strong desynchronizing effects of weak noise in globally coupled systems in assemblies of globally coupled dynamical units weak noise perturbing independently the individual units can cause anomalous dispersion in the synchronized cloud of the units in the phase space when the noise free dynamics of the synchronized assembly is nonperiodic various moments of the linear dimension of the cloud as a function of the noise strength exhibit multiscaling properties with parameter dependent scaling exponents some numerical evidence of this peculiar behavior as well as its interpretation in terms of a multiplicative stochastic process with small additive noise is provided universality of the phenomenon is also discussed

diversity induced coherence resonance in spatially extended chaotic systems the effect of parameter diversity on coupled chua systems is investigated in the absence of diversity the systems jump back and forth between two variable domains of a chaotic attractor and the residence times within a single domain are uncertain by introducing parameter diversity a combined numerical and analytical approach indicates that the systems can jump regularly from one domain to another at an intermediate range of diversity a signature of coherence resonance furthermore the influences of coupling strength and the number of units are also considered our results provide a possibility for the control of chaos in spatially extended chaotic systems by the manipulation of parameter diversity

coherent regimes of mutually coupled chua s circuits we study the dynamical regimes that emerge from the strong coupling between two chua s circuits with parameters mismatch for the region around the perfect synchronous state we show how to combine parameter diversity and coupling in order to robustly and precisely target a desired regime this target process allows us to obtain regimes that may lie outside parameter ranges accessible for any isolated circuit the results are obtained by following a recently developed theoretical technique the order parameter expansion and are verified both by numerical simulations and on electronic circuits the theoretical results indicate that the same predictable change in the collective dynamics can be obtained for large populations of strongly coupled circuits with parameter mismatches

intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators a theory of intrinsic fluctuations is developed of a phase ordering parameter for large populations of weakly and uniformly coupled limit cycle oscillators with distributed native frequencies in particular it is shown that the intensity as well as the correlation time of fluctuations exhibit power law divergence at the onset of mutual entrainment with critical exponents which depend on whether the coupling strength approaches the threshold from below or above this peculiar feature is demonstrated by numerical simulations mainly through finite size scaling analyses in the course of exploring its origin we encounter a new concept termed a correlation frequency which provides a natural interpretation of the finite size scaling laws a comment is given on a recent theory by kuramoto and nishikawa to clarify why it contradicts our results

oscillator death in populations of all to all coupled nonlinear oscillators abstract the behavior of finitely many and infinitely many coupled weakly nonlinear oscillators is studied and it is shown that the uniform rest state can become stable if the coupling is sufficiently strong and the frequencies are sufficiently disparate numerical results show that the phenomenon is robust and occurs when many of the hypotheses are relaxed

variety and generality of clustering in globally coupled oscillators abstract clustering behavior in globally coupled identical oscillators is studied by using the phase model any pair of oscillators is assumed to interact through their phase difference without specifying the coupling function the stability of symmetric cluster states in analyzed where symmetric means that the clusters contain the same number of oscillators it is found that higher harmonics in the coupling function are essential to clustering examples show that various cluster states appear depending on the coupling form including those with oscillating intervals anti phase states and chaotic long transient apart from the phase model we also study a general model for globally coupled limit cycle oscillators in the vicinity of the hopf bifurcation and find that such a system does not exhibit clustering within the lowest order approximation

time delay effects on coupled limit cycle oscillators at hopf bifurcation abstract we present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at hopf bifurcation for a simple model consisting of just two oscillators with a time delayed coupling the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death phase locked and incoherent regions a novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators similar results are obtained for an array of n oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified some general analytic results for the n thermodynamic limit are also obtained and the implications of the time delay effects for physical applications are discussed

a physicist s sandbox we discuss some recent results suggesting that certain spatially extended dynamical systems naturally evolve toward a state characterized by domains of all length scales the analogy with second order phase transitions has prompted the name self organized criticality specific results are available for cellular automaton models which can be thought of as caricatures of a sandpile undergoing avalances the potential generality of the results stems from the very simple nonlinear diffusion dynamics governing the system

collective synchronization in the presence of reactive coupling and shear diversity we analyze the synchronization dynamics of a model obtained from the phase reduction of the mean field complex ginzburg landau equation with heterogeneity we present exact results that uncover the role of dissipative and reactive couplings on the synchronization transition when shears and natural frequencies are independently distributed as it occurs in the purely dissipative case an excess of shear diversity prevents the onset of synchronization but this does not hold true if coupling is purely reactive in this case the synchronization threshold turns out to depend on the mean of the shear distribution but not on all the other distribution s moments

