markov chains of nonlinear markov processes and an application to a winner takes all model for social conformity we discuss nonlinear markov processes defined on discrete time points and discrete state spaces using markov chains in this context special attention is paid to the distinction between linear and nonlinear markov processes we illustrate that the chapman kolmogorov equation holds for nonlinear markov processes by a winner takes all model for social conformity

representation of certain homogeneous hilbertian operator spaces and applications following grothendieck s characterization of hilbert spaces we consider operator spaces f such that both f and f completely embed into the dual of a c algebra due to haagerup musat s improved version of pisier shlyakhtenko s grothendieck inequality for operator spaces these spaces are quotients of subspaces of the direct sum c r of the column and row spaces the corresponding class being denoted by qs c r we first prove a representation theorem for homogeneous f qs c r starting from the fundamental sequences phi c n bigg sum k 1 ne k1 otimes e k bigg c otimes min f 2 quad mbox and quad phi r n bigg sum k 1 ne 1k otimes e k bigg r otimes min f 2 given by an orthonormal basis ek of f under a mild regularity assumption on these sequences we show that they completely determine the operator space structure of f and find a canonical representation of this important class of homogeneous hilbertian operator spaces in terms of weighted row and column spaces this canonical representation allows us to get an explicit formula for the exactness constant of an n dimensional subspace fn of f mathit ex f n sim biggl frac n phi c n phi r bigg frac phi c n phi r n bigg frac n phi r n phi c bigg frac phi r n phi c n bigg biggr 1 2 in the same way the projection injectivity constant of fn is explicitly expressed in terms of c and r too orlicz space techniques play a crucial role in our arguments they also permit us to determine the completely 1 summing maps in effros and ruan s sense between two homogeneous spaces e and f in qs c r the resulting space 1o e f isomorphically coincides with a schatten orlicz class s moreover the underlying orlicz function is uniquely determined by the fundamental sequences of e and f in particular applying these results to the column subspace cp of the schatten p class we find the projection and exactness constants of cpn and determine the completely 1 summing maps from cp to cq for any 1 p q

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

spatial pattern formation in external noise theory and simulation spatial pattern formation in fluctuating media is researched analytically from the point of view of the order parameters concept a reaction diffusion system with external noise is considered as a model of such media stochastic equations for unstable mode amplitudes order parameters the dispersion equation for averaged amplitudes of unstable modes and the fokker planck equation for the order parameters are obtained the theory developed makes it possible to analyze different noise induced effects including the variation of boundaries of ordering and disordering phase transitions depending on the parameters of external noise

finite size fluctuations and stochastic resonance in globally coupled bistable systems the dynamics of a system formed by a finite number n of globally coupled bistable oscillators and driven by external forces is studied focusing on a global variable defined as the arithmetic mean of each oscillator variable several models based on truncation schemes of a hierarchy of stochastic equations for a set of fluctuating cumulant variables are presented this hierarchy is derived using it o stochastic calculus and the noise terms in it are treated using an asymptotic approximation valid for large n in addition a simplified one variable model based on an effective potential is also considered these models are tested in the framework of the phenomenon of stochastic resonance in turn they are used to explain in simple terms the very large gains recently observed in these finite systems

spatially periodic stochastic system with infinite globally coupled oscillators in this paper we study a spatially periodic stochastic system with infinite globally coupled oscillators driven by a constant force f with two typical models we show that when f 0 there is a nonequilibrium transition between the state with zero mean field s 0 and the state with nonzero mean field s not equal 0 for model i the transition is not a phase transition while for the model ii it is second order in addition we find that for coupled oscillators driven only by additive noises when f 0 a transport may emerge if the nonzero mean field breaks the symmetry of the systems with varying f a continuous or discontinuous transition between state s 0 and state s 0 will appear the mean field or current sometimes exhibits hysteresis as a function of f

embedding of the operator space oh and the logarithmic little grothendieck inequality we use voiculescu s concept of free probability to construct a completely isomorphic embedding of the operator space oh in the predual of a von neumann algebra we analyze the properties of this embedding and determine the operator space projection constant of oh n frac 1 108 sqrt frac n 1 ln n le inf p mathcal b ell 2 to oh n p 2 p left p right cb le 288 pi sqrt frac 2n 1 ln n the lower estimate is a recent result of pisier and shlyakhtenko that improves an estimate of order 1 1 lnn of the author the additional factor 1 sqrt 1 ln n indicates that the operator space oh n behaves differently than its classical counterpart ell 2 n we give an application of this formula to positive sesquilinear forms on mathcal b ell 2 this leads to logarithmic characterization of c algebras with the weak expectation property introduced by lance

