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Unstable Periodic Orbits and the Dimensions of Multifractal Chaotic Attractors

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Unstable Periodic Orbits and the Dimensions of Multifractal Chaotic Attractors PHYSICAL REVIE%' A VOLUME 37, NUMBER 5 MARCH 1, 1988 Unstable periodic orbits and the dimensions of mnltifractal chaotic attractors Celso Grebogi Laboratory for Plasma and Fusion Energy Studies, Uniuersity ofMaryland, College Park, Maryland 20742 Edward Ott Laboratory for Plasma and Fusion Energy Studies, Uniuersity ofMaryland, College Park, Maryland 20742 and Department ofElectrical Engineering and Department ofPhysics, Uniuersity ofMaryland, College Park, Maryland 20742 James A. Yorke Institute for Physical Science and Technology and Department ofMathematics, Uniuersity ofMaryland, College Park, Maryland 20742 (Received 28 September 1987) The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily Sne-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singulari- ty scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic or- bits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and chaotic repeBers are considered. I. INTRODUCTION the distribution of energy levels can be related to unstable periodic orbits of the classical Hamiltonian. " ' Anoth- The long time distribution generated by a typical orbit er case where unstable periodic orbits appear' is in of a chaotic nonconservative dynamical system is general- determining the behavior near parameter values where ly highly singular. The subset of phase space to which sudden changes in chaotic attractors occur [the argument the orbit asymptotes with time, the attractor, can be in connection with our Fig. 1 is similar to that for Eq. (2) geometrically fractal. Furthermore, the distribution of of Ref. 10]. orbit points on the attractor can have an arbitrarily 6ne- The organization of this paper is as follows. Section II scaled interwoven structure of hot and cold spots. Sets presents a discussion of the pointwise dimension for at- with such distributions have been called multifractals. tractors and shows that hot and cold spots occur on the By hot and cold spots we mean points on the attractor for unstable manifolds of saddle periodic orbits in the attrac- which the frequency of close approach of typical orbits is tor, Numerical experiments illustrating this are also either much greater than typical (a hot spot) or much less presented. Section III reviews recent work on the dimen- than typical (a cold spot). Recently there has been much sions of attractors including the partition function for- work developing ways of quantitatively characterizing malism. Section IV presents our results relating the dis- how such chaotic orbits distribute themselves on attrac- tribution of typical chaotic orbits on attractors and the tors. ' In particular, the spectrum of fractal dimensions associated fractal dimensions to the unstable periodic or- introduced in Refs. 2-4 are sensitive to the characteris- bits. Section V illustrates the material of Sec. IV with ex- tics of the structure of hot and cold spots on the attrac- amples. Arguments yielding the results stated in Sec. IV tor. In this paper we present results which show that, for are presented in Sec. VI for the case of hyperbolic attrac- a large class of chaotic attractors, the infinite number of tors. Section VII treats the case of chaotic sets which are unstable periodic orbits embedded in the attractor pro- repelling rather than attracting. vide the key to an understanding of such issues. (A brief The dynamical systems to be discussed throughout this preliminary report of some of this work appears in Gre- paper are d-dimensional maps of the form x„+& F(x„), — bogi, Ott, and Yorke. ) where x is a vector in the d-dimensional phase space of The importance of unstable periodic orbits in deter- the system. An attractor A for such a system is a closed mining ergodic properties of chaotic systems has long set, invariant under I', which is the limit set as time goes been recognized in the mathematical literature (e.g., to + Oo for almost every initial condition in some neigh- Bowen and Katok ). For some more recent work see borhood of A. (By "almost every" we mean that the set Refs. 8 and 9 which also illustrate the important point of initial conditions in the neighborhood that do not ap- that information about unstable periodic orbits is readily proach A can be covered by a set of d-dimensional cubes accessible from numerical computation (and perhaps ex- of arbitrarily small total