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

synchronization of integrate and fire oscillators with global coupling we study the behavior of globally coupled assemblies of a large number of integrate and fire oscillators with excitatory pulselike interactions on some simple models we show that the additive effects of pulses on the state of integrate and fire oscillators are sufficient for the synchronization of the relaxations of all the oscillators this synchronization occurs in two forms depending on the system either the oscillators evolve en bloc at the same phase and therefore relax together or the oscillators do not remain in phase but their relaxations occur always in stable avalanches we prove that synchronization can occur independently of the convexity or concavity of the oscillator evolution function furthermore the presence of disorder up to some level is not only compatible with synchronization but removes some possible degeneracy of identical systems and allows new mechanisms towards this state textcopyright 1996 the american physical society

phase locked patterns and amplitude death in a ring of delay coupled limit cycle oscillators we study the existence and stability of phase locked patterns and amplitude death states in a closed chain of delay coupled identical limit cycle oscillators that are near a supercritical hopf bifurcation the coupling is limited to nearest neighbors and is linear we analyze a model set of discrete dynamical equations using the method of plane waves the resultant dispersion relation which is valid for any arbitrary number of oscillators displays important differences from similar relations obtained from continuum models we discuss the general characteristics of the equilibrium states including their dependencies on various system parameters we next carry out a detailed linear stability investigation of these states in order to delineate their actual existence regions and to determine their parametric dependence on time delay time delay is found to expand the range of possible phase locked patterns and to contribute favorably toward their stability the amplitude death state is studied in the parameter space of time delay and coupling strength it is shown that death island regions can exist for any number of oscillators n in the presence of finite time delay a particularly interesting result is that the size of an island is independent of n when n is even but is a decreasing function of n when n is odd

measuring firm size distribution with semi nonparametric densities abstract in this article we propose a new methodology based on a log semi nonparametric log snp distribution that nests the lognormal and enables better fits in the upper tail of the distribution through the introduction of new parameters we test the performance of the lognormal and log snp distributions capturing firm size measured through a sample of us firms in 2004 2015 taking different levels of aggregation by type of economic activity our study shows that the log snp provides a better fit of the firm size distribution we also formally introduce the multivariate log snp distribution which encompasses the multivariate lognormal to analyze the estimation of the joint distribution of the value of the firm s assets and sales the results suggest that sales are a better firm size measure as indicated by other studies in the literature

synchronization and relaxation for a class of globally coupled hamiltonian systems abstract a class of coupled hamiltonian systems is examined in which identical nonlinear oscillators are coupled through a mean field the system is shown to have a steady desynchronized solution which becomes linearly unstable as the coupling strength is increased we observe in the stable case that the order parameter of the system decays to zero for a wide class of initial conditions the decay is exponential onan intermediate time scale and then as t 3 2 as t this system shares many similarities to the vlasov poisson equation and as well as kuramoto s model

company size distribution for developing countries abstract we analyze company size distribution for developing countries using the framework proposed by ramsden and kiss haypal physica a 277 2000 220 although this distribution does not fit developing countries data as good as it does to developed ones the parameters of the distribution and for developing countries are remarkably different to those for developed countries this result supports the hypothesis that parameter plays a role analogous to the temperature of the economy which could be related to the level of economic development as reported previously by saslow am j phys 67 1999 1239 also this supports the hypothesis that is related to the competitive exclusion in economics as tending to zero implies the competition free limit case where company size distribution is predicted to be a power law as reported by takayasu and okuyama fractals 6 1998 67 finally we report the goodness of fit for two functions a finite size scaling and a log normal we found that these functions fit the data better in some cases however this is not in itself sufficient evidence that those functions are an appropriate representation of the phenomenon

