Chaos and generalised multistability in a mesoscopic model of the electroencephalogram

We present evidence for chaos and generalised multistability in a mesoscopic model of the electroencephalogram (EEG). Two limit cycle attractors and one chaotic attractor were found to coexist in a two-dimensional plane of the ten-dimensional volume of initial conditions. The chaotic attractor was found to have a moderate value of the largest Lyapunov exponent (3.4 s⁻¹ base e) with an associated Kaplan-Yorke (Lyapunov) dimension of 2.086. There are two different limit cycles appearing in conjunction with this particular chaotic attractor: one multiperiodic low amplitude limit cycle whose largest spectral peak is within the alpha band (8-13 Hz) of the EEG; and another multiperiodic large-amplitude limit cycle which may correspond to epilepsy. The cause of the coexistence of these structures is explained with a one-parameter bifurcation analysis. Each attractor has a basin of differing complexity: the large-amplitude limit cycle has a basin relatively uncomplicated in its structure while the small-amplitude limit cycle and chaotic attractor each have much more finely structured basins of attraction, but none of the basin boundaries appear to be fractal. The basins of attraction for the chaotic and small-amplitude limit cycle dynamics apparently reside within each other. We briefly discuss the implications of these findings in the context of theoretical attempts to understand the dynamics of brain function and behaviour.     

On spurious and corrupted multifractality: The effects of additive noise, short-term memory and periodic trends

We study the performance of multifractal detrended fluctuation analysis (MF-DFA) applied to long-term correlated and multifractal data records in the presence of additive white noise, short-term memory and periodicities. Such additions and disturbances that can be typically found in the observational records of various complex systems ranging from climate dynamics to physiology, network traffic, and finance. In monofractal records, we find that (i) additive white noise hardly results in spurious multifractality, but causes underestimated generalized Hurst exponents h(q) for all q values; (ii) short-range correlations lead to pronounced crossovers in the generalized fluctuation functions F_q(s) at positions that decrease with increasing moment q, thus causing significantly overestimated h(q) for small q and spurious multifractality; (iii) periodicities like seasonal trends (with standard deviations comparable with the one of the studied process) result in spurious “reversed” multifractality where h(q) increases with increasing q (except for very short time windows). We also show that in multifractal cascades moderate additions of noise, short-range memory, or periodic trends cause flawed results for h(q) with q<2, while h(q) with q>2 remains nearly unchanged.     

Effects of Levy noise and immune delay on the extinction behavior in a tumor growth model

The combined effects of Ltvy noise and immune delay on the extinction behavior in a tumor growth model are explored, The extinction probability of tumor with certain density is measured by exit probability. The expression of the exit probability is obtained using the Taylor expansion and the infinitesimal generator theory. Based on numerical calculations, it is found that the immune delay facilitates tumor extinction when the stability index α〈 1, but inhibits tumor extinction when the stability index α 〉 1. Moreover, larger stability index and smaller noise intensity are in favor of the extinction for tumor with low density. While for tumor with high density, the stability index and the noise intensity should be reduced to promote tumor extinction.     

Influence of strong noise on the decline and propagation of population in the delayed Malthus-Verhulst model

The effects of strong noise on the decline and propagation processes of a population in the Malthus-Verhulst model with time delay are investigated by a stochastic simulation. Time delays in two different processes are concurrent in ecosystems. The simulation results indicate that: The stability of the population is enhanced by the decreasing multiplicative noise intensity and the increasing delay time. The replacement of old individuals with young ones is accelerated by an increasing multiplicative noise intensity, an increasing additive noise intensity and a decreasing delay time. An increasing multiplicative noise intensity will drive the population of species to fluctuate more largely.     

Determinism,realism, and Stapp's 1985 proof of nonlocality

Two recent criticisms of Stapp's 1985 proof of an incompatibility between locality and quantum mechanics are clarified. One of these criticisms (charging that the proof requires determinism) is supported by a detailed logical analysis of the locality conditions in the proof. The other criticism (charging that the proof assumes realism) is shown to be based upon a misinterpretation of the rôle of counterfactuals in the proof. Further, Stapp's own replies to these criticisms are shown to be inadequate.     