mean field stochastic theory for species rich assembled communities a dynamical model of an ecological community is analyzed within a mean field approximation in which one of the species interacts with the combination of all of the other species in the community within this approximation the model may be formulated as a master equation describing a one step stochastic process the stationary distribution is obtained in closed form and is shown to reduce to a log series or log normal distribution depending on the values that the parameters describing the model take on a hyperbolic relationship between the connectance of the matrix of interspecies interactions and the average number of species exists for a range of parameter values the time evolution of the model at short and intermediate times is analyzed using van kampen s approximation which is valid when the number of individuals in the community is large good agreement with numerical simulations is found the large time behavior and the approach to the stationary state is obtained by solving the equation for the generating function of the probability distribution the analytical results which follow from the analysis are also in good agreement with direct simulations of the model

noisy fitzhugh nagumo model from single elements to globally coupled networks we study the noisy fitzhugh nagumo model representative of the dynamics of excitable neural elements and derive a fokker planck equation for both a single element and for a network of globally coupled elements we introduce an efficient way to numerically solve this fokker planck equation especially for large noise levels we show that contrary to the single element the network can undergo a hopf bifurcation as the coupling strength is increased furthermore we show that an external sinusoidal driving force leads to a classical resonance when its frequency matches the underlying system frequency this resonance is also investigated analytically by exploiting the different time scales in the problem

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

ergodic directional switching in mobile insect groups we obtain a fokker planck equation describing experimental data on the collective motion of locusts the noise is of internal origin and due to the discrete character and finite number of constituents of the swarm the stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity this complex behavior arises naturally as a result of the randomness in the system its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models our results suggest that the collective switches of the group s direction of motion might be due to a random ergodic effect and as such they are inherent to group formation

self induced stochastic resonance in excitable systems the effect of small amplitude noise on excitable systems with strong time scale separation is analyzed it is found that vanishingly small random perturbations of the fast excitatory variable may result in the onset of a deterministic limit cycle behavior absent without noise the mechanism termed self induced stochastic resonance combines a stochastic resonance type phenomenon with an intrinsic mechanism of reset and no periodic drive of the system is required self induced stochastic resonance is different from other types of noise induced coherent behaviors in that it arises away from bifurcation thresholds in a parameter regime where the zero noise deterministic dynamics does not display a limit cycle nor even its precursor the period of the limit cycle created by the noise has a non trivial dependence on the noise amplitude and the time scale ratio between fast excitatory variables and slow recovery variables it is argued that self induced stochastic resonance may offer one possible scenario of how noise can robustly control the function of biological systems

stochastic persistence and stationary distribution in a holling tanner type prey predator model abstract in this paper we study a stochastic predator prey model with beddington deangelis type functional response and logistic growth for predators the deterministic model is already well studied and we recall some important results here we construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations for the stochastic model we show that the system admits unique positive global solution starting from the positive initial value then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes finally we show that the system has a stationary distribution under certain parametric restrictions numerical simulations are carried out to substantiate the analytical results

computing with distributed chaos we describe and discuss in detail some recent results by sinha and ditto phys rev lett 81 2156 1998 demonstrating the capacity of a lattice of threshold coupled chaotic maps to perform computations such systems are shown to emulate logic gates encode numbers and perform specific arithmetic operations such as addition and multiplication as well as yield more specialized operations such as the calculation of the least common multiplier of a sequence of numbers furthermore we extend the scheme to multidimensional continuous time dynamics in particular to a system relevant to chaotic lasers

nonlinear markov semigroups and interacting levy type processes semigroups of positivity preserving linear operators on measures of a measurable space x describe the evolutions of probability distributions of markov processes on x their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions b x on x describe the evolutions of averages over the trajectories of these markov processes in this paper we introduce and study the general class of semigroups of non linear positivity preserving transformations on measures that is non linear markov or feller semigroups an explicit structure of generators of such groups is given in case when x is the euclidean space r d or more generally a manifold showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for markov models of interacting particles well posedness results for these equations are given together with applications to interacting particles dynamic law of large numbers and central limit theorem the latter being new already for the standard coagulation fragmentation models

a certain class of diffusion processes associated with nonlinear parabolic equations we introduce a martingale problem to associate diffusion processes with a kind of nonlinear parabolic equation then we show the existence and uniqueness theorems for solutions to the martingale problem

kramers problem in evolutionary strategies we calculate the escape rates of different dynamical processes for the case of a one dimensional symmetric double well potential in particular we compare the escape rates of a smoluchowski process i e a corresponding overdamped brownian motion dynamics in a metastable potential landscape with the escape rates obtained for a biologically motivated model known as the fisher eigen process the main difference between the two models is that the dynamics of the smoluchowski process is determined by local quantities whereas the fisher eigen process is based on a global coupling nonlocal interaction if considered in the context of numerical optimization algorithms both processes can be interpreted as archetypes of physically or biologically inspired evolutionary strategies in this sense the results discussed in this work are utile in order to evaluate the efficiency of such strategies with regard to the problem of surmounting various barriers we find that a combination of both scenarios starting with the fisher eigen strategy provides a most effective evolutionary strategy

trichotomous noise induced catastrophic shifts in symbiotic ecosystems an n species lotka volterra stochastic model of a symbiotic ecological system with the verhulst self regulation mechanism is considered the effect of fluctuating environment on the carrying capacity of a population is modeled as the colored three level markovian trichotomous noise in the framework of the mean field theory an explicit self consistency equation for stationary states is presented stability and instability conditions and colored noise induced discontinuous transitions catastrophic shifts in the model are investigated in some cases the mean field exhibits hysteresis as a function of the noise parameters it is shown that the occurrence of catastrophic shifts can be controlled by noise parameters such as correlation time amplitude and flatness the dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams implications of the results on some modifications of the model are discussed

macroscopic limit cycle via pure noise induced phase transitions bistability generated via a pure noise induced phase transition is reexamined from the view of bifurcations in macroscopic cumulant dynamics it allows an analytical study of the phase diagram in more general cases than previous methods in addition using this approach we investigate spatially extended systems with two degrees of freedom per site for this system the analytic solution of the stationary fokker planck equation is not available and a standard mean field approach cannot be used to find noise induced phase transitions a different approach based on cumulant dynamics predicts a noise induced phase transition through a hopf bifurcation leading to a macroscopic limit cycle motion which is confirmed by numerical simulation

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

addendum to colored noise induced discontinuous transitions in symbiotic ecosystems a symbiotic ecosystem with gompertz self regulation and with adaptive competition between populations is studied by means of a n species lotka volterra stochastic model the influence of fluctuating environment on the carrying capacity of a population is modeled as a dichotomous noise the study is a follow up of previous investigations of symbiotic ecosystems subjected to the generalized verhulst self regulation phys rev e 69 061106 2004 65 051108 2002 in the framework of mean field approximation the behavior of the solutions of the self consistency equation for a stationary system is examined analytically in the full phase space of system parameters depending on the mutual interplay of symbiosis and competition of species variation of noise parameters amplitude correlation time can induce doubly unidirectional discontinuous transitions as well as single unidirectional discontinuous transitions of the mean population size

nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators a mean field model of nonlinearly coupled oscillators with randomly distributed frequencies and subject to independent external white noises is analyzed in the thermodynamic limit when the frequency distribution isbimodal new results include subcritical spontaneous stationary synchronization of the oscillators supercritical time periodic synchronization bistability and hysteretic phenomena bifurcating synchronized states are asymptotically constructed near bifurcation values of the coupling strength and theirnonlinear stability properties ascertained

multivariate ornstein uhlenbeck processes with mean field dependent coefficients application to postural sway we study the transient and stationary behavior of many particle systems in terms of multivariate ornstein uhlenbeck processes with friction and diffusion coefficients that depend nonlinearly on process mean fields mean field approximations of this kind of system are derived in terms of fokker planck equations in such systems multiple stationary solutions as well as bifurcations of stationary solutions may occur in addition strictly monotonically decreasing steady state autocorrelation functions that decay faster than exponential functions are found which are used to describe the erratic motion of the center of pressure during quiet standing

stability analysis of mean field type nonlinear fokker planck equations associated with a generalized entropy and its application to the self gravitating system multidimensional nonlinear fokker planck equations of mean field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions two types of eigenvalue equations are studied the nonlinear fokker planck equations are shown to exhibit an h theorem with a liapunov functional that takes the form of a free energy involving generalized entropies of tsallis the second order variation of the liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean field coupling potential assuming quasiequilibrium for the velocity distribution the reduced eigenvalue equation with space coordinates alone is also obtained the alternative type of eigenvalue equation based on the linearization of the nonlinear fokker planck equations is presented taking the mean field coupling potential to be the gravitational one the nonlinear fokker planck equation in terms of three dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean field model of self gravitating system by solving the eigenvalue equation for the eigenfunction with 0 eigenvalue the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed

chaos from nonlinear markov processes why the whole is different from the sum of its parts abstract nonlinear markov processes have been frequently used to address bifurcations and multistability in equilibrium and non equilibrium many body systems however our understanding of the range of phenomena produced by nonlinear markov processes is still in its infancy we demonstrate that in addition to bifurcations and multistability nonlinear markov processes can exhibit another key phenomena well known in the realm of nonlinear physics chaos it is argued that chaotically evolving process probabilities are a generic feature of many body systems exhibiting nonlinear markov processes even if the isolated subsystems do not exhibit chaos that is when considering a nonlinear markov process as an entity of its own type then the nonlinear markov process in general is qualitatively different from its constituent subprocesses which reflects that the many body system as a whole is different from the sum of its parts

stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations abstract we discuss two fundamental aspects of fokker planck equations that are nonlinear with respect to probability densities first we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions second we demonstrate that nonlinear fokker planck equations can be interpreted as linear fokker planck equations describing nonlinear families of markov diffusion processes we exploit this finding in order to derive complete hierarchies of probability densities from nonlinear fokker planck equations

stationary and dynamical properties of finite n unit langevin models subjected to multiplicative noises abstract we have studied the finite n unit langevin model subjected to multiplicative noises by using the augmented moment method amm as a continuation of our previous paper h hasegawa j phys soc japan 75 2006 033001 effects of couplings on stationary and dynamical properties of the model have been investigated the difference and similarity between the results of diffusive and sigmoid couplings are studied in details time dependences of average and fluctuations in local and global variables calculated by the amm are in good agreement with those of direct simulations dss we also discuss stationary distributions of local and global variables with the use of the fokker planck equation fpe method and dss it is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account

stochastic bifurcation in fitzhugh nagumo ensembles subjected to additive and or multiplicative noises abstract we have studied the dynamical properties of finite n unit fitzhugh nagumo fn ensembles subjected to additive and or multiplicative noises reformulating the augmented moment method amm with the fokker planck equation fpe method h hasegawa j phys soc japan 75 2006 033001 in the amm original 2 n dimensional stochastic equations are transformed to eight dimensional deterministic ones and the dynamics is described in terms of averages and fluctuations of local and global variables the stochastic bifurcation is discussed by a linear stability analysis of the deterministic amm equations the bifurcation transition diagram of multiplicative noise is rather different from that of additive noise the former has the wider oscillating region than the latter the synchronization in globally coupled fn ensembles is also investigated results of the amm are in good agreement with those of direct simulations dss

closure schemes in stochastic nonlinear dynamics a validation case study it is known that randomness in dynamical systems often leads to infinite hierarchies of coupled equations for relevant probabilistic quantities several closure methods for truncation of the hierarchies have been proposed in the literature in the present paper the performance of closure schemes for moment hierarchies is compared by using the well known nonlinear equation of overdamped oscillator with additive gaussian white noise in the case of bistable dynamics it is shown that the closure schemes can give incorrect results despite of their good convergence properties to overcome this deficiency new closure procedures are proposed they are based on using the hermite polynomials and their generalization

additive noise induced turing transitions in spatial systems with application to neural fields and the swift hohenberg equation this work studies the spatio temporal dynamics of a generic integral differential equation subject to additive random fluctuations it introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for fokker planck equations and studies analytically the systems stability near turing bifurcations in addition two types of fluctuation are studied namely fluctuations uncorrelated in space and time and global fluctuations which are constant in space but uncorrelated in time we show that the global fluctuations shift the turing bifurcation threshold this shift is proportional to the fluctuation variance applications to a neural field equation and the swift hohenberg equation reveal the shift of the bifurcation to larger control parameters which represents a stabilization of the system all analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral differential equation to gain some insight into experimental manifestations the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively

regular patterns in dichotomically driven activator inhibitor dynamics we investigate turing pattern formation in the presence of additive dichotomous fluctuations in the context of an extended system with diffusive coupling and fitzhugh nagumo kinetics the fluctuations vary in space and or time depending on the realization of the dichotomous switching the system is at a given time for spatial disorder at a given position in one of two possible excitable dynamical regimes each of the two excitable dynamics for itself does not support pattern formation with proper dichotomous fluctuations however the homogeneous steady state is destabilized via a turing instability we investigate the influence of different switching rates different correlation length of the spatial disorder on pattern formation we find three distinct mechanisms for slow switching existing boundaries become unstable for high rates the system exhibits effective bistability which allows for a turing instability for medium rates the fluctuations create spatial structures via a new mechanism where the influence of the fluctuations is twofold first they produce local inhomogeneities which then grow again caused by fluctuations until the whole space is covered utilizing a nonlinear map approach we show bistability of a period one and a period two orbit being associated with the steady homogeneous and the turing pattern state respectively finally for purely static dichotomous disorder we find destabilization of homogeneous steady states for finite nonzero correlation length of the disorder resulting again in turing patterns