periodicity manifestations in the non locally coupled maps we study how periodicity manifestations recently found in the turbulent globally coupled maps depend on the global feature of the couplings we examine three non locally coupled map models in the first two the all to all interaction is maintained but the coupling decreases with distance in a power and an exponential law in the third the interaction is uniform but cut off sharply we find that in all three and in dimension d 1 3 periodicity manifests universally from turbulence when the same suppression of the local mean field fluctuation is achieved by the non local averaging 2002 published by elsevier science b v

lower critical dimension for populations of oscillators with randomly distributed frequencies a renormalization group analysis it is argued by way of a renormalization group analysis that the lower critical dimension of macroscopic mutual entrainment in a class of populations of oscillators satisfies a certain inequality which is sensitive to the tail of the distribution of native frequencies this result is supported in part by numerical simulations as well as a proof of the absence of long range order in one dimension

phase locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies abstract we study phase locking in a network of coupled nonlinear oscillators with local interactions and random intrinsic frequencies the oscillators are located at the vertices of a graph and interact along the edges they are coupled by sinusoidal functions of the phase differences across the edges and their intrinsic frequencies are independent and identically distributed with finite mean and variance we derive an exact expression for the probability of phase locking in a linear chain of such oscillators and prove that this probability tends to zero as the number of oscillators grows without bound however if the coupling strength increases as the square root of the number of oscillators the probability of phase locking tends to a limiting distribution the kolmogorov smirnov distribution this latter result is obtained by showing that the phase locking problem is equivalent to a discretization of pinned brownian motion the results on chains of oscillators are extended to more general graphs in particular for a hypercubic lattice of any dimension the probability of phase locking tends to zero exponentially fast as the number of oscillators grows without bound we also consider a less stringent type of synchronization characterized by large clusters of oscillators mutually entrained at the same average frequency it is shown that if such clusters exist they necessarily have a sponge like geometry

anomalous lyapunov spectrum in globally coupled oscillators abstract numerical experiments of a globally coupled oscillator system show that one type of collective chaos of high dimension has discrete and continuous parts in its lyapunov spectrum this occurs in a scattered state i e a state in which no two oscillators behave identically it is argued from a consideration of the phase space structure that the discrete exponents are related to in a sense the macroscopic dynamics while the continuous part reflects the microscopic dynamics this type of high dimensional chaos is compared to a second type possessing an apparently continuous part only preceding the appearance of the first type we found a sequence of bifurcations of collective low dimensional behavior in scattered states and their investigation reveals the route to the first type of the high dimensional chaos

collective motions in globally coupled tent maps with stochastic updating we study a generalization of globally coupled maps where the elements are updated with probability p when p is below a threshold p c the collective motion vanishes and the system is the stationary state in the large size limit we present the linear stability analysis

remarks on the mean field dynamics of networks of chaotic elements abstract fluctuations of the mean field of a globally coupled dynamical systems are discussed the origin of hidden coherence is related with the instability of the fixed point solution of the self consistent perron frobenius equation collective dynamics in globally coupled tent maps are re examined both with the help of direct simulation and the perron frobenius equation collective chaos in a single band state and bifurcation against initial conditions in a two band state are clarified with the return maps of the mean field lyapunov spectra and also the newly introduced lyapunov exponent for the perron frobenius equation future problems on the collective dynamics are discussed

collective dynamics of globally coupled phase oscillators under multisite delayed feedback stimulation recently multisite delayed feedback stimulation was proposed as a novel method for mild and effective deep brain stimulation in neurological diseases characterized by pathological cerebral synchronization to develop the mathematical background of this technique we propose a continuous medium model of globally coupled phase oscillators in the form of a delayed integro differential equation this model provides a convenient description of the spatial phase patterns depending on the stimulation topology we describe physically relevant phase locked solutions and analyze their linear stability considering different spatial stimulation profiles as well as delays and polarities we demonstrate that multisite delayed feedback is a robust soft and effective method for the control of spatio temporal synchronization patterns our results can be used for the optimization of parameters in applications of multisite delayed feedback stimulation to the therapy of neurological diseases with pathological synchronization and the therapy of defective central pattern generators

dynamics of a large system of coupled nonlinear oscillators abstract we consider the interaction of a large number of limit cycle oscillators with linear all to all coupling and a distribution of natural frequencies the system exhibits extremely rich dynamics as the coupling strength and the width of the frequency distribution are varied we find a variety of steady behaviors that can be described by a stationary distribution in phase space frequency locking amplitude death incoherence and partial locking an unexpected result is that the system can also exhibit unsteady behavior in which the phase space distribution evolves periodically quasiperiodically or even chaotically the simple form of the model allows us to derive several analytical results the stability boundaries of amplitude death and incoherence are found explicitly rigorous results on the existence and stability of frequency locking are also obtained

effects of microscopic disorder on the collective dynamics of globally coupled maps this paper studies the effect of independent additive noise on the synchronous dynamics of large populations of globally coupled maps our analysis is complementary to the approach taken by teramae and kuramoto j teramae y kuramoto strong desynchronizing effects of weak noise in globally coupled systems phys rev e 63 2001 036210 who pointed out the anomalous scaling properties preceding the loss of coherence we focus on the macroscopic dynamics that remains deterministic at any noise level and differs from the microscopic one using properly defined order parameters an analytical approach is proposed for describing the collective dynamics in terms of an approximate low dimensional system the systematic derivation of the macroscopic equations provides a link between the microscopic features of the population single element dynamics and noise distribution and the properties of the emergent behaviour the macroscopic bifurcations induced by noise are compared to those originating from parameter mismatches within the population

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

size distribution of portuguese firms between 2006 and 2012 this study aims to describe the size distribution of portuguese firms as measured by annual sales and total assets between 2006 and 2012 giving an economic interpretation for the evolution of the distribution along the time three distributions are fitted to data the lognormal the pareto and as a particular case zipf and the simplified canonical law scl we present the main arguments found in literature to justify the use of distributions and emphasize the interpretation of scl coefficients methods of estimation include maximum likelihood modified ordinary least squares in log log scale and nonlinear least squares considering the levenberg marquardt algorithm when applying these approaches to portuguese s firms data we analyze if the evolution of estimated parameters in both lognormal power and scl is in accordance with the known existence of a recession period after 2008 this is confirmed for sales but not for assets leading to the conclusion that the first variable is a best proxy for firm size

dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks abstract we study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single hopf bifurcation oscillator our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips frequency suppression multiple periodic states and chaos such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator we construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms we find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like birhythmicity phase reversals radial trapping phase jumps and spiraling patterns in the amplitude space our results may find useful applications in physical chemical or biological systems

on the size distribution of firms additional evidence from the g7 countries abstract we analyze the average size distribution of a pool of the g7 group s firms over the period 1987 2000 in particular firm sizes are measured employing different proxies and after conditioning on business cycle phases we find that i the empirical distributions are all consistent with a power law ii point estimates suggest that only in limited cases the exponent is equal to 1 i e the resulting size distribution generally is not zipf iii regardless of the variable employed to measure firm sizes firms are distributed more equally during recessions than during expansions

loss of coherence in a system of globally coupled maps we study the formation of symmetric i e equally sized or nearly symmetric clusters in an ensemble of globally coupled identical chaotic maps it is shown that the loss of synchronization for the coherent state and the emergence of subgroups of oscillators with synchronized behavior are two distinct processes and that the type of behavior that arises after the loss of total synchronization depends sensitively on the dynamics of the individual map for our system of globally coupled logistic maps symmetric two cluster formation is found to proceed through a periodic state associated with the stabilization either of an asynchronous period 2 cycle or of an asynchronous period 4 cycle with further reduction of the coupling strength each of the principal clustering states undergoes additional bifurcations leading to cycles of higher periodicity or to quasiperiodic and chaotic dynamics if desynchronization of the coherent chaotic state occurs before the formation of stable clusters becomes possible high dimensional chaotic motion is observed in the intermediate parameter interval

from collective oscillations to collective chaos in a globally coupled oscillator system abstract various types of collective behaviors are discovered in globally coupled ginzburg landau oscillators when the coupling is sufficiently weak the oscillators are either in complete synchrony or their phases are scattered completely randomly for finite coupling new collective behaviors emerge such as splitting of the population into a small number of clusters or their fusion into a continuous stringlike distribution in the phase plane low dimensional chaotic dynamics arises from the coupled motion of 3 point clusters chaotic motion is also exhibited by fused clusters which is of extremely high dimension possibly proportional to the system size as is implied from its lyapunov analysis in the latter type of chaos the motion of the string is in some cases characterized by repeated stretching and foldings it is argued how this kind of coherent behavior seen on a collective level does not contradict the high dimensionality of the corresponding chaotic attractor

nonstandard scaling law of fluctuations in finite size systems of globally coupled oscillators universal scaling laws form one of the central issues in physics a nonstandard scaling law or a breakdown of a standard scaling law on the other hand can often lead to the finding of a new universality class in physical systems recently we found that a statistical quantity related to fluctuations follows a nonstandard scaling law with respect to the system size in a synchronized state of globally coupled nonidentical phase oscillators i nishikawa et al chaos 22 013133 2012 however it is still unclear how widely this nonstandard scaling law is observed in the present paper we discuss the conditions required for the unusual scaling law in globally coupled oscillator systems and validate the conditions by numerical simulations of several different models

amplitude response of coupled oscillators abstract we investigate the interaction of a pair of weakly nonlinear oscillators eg each near a hopf bifurcation when the coupling strength is comparable to the attraction of the limit cycles changes in amplitude cannot then be ignored and there are new phenomena we show that even for simple forms of these equations there are parameter regimes in which the interaction causes the system to stop oscillating and the rest state at zero stabilized by the interaction is the only stable solution when the uncoupled oscillators have a local frequency that is independent of amplitude zero shear and the coupling is scalar we give a complete description of the behavior we then analyze more complicated equations to show the effects of amplitude dependent frequency in the uncoupled equations nonzero shear and nonscalar coupling for example when there is shear there can be bistability between a phase locked solution and an unlocked drift solution in which the oscillators go at different average frequencies there can also be bistability between a phase locked solution and an equilibrium nonoscillatory solution the equations for the zero shear case have a pair of degenerate bifurcation points and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities when the coupling is nonscalar there are a variety of new behaviors including bistability of an in phase and an anti phase solution for two identical oscillators parameters for which only the anti phase solution is stable and phase trapping i e nonlocked solutions in which the oscillators still have the same average frequency it is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system

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

order parameter expansion and finite size scaling study of coherent dynamics induced by quenched noise in the active rotator model we use a recently developed order parameter expansion method to study the transition to synchronous firing occurring in a system of coupled active rotators under the exclusive presence of quenched noise the method predicts correctly the existence of a transition from a rest state to a regime of synchronous firing and another transition out of it as the intensity of the quenched noise increases and leads to analytical expressions for the critical noise intensities in the large coupling regime it also predicts the order of the transitions for different probability distribution functions of the quenched variables using numerical simulations and finite size scal ing theory to estimate the critical exponents of the transitions we found values which are consistent with those reported in other scalar systems in the exclusive presence of additive static disorder full detail a particular simple analysis was developed in 4 where the authors used an expansion of the dynamical equa tions of the model up to first order in the deviation of the quenched variables from their mean value and identified a self consistency equation which had to be solved numeri cally the order parameter expansion used in the present pa per expands consistently this analysis up to terms of second order thus reaching a higher accuracy the resulting closed system of three differential equations reproduces the order ing abilities of quenched noise in this system and further more predicts a sharp transition back into the disordered state where no macroscopic order is observed the presented method gives an intuition about the different regimes the system is capable of and how the phase borders depend on the system parameters with its use the change of the nullclines due to diversity in the fitzhugh nagumo model 24 and phase transitions induced by diversity 10 have been investigated to complete the study we have done ex tensive numerical calculations of the full dynamical system in the vicinity of the transitions the order of the transitions has been determined and furthermore we present a detailed finite size scaling analysis of the critical exponents the paper is organized as follows first in sec ii we will define the active rotator model and summarize its main prop erties macroscopic observables that describe the collective behavior are introduced then in sec iii the approximation method is applied and conclusions about steady states are drawn in sec iv we present numerical results that support the previous findings and use the theory of finite size scaling to determine some of the critical exponents characterizing the transitions

