Takehiko Horita

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.

Stochastic Resonance in the Hodgkin-Huxley Network

Stochastic resonance in a coupled Hodgkin-Huxley equation is investigated. The dependence of signal to noise ratio on the frequencies of the periodic input signals is examined by numerical experiments. Two or three Hodgkin-Huxley equations are coupled with a propagational time delay to compose a network. For a network with two elements, an enhancement of the stochastic resonance for the periodic input signals with particular frequencies is found. It is also found that a network with three elements is capable of distinguishing periodic input signals by those frequencies.

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.

Opening a closed Hamiltonian map

A closed Hamiltonian map is opened by introducing an interaction with the outside of the system. The resulting open Hamiltonian system possesses an exit with escaping orbits through it. For such a system equipped with two or three exits, the exit basin structure of the escaping orbits is observed to have a fractal boundary and a boundary shared by the three basins, i.e., a Wada basin boundary. In the small size limit of the exits, a complete fractalization of the phase space, where the predictability of the future state is almost lost, is also observed.

Stochastic resonance for the superimposed periodic pulse train

Stochastic Resonance in a coupled FitzHugh-Nagumo equation is investigated. The optimal noise intensity and the optimal input frequency, which maximize the signal to noise ratio of the output signal, are studied numerically, and their dependence on system parameters and connection coefficients is examined. It is found that a network composed of six elements can separate a superimposed periodic pulse train by controlling the noise intensity.

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

Mapping Model of Chaotic Phase Synchronization

A coupled map model for chaotic phase synchronization and desynchronization phenomena is proposed. The model is constructed by integrating the coupled kicked oscillator system, with the kicking strength depending on the complex state variables. It is shown that the proposed model clearly exhibits the chaotic phase synchronization phenomenon. Furthermore, we numerically prove that in the region where the phase synchronization is weakly broken, anomalous scaling of the phase difference rotation number is observed. This proves that the present model belongs to the universality class discovered by Pikovsky et al. Furthermore, the phase diffusion coefficient in the de-synchronization state is analyzed.

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)