fractional brownian motion as a nonstationary process an alternative paradigm for dna sequences article discussing the fractional brownian motion fbm as a nonstationary process and an alternative paradigm for dna sequences

dynamics of moments of fitzhugh nagumo neuronal models and stochastic bifurcations for the study of the behavior of noisy neuronal models rodriguez and tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means variances and covariance of the state variables r rodriguez and h c tuckwell phys rev e 54 5585 1996 in this work we first report a modification of their method in the case of the fitzhugh nagumo model which enhances the accuracy of the approximation without including higher order moments this method is then combined with a self consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto and cross correlation functions of the state variables finally we argue that the moments equations can also reveal the existence of stochastic bifurcations i e qualitative changes in the dynamics of stochastic systems

incidence of nonextensive thermodynamics in temporal scaling at feigenbaum points abstract recently in phys rev lett 95 2005 140601 grassberger addresses the interesting issue of the applicability of q statistics to the renowned feigenbaum attractor he concludes there is no genuine connection between the dynamics at the critical attractor and the generalized statistics and argues against its usefulness and correctness yet several points are not in line with our current knowledge nor are his interpretations we refer here only to the dynamics on the attractor to point out that a correct reading of recent developments invalidates his basic claim

the effect of noise on β cell burst period bursting electrical behavior is commonly observed in a variety of nerve and endocrine cells including that in electrically coupled cells located in intact pancreatic islets however individual cells usually display either spiking or very fast bursting behavior and the difference between isolated and coupled cells has been suggested to be due to stochastic fluctuations of the plasma membrane ion channels which are supposed to have a stronger effect on single cells than on cells situated in clusters the channel sharing hypothesis this effect of noise has previously been studied using numerical simulations we show here how the application of two recent methods allows an analytic treatment of the stochastic effects on the location of the saddle node and homoclinic bifurcations which determine the burst period thus the stochastic system can be analyzed similarly to the deterministic system but with a quantitative description of the effect of noise this approach supports previous investigations

statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks dynamical stochastic models of single neurons and neural networks often take the form of a system of n 2 coupled stochastic differential equations we consider such systems under the assumption that third and higher order central moments are relatively small in the general case a system of 1 n n13 generally nonlinear coupled ordinary differential equations holds for the approximate means variances and covariances for the general linear system the solutions of these equations give exact results this is illustrated in a simple case generally the moment equations can be solved numerically results are given for a spiking fitzhugh nagumo model neuron driven by a current with additive white noise differential equations are obtained for the means variances and covariances of the dynamical variables in a network of n connected spiking neurons in the presence of noise s1063 651x 96 03511 8 pacs number s 87 10 1e 05 40 1j 02 50 2r

asymptotic scaling behavior of block entropies for an intermittent process intermittent systems play a prominent role in the field of dynamical phase transitions their extraordinary characteristics also show up when applying the concept of block entropies to symbol sequences which are generated through the method of symbolic dynamics we investigate the dependence of those dynamical entropies on the block length n in the asymptotic limit i e for n ensuremath rightarrow ensuremath infty an analytic treatment is possible when starting from the assumption of independent laminar laps the results are important for a refined characterization of intermittent systems moreover they bring intermittent systems in contact with information carrying sequences which exhibit a very special scaling behavior of dynamical entropies textcopyright 1996 the american physical society

dynamical model for dna sequences we address the problem of dna sequences developing a dynamical method based on the assumption that the statistical properties of dna paths are determined by the joint action of two processes one deterministic with long range correlations and the other random and ensuremath delta function correlated the generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos that are responsible for the birth of anomalous diffusion it is assumed that the deterministic process corresponds to unknown biological rules that determine the dna path whereas the noise mimics the influence of an infinite dimensional environment on the biological process under study we prove that the resulting diffusion process if the effect of the random process is neglected is an ensuremath alpha stable l evy process with 12 we also show that if the diffusion process is determined by the joint action of the deterministic and the random process the correlation effects of the deterministic dynamics are cancelled on the short range scale but show up in the long range one we denote our prescription to generate statistical sequences as the copying mistake map cmm we carry out our analysis of several dna sequences and their cmm realizations with a variety of techniques and we especially focus on a method of regression to equilibrium which we call the onsager analysis with these techniques we establish the statistical equivalence of the real dna sequences with their cmm realizations we show that long range correlations are present in exons as well as in introns but are difficult to detect since the exon dynamics is shown to be determined by the entanglement of three distinct and independent cmm s

