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arXiv1999

Multi-scale Correlation Functions in Strong Turbulence

Under the framework of V. Yakhot [Phys.Rev.E, {\bf57}, 1737 (1998)] modelling of intermittent structure functions in fully developed turbulence and based on the experimentally supported Markovian nature of turbulence cascades [R.Friedrich and J.Peinke, Phys.Rev.Lett, {\bf 78}, 863 (1997)], we calculate the multiscaling correlation functions. Fusion rules [V.S.L'vov and I. Procaccia, Phys.Rev.Lett. {\bf 76}, 2898 (1996)] which are experimentally tested [R. Benzi, L. Biferale, and F. Toschi, Phys.Rev.Lett. {\bf 80}, 3244 (1998)] to be compatible with almost uncorrelated multiplicative process can analyticaly be supported by direct calculations.

cond-mat↗

arXiv1999

Inverse cascade in two-dimensional turbulence: deviations from Gaussianity

High resolution numerical simulations of stationary inverse energy cascade in two-dimensional turbulence are presented. Deviations from Gaussianity of velocity differences statistics are quantitatively investigated. The level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd-order moments, which bring the signature of the inverse energy flux. Their scaling laws do not present any measurable intermittency correction. The seventh order skewness is found to increase by almost two orders of magnitude with respect to the third, thus becoming of order unity.

chao-dyn↗

arXiv1999

Statistics of Wave Functions in Coupled Chaotic Systems

Using the supersymmetry technique, we calculate the joint distribution of local densities of electron wavefunctions in two coupled disordered or chaotic quantum billiards. We find novel spatial correlations that are absent in a single chaotic system. Our exact result can be interpreted for small coupling in terms of the hybridization of eigenstates of the isolated billiards. We show that the presented picture is universal, independent of microscopic details of the coupling.

cond-mat.mes-hall↗

arXiv1999

Fixed points, stability and intermittency in a shell model for advection of passive scalars

We investigate the fixed points of a shell model for the turbulent advection of passive scalars introduced Jensen, Paladin and Vulpiani. The passive scalar field is driven by the velocity field of the popular GOY shell model. The scaling behavior of the static solutions is found to differ significantly from Obukhov-Corrsin scaling theta_n ~ k_n^(-1/3) which is only recovered in the limit where the diffusivity vanishes, D -> 0. From the eigenvalue spectrum we show that any perturbation in the scalar will always damp out, i.e. the eigenvalues of the scalar are negative and are decoupled from the eigenvalues of the velocity. Furthermore we estimate Lyapunov exponents and the intermittency parameters using a definition proposed by Benzi et al. The full model is as chaotic as the GOY model, measured by the maximal Lyapunov exponent, but is more intermittent.

chao-dyn↗

arXiv2000

Quantum Chaos Border for Quantum Computing

We study a generic model of quantum computer, composed of many qubits coupled by short-range interaction. Above a critical interqubit coupling strength, quantum chaos sets in, leading to quantum ergodicity of the computer eigenstates. In this regime the noninteracting qubit structure disappears, the eigenstates become complex and the operability of the computer is destroyed. Despite the fact that the spacing between multi-qubit states drops exponentially with the number of qubits $n$, we show that the quantum chaos border decreases only linearly with $n$. This opens a broad parameter region where the efficient operation of a quantum computer remains possible.

quant-ph↗

arXiv1999

Unstable dimension variability and synchronization of chaotic systems

An aspect of the synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits unstable dimension variability. Unstable dimension variability is a cause of severe modeling difficulty for physical phenomena, since trajectories obtained from the mathematical model may not be related to trajectories of the actual system. We present and example of unstable dimension variability occurring in a system of two coupled chaotic maps, considering the dynamics in the synchronization manifold and its corresponding transversal direction, where a tongue-like structure is formed. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values.

chao-dyn↗

arXiv2000

The statistics of velocity fluctuations arising from a random distribution of point vortices: the speed of fluctuations and the diffusion coefficient

