ␣ tive drive exponents that correspond to eigendirections that ␣ n n locally intersect the attractor at more than a single point x͑2͒ ϭ ͑17b͒ nϩ1 1 ͑‘‘nonempty’’ eigendirections͒, and we take the most nega- ͑x͑2͒Ϫ␣͒,ifx͑2͒Ͻ␣ ͭ n n tive of those, which we now denote by Ϫhd͑x͒. In terms of this new designation of h ͑x͒, our statements ͑i͒ and ͑ii͒ are d where ϩ ϭ1 and we also take ϩ ϭ1. See Fig. 1. The still expected to apply. The reason for taking the most nega- ␣  a b response system assumed to be of the form tive intersecting drive exponent is that the definition of ␥͑x͒, Eq. ͑3͒, specifies a limit inferior over ␦, and ␥͑x͒ from Eq. ͑1͒ y nϩ1ϭy nϩxn , ͑18͒ ͑4͒ is smallest for larger hd͑x͒. To see that there may be contracting directions that are empty in the sense that they can be considered as a discrete version of a low-pass filter intersect the attractor only at a single point, recall that high- ͓10͔. The attractor for Eq. ͑17͒ has a natural measure that is dimensional systems can often be shown to possess lower- uniform in x͑2͒ and varies wildly in the x͑1͒ direction pro- dimensional ‘‘inertial manifolds’’ ͓8͔ such that there is a vided that aÞ␣. The box-counting dimension of the attrac- dynamical system for state points in the inertial manifold, tor is D ϭ2, since typical trajectories are dense in the unit and this dynamical system yields all the ergodic invariant 0 square by virtue of aϩbϭ1. The past-history Lyapunov sets of the original higher-dimensional system ͑in particular, exponents for the drive system evaluated at a point its chaotic attractors͒. In that case contracting eigendirections xϭ(x(1),x(2)) are ͓9͔ transverse to the inertial manifold are clearly empty. In the absence of knowledge as to which of the contracting-drive 1 1 h ͑x͒ϭa͑x͒ln ϩb͑x͒ln Ͼ0 ͑19a͒ eigenvalues are empty, statements ͑i͒ and ͑ii͒ are still useful 1 ␣͑x͒ ͑x͒ in that use of the most contracting-drive Lyapunov exponent in place of h ͑x͒ provides a lower bound on ␥͑x͓͒statement d h ͑x͒ϭa͑x͒ln ϩb͑x͒ln Ͻ0, ͑19b͒ ͑i͔͒, and a sufficient condition for DGS ͓Eq. ͑5͔͒. 2 a b Finally, we remark that the condition for DGS can some- where times be verified in terms of finite-time Lyapunov exponents. The time-T Lyapunov exponents of a system are defined to na͑x͒ be 1/T times the logarithms of the singular values of the a͑x͒ϭ lim , ͑20a͒ Jacobian matrix of the time-T map of the system. ͑The sin- n ϱ n → gular values of a matrix M are the square roots of the eigen- values of MMT, where MT is the transpose of M. If, for ͒ nb͑x͒ some T and for all x on the attractor, the most negative b͑x͒ϭ lim . ͑20b͒ n ϱ n time-T Lyapunov exponent of the drive system at x exceeds → the least-negative time-T Lyapunov exponent of the response system at ͓x,͑x͔͒, then we can show that condition ͑5͒, and In Eq. ͑20͒, na͑x͓͒nb͑x͔͒ is the number of times the first n ͑2͒ ͑2͒ hence DGS, holds. ͑It may sometimes be possible to verify preiterates from x are in x Ͻ␣ ͓x у␣͔. ͓Note that a͑x͒ this finite-time condition for a single iteration in the case of ϩb͑x͒ϭ1.͔ By virtue of the symbolic dynamics for Eq. ͑17͒, ͑2͒ ͑2͒ a map or an infinitesimal time step in the case of a flow.͒ there are orbits that visit the regions x Ͻ␣ and x у␣ in any order. Thus any value for a͑x͒ in ͓0,1͔ can be attained by proper choice of x. On the other hand, if x is randomly cho- ͑1,2͒ III. AN EXAMPLE sen in the area of the unit square ͑0рx р1͒, we have a͑x͒ϭ␣ and b͑x͒ϭ with probability one ͓i.e., the natural We now consider a simple example. The drive system is measure of the region x͑2͒Ͻ␣ ͑x͑2͒у␣͒ is ␣ ͔͑͒. Thus the given by a generalized baker’s map ͓9͔, which takes the unit Lyapunov exponents for typical points on the drive attractor square, 0рx͑1͒Ͻ1, 0рx͑2͒Ͻ1, to itself, are 55 DIFFERENTIABLE GENERALIZED SYNCHRONIZATION... 4033
FIG. 2. y vs x͑1͒ for Eqs. ͑17͒ and ͑18͒.
