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Differentiable Generalized Synchronization of Chaos

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Differentiable Generalized Synchronization of Chaos 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 1996! 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 ~DGS!. 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. @S1063-651X~97!02704-9# PACS number~s!: 05.45.1b I. INTRODUCTION stable if there is a region of y space B such that, for any two ~1! ~2! ~1! initial y vectors y0 ,y0 PB, we have limt `iy~t,x0 ,y0 ! ~2! ~1! →~2! Recently, the concept of generalized synchronization has 2y~t,x0 ,y0 !i50, and y~t,x0 ,y ! and y~t,x0 ,y ! are in the been introduced @1# 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 ~GS! 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, y5f~x!, GS. The most natural such situation occurs in the one-way where f 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 bounded!.If of the dynamics ~1b! 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,h~x!…5f~y!1C•~x2y!, ~2! 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 @2#.! a coupling matrix, and ˜f and f are close. In cases where the x In the case of continuous time ~flows! we can write the dynamics is chaotic and C is properly chosen, exact stable combined drive-response system as synchronism, y(t)5x(t), occurs provided f5˜f ~see @1,3,4# and references therein!. Experimentally, one cannot expect dx/dt5f~x!, ~1a! exact equality of f and ˜f, and hence one cannot expect exact synchronism. Even so, GS might apply, with essentially the dy/dt5g y,h x , 1b „ ~ !… ~ ! same useful practical consequences as exact synchronism. where xPRk, yPRl, and f:Rk Rk, h:Rk Rm, and g:Rl Another situation occurs when we cannot observe the sys- 3Rm Rl are continuously differentiable→ functions.→ In the tem state x directly, and Eq. ~1b! 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 xt115F~xt!, ~1a8! the input ~e.g., see @5#!. More generally, we can expand this viewpoint to regard Eq. ~1b! as a nonlinear filter. While knowledge of the existence of a relation of the yt115G„yt ,H~xt!…, ~1b8! form y5f~x! is useful, it is often important to also consider where we assume F is invertible, G~y,H! is invertible in y, the continuity and smoothness of the function f. For ex- and F, G, and H are continuously differentiable. ample, it is known in the context of filtering @5# that the Kocarev and Parlitz @3# formulated a condition for the relationship between the filtered signal and the system state, occurrence of GS for the system ~1!, which, after a slight although expressible @6# in the form y5f~x!, 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 f 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 f 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 response!. 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 ~DGS!. By this we simply College Park, MD 20742. mean that there is GS and the function f~x! is continuously 1063-651X/97/55~4!/4029~6!/$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 f~x! 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 k21 positive exponents and one nega- define differentiability of f as follows. If there exists a ma- tive exponent, denoted by 2hd~x!. In this case we expect the trix “xf, such that for small d with x1d on the attractor, attractor structure at each point x to be smooth in the k21 expanding directions and to be fractal in the contracting di- f~x1d!5f~x!1d•“xf1o~idi!, rection corresponding to the Lyapunov exponent 2hd~x!. This situation applies for many examples encountered in then we say f 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 f 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 2hr~x! denote the ture of the attractor. least-negative response-system past-history Lyapunov expo- nent corresponding to the point „x,y5f~x!… ~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 g~x! of the function f~x! evaluated at the point ~x,y!, are negative!. where x is on the drive attractor and y5f~x!. For points x Our principal results for the case where the drive has only and x1d on the drive attractor, we define the Ho¨lder expo- one negative exponent are the following. Their application to nent g~x! of f~x! at x as the general case, in which there may be more than one nega- tive exponent, is discussed later. ~i! The Ho¨lder exponent of the function f~x! at a point x g~x!5 liminf$logif~x1d!2f~x!i/logidi%, ~3! d 0 on the drive attractor is one if hr~x!>hd~x!. For typical sys- → tems, if hr~x!,hd~x! the Ho¨lder exponent is if the right-hand side is less than one, and x 51ifthe g~ ! g~x!5h ~x!/h ~x!. ~4! right-hand side is greater than one. r d It can be easily shown that g~x!.0 implies that f is con- We discuss the meaning of the phrase ‘‘typical system’’ sub- tinuous at x, and that f is not differentiable at x for g~x!,1. sequently. @Including the atypical cases, we have that, in If f~x! is differentiable at x, then g~x!51. If f~x! is discon- general, if hr~x!,hd~x!, then g~x!>hr~x!/hd~x!.# tinuous at x, then g~x!50. On the other hand, g~x!51 does ~ii! The function f~x! is differentiable for all x on the not necessarily imply that f~x! is differentiable at x, nor drive attractor ~i.e., DGS applies!,if does g~x!50 necessarily imply that f~x! is discontinuous at x. hr~x!.hd~x! ~5! We proceed to determine g~x! in terms of the dynamics of the drive and response systems.
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