perturbation of homoclinics and subharmonics in duffing s equation abstract for lambda mu and g f small the equation x x 2x cubed g x lambda x mu cos t f t is considered near the separatrix s of the unperturbed equation lambda mu 0 0 for g f in a neighborhood of the zero functions a complete description is given of the bifurcation curves in lambda mu space to homoclinic points the perturbation of subharmonics outside s is also considered in a symmetric and nonsymmetric case

non gaussian statistics of anomalous diffusion the dna sequences of prokaryotes this article discusses a non gaussian indicator to measure the deviation from gaussian statistics of a diffusion process generated by dichotomous fluctuations with infinite memory

homoclinic finger rings in rn abstract in this paper we investigate bifurcations of a degenerate homoclinic loop in r n we prove that a homoclinic finger ring an invariant manifold of a definite dimension textured with homoclinic orbits arises from the degenerate homoclinic orbit the size of the homoclinic finger ring is decided by not only its dimension of manifold but also its width for the rise of homoclinic finger rings of different dimensions we give conditions which are proved to form bifurcation manifolds in the parameter space we further estimate the width for the homoclinic finger ring and give a method to compute the bifurcation manifolds approximately

fast subsystem bifurcations in a slowly varying lie nard system exhibiting bursting a perturbed lienard differential system is examined using local stability and hopf bifurcation analyses asymptotic techniques and melnikov s method the results of these analyses are applied to a simple cubic model that exhibits a variety of different oscillatory behaviors for different parameter values for a bounded region in fast parameter space the model exhibits square wave bursting patterns analogous to the bursting electrical activity observed in pancreatic beta cells under certain hypotheses solutions of the cubic model are known to have square wave patterns by using the theory developed for the more general lienard system each hypothesis is shown to correspond to a curve in parameter space together the curves bound a region in which the model exhibits square wave bursting patterns since the model is simple the curves that bound this region can all be determined analytically

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

entropy and information in processes of self organization uncertainty and predictability abstract the mean uncertainties of probability distributions of discrete sets of events and those of dynamic sequences of events are investigated on the macroscopic and on the microscopic level first one time distributions are studied the key point is the specification of the state space the coarse grained physical phase space leads to the thermodynamical entropy and the order parameter space to the informational entropy in a similar way the uncertainty of the state after one step ahead in time introduced by shannon mcmillan and khinchin is related to the dynamic entropies studied by kolmogorov and sinai and to the gain of information the non equilibrium entropies and their relation to properties of attractors are discussed for several examples taken from physical and non physical self organization processes

subharmonic bifurcations in a perturbed nonlinear oscillation abstract in this paper we discuss subharmonic bifurcations inside the homoclinic orbits and outside for a general form of perturbed two dimensional hamiltonian systems with a pair of real parameters and a pair of function parameters g 1 f 1 by approximating periodic orbits to homoclinic orbits we determine bifurcation curves for in r 2 and bifurcation manifolds for g 1 f 1 in the corresponding space of functions furthermore we determine the codimensions of those bifurcation manifolds all results on subharmonic bifurcations are given with conditions on homoclinic orbits our results generalize hale and spezamiglio s in nonlinear anal 9 1985 181