nonlinear fokker planck equations whose stationary solutions make entropy like functionals stationary abstract the present study extends the correspondence principle of martinez et al that establishes a link between nonlinear fokker planck equations nlfpes and the variational principle approach of the theory of canonical ensembles by virtue of the extended correspondence principle we reobtain results of kaniadakis and quarati for bose and fermi systems and find relations similar to those derived by plastino and plastino and recently by borland concerning the entropy of nonextensive thermostatistics introduced in physics by tsallis moreover we propose nlfpes related to the renyi entropy and to the entropy proposed by sharma and mittal the latter comprises tsallis entropy and the renyi entropy

mean field limit of systems with multiplicative noise a detailed study of the mean field solution of langevin equations with multiplicative noise is presented three different regimes depending on noise intensity weak intermediate and strong noise are identified by performing a self consistent calculation on a fully connected lattice the most interesting strong noise regime is shown to be intrinsically unstable with respect to the inclusion of fluctuations as a ginzburg criterion shows on the other hand the self consistent approach is shown to be valid only in the thermodynamic limit while for finite systems the critical behavior is found to be different in this last case the self consistent field itself is broadly distributed rather than taking a well defined mean value its fluctuations described by an effective zero dimensional multiplicative noise equation govern the critical properties these findings are obtained analytically for a fully connected graph and verified numerically both on fully connected graphs and on random regular networks the results presented here shed some doubt on what is the validity and meaning of a standard mean field approach in systems with multiplicative noise in finite dimensions where each site does not see an infinite number of neighbors but a finite one the implications of all this on the existence of a finite upper critical dimension for multiplicative noise and kardar parisi zhang problems are briefly discussed

embedding of cq and rq into noncommutative lp spaces 1 p q 2 we prove that a quotient of a subspace of cp prp 1 p 2 embeds completely isomorphically into a noncommutative lp space where cp and rp are respectively the p column and p row hilbertian operator spaces we also represent cq and rq p q 2 as quotients of subspaces of cp prp consequently cq and rq embed completely isomorphically into a noncommutative lp m we further show that the underlying von neumann algebra m cannot be semifinite

noise controlled oscillations and their bifurcations in coupled phase oscillators we derive in gaussian approximation dynamical equations for the first two cumulants of the mean field fluctuations in a system of globally coupled stochastic phase oscillators in these equations the intensity of noise serves as an explicit control parameter its variation generates transitions between three dynamical regimes i stationary ii rotatory and iii locally oscillatory breathing the latter regime has previously not been reported in studies of globally coupled noisy phase oscillators our detailed bifurcation analysis is supported by numerical simulations of an ensemble of coupled stochastic phase oscillators similar regimes are also found in simulations of globally coupled stochastic fitzhugh nagumo elements

nonextensive aspects of small world networks abstract nonextensive aspects of the degree distribution in watts strogatz ws small world networks p sw k have been discussed in terms of a generalized gaussian referred to as q gaussian which is derived by the three approaches the maximum entropy method mem stochastic differential equation sde and hidden variable distribution hvd in mem the degree distribution p q k in complex networks has been obtained from q gaussian by maximizing the nonextensive information entropy with constraints on averages of k and k 2 in addition to the normalization condition in sde q gaussian is derived from langevin equations subject to additive and multiplicative noises in hvd q gaussian is made by a superposition of gaussians for random networks with fluctuating variances in analogy to superstatistics interestingly a single p q k may describe with an accuracy of p sw k p q k 10 2 main parts of degree distributions of sw networks within which about 96 99 of all k states are included it has been demonstrated that the overall behavior of p sw k including its tails may be well accounted for if the k dependence is incorporated into the entropic index in mem which is realized in microscopic langevin equations with generalized multiplicative noises