on the bifurcation structure of the mean field fluctuation in the globally coupled tent map systems abstract a theoretical analysis on the behavior of the globally coupled tent map systems based on the kinetics of the mean field value is presented we investigated the nature of the mean field fluctuation in large system size limit and uncovered the existence of a complicated bifurcation structure that leads the appearance of collective motion with the increase of the coupling strength the way to the appearance of the collective motion sensitively depends on the gradient a of individual tent maps for a belonging to a set which is dense in 2 2 the collective motion appears when the coupling strength exceeds a certain threshold value while for a belonging to another dense set in 2 2 any non zero coupling will induce the collective motion the dependence of the amplitude of mean field fluctuation on the coupling strength is also obtained as a function of the parameter a and the coupling strength

macroscopic chaos in globally coupled maps abstract we study the coherent dynamics of globally coupled maps showing macroscopic chaos with this term we indicate the hydrodynamical like irregular behavior of some global observables with typical times much longer than the times related to the evolution of the single or microscopic elements of the system the usual lyapunov exponent is not able to capture the essential features of this macroscopic phenomenon using the recently introduced notion of finite size lyapunov exponent we characterize in a consistent way these macroscopic behaviors basically at small values of the perturbation we recover the usual microscopic lyapunov exponent while at larger values a sort of macroscopic lyapunov exponent emerges which can be much smaller than the former a quantitative characterization of the chaotic motion at hydrodynamical level is then possible even in the absence of the explicit equations for the time evolution of the macroscopic observables

linear reformulation of the kuramoto model of self synchronizing coupled oscillators the present paper introduces a linear reformulation of the kuramoto model describing a self synchronizing phase transition in a system of globally coupled oscillators that in general have different characteristic frequencies the reformulated model provides an alternative coherent framework through which one can analytically tackle synchronization problems that are not amenable to the original kuramoto analysis it allows one to solve explicitly for the synchronization order parameter and the critical point of 1 the full phase locking transition for a system with a finite number of oscillators unlike the original kuramoto model which is solvable implicitly only in the mean field limit and 2 a new class of continuum systems it also makes it possible to probe the system s dynamics as it moves toward a steady state while discussion in this paper is restricted to systems with global coupling the formalism introduced by the linear reformulation also lends itself to solving systems that exhibit local or asymmetric coupling

integrable order parameter dynamics of globally coupled oscillators we study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter mean field and amplitude frequency correlator reduction the main result of the paper is the exact solution of a corresponding nonlinear system on a two dimensional invariant manifold we present a complete classification of phase portraits and bifurcations obtain explicit expressions for invariant manifolds a limit cycle among them and derive analytical solutions for arbitrary initial data and different regimes

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

disorder can eliminate oscillator death we use an array of diffusively coupled limit cycle oscillators with a regular monotonic trend of natural frequencies to demonstrate that the disorder introduced in the form of random deviations from a linear trend of frequencies can weaken considerably desynchronization induced oscillator death and as a result increase oscillation intensity substantially there exist definite optimal levels of magnitude of spatial disorder at which maximal oscillatory energy is attained in the array

theory of self synchronization in the presence of native frequency distribution and external noises abstract a general scheme is presented for the theoretical treatment of self synchronization of many body oscillators with variable amplitudes close to harmonic ones with small nonlinearity and dissipative interactions in the presence of external noises and a native frequency distribution applications of the theory are given to interacting van der pol oscillators with various frequency distributions the critical condition which leads to a nonvanishing order parameter is studied in detail it turns out that the synchronization is described in terms of two kinds of bifurcation mechanism mechanical and thermal ones new strong slaving emerges in the case of the mechanical bifurcation multifarious dependences of the order parameter on the width of native frequency distribution appears in some cases because of the presence of these two bifurcation mechanisms however only one bifurcation of landau type is observable since hybridization occurs when the two mechanisms coexist

theory of collective firing induced by noise or diversity in excitable media large variety of physical chemical and biological systems show excitable behavior characterized by a nonlinear response under external perturbations only perturbations exceeding a threshold induce a full system response firing it has been reported that in coupled excitable identical systems noise may induce the simultaneous firing of a macroscopic fraction of units however a comprehensive understanding of the role of noise and that of natural diversity present in realistic systems is still lacking here we develop a theory for the emergence of collective firings in nonidentical excitable systems subject to noise three different dynamical regimes arise subthreshold motion where all elements remain confined near the fixed point coherent pulsations where a macroscopic fraction fire simultaneously and incoherent pulsations where units fire in a disordered fashion we also show that the mechanism for collective firing is generic it arises from degradation of entrainment originated either by noise or by diversity

dynamical movement primitives learning attractor models for motor behaviors nonlinear dynamical systems have been used in many disciplines to model complex behaviors including biological motor control robotics perception economics traffic prediction and neuroscience while often the unexpected emergent behavior of nonlinear systems is the focus of investigations it is of equal importance to create goal directed behavior e g stable locomotion from a system of coupled oscillators under perceptual guidance modeling goal directed behavior with nonlinear systems is however rather difficult due to the parameter sensitivity of these systems their complex phase transitions in response to subtle parameter changes and the difficulty of analyzing and predicting their long term behavior intuition and time consuming parameter tuning play a major role this letter presents and reviews dynamical movement primitives a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques the essence of our approach is to start with a simple dynamical system such as a set of linear differential equations and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term both point attractors and limit cycle attractors of almost arbitrary complexity can be generated we explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics

domain synchronization of a cyclic lotka volterra reaction diffusion mechanism abstract the domain synchronization of a cyclic lotka volterra reaction diffusion mechanism is studied in a dilute solution this is done by performing a number of molecular dynamics simulations and by setting up a mathematical model that can account for the dynamics seen in the simulations the simulations show that below a certain critical length the domain is synchronized but for larger domains the system quickly becomes unsynchronized a mathematical model based on the law of mass action and fick s law will not reveal this critical length however if an ad hoc gaussian noise term is added to the model it resemble the synchronization phenomena seen in the molecular dynamics simulations

periodically forced finite networks of heterogeneous globally coupled oscillators a low dimensional approach we study a network of 500 globally coupled modified van der pol oscillators the value of a parameter associated with each oscillator is drawn from a normal distribution giving a heterogeneous network for strong enough coupling the oscillators all have the same period and we consider periodic forcing of the network when it is in this state by exploiting the correlations that quickly develop between the state of an oscillator and the value of its parameter we obtain an approximate low dimensional description of the system in terms of the first few coefficients in a polynomial chaos expansion standard bifurcation analysis can then be performed on the low dimensional system which results from this computational coarse graining and the results obtained from this predict very well the behaviour of the high dimensional system for any set of realisations of the random parameter situations in which the method begins to fail are also discussed

collective chaos and noise in the globally coupled complex ginzburg landau equation abstract we study a globally coupled version of the complex ginzburg landau equation gc cgle which consists of a large number n of identical two dimensional oscillators coupled through their mean amplitude depending on parameter values different dynamical regimes are attained we focus particularly on an interesting regime where the individual oscillators follow erratic motion but in a sufficiently coherent way so that the average motion does not vanish when n becomes large and is also chaotic a simple description of this state is proposed by considering the motion of a single forced two dimensional system which has both a limit cycle and a fixed point as stable attractors determining which of these two deterministic attractors is selected by a weak noise and how this depends on the parameter of the reduced system allows us to determine self consistently the average amplitude and dominant frequency of the collective behaviour of the full system finally we show that adding a small noise to the gc cgle transforms the chaotic collective behaviour into a purely periodic one

phase transitions induced by microscopic disorder a study based on the order parameter expansion based on the order parameter expansion we present an approximate method which allows us to reduce large systems of coupled differential equations with diverse parameters to three equations one for the global mean field variable and two which describe the fluctuations around this mean value the method is based on a systematic perturbation expansion and can be applied around the vicinity of the homogeneous state with this tool we analyze phase transitions induced by microscopic disorder in three prototypical models of phase transitions which have been studied previously in the presence of thermal noise we study how macroscopic order is induced or destroyed by time independent local disorder and analyze the limits of the approximation by comparing the results with the numerical solutions of the self consistency equation which arises from the property of self averaging finally we carry on a finite size analysis of the numerical results and calculate the corresponding critical exponents