a non extensive approach to the entropy of symbolic sequences symbolic sequences with long range correlations are expected to result in a slow regression to a steady state of entropy increase however we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained provided that the extensive entropy of tsallis with entropic index q is adopted thereby resulting in a new form of entropy that we shall refer to as kolmogorov sinai tsallis kst entropy we assume that the same symbols either 1 or 1 are repeated in strings of length l with the probability distribution p l 1 l the numerical evaluation of the kst entropy suggests that at the value 2 a sort of abrupt transition might occur for the values of in the range 1 3 it is argued that this phase transition like property signals the onset of the thermodynamical regime at 2

probability densities for noisy delay bifurcations the delay of a transition in a nonlinear system due to a slowly varying control parameter can be significantly reduced by very small noise a new asymptotic approximation for the time dependent probability density function gives a complete description of the process into the transition region and is easily interpreted in terms of the noisy dynamics it is also used to calculate mean transition times the method is applied to two nonlinear systems with noise a one dimensional canonical model for a steady bifurcation and the noisy fitzhugh nagumo model

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

dynamical approach to levy processes we derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a liouville like equation by means of a projection procedure this approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable f y t of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d 2 f y t dt 2 furthermore in the special case where f y t has an inverse power law with the index b ranging from 0 to 1 thus making it nonintegrable we show analytically that the

self similar sequences and universal scaling of dynamical entropies symbol sequences play a prominent role in the context of symbolic dynamics important features of a dynamical system are reflected by related statistics of subsequences a dynamical behavior giving rise to a self similar attractor and universal scaling relations expressed by critical exponents will lead to self similar statistics of subsequences in the present paper we show how self similar distributions of subsequences i e temporal self similarity can be connected with a scaling relation for dynamical entropies moreover the effect of slightly perturbing perfectly self similar sequences by contaminating them with noise is investigated the achieved results are of importance for physical processes marking the borderline between order and chaos textcopyright 1996 the american physical society

asymmetric unimodal maps some results from q generalized bit cumulants in this study using q generalized bit cumulants q is the nonextensivity parameter of the recently introduced tsallis statistics we investigate the asymmetric unimodal maps x t 1 1 ensuremath a x t z i i 1 2 correspond to x t g0 and x t l0 respectively z i g1 0lal 2 t 0 1 2 dots the study of the q generalized second cumulant c 2 q of these maps allows us to determine the dependence of the nonextensivity parameter q on the inflection parameter pairs z 1 z 2 the slope of the c 2 q versus c 2 1 plot where c 2 1 is the standard second cumulant provides the necessary tool to accomplish this task the slope behaves exactly the same as the proper q values say q that were obtained for logisticlike maps z 1 z 2 z by costa et al phys rev e 56 245 1997 it appears that as z 2 ensuremath z 1 ensuremath rightarrow ifmmode pm else textpm fi ensuremath infty this slope approaches unity this behavior is very similar to the behavior of q as a function of the inflection parameter for some z dependent maps namely as stackrel ensuremath rightarrow z ensuremath infty q approaches 1

dynamical approach to anomalous diffusion response of levy processes to a perturbation l evy statistics are derived from a dynamical system which can be either hamiltonian or not using a master equation approach we compare these predictions to the random walk approach recently developed by zumofen and klafter for both the nonstationary phys rev e 47 851 1993 and stationary physica a 196 102 1993 case we study the unperturbed dynamics of the system analytically and numerically and evaluate the time evolution of the second moment of the probability distribution we also study the response of the dynamical system undergoing anomalous diffusion to an external perturbation and show that if the slow regression to equilibrium of the variable velocity is triggered by the perturbation the process of diffusion of the space variable takes place under nonstationary conditions and a conductivity steadily increasing with time is generated in the early part of the response process in the regime of extremely long times the conductivity becomes constant with a value though that does not correspond to the prescriptions of the ordinary green kubo treatments

linearly independent homoclinic bifurcations parameterized by a small function abstract in this paper we discuss a small nonautonomous perturbation of an autonomous system on r n which has a homoclinic solution regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co existence of homoclinic orbits those conditions of various co existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations