Self-Organized Criticality in a Simple Neuron Model Based on Scale-Free Networks

A simple model for a set of interacting idealized neurons in scale-free networks is introduced. The basic elements of the model are endowed with the main features of a neuron function. We find that our model displays powerlaw behavior of avalanche sizes and generates long-range temporal correlation. More importantly, we find different dynamical behavior for nodes with different connectivity in the scale-free networks.     

Self-Organized Criticality in a Simple Neuron Model Based on Scale-Free Networks

A simple model for a set of interacting idealized neurons in scale-free networks is introduced. The basic elements of the model are endowed with the main features of a neuron function. We find that our model displays powerlaw behavior of avalanche sizes and generates long-range temporal correlation. More importantly, we find different dynamical behavior for nodes with different connectivity in the scale-free networks.     

Hausdorff dimensions in two-dimensional maps and thermodynamic formalism

We compute numerically the Hausdorff dimensions of the Gibbs measures on the invariant sets of Axiom A systems. In particular, we stress the existence of a measure which has maximal dimension and can be relevant for the ergodic properties of the system. For hyperbolic maps of the plane with constant Jacobianj, we apply the Bowen-Ruelle formula, using the relationF(=d _H–1)=lnj, which links the Hausdorff dimensiond _H of an attractor to a free energy functionalF() defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map.     

A Modified Earthquake Model of Self-Organized Criticality on Small World Networks

A modified Olami-Feder-Christensen model of self-organized criticality on a square lattice with the properties of small world networks has been studied. We find that our model displays power-law behavior and the exponent τ of the model depends on φ, the density of long-range connections in our network.     

Hausdorff Dimension in Stochastic Dispersion

We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order away from the origin, there is an uncountable set of measure zero of points, which escape to infinity at the linear rate. In this paper we prove that this set of linear escape points has full Hausdorff dimension.     

Avalanches and Self-Organized Criticality in superconductors

We review the use of superconductors as a playground for the experimental study of front roughening and avalanches. Using the magneto-optical technique, the spatial distribution of the vortex density in the sample is monitored as a function of time. The roughness and growth exponents corresponding to the vortex `landscape' are determined and compared to the exponents that characterize the avalanches in the framework of Self-Organized Criticality. For those situations where a thermo-magnetic instability arises, an analytical non-linear and non-local model is discussed, which is found to be consistent to great detail with the experimental results. On anisotropic substrates, the anisotropy regularizes the avalanches.     

Principle of Maximum Entropy and Reduced Dynamics

A new method to obtain a series of reduced dynamics at various stages of coarse-graining is proposed. This ranges from the most coarse-grained one which agrees with the deterministic time evolution equation for averages of the relevant variables to the least coarse-grained one which is the generalized Fokker-Planck equation for the probability distribution function of the relevant variables. The method is based on the extention of the Kawasaki-Gunton operator with the help of the principle of maximum entropy.     

Differential Entropy and Dynamics of Uncertainty

We analyze the functioning of Gibbs-type entropy functionals in the time domain, with emphasis on Shannon and Kullback-Leibler entropies of time-dependent continuous probability distributions. The Shannon entropy validity is extended to probability distributions inferred from L ²(R ⁿ ) quantum wave packets. In contrast to the von Neumann entropy which simply vanishes on pure states, the differential entropy quantifies the degree of probability (de)localization and its time development. The associated dynamics of the Fisher information functional quantifies nontrivial power transfer processes in the mean, both in dissipative and quantum mechanical cases. PACS NUMBERS: 05.45.+b, 02.50.-r, 03.65.Ta, 03.67.-a     

各向异性光子晶体中熵压缩动力学行为

本文详细分析了各向异性光子晶体环境中量子弱测量和测量反转对原子熵压缩的影响,得到一些重要结论。研究发现,量子比特和光子晶体带隙边缘频率之间的失谐量δ以及量子弱测量和测量反转强度对熵压缩起着重要作用,通过调节它们,可以获得更好的熵压缩。     

Dynamics and Entropy Analysis of a Frictionally Loaded Pendulum

We use friction to simultaneously damp and excite a pendulum system. A Froude pendulum attached to a suspension shaft is subjected to a frictional load. We investigate two types of response of the system: regular and chaotic responses, depending on the excitation frequency. A transient chaotic solution was also obtained. We identify the motions using phase portraits, Poincaré maps, and Fourier spectra. Finally, the composite multiscaled entropy was estimated for the specified cases to confirm the preliminary classification.     

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.     

Positive Lyapunov exponents in the Kramers oscillator

The maximum Lyapunov exponent is computed numerically for the double-well oscillator in a heat bath. Positive exponents are found in a wide range of friction coefficients in the low-damping regime.     

Coherent Transport and Dynamical Entropy for Fermionic Systems

This paper consists of two parts. First we set up a general scheme of local traps in a homogeneous deterministic quantum system. The current of particles caught by the trap is linked to the dynamical behaviour of the trap states. In this way, transport properties in a homogeneous system are related to spectral properties of a coherent dynamics. Next we apply the scheme to a system of Fermions in the one-particle approximation. We obtain in particular lower bounds for the dynamical entropy in terms of the current induced by the trap.     

Entropy, Lyapunov exponents, and mean free path for billiards

We review known results and derive some new ones about the mean free path, Kolmogorov-Sinai entropy, and Lyapunov exponents for billiard-type dynamical systems. We focus on exact and asymptotic formulas for these quantities. The dynamical systems covered in this paper include the priodic Lorentz gas, the stadium and its modifications, and the gas of hard balls. Some open questions and numerical observations are discussed.     

Computing the pressure for Axiom-A attractors by time series and large deviations for the Lyapunov exponent

For the Axiom-A attractors a relation is given between the topological pressure and the spectrum of the generalized Lyapunov exponents. As a consequence, a simple formula is found to compute the topological entropy of the attractor by means of a time series. The results are used to compute the large deviations for positive Lyapunov exponents.     

Multidimensional mappings in antiferromagnetic models and Lyapunov exponents with the magnetization plateau

We consider the Heisenberg model with two- and three-spin exchange interactions on a recursive ladder in a strong magnetic field. Recurrent relations for branches of the partition function of the Ising model with two- and three-spin exchange interactions are deduced. As a recursive lattice the zigzag ladder is chosen. In the antiferromagnetic case magnetization plateau are observed at low temperatures. Lyapunov exponents for the three-dimensional mapping at low temperatures are calculated. It is shown that for some values of two- and three-spin exchange parameters in the antiferromagnetic case the maximum of the Lyapunov exponent approaches zero.     

On the multifractal analysis of Bernoulli convolutions. II. Dimensions

We show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved Bernoulli measures for the golden mean. In this second part we show how the Hausdorff dimension of the set where the measure has a power law singularity of strength is related to the large-deviation function given in Part I.     

Dimensions and Waiting Times for Gibbs Measures

For shifts of finite type, we relate the waiting time between two different orbits, one chosen according to an ergodic measure, the other according to a Gibbs measure, to Billingsley dimensions of generic sets. This is achieved by computing Billingsley dimensions of saturated sets in terms of a relative entropy which satisfies a pointwise ergodic result. As a by-product, a similar result is obtained for match lengths that are dual quantities of waiting times.     

Hausdorff Dimensions of Zero-Entropy Sets of Dynamical Systems with Positive Entropy

Suppose that (X,T) is a compact positive entropy dynamical system which we mean that X is a compact metric space and T: X→ X is a continuous transformation of X and the topological entropy h(T)>0. A point x ∈ X is called a zero-entropy point provided , where is the forward orbit of x under T and Orb₊(x) is the closure. Let ε⁰(X, T) denote the set of all zero-entropy points. Naturally, one would like to ask the following important question: How big is ε⁰(X, T) for a dynamical system? In this paper, we answer this question. More precisely, we prove that if, furthermore, (X, T) is locally expanding, then the Hausdorff dimension of ε⁰(X, T) vanishes.     

Effects of Interactive Function Forms in a Self-Organized Critical Model Based on Neural Networks

Based on the standard self-organizing map neural network model and an integrate-and-fire mechanism, we introduce a kind of coupled map lattice system to investigate scale-invariance behavior in the activity of model neural populations. We let the parameter β, which together with α represents the interactive strength between neurons, have different function forms, and we find the function forms and their parameters are very important to our model‘s avalanche dynamical behaviors, especially to the emergence of different avalanche behaviors in different areas of our system.