Terumitsu Morita

On Partial Dimensions and Spectra of Singularities of Strange Attractors

New relations of thegenerClIized dimensions and entropies of strange attractors to the·fluctua· tions of divergence rates of nearby orbits and to the eigenvalues of the Jacobian matrices of unstable periodic points are obtained in order to derive the spectra of singularities from the dynamical viewpoints. A strange attractor often has a multifractal structure. The probability measure on such an attractor is highly concentrated in some regions and very rarefied in other regions. Repeated magnifications of a small piece of each region lead to a hierarchy of similar structures. Such a complicated structure of the probability measure can be described by the spectrum of scaling indices j(a),!) which can be obtained as follows. Cover the attractor with boxes and estimate the partition function r(q, r)=~i(Piq Il/), where li and Pi are the size and the probability of the ith box, respectively. As max(tJ~O, r goes to infinity for r>r(q) and to zero for r< r(q). This defines r(q) which are related to the generalized dimensions D(q) by r(q)=(q-1)D(q).2) The Legendre transformation of r(q),

q-Phase Transitions in Chaotic Attractors of Differential Equations at Bifurcation Points

Most nonlinear ordinary differential equations exhibit chaotic attractors which have singular local structures at their bifurcation points. By taking the driven damped pendulum and the Duffing equation, such chaotic attractors are studied in terms of the q-phase transitions of a q-weighted average A(q), (-oo<q<oo) of the coarse-grained expansion rates A of nearby orbits along the unstable manifolds. We take their Poincare maps in order to obtain the expansion rates A and their spectrum g\(A) explicitly. It is shown that q-phase transitions occur at crises in the differential equations. Just before the crises, qp-phase transitions occur due to the collisions of the attractors with unstable periodic orbits. Numerical values of the transition points qp thus obtained agree fairly well with theoretical predictions. q.-phase transitions occur just after the crises where the chaotic . attractors are suddenly spread over chaotic repellers. Thus it turns out that the q-phase transitions of A(q) are useful for characterizing the chaotic attractors of differential equations at their bifurcation points.

Spatial and Temporal Scaling Properties of Strange Attractors and Their Representations by Unstable Periodic Orbits

Scaling properties of chaos are studied from a dynamic viewpoint by taking invertible two· dimensional maps. A fundamental role is played by an evolution equation for the probability of a small box along a chaotic orbit. A new relation between the partial dimen~ions of strange attractors in the expanding and contracting direction is derived in terms of the local expansion rates of nearby orbits. When the lacobians of the maps are constant, this relation can be written in terms of a potential rJJ(q) for the fluctuations of the local expansion rate in the expanding direction. For hyperbolic attractors, this potential rJJ(q) is related to the generalized entropies K(q) by rJJ(q)=(q -l)K(q), and the above relation reduces to a simple relation between the generalized dimensions and the generalized entropies. Furthermore, the generalized dimensions, the potential rJJ(q) and the probability densities are expressed in terms of the local expansion rates of unstable periodic orbits within the attractors, leading to a new method of studying chaos.

Scaling Structures of Chaotic Attractors and q-Phase Transitions at Crises in the Hénon and the Annulus Maps

Singular local structures of the chaotic attractors at the bifurcation points for three types of crises in the Henon and the annulus maps are studied theoretically and numerically in terms of the spectrum heAl of the coarse· grained local expansion rates A of nearby orbits along the local unstable manifold, and are shown to produce linear parts in heAl with slopes qa=2, qp< 1/2 and qT= 1. These linear slopes bring about three types of discontinuous phase transitions of the q-weighted average A(q), (-oo<q<oo) of A at q=qa, qp, qT, respectively, as q is varied. The q-phase transition at q = qa is due to the homoclinic tangencies, whereas that at q = qp is caused by a collision of the chaotic attractor with an unstable periodic orbit, where the slope qp and the singularity exponent a of the natural invariant measure at the periodic orbit are given in terms of the eigenvalues of certain periodic orbits. Just after the merging of two chaotic attractors into one chaotic attractor, a q-phase transition occurs at q=qT due to the intermittent hopping motions between two phase-space regions formerly occupied by the old attractors.

A scaling method for deriving kinetic equations from the BBGKY hierarchy

On the basis of the scale covariance of correlation functions under a coarsegraining in space and time, the Boltzmann equation for neutral gases, the Balescu-Lenard-Boltzmann-Landau equation for dilute plasmas, and linear equations for the variances of fluctuations are derived from the BBGKY hierarchy equations with no short-range correlations at the initial time. This is done by using Moris scaling method in an extended form. Thus it is shown that the scale invariance of macroscopic features affords a useful principle in nonequilibrium statistical mechanics. It is also shown that there existtwo kinds of correlation functions, one describing the interlevel correlations of the kinetic level with its sublevels and the other representing the fluctuations in the kinetic level.

Classification of collisions of chaotic attractors with unstable periodic orbits in terms of the q-phase transitions

Les collisions des attracteurs chaotiques ayant des orbites periodiques instables sont classees en trois types en termes de transition a q phases juste avant les collisions: 1) le type avec un point de transition qβ qβ>1

A scaling method for obtaining kinetic equations from the BBGKY hierarchy

Abstract Kinetic equations for the one-particle distribution function of dilute gases and plasmas and evolution equations for the variance of their fluctuations are obtained from the BBGKY hierarchy equations by using Moris scaling method in an extended form.