volume. ) The basin of attraction perimentally ' ) and can be used for determining ergodic for the attractor is the closure of the set of points which properties. In addition, in the theory of quantum chaos, asymptote to the attractor as time goes to + 00. In the 1988 The American Physical Society CELSO GREBOGI„EDWARD OTT, AND JAMES A. YORKE case of continuous time systems (ffows), we can think of sure, the attractor has been decomposed into two disjoint F(x ) as arising from a Poincare surface of section. F invariant sets. We conclude that D~(x) must be the same for almost every x with respect to the natural mea- sure on the attractor. Returning now to consideration of the zero measure sei For most purposes w'e may think of the natural mea- of points x for which Dz(x ) is not typical (i.e., is not the sure an attractor as follows: For a subset of the of 5 common value assumed at almost every x on the attrac- phase space and an initial condition x in the basin of at- tor) and taking the map to be two dimensional (d =2), we traction of the attractor, we define S ) as the fraction p(x, will obtain the following result. Let be an index label- of time the trajectory originating at x spends in S in the j ing the fixed points of the n times iterated map F". (The limit that the length of the trajectory goes to infinity. If components of a period n orbit are fixed points of F".} is the same for almost every x in the basin of at- p(x, S) We assume that the Jacobian matrix of I"at fixed point traction, then we denote this value and say that is j p(S} p has one unstable direction and one stable direction. Then the natural measure of the attractor (cf. Appendix). for any point x on the unstable manifold of fixed point j Henceforth, we assume that the attractor has a natural of I'", measure. In particular, this means that the attractor is ergodic (i.e., it cannot be split into two disjoint pieces logk, iJ. that each have positive natural measure and are invariant D~(x ) =1 logA, under application of F). » Let 8 (l,x ) denote a d-dimensional ball of radius I cen- where lL, , & 1 and A,» & 1 are the magnitudes of the unsta- tered at a point x on an attractor embedded in the d- ble and stable eigenvalues of the Jacobian matrix of F". dimensional phase space of the dynamical system being Since points on different periodic orbits typically have considered. Then the pointwise dimension (at the point diff'erent eigenvalues, Dz(x) will clearly be different for x) of the attractor is defined as different periodic orbits and hence will not be the typical ). logy. (8(l,x }) Dp(x (2.1) To obtain (2.2) consider a point on the unstable I-o logl xp manifold of a saddle periodic point and two small circu- D (x) or p(B(l,x))-l ~ . For almost every point with lar disks centered at xo with radii l, and l2, where respect to the natural measure on the attractor, D~(x) I, /12 —A, 2J'. We iterate the two disks backward a large takes on a common value and is equal to the information integral number of periods so that the two disks are now dimension (defined in Sec. III). That is, the set of points similar ellipses close to the saddle and with their major on the chaotic attractor for which D (x ) is not this com- axes parallel to the stable manifold of the saddle (cf. Fig. mon value may be covered with a set of d-dimensional 1). We now iterate the 12 ellipse backward one more cubes of varying sizes which together contain an arbi- period. Since it is close to the saddle, its backward itera- trarily small amount of the natural measure of the attrac- tion by one period is governed by the linearized map at tor. [Points x where D (x ) is greater than (less than) the the saddle (i.e., by the eigenvalues A, ,J and A, »). Thus, common value it assumes at almost every point with since we choose l, /12 —A, 2J', the major diameter of the l2 respect to the natural measure are the hot (cold) spots re- ellipse is now the same as that for the 1, ellipse, while its ferred to in Sec. I.] For example, a chaotic attractor typi- minor diameter is smaller than that for the !i ellipse by cally has a dense set of unstable periodic orbits embedded the factor A,z /A, tJ. The inverse images of the disks con- within it, and, as we shall see, Dr(x) with x on one of tain the same natural measure as the original disks. these periodic orbits does not take on the typical values. Thus, treating the attractor measure as if it were smooth The periodic points, however, are countable and so have along the unstable direction, we have p{B(lt xp)}/ zero measure.
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