onset of cooperative entrainment in limit cycle oscillators with uniform all to all interactions bifurcation of the order function abstract the concept of order function was previously proposed as the key to a general theory of macroscopic mutual or cooperative entrainment in large populations of limit cycle oscillators with weak interactions such that every element is linked to all the other as well as with weakly dispersed intrinsic frequencies that is limit cycle oscillators that can be modeled by globally coupled phase oscillators with distributed frequencies following previous work a bifurcation theory of the order function is developed on the basis of its self consistent functional equation to elucidate in particular generic scaling behavior of such systems at the onset of cooperative entrainment among other results when the onset is not abrupt the critical exponent of fundamental order parameters turns out to generically differ from the conventional value 1 2 taken by the well studied sinusoidal coupling model as well as by typical mean fields models of thermodynamic phase transitions to which coupled oscillator models investigated here are analogous the theory also reveals what happens in nongeneric cases moreover a criterion is found of whether the bifurcation is normal or inverted all these analytical results and predictions are verified not only by numerically solving the equation of the order function but also by numerical simulations although this paper is mainly concerned with the critical behaviors noncritical regimes are also explored to demonstrate overall power of the order function theory by reproducing simulation results such as average frequency spectra the theory however keeps some room to be further generalized a finding which suggests this is put forth

asymptotic formation and orbital stability of phase locked states for the kuramoto model abstract we discuss the asymptotic formation and nonlinear orbital stability of phase locked states arising from the ensemble of non identical kuramoto oscillators we provide an explicit lower bound for a coupling strength on the formation of phase locked states which only depends on the diameters of natural frequencies and initial phase configurations we show that when the phases of non identical oscillators are distributed over the half circle and the coupling strength is sufficiently large the dynamics of kuramoto oscillators exhibits two stages transition and relaxation stages in a transition stage initial configurations shrink to configurations whose diameters are strictly less than 2 in a finite time and then the configurations tend to phase locked states asymptotically this improves previous results on the formation of phase locked states by chopra spong 2009 26 and ha ha kim 2010 27 where their attention were focused only on the latter relaxation stage we also show that the kuramoto model is l 1 contractive in the sense that the l 1 distance along two smooth kuramoto flows is less than or equal to that of initial configurations in particular when two initial configurations have the same averaged phases the l 1 distance between them decays to zero exponentially fast for the configurations with different phase averages we use the method of average adjustment and translation invariant of the kuramoto model to show that one solution converges to the translation of the other solution exponentially fast this establishes the orbital stability of the phase locked states our stability analysis does not employ any standard linearization technique around the given phase locked states but instead we use a robust l 1 metric functional as a lyapunov functional in the formation process of phase locked states we estimate the number of collisions between oscillators and lower upper bounds of the transversal phase differences

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

on an unconditional basis of generalized eigenvectors of the nonself adjoint gribov operator in bargmann space abstract in this article we prove that the system of generalized eigenvectors of the gribov operator is an unconditional basis in bargmann space we also give a generalized diagonalization in the abel sense of the semigroups respectively associated to this operator and to its square root finally some open problems are pointed out

diagonalization of non selfadjoint analytic semigroups and application to the shape memory alloys operator abstract to a densely defined but not necessarily selfadjoint operator a on a hilbert space h we consider on r h the following abstract elliptic problem of dirichlet type formula then in this paper we establish that for every t 0 the solution formula can be expanded into a series of generalized eigenvectors of the operator a provided that its resolvent belongs to carleman class c p for some p 0 1 2 a similar result holds for t large enough if the inverse a 1 belongs to carleman class c p for every p 1 2 see theorem 3 1 and theorem 3 2 furthermore we apply these obtained results to the shape memory alloys non selfadjoint operator formula and d n n x n when acting on an appropriate hilbert space e of functions on the interval 0 1 by establishing that the inverse c 1 belongs to the carleman class c p for every p 1 2 so that we get in this case more regularity in the sense that the operatorial series k 1 e tc p k converges strongly in e to the analytic semigroup e tc for every t 0 the p k are the projectors into the root subspaces of c a similar result holds for formula provided that t is large enough see theorem 4 1 and theorem 4 2 in section 4 for the precise result

the onset and end of the gunn effect in extrinsic semiconductors a hopf bifurcation analysis of the spontaneous current oscillation in direct current dc voltage biased extrinsic semiconductors is given for the classical model of the gunn effect in n gaas for semiconductor lengths l larger than a certain minimal value the steady state is linearly unstable for voltages in an interval phi alpha phi omega as l increases the branch of time periodic solutions bifurcating at simple eigenvalues when phi phi alpha turns from subcritical to supercritical and then back to subcritical again for very long semiconductors a quasi continuum of oscillatory modes bifurcates from the steady state at the onset of the instability the bifurcating branch is then described by a scalar reaction diffusion equation with cubic nonlinearity subject to antiperiodic boundary conditions on a subinterval of 0 l for the electron velocity curve we have considered the bifurcation is subcritical which may agree with experimental observations in n gaas an extension o

hyperbolic chaos at blinking coupling of noisy oscillators we study an ensemble of identical noisy phase oscillators with a blinking mean field coupling where onecluster and two cluster synchronous states alternate in the thermodynamic limit the population is described by a nonlinear fokker planck equation we show that the dynamics of the order parameters demonstrates hyperbolic chaos the chaoticity manifests itself in phases of the complex mean field which obey a strongly chaotic bernoulli map hyperbolicity is confirmed by numerical tests based on the calculations of relevant invariant lyapunov vectors and lyapunov exponents we show how the chaotic dynamics of the phases is slightly smeared by finite size fluctuations doi 10 1103 physreve 87 032912

analyse de scattering d un operateur cubique de heun dans l espace de bargmann the boundary conditions at infinity are used in a description of all maximal dissipative extensions in bargmann space of the minimal heun s operator the characteristic functions of the dissipative extensions are computed completeness theorems are obtained for the system of generalized eigenvectors

periodic behavior of the stochastic brusselator in the mean field limit we prove a propagation of chaos result for the mean field limit of a model for a trimolecular chemical reaction called brusselator then we show that the pair of nonlinear i e law dependent stochastic differential equations describing the evolution of the concentration of the molecules at a given site in the mean field limit has a solution with a periodic law in t finally we identify the limit and establish a central limit theorem for the periodic law in the case where the noise tends to zero

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

on an unconditional basis of generalized eigenvectors of an analytic operator and application to a problem of radiation of a vibrating structure in a light fluid abstract in this paper we prove that the system of generalized eigenvectors of the perturbed operator t e t 0 e t 1 e 2 t 2 e k t k forms an unconditional basis with parentheses in a separable hilbert space x where e c t 0 is a closed densely defined linear operator on x with domain d t 0 having compact resolvent while t 1 t 2 are linear operators on x with the same domain d d t 0 satisfying a specific growing inequality an application to a problem of radiation of a vibrating structure in a light fluid is presented

densite des vecteurs propres generalises d une classe d operateurs compacts non auto adjoints et applications we consider a closed densely defined linear operatort in a hilbert spacee and assume the existence of 0 t such thatk t 0i 1 is compact and the existence ofp 0 such thats n k o n 1 p wheres n k denotes the sequence of non zero eigenvalues of the compact hermitian operator sqrt k k in this work sufficient conditions announced in 1 are introduced to assure that the closed subspace ofe spanned by the generalized eigenvectors oft coincides withe these conditions are in particular verified by a family of non self adjoint operators arising in reggeon s field theory

an ode model of the motion of pelagic fish a system of ordinary differential equations odes is derived from a discrete system of vicsek czirok et al phys rev lett 75 6 1226 1229 1995 describing the motion of a school of fish classes of linear and stationary solutions of the odes are found and their stability explored using equivariant bifurcation theory the existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found applications of the model to the migration of the capelin a pelagic fish that undertakes an extensive migration in the north atlantic are discussed and simulation of the odes presented

phase diagram of a generalized winfree model we study the phase diagram of a generalized winfree model the modification is such that the coupling depends on the fraction of synchronized oscillators a situation which has been noted in some experiments on coupled josephson junctions and mechanical systems we let the global coupling k be a function of the kuramoto order parameter r through an exponent z such that z 1 corresponds to the standard winfree model zl1 strengthens the coupling at low r low amount of synchronization and at zg1 the coupling is weakened for low r using both analytical and numerical approaches we find that z controls the size of the incoherent phase region and that one may make the incoherent behavior less typical by choosing zl1 we also find that the original winfree model is a rather special case indeed the partial locked behavior disappears for zg1 at fixed k and varying ensuremath gamma the stability boundary of the locked phase corresponds to a transition that is continuous for zl1 and first order for zg1 this change in the nature of the transition is in accordance with a previous study of a similarly modified kuramoto model

remarks on the nonlinear stability of the kuramoto sakaguchi equation we present a nonlinear stability estimate of the incoherent solution to the kuramoto sakaguchi equation using energy estimates our global stability analysis does not require the amplitude of the solution to be small as long as the ratio of diffusion strength to coupling strength is sufficiently large we also show that the kuramoto sakaguchi equation is l 2 contractive in the perturbed regime of the incoherent solution moreover we establish the l infinity convergence in any finite time interval from the kuramoto sakaguchi equation to the kuramoto equation as the diffusion coefficient goes to zero c 2015 elsevier inc all rights reserved

on the finiteness of collisions and phase locked states for the kuramoto model synchronization phenomenon is ubiquitous in our complex systems and many phenomenological models have been proposed and studied analytically and numerically among them the kuramoto model serves as a prototype model for the phase synchronization of weakly coupled oscillators in this paper we study the finiteness of collisions crossings among kuramoto oscillators in the relaxation process toward the phase locked states and the total number of phase locked states with positive kuramoto order parameters for identical oscillators it is well known that collisions between distinct oscillators cannot occur in finite time and we show that there are only a finite number of phase locked states with positive order parameters however for non identical oscillators oscillators with different natural frequencies can cross each other in their relaxation process and estimating the total number of phase locked states is a nontrivial matter we show that for the non identical case asymptotic phase locking is equivalent to the finiteness of collisions and the total number of phase locked states with positive order parameters is bounded above by 2 n where n is the number of oscillators

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

stability of incoherence in a population of coupled oscillators we analyze a mean field model of coupled oscillators with randomly distributed frequencies this system is known to exhibit a transition to collective oscillations for small coupling the system is incoherent with all the oscillators running at their natural frequencies but when the coupling exceeds a certain threshold the system spontaneously synchronizes we obtain the first rigorous stability results for this model by linearizing the fokker planck equation about the incoherent state an unexpected result is that the system has pathological stability properties the incoherent state is unstable above threshold butneutrally stable below threshold we also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise

spectral mackey functors and equivariant algebraic k theory i abstract spectral mackey functors are homotopy coherent versions of ordinary mackey functors as defined by dress we show that they can be described as excisive functors on a suitable category and we use this to show that universal examples of these objects are given by algebraic k theory more importantly we introduce the unfurling of certain families of waldhausen categories bound together with suitable adjoint pairs of functors this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic k theory of such objects as spectral mackey functors finally we employ this technology to introduce fully functorial versions of a theory upside down a theory and the algebraic k theory of derived stacks we use this to give what we think is the first general construction of 1 et equivariant algebraic k theory for profinite etale fundamental groups this is key to our approach to the mackey functor case of a sequence of conjectures of gunnar carlsson

nonlinear instability of the incoherent state for the kuramoto sakaguchi fokker plank equation we study the nonlinear instability of the incoherent solution to the kuramoto sakaguchi fokker plank ksfp equation in a large coupling strength regime for our instability analysis we construct an approximate exponentially growing perturbation mode using an elementary energy method this method does not require spectral information from the linearized ksfp equation or an explicit growing solution for the corresponding linear equation when the distribution function of oscillator s natural frequencies is either a dirac measure or a bounded function with a compact support in a small interval around the origin the incoherent solution is nonlinearly unstable depending on the relative sizes of the coupling strength and diffusion coefficient

regularized trace formula of magic gribov operator on bargmann space abstract in this article we obtain a regularized trace formula for magic gribov operator h g h acting on bargmann space where g a 3 a 3 and h a a i a a a a here a and a are the standard bose annihilation and creation operators and in reggeon field theory the real parameters is the magic coupling of pomeron is pomeron intercept is the triple coupling of pomeron and i 2 1 by applying some abstract results of sadovnichii podol skii 2002 23 we give the number of corrections sufficient for the existence of finite formula of the trace of concrete magic gribov s operator

sur une valeur propre d un operateur the studied eigenvalue is the smallest one of the operator h mu mu a i lambda a a a a in the orthogonal complement of the vacuum wherea anda are the creation and annihilation operators call this eigenvaluee as attention is focused on the dependence on this eigenvalue exists and is a positive number for positive values of using the fact that the inverse ofh is a positive operator it is proved thate extends to a positive increasing analytic function on the whole real line in particular e 0 0 contrary to what might have been expected from the fact thata a a a is formally self adjoint

