Hiroki Hata

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.

On-off intermittency in oscillatory media

Abstract On-off intermittency observed in three types of spatially extended dynamical systems is reported. This is done by examining the linear stability of spatially synchronized state under a spatially inhomogeneous fluctuation. When the system size slightly exceeds the critical size and a single inhomogeneous mode becomes unstable, systems exhibit typical on-off intermittency, whose statistical characteristics such as distribution of burst amplitude, Fourier spectrum and distribution of laminar duration turn out to be in agreement with those known in the on-off intermittency for small degrees of freedom system.

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

Dynamics on Critical Tori at the Onset of Chaos and Critical KAM Tori

The onset of chaos from periodic and quasiperiodic motion is now well understood. The description of chaos, however, is still unsatisfactory. For example, the spectrum lea) of singularities of the natural invariant measurel) cannot characterize the bifurcations of chaotic attractors such as the band splittings and crises in 1d (i.e., one-dimensional) maps.2) The spectrum «P(A) of coarse-grained expansion rates of nearby orbits along the unstable manifold2),3) cannot capture the critical attractors at the onset of chaos. Therefore lea) and «P(A) are not enough to describe the successive bifurcations of chaotic attractors to the critical attractors. In the present paper, we shall discuss how to capture the critical attractors in terms of the local expansion rates in order to find a unified description of the critical attractors and chaotic attractors. We shall also discuss the critical KAM tori of Hamiltonian systems. Let us consider an orbit {Xt}, (t=0, 1, 2, .. _) on an invariant torus generated by a 2d map Xt+l = F(Xt ). The dynamic approach starts with the sum of the local expansion rates Sn(Xo)=~7;;;Jih(Xt), where ih(Xt) is the local expansion rate of nearby orbits at X t along the invariant torus, i.e., ,MXt) =logIDF(Xt) ul(Xt)1 with Ul(Xt) being the unit vector tangent to the invariant torus at X t.2),3) Recently Anania and Politi) suggested that for the critical 2= cycle of period doubling, Sn(Xo) grows with n as logn for large n. !3n(Xo)=Sn(Xo)/logn is plotted against n in Fig. 1 for an orbit on the critical torus of the sine circle map5)

On-off diffusion: onset and statistics

A particle motion in a 2D periodic potential with the symmetry such that a particular motion can be restricted on the x-axis, subject to the external periodic force in the x-direction, is studied. It is found that by changing the amplitude of the external force, the 1D diffusive motion in the x-direction undergoes the instability at an amplitude, above which the diffusive motion in the y-direction, showing on-off intermittency, is observed. We call it on-off diffusion. By introducing a simple mapping model, the diffusion coefficient in the y-direction is found to take the scaling form slightly above the instability point where and . and L, respectively, are the transverse Lyapunov exponent evaluating the magnitude of instability and its fluctuation in the y-direction. The scaling function h(z) takes the asymptotic form, for and for .

Chaos-Induced Diffusion in a Nonlinear Dissipative Mathieu Equation for a Charged Fine Particle in an AC Trap

Charged fine particles confined in an AC trap exhibit either periodic motion or irregular motion, depending on the frequency and amplitude of the AC electric field. This motion was analyzed using an idealized electric field model with a nonlinear term in the radial direction ( r ) and an angular (θ-dependent) term. The potential U ( r ,θ, z , t ) generates a rotational diffusion of chaotic orbits, and a transition from ballistic motion to diffusive motion was observed in the mean square displacement (MSD) of θ. The distribution function f (τ) for the lifetime of angular unidirectional motion is exponential. This exponential distribution is produced by the chaotic switching between clockwise and anticlockwise rotations of orbits on the x y -plane. The time-correlation function C (τ) of v θ also has an exponential decay form as a result of the lifetime distribution function f (τ). The scaling function of the MSD of θ(τ) is derived using the correlation time τ c of C (τ).

Structural Criticality of Dynamical Glass State in a Spatially Coupled Map

Understanding complex phenomena in spatially extended dynamical systems is one of the most important problems in nonlinear science. Efforts to classify and characterize various phenomena have produced interesting new concepts, e.g., spatiotemporal intermittency,I>·> self-organized criticality,> chaotic itineracy> and so on. Recently, temporally periodic and spatially irregular motion in coupled map systems was reported by Kaneko> and Fujisaka, Egami and Yamada.> Fujisaka et al. found that extremely many attractors coexist in phase space and that the relaxation process from an initial state to one of the attractors is chaotic and anomalously slow. This is the reason why the state is called a dynamical glass state.> In this paper, we study the spatially one-dimensional extended dynamical system