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

analytical results for coupled map lattices with long range interactions we obtain exact analytical results for lattices of maps with couplings that decay with distance as r ensuremath ensuremath alpha we analyze the effect of the coupling range on the system dynamics through the lyapunov spectrum for lattices whose elements are piecewise linear maps we get an algebraic expression for the lyapunov spectrum when the local dynamics is given by a nonlinear map the lyapunov spectrum for a completely synchronized state is analytically obtained the critical line characterizing the synchronization transition is determined from the expression for the largest transversal lyapunov exponent in particular it is shown that in the thermodynamical limit such transition is only possible for sufficiently long range interactions namely for ensuremath alpha l ensuremath alpha c d where d is the lattice dimension

optimized analysis of the critical behavior in polymer mixtures from monte carlo simulations a complete outline is given for how to determine the critical properties of polymer mixtures with extrapolation methods similar to the ferrenberg swendsen techniques recently devised for spin systems by measuring not only averages but the whole distribution of the quantities of interest it is possible to extrapolate the data obtained in only a few simulations neart c over the entire critical region thereby saving at least 90 of the computer time normally needed to locate susceptibility peaks or cumulant intersections and still getting more precise results a complete picture of the critical properties of polymer mixtures in the thermodynamic limit is then obtained with finite size scaling functions since the amount of information extracted from a simulation in this way is drastically increased as compared to conventional methods the investigation of mixtures with long chains or built in asymmetries is now possible as an example the critical points exponents and amplitudes of dense symmetric polymer mixtures with chain lengths ranging fromn 16 up ton 256 are determined within the framework of the 3d bond fluctuation model using grand canonical simulation techniques as an example for an asymmetry the generalization of the method to asymmetric monomer potentials is briefly discussed

factorization with genus 2 curves the elliptic curve method ecm is one of the best factorization methods available it is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower we use special hyperelliptic curves and kummer surfaces to reduce the complexity of the algorithm our implementation gmp hecm is faster than gmp ecm for factoring large numbers

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

averaging of globally coupled oscillators abstract we study a specific system of symmetrically coupled oscillators using the method of averaging the equations describe a series array of josephson junctions we concentrate on the dynamics near the splay phase state also known as the antiphase state ponies on a merry go round or rotating wave we calculate the floquet exponents of the splay phase periodic orbit in the weak coupling limit and find that all of the floquet exponents are purely imaginary in fact all the floquet exponents are zero except for a single complex conjugate pair thus nested two tori of doubly periodic solutions surround the splay phase state in the linearized averaged equations we numerically integrate the original system and find startling agreement with the averaging results on two counts the observed ratio of frequencies is very close to the prediction and the solutions of the full equations appear to be either periodic or doubly periodic as they are in the averaged equations such behavior is quite surprising from the point of view of generic dynamical systems theory one expects higher dimensional tori and chaotic solutions we show that the functional form of the equations and not just their symmetry is responsible for this nongeneric behavior

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method

collective synchronisation in lattices of nonlinear oscillators with randomness the authors study mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies it is shown rigorously that the model exhibits no long range order in one dimension and that in higher dimensional lattices large clusters of synchronised oscillators necessarily have a sponge like structure surprisingly the phase locking behaviour of the mean field model is completely different from that of any finite dimensional lattice indicating that d infinity is the upper critical dimension for phase locking

a tale of two tails do power law and lognormal models fit firm size distributions in the mid victorian era abstract the paper explores the frequency and size distributions of firm size in a novel dataset for the mid victorian era from a recent extraction of the england and wales population censuses of 1851 1861 1871 and 1881 the paper contrasts the hypothesis of the power laws against the lognormal model for the tails of the distributions using maximum likelihood estimation log likelihood ratio clipped sample coefficient of variation umpu wilks test kolmogorov smirnov statistic among other state of the art statistical methods our results show that the power law hypothesis is accepted for the size distribution for the years 1851 and 1861 while 1871 is marginally non significant but for 1881 the test is inconclusive the paper discusses the process that generates these distributions citing recent literature that shows how after adding an i i d noise to the gibrat s multiplicative model one can recreate a power law behaviour overall the paper provides describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a pareto model and that the lognormal model is firmly rejected