singular unlocking transition in the winfree model of coupled oscillators the winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies as the coupling strength and the spread of natural frequencies are varied the various stable states of the model can undergo bifurcations nearly all of which have been characterized previously the one exception is the unlocking transition in which the frequency locked state disappears abruptly as the spread of natural frequencies exceeds a critical width viewed as a function of the coupling strength this critical width defines a bifurcation curve in parameter space for the special case where the frequency distribution is uniform earlier work had uncovered a puzzling singularity in this bifurcation curve here we seek to understand what causes the singularity using the poincar e lindstedt method of perturbation theory we analyze the locked state and its associated unlocking transition first for an arbitrary distribution of natural frequencies and then for discrete systems of n oscillators we confirm that the bifurcation curve becomes singular for a continuum uniform distribution yet find that it remains well behaved for any finite n suggesting that the continuum limit is responsible for the singularity

quelques nouvelles proprietes de regularite de l operateur de gribov in this work we establish new regularity properties for gribov s operator h a a i a a a a 2 wherea anda are the creation and annihilation operators particularly we prove that for all e 0 h 1 is in the class of carleman s operatorl1 e

a shocking display of synchrony abstract this article explores the kuramoto model describing the synchronization of a population of coupled oscillators two versions of this model are considered a discrete version suitable for a population with a finite number of oscillators and a continuum model found in the limit of an infinite population when the strength of the coupling between the oscillators exceeds a threshold the oscillators partially synchronize we explore the transition in the continuum model which takes the form of a bifurcation of a discrete mode from a continuous spectrum we use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition and compare with the synchronization predicted by the discrete model there are similarities with instabilities in ideal plasmas and inviscid fluids but these are superficial

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

pro unitality and pro excision in algebraic k theory and cyclic homology the purpose of this paper is to study pro excision in algebraic k theory and cyclic homology after suslin wodzicki cuntz quillen cortinas and geisser hesselholt as well as continuity properties of andre quillen and hochschild homology a key tool is first to establish the equivalence of various pro tor vanishing conditions which appear in the literature this allows us to prove that all ideals of commutative noetherian rings are pro unital in a suitable sense we show moreover that such pro unital ideals satisfy pro excision in derived hochschild and cyclic homology it follows hence and from the suslin wodzicki criterion that ideals of commutative noetherian rings satisfy pro excision in derived hochschild and cyclic homology and in algebraic k theory in addition our techniques yield a strong form of the pro hochschild kostant rosenberg theorem an extension to general base rings of the cuntz quillen excision theorem in periodic cyclic homology a generalisation of the f igin tsygan theorem a short proof of pro excision in topological hochschild and cyclic homology and new artin rees and continuity statements in andre quillen and hochschild homology msc 19d55 primary 16e40 13d03 secondary

witt vectors and tambara functors abstract we give a combinatorial description of the ring of g witt vectors on a polynomial algebra over the integers for every finite group g using this description we show that the functor which takes a commutative ring with trivial action of g to its ring of witt vectors coincides with the left adjoint of the algebraic functor from the category of g tambara functors to the category of commutative rings with an action of g

amplitude death in an array of limit cycle oscillators we analyze a large system of limit cycle oscillators with mean field coupling and randomly distributed natural frequencies we prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance the system undergoes amplitude death the oscillators pull each other off their limit cycles and into the origin which in this case is astable equilibrium point for the coupled system we determine the region in couplingvariance space for which amplitude death is stable and present the first proof that the infinite system provides an accurate picture of amplitude death in the large but finite system

uncertainty in phase frequency synchronization of large populations of globally coupled nonlinear oscillators abstract numerical evidence as well as an analytical description is given for the existence of a kind of uncertainty principle in the mean field model of the kuramoto type of large populations of nonlinearly coupled oscillators here the noise plays the role of the planck constant this means that there is a limit to the possibility of synchronizing both phases and frequencies the collective phase synchronization competes in fact within such a model with the frequency synchronization the explanation rests on the effect of the noise which is responsible for loss of accuracy comparison is made with a more sophisticated model where such uncertainty is absent when the coupling parameter is sufficiently large

towards a comprehensive theory of brain activity coupled oscillator systems under external forces abstract recently jirsa et al and haken discussed a theory comprising the brain wave equation proposed by nunez the wilson cowan ermentrout cowan model and hopfield networks this theory was applied to model findings obtained in an experiment that relates brain activity and behavior the so called julliard experiment in previous works of jirsa et al and frank et al the focus was on the brain wave aspect recently we conducted similar experiments the results obtained are modeled in this paper in terms of coupled oscillator systems coupled oscillator systems are considered to represent the wilson cowan ermentrout cowan model aspect of the unifying theory building on a model proposed by haken kelso and bunz and the theory of weakly coupled oscillators established by winfree and by kuramoto we derive a nonlinear fokker planck equation whose stationary solutions mimic the neocortical brain activity observed in our experiments