Multiplicative non-Gaussian noise and additive Gaussian white noise induced transition in a piecewise nonlinear model

We study the transition problems in a piecewise nonlinear model induced by correlated multiplicative non-Gaussian noise and additive Gaussian white noise. Firstly, applying the path integral approach, the unified colored noise approximation, the analytical expression of the steady-state probability density function (SPD) is derived. Then the change regulation of the SPD is analyzed with the change of the strength and relevance of multiplicative noise and additive noise. From numerical computations we obtain some new nonlinear phenomena: the transition can be induced by the cross-correlation strength between noises, the non-Gaussian noise intensity and the Gaussian noise intensity as well as the non-Gaussian noise deviation parameter. This indicates that the effect of the non-Gaussian noise intensity on SPD is the same as that of the Gaussian noise intensity. Moreover, we also find the correlation time of the non-Gaussian noise can not induce the transition.     

Investigation of rail noise and bridge noise using a combined 3D dynamic model and 2.5D acoustic model

Bridge noise and rail noise are two major sources of an elevated rail transit bridge in the low and medium frequency range (20–1000 Hz). However, in most of the existing literature, the noise radiated from the bridge and rail was investigated separately or using a simplified source model. In this study, an accurate method is proposed to simulate both the rail noise and bridge noise simultaneously. First, the dynamic responses of the rail and multi-span bridge are obtained using a three-dimensional (3D) vehicle-track-bridge interaction analysis model. Then, the two-dimensional (2D) infinite element model is used to calculate 3D modal acoustic transfer vectors of the rail and bridge based on the wavenumber transformation, in order to overcome the singularity and non-uniqueness of the conventional boundary element method and reduce the computation cost. Third, a field test is conducted, and the accuracy of the proposed simulation procedure is verified. Finally, the contribution of the rail and bridge noise to the total noise level is investigated in the whole space near the bridge. Generally the bridge noise occupies a higher contribution in the space beneath the girder due to the shielding effect of the bridge shape on the rail noise, while the rail noise is dominant in the upper space above the bridge. It is found the presence of the vehicle bodies has considerable effect on the rail noise but little influence on the bridge noise. The slope of the roughness level spectrum has significant influence on the dominant field of bridge noise and rail noise. For the excitation of the assumed ISO roughness level used in this study, the difference between the rail noise and bridge noise is only about 3 dB at field points 15–30 m away from the track center, which indicates both the bridge and rail noise should be included in the noise prediction for an elevated rail transit bridge.     

On a Markov chain model for population growth subject to rare catastrophic events

We consider a Markov chain model for population growth subject to rare catastrophic events. In this model, the moves of the process are getting algebraically rare (as from $x^{?\lambda }$) when the process visits large heights $x$, and given a move occurs and the height is large, the chain grows by one unit with large probability or undergoes a rare catastrophic event with small complementary probability $～\gamma /x$. We assume pure reflection at the origin. This chain is irreducible and aperiodic; it is always recurrent, either positive or null recurrent.Estimates are obtained for first-return time probabilities to the origin (excursion length), eventual return (contact) probability and excursion height. All exhibit power-law decay in some range of the parameters $(\gamma ,\lambda )$. We show a scaling relationship between heights and lengths of the excursions. From this, the mean and median of both the empirical average and sample maximum are shown to grow algebraically with exponents being identified in terms of $(\gamma ,\lambda )$.     

Non-homogeneous random walks, generalised master equations, fractional Fokker-Planck equations, and the generalised Kramers-Moyal expansion

A generalised random walk scheme for random walks in an arbitrary external potential field is investigated. From this concept which accounts for the symmetry breaking of homogeneity through the external field, a generalised master equation is constructed. For long-tailed transfer distance or waiting time distributions we show that this generalised master equation is the genesis of apparently different fractional Fokker-Planck equations discussed in literature. On this basis, we introduce a generalisation of the Kramers-Moyal expansion for broad jump length distributions that combines multiples of both ordinary and fractional spatial derivatives. However, it is shown that the nature of the drift term is not changed through the existence of anomalous transport statistics, and thus to first order, an external potential Φ(x) feeds back on the probability density function W through the classical term ∝/ x (x)W(x, t), i.e., even for Lévy flights, there exists a linear infinitesimal generator that accounts for the response to an external field. Received 30 June 2000 and Received in final form 12 November 2000     