1 1 attractor at which hrϽ͉h2͑x͉͒. In particular, if aϽb , then ¯h ϭ␣ ln ϩ ln Ͼ0, ͑21a͒ 1 ␣  by Eq. ͑19b͒, we see that, depending on x, ͉h2͑x͉͒ can attain any value in the range ¯ h2ϭ␣ lnaϩ lnbϽ0. ͑21b͒ ln͑1/a͒у͉h2͑x͉͒уln͑1/b͒. ͑23͒ Hence, although the box-counting dimension of the drive attractor is D0ϭ2, by the Kaplan-Yorke formula ͓9,11͔, its information dimension is between 1 and 2, We now consider the case where ¯ ¯ ¯ ln͑1/a͒ϾhrϾ͉h2͉. ͑24͒ D1ϭ1ϩ͑h1 /͉h2͉͒. ͑22͒ In this case there are points x at which the Ho¨lder exponent Application of the Kaplan-Yorke formula to the combined predicted by Eq. ͑4͒ is less than one. The natural measure of drive-response system Eqs. ͑17͒ and ͑18͒ again yields Eq. these points is zero, but they are dense in the attractor. Thus, ¯ Ϫ1 ͑22͒ provided that hrϾ͉h2͉ ͑where hrϭln ͒, but yields a although the information dimension is preserved, the surface ¯ larger value of the dimension if hrϽ͉h2͉. Figure 2͑a͒ shows yϭ͑x͒ is still nonsmooth. Figures 2͑b͒ and 2͑c͒ show the a numerical computation of the surface yϭ͑x͒ for a case results of numerical computations of the surface yϭ͑x͒ for Ϫ1 ¯ satisfying hrϭln Ͻ͉h2͉ ͓aϭ0.2, ϭ0.8͔. The resulting two cases satisfying Eq. ͑24͓͒aϭ0.2, ϭ0.6 for Fig. 2͑b͒, curve is fractal as indicated by its very wrinkled appearance. aϭ0.2, ϭ0.4 for Fig. 2͑c͒, and we assume ␣р0.2 in both ͑Note that since is independent of x͑2͒ and ␣, it suffices to cases ͓12͔͔. The effect of the dense set where ␥͑x͒Ͻ1 clearly plot y versus x͑1͒ and Fig. 2͑a͒ is valid for all 0Ͻ␣Ͻ1.͒ manifests itself in the plot shown in Fig. 2͑b͒ giving the Ϫ1 ¯ Now consider the case where hrϭln Ͼ͉h2͉. In this surface an extremely wrinkled appearance. The other case case the filter does not change the Kaplan-Yorke dimension; satisfying Eq. ͑24͒, Fig. 2͑c͒ appears less wrinkled since hr is i.e., the attractor of the combined drive-response system, closer to ln͑1/a͒. Finally Fig. 2͑d͒ shows a case where Eqs. ͑17͒ and ͑18͒, still has the information dimension given hrϾln͑1/a͒͑aϭ0.2, ϭ0.1͒ in which case the function ¯ by Eq. ͑22͒. On the other hand, even though hrϾ͉h2͉, there yϭ͑x͒ is predicted to be differentiable everywhere. As pre- is still the possibility that there are points x on the drive dicted, the curve in Fig. 2͑d͒ appears to be smooth. 4034 BRIAN R. HUNT, EDWARD OTT, AND JAMES A. YORKE 55
Note that, as we increase from zero, ͑x͒ first loses tractor ͓13͔, and the critical is just this minimum Lyapunov differentiability as passes through a , which is the number. Lyapunov number of the period one unstable periodic orbit (1) (2) (x ,x )ϭ͑0,0͒. More generally, consider the system ACKNOWLEDGMENTS ͑1aЈ͒,͑1bЈ͒, where the k-dimensional drive system is uni- formly expanding in k-1 directions and is uniformly con- We thank L. M. Pecora for useful discussions and for tracting in one direction, and the response system is a linear providing us with a preprint of his paper with T. L. Carroll filter of the form ynϩ1ϭ⌳ynϩAxn , where ⌳ and A are ma- ͓5͔. This research was supported by the Office of Naval Re- trices. Let Ͻ1 denote the magnitude of the largest eigen- search, by the U.S. Department of Energy ͑Mathematical, value of ⌳. Now say that the parameters of the filter are Information, and Computational Sciences Division, High varied. We conjecture that the bifurcation at which ͑x͒ first Performance Computing and Communications Program͒, becomes nondifferentiable typically occurs as increases and by the National Science Foundation ͑Divisions of Math- through a critical value that is determined by a special low- ematical and Physical Sciences͒. The numerical computa- period periodic orbit which has the smallest ‘‘contracting’’ tions reported in this paper were made possible by a grant Lyapunov number among all the periodic orbits on the at- from the W. M. Keck Foundation.