stability of synchronized states in networks of phase oscillators the stability of synchronized states frequency locked states in networks of phase oscillators is investigated for several network topologies it is shown that for some topologies there is more than one stable synchronized state according to the sign of coupling strength between oscillators it is also shown that in some cases the synchronized state corresponds to zero order parameter

asymptotic description of transients and synchronized states of globally coupled oscillators abstract a two timescale asymptotic method has been introduced to analyze the multimodal mean field kuramoto model of oscillator synchronization in the high frequency limit the method allows to uncouple the probability density in different components corresponding to the different peaks of the oscillator frequency distribution each component evolves towards a stationary state in a comoving frame and the overall order parameter can be reconstructed by combining them synchronized phases are a combination of traveling waves and incoherent solutions depending on parameter values our results agree very well with direct numerical simulations of the nonlinear fokker planck equation for the probability density numerical results have been obtained by finite differences and a spectral method in the particular case of bimodal symmetric and asymmetric frequency distribution with or without external field we also recover in a very easy and intuitive way the only other known analytical results those corresponding to reflection symmetric bimodal frequency distributions near bifurcation points

formation of phase locked states in a population of locally interacting kuramoto oscillators abstract in this paper we present an asymptotic formation of phase locked states from the ensemble of kuramoto oscillators with a symmetric and connected interaction topology for a limited interaction topology that does not have an all to all interaction lyapunov type approaches based on phase and frequency diameters do not work due to the lack of completeness thus we employ an energy method together with the connectedness of underlying interaction topologies to determine the complete synchronization estimates our synchronization estimation method consists of two parts first we establish that the uniform boundedness of fluctuations yields the asymptotic formation of phase locked states using ojasiewicz gradient inequality second we show that for the initial configurations lying in the half circle the uniform boundedness of fluctuations can be derived by a comparison with solutions to the linear gronwall s differential inequality for the total phase variance

approximation of the semigroup generated by the hamiltonian of reggeon field theory in bargmann space abstract the reggeon field theory is governed by a non self adjoint operator constructed as a polynomial in a a the standard bose annihilation and creation operators in zero transverse dimension this hamiltonian acting in bargmann space is defined by h a 2 a 2 a a i a a a a where i 2 1 and are real numbers and the operators a a satisfy the commutation relation a a i as the quantum mechanical system described by h has a velocity dependent potential containing powers of momentum up to the fourth the problem of existence of hamiltonian path integral for the evolution operator e t h of this theory is of interest on its own in particular can we express e t h as a limit of integral operators in this article one considerably reduces the difficulty by studying the trotter product formula of h to reach two objectives the first objective is to prove a very specific error estimate for the error in a trotter product formula in trace norm for h viewed as the sum of the operators a 2 a 2 and a a i a a a a the second objective of this work is to give a approximation of the semigroup generated by h when h is split in the sum of a 2 a 2 a a and i a a a a we note that this case is entirely different in fact the usual trotter product formula is not defined because the interaction operator a a a a is not the infinitesimal generator of a semigroup on bargmann space for 0 and 0 we choose an approximation operator i i a a a a e a 2 a 2 a a and we give a connection between and e h this choice allows us to give in a intissar note on the path integral formulation of reggeon field theory preprint a generalized trotter product formula for t a a i a a a a i e for limit case as 0 and answers to the above question

a moment based approach to the dynamical solution of the kuramoto model we examine the dynamics of the kuramoto model with a new analytical approach by defining an appropriate set of moments the dynamical equations can be exactly closed we discuss some applications of the formalism such as the existence of an effective hamiltonian for the dynamics we also show how this approach can be used to numerically investigate the dynamical behaviour of the model without finite size effects

existence of hysteresis in the kuramoto model with bimodal frequency distributions we investigate the transition to synchronization in the kuramoto model with bimodal distributions of the natural frequencies previous studies have concluded that the model exhibits a hysteretic phase transition if the bimodal distribution is close to a unimodal one due to the shallowness of the central dip here we show that proximity to the unimodal bimodal border does not necessarily imply hysteresis when the width but not the depth of the central dip tends to zero we draw this conclusion from a detailed study of the kuramoto model with a suitable family of bimodal distributions