Quenched noise in surface growth

The analytical continuous equations for the Tang and Leschhorn (Phys. Rev. A 45 (1992) R8309) and the Buldyrev et al. (Phys. Rev. A 45 (1992) R8313) models are derived from the microscopic rules using a regularization procedure. As was shown in a previous paper (Phys. Rev. E 62 (2000) 6970) the continuous equation for the Tang and Leschhorn model is formally different to the Kardar–Parisi–Zhang equation (Phys. Rev. Lett. 56 (1986) 889) with quenched noise (QKPZ). Nevertheless, after expanding the multiplicative noise, it is shown that we recover the usual QKPZ, demonstrating analytically that both equations belong to the same universality class.     

Bistable generalised Langevin dynamics driven by correlated noise possessing a long jump distribution: barrier crossing and stochastic resonance

The generalised Langevin equation with a retarded friction and a double-well potential is solved. The random force is modelled by a multiplicative noise with long jumps. Probability density distributions converge with time to a distribution similar to a Gaussian but tails have a power-law form. Dependence of the mean first passage time on model parameters is discussed. Properties of the stochastic resonance, emerging as a peak in the plot of the spectral amplification against the temperature, are discussed for various sets of the model parameters. The amplification rises with the memory and is largest for the cases corresponding to the large passage time.     

A generalised Davydov-Scott model for polarons in linear peptide chains

We present a one-parameter family of mathematical models describing the dynamics of polarons in periodic structures, such as linear polypeptides, which, by tuning the model parameter, can be reduced to the Davydov or the Scott model. We describe the physical significance of this parameter and, in the continuum limit, we derive analytical solutions which represent stationary polarons. On a discrete lattice, we compute stationary polaron solutions numerically. We investigate polaron propagation induced by several external forcing mechanisms. We show that an electric field consisting of a constant and a periodic component can induce polaron motion with minimal energy loss. We also show that thermal fluctuations can facilitate the onset of polaron motion. Finally, we discuss the bio-physical implications of our results.     

A three-dimensional physical model of MRI noise based on current noise sources in a conductor

Magnetic flux noise is generated by any conductor in equilibrium with a bath as a result of random fluctuating currents. A physical model of this flux noise is proposed, based on allowable current patterns in the conductor, which we describe as natural current modes. This model gives insight into the spatial characteristics of the magnetic noise which is encountered in a variety of magnetic measurements and imaging modalities such as magnetic resonance imaging (MRI).     

Effect of noise on defect chaos in a reaction-diffusion model

The influence of noise on defect chaos due to breakup of spiral waves through Doppler and Eckhaus instabilities is investigated numerically with a modified Fitzhugh-Nagumo model. By numerical simulations we show that the noise can drastically enhance the creation and annihilation rates of topological defects. The noise-free probability distribution function for defects in this model is found not to fit with the previously reported squared-Poisson distribution. Under the influence of noise, the distributions are flattened, and can fit with the squared-Poisson or the modified-Poisson distribution. The defect lifetime and diffusive property of defects under the influence of noise are also checked in this model.     

Role of noise in a market model with stochastic volatility

We study a generalization of the Heston model, which consists of two coupled stochastic differential equations, one for the stock price and the other one for the volatility. We consider a cubic nonlinearity in the first equation and a correlation between the two Wiener processes, which model the two white noise sources. This model can be useful to describe the market dynamics characterized by different regimes corresponding to normal and extreme days. We analyze the effect of the noise on the statistical properties of the escape time with reference to the noise enhanced stability (NES) phenomenon, that is the noise induced enhancement of the lifetime of a metastable state. We observe NES effect in our model with stochastic volatility. We investigate the role of the correlation between the two noise sources on the NES effect.     

一个特殊模型中的相干布居俘获

外场作用下,对称双势阱中将发生相干布居俘获现象. 在氨分子模型中,当分子初始处在较低的本征态时,在一定的条件下激发态上将没有粒子数布居,外加光场强烈地使较低的双重态耦合在一起,且粒子数布居总保持为1,尽管此时外场与|2〉→|3〉接近共振. 这是一般模型所不能得到的结论. 关键词： 相干布居俘获 Ξ型三能级系统 双势阱