͓1͔ N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. exponent-based conditions analogous to ͑ii͒ for the persistence Abarbanel, Phys. Rev. E 51, 980 ͑1996͒. Generalized synchro- of smooth invariant manifolds under perturbation; these results nism in the context of two mutually coupled systems is con- imply ͑ii͒ in the case where the drive attractor is a compact sidered in V. S. Afraimovich, N. N. Verichev, and M. I. manifold and the response system is linear in y. M. W. Hirsch Rabinovich, Radiophys. Quantum Electron 29, 795 ͑1986͒. and C. C. Pugh ͓Proc. Symp. Pure Math. 14, 133 ͑1970͔͒ and ͓2͔ T. Sauer, J. A. Yorke, and M. Casdagli, J. Stat. Phys. 65, 579 M. W. Hirsch, C. C. Pugh, and M. Shub ͓Invariant Manifolds, ͑1991͒; E. Ott, T. Sauer, and J. A. Yorke, Coping with Chaos Lecture Notes in Mathematics Vol. 583 ͑Springer-Verlag, New ͑Wiley, New York, 1994͒, Chap. 5; F. Takens, in Dynamical York, 1977͔͒ give results like the lower bound on ␥͑x͒ in ͑i͒ Systems and Turbulence, edited by D. Rand and L. S. Young and ͑ii͒, but requiring uniform ͑Lipschitz-like͒ bounds on the ͑Springer-Verlag, Berlin, 1981͒, p. 366. contraction rates, again assuming in the case of ͑ii͒ that the ͓3͔ L. Kocarev and V. Parlitz, Phys. Rev. Lett. 76, 1816 ͑1996͒. drive attractor is a compact manifold. More recently J. Stark ͓4͔ H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69,32͑1983͒; has considered the case of general drive attractors and has T. L. Carroll and L. M. Pecora, Physica D 67, 126 ͑1993͒;K. independently obtained results like ͑i͒ and ͑ii͒ concerning the M. Cuomo and A. V. Oppenheim, Phys. Rev. Lett. 71,65 regularity of at ‘‘almost every’’ point; we, on the other ͑1993͒; M. Ding and E. Ott, Phys. Rev. E 49, R945 ͑1994͒;L. hand, consider the regularity at all points. Kocarev and V. Parlitz, Phys. Rev. Lett. 74, 5028 ͑1995͒. ͓8͔ For review articles on inertial manifolds see R. Temam, Math. ͓5͔ R. Badii et al., Phys. Rev. Lett. 60, 979 ͑1988͒; A. Chennaoui, Intelligencer 12,68͑1990͒; and I. D. Chueshov, Uspekhi Mat. J. Liebler, and H. G. Schuster, J. Stat. Phys. 59, 1311 ͑1990͒; Nauk. 48, 135 ͑1993͓͒or in English translation Russ. Math. F. Mitschke, Phys. Rev. A 41, 1169 ͑1990͒; D. S. Broomhead, Surveys 48, 133 ͑1993͔͒. J. P. Huke, and M. R. Muldoon, J. R. Stat. Soc. B 54, 373 ͓9͔ J. D. Farmer, E. Ott and J. A. Yorke, Physica D 7, 153 ͑1983͒; ͑1992͒; L. M. Pecora and T. L. Carroll, Chaos 6, 432 ͑1996͒; E. Ott, Chaos in Dynamical Systems ͑Cambridge University B. R. Hunt, E. Ott, and J. A. Yorke, Phys. Rev. E 54, 4819 Press, Cambridge, 1993͒, pp. 75–88. ͑1996͒. ͓For an earlier work addressing dimension computa- ͓10͔ P. Paoli, A. Politi, G. Broggi, M. Ravani, and R. Badii, Phys. tions related to filtering theory, see J. L. Kaplan, J. Mallet- Rev. Lett. 62, 2429 ͑1989͒. Paret, and J. A. Yorke, Ergodic Theory and Dynamical Sys- ͓11͔ J. L. Kaplan and J. A. Yorke, Functional Differential Equa- tems 4, 261 ͑1984͒.͔ tions and Approximations of Fixed Points, Lecture Notes in ͓6͔ In the case of a discrete-time, causal, low-pass filter Mathematics Vol. 730 ͑Springer-Verlag, Berlin, 1979͒, p. 204. y nϩ1ϭ͑y nϩk•xn͒, where ͉͉Ͻ1 and k is a constant vector, ͓12͔ For some values of ␣Ͼ0.2, Eq. ͑24͒ does not hold, and the ϱ m we can write y n as y nϭ͚ mϭ1 k•xnϪm . Since xn evolves information dimension of the attractor is not preserved, though from an invertible dynamical system, xnϩ1ϭF͑xn͒, we have Figs. 2͑b͒ and 2͑c͒ are unaffected. ϱ m Ϫm ͑x͒ϭ͚ mϭ1 k•F ͑x͒. This sum converges since ͉͉Ͻ1, ͓13͔ That the attainment of the minimum-contracting Lyapunov and it is a continuous function of x if F͑x͒ is continuous. number typically occurs on a low-period periodic orbit is im- ͓7͔ R. J. Sacker ͓J. Math. Mech. 18, 705 ͑1969͔͒ and N. Fenichel plied by the work in B. R. Hunt and E. Ott, Phys. Rev. Lett. ͓Indiana Univ. Math. J. 21, 193 ͑1971͔͒ obtain Lyapunov 76, 2254 ͑1996͒.