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

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

statistics of extinction and survival in lotka volterra systems we analyze purely competitive many species lotka volterra systems with random interaction matrices focusing the attention on statistical properties of their asymptotic states generic features of the evolution are outlined from a semiquantitative analysis of the phase space structure and extensive numerical simulations are performed to study the statistics of the extinctions we find that the number of surviving species depends strongly on the statistical properties of the interaction matrix and that the probability of survival is weakly correlated to specific initial conditions s1063 651x 98 03204 8

a mini tutorial on measure valued markov processes and nonlinear martingale problems as a reply to mccauley s comment abstract one goal of this mini tutorial is to provide an introduction into the theory of measure valued markov processes and nonlinear martingales defined by strongly nonlinear fokker planck equations and to discuss the physical relevance of the associated processes another goal is to reply to mccauley s comment on t d frank physica a 331 391 2004 the tutorial addresses in detail two approaches found in physics and mathematics the first approach exploits a mapping between linear and nonlinear fokker planck equations the second approach exploits martingale theory several examples of markov processes and martingales in quantum mechanical nonextensive and self organizing systems defined by nonlinear fokker planck equations are discussed

stochastic bifurcations in a bistable duffing van der pol oscillator with colored noise this paper aims to investigate gaussian colored noise induced stochastic bifurcations and the dynamical influence of correlation time and noise intensity in a bistable duffing van der pol oscillator by using the stochastic averaging method theoretically one can obtain the stationary probability density function of amplitude for the duffing van der pol oscillator and can reveal interesting dynamics under the influence of gaussian colored noise stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution which indicates that system parameters noise intensity and noise correlation time respectively can be treated as bifurcation parameters they also imply that the effects of multiplicative noise are rather different from that of additive noise the results of direct numerical simulation verify the effectiveness of the theoretical analysis moreover the largest lyapunov exponent calculations indicate that p and d bifurcations have no direct connection

resonant behavior of a fractional oscillator with fluctuating frequency the long time behavior of the first moment for the output signal of a fractional oscillator with fluctuating frequency subjected to an external periodic force is considered colored fluctuations of the oscillator eigen frequency are modeled as a dichotomous noise the viscoelastic type friction kernel with memory is assumed as a power law function of time using the shapiro loginov formula exact expressions for the response to an external periodic field and for the complex susceptibility are presented on the basis of the exact formulas it is demonstrated that interplay of colored noise and memory can generate a variety of cooperation effects such as multiresonances versus the driving frequency and the friction coefficient as well as stochastic resonance versus noise parameters the necessary and sufficient conditions for the cooperation effects are also discussed particularly two different critical memory exponents have been found which mark dynamical transitions in the behavior of the system

on non semisplit extensions tensor products and exactness of group c algebras we show the existence of a block diagonal extensionb of the suspensions a of the reduced groupc algebraa c sl 2 such that there is only onec norm on the algebraic tensor productb op b butb is not nuclear even not exact thus the class of exactc algebras is not closed under extensions the existence comes from a new established tensorial duality between the weak expectation property wep of lance and the local variant llp of the lifting property we characterize the local lifting property of separable unitalc algebrasa as follows a has the local lifting property if and only if ext s a is a group wheres a is the suspension ofa if moreovera is the quotient algebra of ac algebra withwep for brevity a isqwep but does not satisfyllp then there exists a quasidiagonal extensionb of the suspensions a by the compact operators such that on the algebraic tensor productb op b there is only onec norm the question if everyc algebra isqwep remains open but we obtain the following results onqwep ac algebrac isqwep if and only ifc isqwep a von neumannii 1 factorn with separable predualn isqwep if and only ifn is a von neumann subfactor of the ultrapower of the hyperfiniteii 1 factor ifg is a maximally almost periodic discrete non amenable group with haagerup s herz schur multiplier constant g 1 then the universal groupc algebrac g is not exact but the reduced groupc albegrac g is exact and isqwep but does not satisfywep andllp we study functiorial properties of the classes ofc algebras satisfyingwep llp resp beingqwep as applications we obtain some unexpected relations between some open questions onc algebras

coherence enhancement in nonlinear systems subject to multiplicative ornstein uhlenbeck noise we show that for two biologically relevant models with self sustained oscillations under the action of a multiplicative ornstein uhlenbeck process their coherence response behaves nonmonotonically with the process correlation time there is a correlation time for which the quality factor is optimized this phenomenon is a consequence of the interplay between the correlation time and the system s periodicity this relation is evidenced through a power law relation with an exponent close to 1 2

colored noise induced hopf bifurcations in predator prey communities a broad class of n 1 species ratio dependent predator prey stochastic models which consist of one predator population and n prey populations is considered the effect of a fluctuating environment on the carrying capacities of prey populations is taken into account as colored noise in the framework of the mean field theory approximate self consistency equations for prey populations mean density and for predator population density are derived to the first order in the noise variance in some cases the mean field exhibits hopf bifurcations as a function of noise correlation time the corresponding transitions are found to be reentrant e g the periodic orbit appears above a critical value of the noise correlation time but disappears again at a higher value of the noise correlation time the nonmonotonous dependence of the critical control parameter on the noise correlation time is found and the conditions for the occurrence of hopf bifurcations are presented our results provide a possible scenario for environmental fluctuations induced transitions between the oscillatory regime and equilibrium state of population sizes observed in nature