This paper is devoted to a statistical analysis of the fluctuations of velocity and acceleration produced by a random distribution of point vortices in two-dimensional turbulence. We show that the velocity probability density function (p.d.f.) behaves in a manner which is intermediate between Gaussian and Lévy laws while the distribution of accelerations is governed by a Cauchy law. Our study accounts properly for a spectrum of circulations among the vortices. In the case of real vortices (with a finite core) we show analytically that the distribution of accelerations makes a smooth transition from Cauchy (for small fluctuations) to Gaussian (for large fluctuations) passing probably through an exponential tail. We introduce a function $T(V)$ which gives the typical duration of a velocity fluctuation $V$; we show that $T(V)$ behaves like $V$ and $V^{-1}$ for weak and large velocities respectively. These results have a simple physical interpretation in the nearest neighbor approximation and in Smoluchowski theory concerning the persistence of fluctuations. We discuss the analogies with respect to the fluctuations of the gravitational field in stellar systems. As an application of these results, we determine an approximate expression for the diffusion coefficient of point vortices. When applied to the context of freely decaying two-dimensional turbulence, the diffusion becomes anomalous and we establish a relationship $ν=1+{ξ\over 2}$ between the exponent of anomalous diffusion $ν$ and the exponent $ξ$ which characterizes the decay of the vortex density. This relation is in good agreement with laboratory experiments and numerical simulations.

cond-mat.stat-mech↗

arXiv1999

Characteristic Relations of Type-I Intermittency in the Presence of Noise

Near the point of tangent bifurcation, the scaling properties of the laminar length of type-I intermittency are investigated in the presence of noise. Based on analytic and numerical studies, we show that the scaling relation of the laminar length is dramatically deformed from $\frac{1}{\sqrtε}$ for $ε>0$ to $\exp\{\frac{1}{D}|ε|^{3/2}\}$ for $ε<0$ as $ε$ passes the bifurcation point $(ε=0)$. The results explain why two coupled Rössler oscillators exhibit deformation of the scaling relation of the synchronous length in the nearly synchronous regime.

chao-dyn↗

arXiv1999

Kullback-Leibler and Renormalized Entropy: Applications to EEGs of Epilepsy Patients

Recently, renormalized entropy was proposed as a novel measure of relative entropy (P. Saparin et al., Chaos, Solitons & Fractals 4, 1907 (1994)) and applied to several physiological time sequences, including EEGs of patients with epilepsy. We show here that this measure is just a modified Kullback-Leibler (K-L) relative entropy, and it gives similar numerical results to the standard K-L entropy. The latter better distinguishes frequency contents of e.g. seizure and background EEGs than renormalized entropy. We thus propose that renormalized entropy might not be as useful as claimed by its proponents. In passing we also make some critical remarks about the implementation of these methods.

physics.bio-ph↗

arXiv2001

Phase Synchronization with Type-II Intermittency in Chaotic Oscillators

We study the phase synchronization (PS) with type-II intermittency showing $\pm 2 π$ irregular phase jumping behavior before the PS transition occurs in a system of two coupled hyperchaotic Rössler oscillators. The behavior is understood as a stochastic hopping of an overdamped particle in a potential which has $2 π$-periodic minima. We characterize it as type-II intermittency with external noise through the return map analysis. In $ε_{t} < ε< ε_{c}$ (where $ε_{t}$ is the bifurcation point of type-II intermittency and $ε_{c}$ is the PS transition point in coupling strength parameter space), the average length of the time interval between two successive jumps follows $ \sim \exp(\midε_{t} - ε\mid^{2})$, which agrees well with the scaling law obtained from the Fokker-Planck equation.

chao-dyn↗

arXiv1999

Synchronization of chaotic oscillations in doped fiber ring lasers

We investigate synchronization and subsequently communication using chaotic rare-earth-doped fiber ring lasers, represented by a physically realistic model. The lasers are coupled by transmitting a fraction c of the circulating electric field in the transmitter and injecting it into the optical cavity of the receiver. We then analyze a coupling strategy which relies on modulation of the intensity of the light alone. This avoids problems associated with the polarization and phase of the laser light. We study synchronization as a function of the coupling strength and see excellent convergence, even with small coupling constants. We prove that in an open-loop configuration (c=1) synchronization is guaranteed due to the particular structure of our equations and of the injection method we use for these coupled laser systems. We also analyze the generalized synchronization of these model lasers when there is parameter mismatch between the transmitter and the receiver. We then address communicating information between the transmitter and receiver lasers by modulation of the chaotic optical waveform. Finally we comment on the cryptographic setting of our algorithms, especially the open-loop strategy at c=1, and hope this may lead others to perform the cryptographic analyses to determine which, if any, of the communications strategies we investigate are secure.

chao-dyn↗

arXiv1997

Quantum mechanical time-delay matrix in chaotic scattering

We calculate the probability distribution of the matrix Q = -i \hbar S^{-1} dS/dE for a chaotic system with scattering matrix S at energy E. The eigenvalues τ_j of Q are the so-called proper delay times, introduced by E. P. Wigner and F. T. Smith to describe the time-dependence of a scattering process. The distribution of the inverse delay times turns out to be given by the Laguerre ensemble from random-matrix theory.

chao-dyn↗

arXiv1997

Semiclassical spectra from periodic-orbit clusters in a mixed phase space

We determine semiclassical quasienergy spectra from periodic orbits for a system with a mixed phase space, the kicked top. Throughout the transition from integrability to well developed chaos the standard error incurred for the quasienergies is a small percentage of their mean spacing. Such fine accuracy does not even require the angular momentum quantum number $j\propto 1/\hbar$ to be large; it already prevails for $j=1,2,3$. To open the way towards a reliable spectrum the conventional trace formula à la Gutzwiller has to be extended by including (i) ghosts, i.e., complex predecessors of bifurcating orbits and the possibility of Stokes transitions undergone by such complex orbits and (ii) collective contributions of clusters of periodic orbits near bifurcations. Even bifurcations of codimension higher than one must be reckoned with by accounting for the clusters involved through the appropriate diffraction integrals.

chao-dyn↗

arXiv1997

Avoided Band Crossings: Tuning Metal-Insulator Transitions in Chaotic Systems

We show that avoided crossings of energy bands may give rise to a variety of phenomena such as transitions from metal to insulator and vice versa, changes in localization lengths, and changes in the fractal dimension of energy spectra. We explain the occurrence of these phenomena in the kicked Harper model under classically chaotic conditions and predict them to occur in other systems.

cond-mat↗

arXiv1998

Statistical mechanics of voting

Decision procedures aggregating the preferences of multiple agents can produce cycles and hence outcomes which have been described heuristically as `chaotic'. We make this description precise by constructing an explicit dynamical system from the agents' preferences and a voting rule. The dynamics form a one dimensional statistical mechanics model; this suggests the use of the topological entropy to quantify the complexity of the system. We formulate natural political/social questions about the expected complexity of a voting rule and degree of cohesion/diversity among agents in terms of random matrix models---ensembles of statistical mechanics models---and compute quantitative answers in some representative cases.

cond-mat.stat-mech↗

arXiv1998

Excitation spectrum of Andreev billiards with a mixed phase space

We present a semiclassical theory for the excitation spectrum of a ballistic quantum dot weakly coupled to a superconductor, for the generic situation that the classical motion gives rise to a phase space containing islands of regularity in a chaotic sea. The density of low-energy excitations is determined by quantum energy scales that are related in a simple way to the morphology of the mixed phase space. An exact quantum mechanical computation for the annular billiard shows good agreement with the semiclassical predictions, in particular for the reduction of the excitation gap when the coupling to the regular regions is maximal.

cond-mat.mes-hall↗

arXiv1999

Better Nonlinear Models from Noisy Data: Attractors with Maximum Likelihood

A new approach to nonlinear modelling is presented which, by incorporating the global behaviour of the model, lifts shortcomings of both least squares and total least squares parameter estimates. Although ubiquitous in practice, a least squares approach is fundamentally flawed in that it assumes independent, normally distributed (IND) forecast errors: nonlinear models will not yield IND errors even if the noise is IND. A new cost function is obtained via the maximum likelihood principle; superior results are illustrated both for small data sets and infinitely long data streams.

chao-dyn↗

arXiv1999

Correlations in the Adiabatic Response of Chaotic Systems

Adiabatic variation of the parameters of a chaotic system results in a fluctuating reaction force. In the leading order in the adiabaticity parameter, a dissipative force, that is present in classical mechanics was found to vanish in quantum mechanics. On the time scale t, this force is proportional to I(t), the integral of the force-force correlation function over time t. In order to understand the crossover between the classical and the quantum mechanical behavior we calculated I(t) in random matrix theory. We found that for systems belonging to the Gaussian unitary ensemble this crossover takes place at a characteristic time (proportional to the Heisenberg time) and for longer times I(t) practically vanishes, resulting in vanishing dissipation. For systems belonging to the Gaussian orthogonal ensemble I(t) drops like 1/t and there is no such characteristic time. I(t) is calculated for various models and the relation to experiment is discussed.

chao-dyn↗