on a test statistic for the kuramoto order parameter of synchronization an illustration for group synchronization during rocking chairs we discuss a quantitative approach to detect phase synchronization in noisy experimental multivariate data to this end we derive a test statistic based on the kuramoto order parameter we derive criteria for rejecting the null hypothesis of zero phase synchronization at given significance levels we apply the test statistic to a recently conducted study on group behavior that used a rocking chair paradigm in this context we quantify the emergence of phase synchronization within small groups of participants

glassy synchronization in a population of coupled oscillators a phase model for a population of oscillators with random excitatory and inhibitory mean field coupling and subject to external white noise random forces is proposed and studied in the thermodynamic limit different stable phases for the oscillator population may be found i an incoherent state where all possible values of an oscillator phase are equally probable ii a synchronized state where the population has a nonzero collective phase iii a glassy phase where the global synchronization is zero but the oscillators are in phase with the random disorder and iv a mixed phase where the oscillators are partially synchronized and partially in phase with the disorder these predictions are based upon bifurcation analysis of the reduced equation valid at the thermodynamic limit and confirmed by brownian simulation

on the relation between vicsek and kuramoto models of spontaneous synchronization abstract the vicsek model for self propelling particles in 2d is investigated with respect to the addition of the stochastic perturbation of dynamic equations we show that this model represents in essence the same type of bifurcations under a different type of noise as the celebrated kuramoto model of spontaneous synchronization these models demonstrate similar behavior at least within the mean field approach to prove this we consider two types of noise for the vicsek model which are commonly considered in the literature the intrinsic and the extrinsic ones according to the terminology of pimentel et al j a pimentel m aldana c huepe h larralde intrinsic and extrinsic noise effects on phase transitions of network models with applications to swarming systems physical review e statistical nonlinear and soft matter physics 77 6 2008 doi 10 1103 physreve 77 061138 url http dx doi org 10 1103 physreve 77 061138 the qualitative correspondence with the bifurcation of stationary states in the kuramoto model is stated a new type of stochastic perturbation the mixed noise is proposed it is constructed as the weighted superposition of the intrinsic and the extrinsic noises the corresponding phase diagram noise amplitude vs interaction strength is obtained the possibility of the tricritical behavior for the vicsek model is predicted

thermodynamic limit of the first order phase transition in the kuramoto model in the kuramoto model a uniform distribution of the natural frequencies leads to a first order i e discontinuous phase transition from incoherence to synchronization at the critical coupling parameter k c we obtain the asymptotic dependence of the order parameter above criticality r r c alpha k k c 2 3 for a finite population we demonstrate that the population size n may be included into a self consistency equation relating r and k in the synchronized state we analyze the convergence to the thermodynamic limit of two alternative schemes to set the natural frequencies other frequency distributions different from the uniform one are also considered

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

the witt vectors for green functors abstract we define twisted hochschild homology for green functors this construction is the algebraic analogue of the relative topological hochschild homology thh c n and under flatness conditions it describes the e 2 term of the kunneth spectral sequence for relative thh applied to ordinary rings we obtain new algebraic invariants extending hesselholt s construction of the witt vectors of noncommutative rings we interpret our construction as providing witt vectors for green functors

collective behavior in ensembles of globally coupled maps abstract coherent collective behavior in an ensemble of globally coupled maps is investigated in the limit of infinite number of elements a nonlinear frobenius perron equation is derived for this system and it is shown that it can have quasiperiodic and chaotic solutions for the description of finite ensembles we propose a noisy nonlinear frobenius perron equation and show that it gives the correct power spectrum of mean field fluctuations

on the riesz basis of a family of analytic operators in the sense of kato and application to the problem of radiation of a vibrating structure in a light fluid abstract in this paper we prove that the system formed by some of generalized eigenvectors of the operator t 0 e t 1 e 2 t 2 which are analytic on e forms a riesz basis of the separable hilbert space h where e c and t 0 t 1 t 2 some linear transformations on h which have the same domain d h after that we give an application for a problem concerning the radiation of a vibrating structure in a light fluid

on the nonexistence of elements of kervaire invariant one we show that the kervaire invariant one elements j 2j 1 2s exist only for j 6 by browder s theorem this means that smooth framed manifolds of kervaire invariant one exist only in dimensions 2 6 14 30 62 and possibly 126 except for dimension 126 this resolves a longstanding problem in algebraic topology