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

proof of the existence and uniqueness of a solution for the haissinski equation with a capacitive wake function the existence and uniqueness of a solution for the haissinski equation with a capacitive wake function has been analytically proved

clustering of diffusing organisms recently a brownian bugs model has been proposed as a possible explanation for the clustering of organisms such as planktons w r young a j roberts and g stuhne nature london 412 328 2001 in this model diffusing organisms die and give birth with equal probabilities and are independent of each other the clustering phenomenon is due to the discreteness of birth and death events and cannot be explained in the framework of the continuous model of population dynamics we show by exact calculation that at dimensions smaller than or equal to 2 this simple mechanism is enough to provoke aggregations of the organisms at dimensions higher than 2 the clustering phenomenon disappears but reproductive fluctuations can still be large compared to diffusive ones

memory enhanced energetic stability for a fractional oscillator with fluctuating frequency the long time limit behavior of the variance and the correlation function for the output signal of a fractional oscillator with fluctuating eigenfrequency subjected to a periodic force is considered the influence of a fluctuating environment is modeled by a multiplicative white noise and by an additive noise with a zero mean the viscoelastic type friction kernel with memory is assumed as a power law function of time the exact expressions of stochastic resonance sr characteristics such as variance and signal to noise ratio snr have been calculated it is shown that at intermediate values of the memory exponent the energetic stability of the oscillator is significantly enhanced in comparison with the cases of strong and low memory a multiresonancelike behavior of the variance and snr as functions of the memory exponent is observed and a connection between this effect and the memory induced enhancement of energetic stability is established the effect of memory induced energetic stability encountered in case the harmonic potential is absent is also discussed

le chatelier s principle in replicator dynamics the le chatelier principle states that physical equilibria are not only stable but they also resist external perturbations via short time negative feedback mechanisms a perturbation induces processes tending to diminish its results the principle has deep roots e g in thermodynamics it is closely related to the second law and the positivity of the entropy production here we study the applicability of the le chatelier principle to evolutionary game theory i e to perturbations of a nash equilibrium within the replicator dynamics we show that the principle can be reformulated as a majorization relation this defines a stability notion that generalizes the concept of evolutionary stability we determine criteria for a nash equilibrium to satisfy the le chatelier principle and relate them to mutualistic interactions game theoretical anticoordination showing in which sense mutualistic replicators can be more stable than say competing ones there are globally stable nash equilibria where the le chatelier principle is violated even locally in contrast to the thermodynamic equilibrium a nash equilibrium can amplify small perturbations though both types of equilibria satisfy the detailed balance condition

completely p summing maps on the operator hilbert space oh abstract we prove the completely p summing ideals of oh are all equal as sets for 1 p 2 a phase transition then occurs at p 2 as we also show for p 2 the completely p summing ideals of oh turn out as sets to be schatten ideal classes with the limiting case being the schatten 4 class ideal s 4 when p

on the linear discrepancy model and risky shifts in group behavior a nonlinear fokker planck perspective using a nonlinear fokker planck perspective we re formulate the linear discrepancy model proposed by boster and colleagues that describes the emergence of risky shifts during group decision making analytical expressions for the stationary case are derived and risky shifts are obtained by monte carlo simulations striking similarities with the kuramoto model for group synchronization are pointed out

effect of noise and perturbations on limit cycle systems abstract this paper demonstrates that the influence of noise and of external perturbations on a nonlinear oscillator can vary strongly along the limit cycle and upon transition from limit cycle to stationary point behaviour for this purpose we consider the role of noise on the bonhoeffer van der pol model in a range of control parameters where the model exhibits a limit cycle but the parameters are close to values corresponding to a stable stationary point our analysis is based on the van kampen approximation for solutions of the fokker planck equation in the limit of weak noise we investigate first separately the effect of noise on motion tangential and normal to the limit cycle the key result is that noise induces diffusion like spread along the limit cycle but quasistationary behaviour normal to the limit cycle we then describe the coupled motion and show that noise acting in the normal direction can strongly enhance diffusion along the limit cycle we finally argue that the variability of the system s response to noise can be exploited in populations of nonlinear oscillators in that weak coupling can induce synchronization as long as the single oscillators operate in a regime close to the transition between oscillatory and excitatory modes

