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At least 811 records · Page 45Linked to original sources

arXiv1998

Modelling Sonoluminescence

In single-bubble sonoluminescence, a bubble trapped by a sound wave in a flask of liquid is forced to expand and contract; exactly once per cycle, the bubble emits a very sharp ($< 50 ps$) pulse of visible light. This is a robust phenomenon observable to the naked eye, yet the mechanism whereby the light is produced is not well understood. One model that has been proposed is that the light is "vacuum radiation" generated by the coupling of the electromagnetic fields to the surface of the bubble. In this paper, we simulate vacuum radiation by solving Maxwell's equations with an additional term that couples the field to the bubble's motion. We show that, in the static case originally considered by Casimir, we reproduce Casimir's result. In a simple purely time-dependent example, we find that an instability occurs and the pulse of radiation grows exponentially. In the more realistic case of spherically-symmetric bubble motion, we again find exponential growth in the context of a small-radius approximation.

hep-ph↗

arXiv1998

Long-term properties of time series generated by a perceptron with various transfer functions

We study the effect of various transfer functions on the properties of a time series generated by a continuous-valued feed-forward network in which the next input vector is determined from past output values. The parameter space for monotonic and non-monotonic transfer functions is analyzed in the unstable regions with the following main finding; non-monotonic functions can produce robust chaos whereas monotonic functions generate fragile chaos only. In the case of non-monotonic functions, the number of positive Lyapunov exponents increases as a function of one of the free parameters in the model, hence, high dimensional chaotic attractors can be generated. We extend the analysis to a combination of monotonic and non-monotonic functions.

cond-mat.dis-nn↗

arXiv1999

Border Collision Bifurcations in Two Dimensional Piecewise Smooth Maps

Recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps which can not be classified among the generic cases like saddle-node, pitchfork or Hopf bifurcations occurring in smooth maps. In this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision. We then present a systematic analysis of such bifurcations by deriving a normal form --- the piecewise linear approximation in the neighborhood of the border. We show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form, and these are classified under six cases. We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. This theoretical framework will help in explaining bifurcations in all systems which can be represented by two dimensional piecewise smooth maps.

chao-dyn↗

arXiv1998

Traces and determinants of strongly stochastic operators

Periodic orbit theory allows calculations of long time properties of chaotic systems from traces, dynamical zeta functions and spectral determinants of deterministic evolution operators, which are in turn evaluated in terms of periodic orbits. For the case of stochastic dynamics a direct numerical evaluation of the trace of an evolution operator is possible as a multidimensional integral. Techniques for evaluating such path integrals are discussed. Using as an example the logistic map f(x) = lambda x(1-x) with moderate to strong additive Gaussian noise, rapid convergence is demonstrated for all values of lambda with strong noise as well as at fixed lambda = 5 for all noise levels.

chao-dyn↗

arXiv1998

Influence of phase space localization on the energy diffusion in a quantum chaotic billiard

The quantum dynamics of a chaotic billiard with moving boundary is considered in this work. We found a shape parameter Hamiltonian expansion which enables us to obtain the spectrum of the deformed billiard for deformations so large as the characteristic wave length. Then, for a specified time dependent shape variation, the quantum dynamics of a particle inside the billiard is integrated directly. In particular, the dispersion of the energy is studied in the Bunimovich stadium billiard with oscillating boundary. The results showed that the distribution of energy spreads diffusively for the first oscillations of the boundary (${< Δ^2 E}> =2 D t$). We studied the diffusion contant $D$ as a function of the boundary velocity and found differences with theoretical predictions based on random matrix theory. By extracting highly phase space localized structures from the spectrum, previous differences were reduced significantly. This fact provides the first numerical evidence of the influence of phase space localization on the quantum diffusion of a chaotic system.

chao-dyn↗

arXiv1999

Bicritical Behavior of Period Doublings in Unidirectionally-Coupled Maps

We study the scaling behavior of period doublings in two unidirectionally-coupled one-dimensional maps near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. Note that the bicritical point corresponds to a border of chaos in both subsystems. For this bicritical case, the second response subsystem exhibits a new type of non-Feigenbaum critical behavior, while the first drive subsystem is in the Feigenbaum critical state. Using two different methods, we make the renormalization group analysis of the bicritical behavior and find the corresponding fixed point of the renormalization transformation with two relevant eigenvalues. The scaling factors obtained by the renormalization group analysis agree well with those obtained by a direct numerical method.

chao-dyn↗

arXiv1998

Theory of Circle Maps and the Problem of One-Dimensional Optical Resonator with a Periodically Moving Wall

We consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions. Notably, well-known results in the theory of circle maps (which we review briefly) imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially. Even if these intervals are dense for typical motions of the reflecting boundary, in the complement there is a positive measure set of parameters where the energy remains bounded.

physics.optics↗

arXiv1998

Correlation functions in isotropic and anisotropic turbulence: the role of the symmetry group

The theory of fully developed turbulence is usually considered in an idealized homogeneous and isotropic state. Real turbulent flows exhibit the effects of anisotropic forcing. The analysis of correlation functions and structure functions in isotropic and anisotropic situations is facilitated and made rational when performed in terms of the irreducible representations of the relevant symmetry group which is the group of all rotations SO(3). In this paper we firstly consider the needed general theory and explain why we expect different (universal) scaling exponents in the different sectors of the symmetry group. We exemplify the theory context of isotropic turbulence (for third order tensorial structure functions) and in weakly anisotropic turbulence (for the second order structure function). The utility of the resulting expressions for the analysis of experimental data is demonstrated in the context of high Reynolds number measurements of turbulence in the atmosphere.

chao-dyn↗

arXiv1999

Anomalous transport and quantum-classical correspondence

We present evidence that anomalous transport in the classical standard map results in strong enhancement of fluctuations in the localization length of quasienergy states in the corresponding quantum dynamics. This generic effect occurs even far from the semiclassical limit and reflects the interplay of local and global quantum suppression mechanisms of classically chaotic dynamics. Possible experimental scenarios are also discussed.

chao-dyn↗

arXiv1999

Synchronization and Maximum Lyapunov Exponents of Cellular Automata

We study the synchronization of totalistic one dimensional cellular automata (CA). The CA with a non zero synchronization threshold exhibit complex non periodic space time patterns and conversely. This synchronization transition is related to directed percolation. We study also the maximum Lyapunov exponent for CA, defined in analogy with continuous dynamical systems as the exponential rate of expansion of the linear map induced by the evolution rule of CA, constructed with the aid of the Boolean derivatives. The synchronization threshold is strongly correlated to the maximum Lyapunov exponent and we propose approximate relations between these quantities. The value of this threshold can be used to parametrize the space time complexity of CA.

cond-mat.stat-mech↗

arXiv1999

Randomly Amplified Discrete Langevin Systems

A discrete stochastic process involving random amplification with additive noise is studied analytically. If the non-negative random amplification factor $b$ is such that $ =1$ where $β$ is any positive non-integer, then the steady state probability density function for the process will have power law tails of the form $p(x) \sim 1/x^{β+1}$. This is a generalization of recent results for $0 < β< 2$ obtained by Takayasu et al. in Phys. Rev. lett. 79, 966 (1997). It is shown that the power spectrum of the time series $x$ becomes Lorentzian, even when $1 < β< 2$, i.e., in case of divergent variance.

chao-dyn↗

arXiv1999

Thermostating by Deterministic Scattering: Heat and Shear Flow

We apply a recently proposed novel thermostating mechanism to an interacting many-particle system where the bulk particles are moving according to Hamiltonian dynamics. At the boundaries the system is thermalized by deterministic and time-reversible scattering. We show how this scattering mechanism can be related to stochastic boundary conditions. We subsequently simulate nonequilibrium steady states associated to thermal conduction and shear flow for a hard disk fluid. The bulk behavior of the model is studied by comparing the transport coefficients obtained from computer simulations to theoretical results. Furthermore, thermodynamic entropy production and exponential phase-space contraction rates in the stationary nonequilibrium states are calculated showing that in general these quantities do not agree.

chao-dyn↗

arXiv1999

Noise-Induced Phase Space Transport in Two-Dimensional Hamiltonian Systems

First passage time experiments were used to explore the effects of low amplitude noise as a source of accelerated phase space diffusion in two-dimensional Hamiltonian systems, and these effects were then compared with the effects of periodic driving. The objective was to quantify and understand the manner in which ``sticky'' chaotic orbits that, in the absence of perturbations, are confined near regular islands for very long times, can become ``unstuck'' much more quickly when subjected to even very weak perturbations. For both noise and periodic driving, the typical escape time scales logarithmically with the amplitude of the perturbation. For white noise, the details seem unimportant: Additive and multiplicative noise typically have very similar effects, and the presence or absence of a friction related to the noise by a Fluctuation-Dissipation Theorem is also largely irrelevant. Allowing for colored noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time becomes so large that there is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. Similarly, periodic driving is relatively inefficient when the driving frequency is not comparable to these natural frequencies. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. The logarithmic dependence of the escape time on amplitude reflects the fact that the time required for perturbed and unperturbed orbits to diverge a given distance scales logarithmically in the amplitude of the perturbation.

astro-ph↗

arXiv1998

Classical and quantum chaos in a circular billiard with a straight cut

We study classical and quantum dynamics of a particle in a circular billiard with a straight cut. This system can be integrable, nonintegrable with soft chaos, or nonintegrable with hard chaos, as we vary the size of the cut. We use a quantum web to show differences in the quantum manifestations of classical chaos for these three different regimes.

chao-dyn↗

arXiv1999

Semiclassical Non-Trace Type Formulas for Matrix Element Fluctuations and Weighted Densities of States

Densities of states weighted with the diagonal matrix elements of two operators A and B, i.e., rho^(A,B)(E) = sum_n delta(E-E_n) cannot, in general, be written as a trace formula, and therefore no simple extension of semiclassical trace formulas is known for this case. However, from the high resolution analysis of quantum spectra in the semiclassical regime we find strong evidence that weighting the delta-functions in the quantum mechanical density of states with the product of diagonal matrix elements, , is equivalent to weighting the periodic orbit contributions in the semiclassical periodic orbit sum with the product of the periodic orbit means, _p _p, of the classical observables A and B. Results are presented for the hydrogen atom in a magnetic field for both the chaotic and near-integrable regime, and for the circle billiard.

chao-dyn↗

arXiv1999

Non-Universal Quantities from Dual RG Transformations

Using a simplified version of the RG transformation of Dyson's hierarchical model, we show that one can calculate all the non-universal quantities entering into the scaling laws by combining an expansion about the high-temperature fixed point with a dual expansion about the critical point. The magnetic susceptibility is expressed in terms of two dual quantities transforming covariantly under the RG transformation and has a smooth limit in the high-temperature limit. Using the analogy with hamiltonian systems, the simplified example discussed here is similar to the anharmonic oscillator, while more realistic examples can be tought of as coupled oscillators allowing resonance phenomena.

hep-lat↗

arXiv1999

Correlations of Eigenvectors for Non-Hermitian Random-Matrix Models

We establish a general relation between the diagonal correlator of eigenvectors and the spectral Green's function for non-hermitian random-matrix models in the large-N limit. We apply this result to a number of non-hermitian random-matrix models and show that the outcome is in good agreement with numerical results.

cond-mat↗

arXiv1999

Characteristic distributions of finite-time Lyapunov exponents

We study the probability densities of finite-time or \local Lyapunov exponents (LLEs) in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are significant finite-size corrections which are coordinate dependent. Depending on the nature of the dynamical state, the distribution of local Lyapunov exponents has a characteristic shape. For intermittent dynamics, and at crises, dynamical correlations lead to distributions with stretched exponential tails, while for fully-developed chaos the probability density has a cusp. Exact results are presented for the logistic map, $x \to 4x(1-x)$. At intermittency the density is markedly asymmetric, while for `typical' chaos, it is known that the central limit theorem obtains and a Gaussian density results. Local analysis provides information on the variation of predictability on dynamical attractors. These densities, which are used to characterize the {\sl nonuniform} spatial organization on chaotic attractors are robust to noise and can therefore be measured from experimental data.

chao-dyn↗