h theorem for a mean field model describing coupled oscillator systems under external forces this article studies the asymptotic behavior of solutions of fokker planck equations describing mean field approximations of weakly coupled oscillator systems subjected to external forces using an h theorem we show that transient probability densities converge to stationary ones furthermore stability criteria are derived for the stationary solutions of these fokker planck equations the obtained results are applied to a model that combines the haken kelso bunz model and the models of weakly coupled oscillators proposed by winfree and kuramoto the stability criteria based on the h theorem agree with those derived in our earlier analyses

a maurey type result for operator spaces the little grothendieck theorem for banach spaces says that every bounded linear operator between c k and l2 is 2 summing however it is shown in m junge embedding of the operator space oh and the logarithmic little grothendieck inequality invent math 161 2 2005 225 286 that the operator space analogue fails not every cb map v k oh is completely 2 summing in this paper we show an operator space analogue of maurey s theorem every cb map v k oh is q cb summing for any q 2 and hence admits a factorization v x c q v cb axb q with a b in the unit ball of the schatten class s2q

stochastic analysis of a pulse type prey predator model a stochastic lotka volterra model a so called pulse type model for the interaction between two species and their random natural environment is investigated the effect of a random environment is modeled as random pulse trains in the birth rate of the prey and the death rate of the predator the generalized cell mapping method is applied to calculate the probability distributions of the species populations at a state of statistical quasistationarity the time evolution of the population densities is studied and the probability of the near extinction time from an initial state to a critical state is obtained the effects on the ecosystem behaviors of the prey self competition term and of the pulse mean arrival rate are also discussed our results indicate that the proposed pulse type model shows obviously distinguishable characteristics from a gaussian type model and may confer a significant advantage for modeling the prey predator system under discrete environmental fluctuations

derivation of nonlinear fokker planck equations by means of approximations to the master equation nonlinear fokker planck equations fpes are derived as approximations to the master equation in cases of transitions among both discrete and continuous sets of states the nonlinear effects introduced through the transition probabilities are argued to be relevant for many real phenomena within the class of anomalous diffusion problems the nonlinear fpes obtained appear to be more general than some previously proposed on a purely phenomenological basis ones in spite of this the same kind of solution applies i e it is shown that the time dependent tsallis s probability distribution is a solution of both equations obtained either from discrete or continuous sets of states and that the corresponding stationary solution is in the infinite time limit a stable solution

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

permanence and asymptotical behavior of stochastic prey predator system with markovian switching in this paper we investigate the stochastic permanence and extinction of a stochastic ratio dependent prey predator model controlled by a markov chain in the permanent case we estimate the superior limit and the inferior limit of the average in time of the sample path of the solution the boundaries are related to the stationary probability distribution of the markov chain and the parameters of the subsystems finally we illustrate our main results by two examples and some numerical experiments

experiments on coherence resonance noisy precursors to hopf bifurcations experimental and numerical evidence of coherence resonance in an electrochemical system is reported external noise with a gaussian distribution is superimposed on the system when the anodic current is exhibiting stationary fixed point dynamics below a supercritical hopf bifurcation the amplitude of the added stochastic perturbations is increased monotonically and the induced oscillatory behavior is analyzed it is observed both in experiments and in simulations that the regularity of the noise induced current oscillations reaches a maximum value for an optimum noise level this is indicative of coherence resonance and can be explained with a mechanism based on noisy precursors to a hopf bifurcation

from individual choice to group decision making some universal features are independent of both the social nature of the individuals making the decision and the nature of the decision itself on this basis a simple magnet like model is built pair interactions are introduced to measure the degree of exchange among individuals while discussing an external uniform field is included to account for a possible pressure from outside individual biases with respect to the issue at stake are also included using local random fields a unique postulate of minimum conflict is assumed the model is then solved with emphasis on its psycho sociological implications counter intuitive results are obtained at this stage no new physical technicality is involved instead the full psycho sociological implications of the model are drawn few cases are then detailed to enlight them in addition several numerical experiments based on our model are shown to give both an insight on the dynamics of the model and suggest further research directions

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

colored noise induced discontinuous transitions in symbiotic ecosystems a symbiotic ecosystem is studied by means of the lotka volterra stochastic model using the generalized verhulst self regulation the effect of fluctuating environment on the carrying capacity of a population is taken into account as dichotomous noise the study is a follow up of our investigation of symbiotic ecosystems subjected to three level trichotomous noise r mankin a ainsaar a haljas and e reiter phys rev e 65 051108 2002 relying on the mean field theory an exact self consistency equation for stationary states is derived in some cases the mean field exhibits hysteresis as a function of noise parameters it is established that random interactions with the environment can cause discontinuous transitions the dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams predictions from the mean field theory are compared with the results of numerical simulations our results provide a possible scenario for catastrophic shifts of population sizes observed in nature