bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

cluster splitting bifurcation in a system of coupled maps abstract we consider cluster splitting bifurcations in a system of globally coupled maps as coupling parameter decreases at these transitions the number of clusters i e groups of elements with identical dynamics increases we demonstrate that different cascades of cluster splitting can occur depending on statistics of redistribution of the oscillators between new born clusters

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

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

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

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

large deviations for the one dimensional edwards model in this article we prove a large deviation principle for the empirical drift of a one dimensional brownian motion with self repellence called the edwards model our results extend earlier work in which a law of large numbers and a central limit theorem were derived in the edwards model a path of length t receives a penalty e ht w hereht is the selfintersection local time of the path and 0 is a parameter called the strength of self repellence we identify the rate function in the large deviation principle for the endpoint of the path as 2 3 i 1 3 with i given in terms of the principal eigenvalues of a one parameter family of sturm liouville operators we show that there exist numbers 0 b b such that 1 i is linearly decreasing on 0 b 2 i is real analytic and strictly convex on b 3 i is continuously differentiable at b and 4 i has a unique zero at b the latter fact identifies b as the asymptotic drift of the endpoint the critical drift b is associated with a crossover in the optimal strategy of the path for b b the path assumes local drift b during the full time t while for 0 b b it assumes local drift b during time b b 2b t and local drift b during the remaining time b b 2b t thus in the second regime the path makes an overshoot of size b b 2 t so as to reduce its intersection local time

globally coupled circle maps abstract a network of circle maps is investigated successive transitions are found among coherent ordered partially ordered po and turbulent phases in the ordered phase two cluster attractors with similarity have a large basin volume which are reduced to a unique circle map through a scaling transformation with two parameters the po phase is characterized by the increase of partition complexity in the phase chaotic itinerancy is observed whose dynamics consists of quasistationary high dimensional chaotic motion low dimensional attractor ruins and switching among them dependence of the diffusion constant on the size and the nonlinearity clearly deviates from that by a random phase approximation suggesting remaining correlation among elements even in the turbulent phase

central limit theorem for the edwards model the edwards model in one dimension is a transformed path measure for standard brownian motion discouraging self intersections we prove a central limit theorem for the endpoint of the path extending a law of large numbers proved by westwater the scaled variance is characterized in terms of the largest eigenvalue of a one parameter family of differential operators introduced and analyzed by van der hofstad and den hollander interestingly the scaled variance turns out to be independent of the strength of self repellence and to be strictly smaller than one the value for free brownian motion

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

large deviations and exponential decay for the magnetization in a gaussian random field we consider a continuous model for transverse magnetization of spins diffusing in a homogeneous gaussian random longitudinal field lambda v x x in bbb r d where lambda is the coupling constant giving the intensity of the random field in this setting the transverse magnetization is given by the formula m t bbb e exp lambda 2 int 0 t int 0 t k b r b s ds dr where b t t ge 0 is the standard process of brownian motion and k x is the covariance function of the original random field v x we use large deviation techniques to show that the limit s lambda lim t rightarrow infty frac 1 t ln m t exists we also determine the small lambda behavior of the rate s lambda and show that it is indeed decaying as conjectured in the physics literature

partition complexity in a network of chaotic elements a network of chaotic elements is investigated with the use of globally coupled maps elements can split into clusters with synchronized oscillation in a partially ordered phase the clustering has an inhomogeneous tree structure like replica symmetry breaking in spin glass the clustering has large variety by attractors and strongly depends on initial conditions the variety of clusterings is characterized by the distribution of partitions originated in spin glass theory the qualitative similarity and quantitative disagreement of the given attractors with a spin glass are clarified

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

the constants in the central limit theorem for the one dimensional edwards model the edwards model in one dimension is a transformed path measure for one dimensional brownian motion discouraging self intersections we study the constants appearing in the central limit theorem clt for the endpoint of the path which represent the mean and the variance and the exponential rate of the normalizing constant the same constants appear in the weak interaction limit of the one dimensional domb joyce model the domb joyce model is the discrete analogue of the edwards model based on simple random walk where each self intersection of the random walk path recieves a penalty e 2 we prove that the variance is strictly smaller than 1 which shows that the weak interaction limits of the variances in both clts are singular the proofs are based on bounds for the eigenvalues of a certain one parameter family of sturm liouville differential operators obtained by using monotonicity of the zeros of the eigen functions in combination with computer plots

weak interaction limits for one dimensional random polymers in this paper we present a new and flexible method to show that in one dimension various self repellent random walks converge to self repellent brownian motion in the limit of weak interaction after appropriate space time scaling our method is based on cutting the path into pieces of an appropriately scaled length controlling the interaction between the different pieces and applying an invariance principle to the single pieces in this way we show that the self repellent random walk large deviation rate function for the empirical drift of the path converges to the self repellent brownian motion large deviation rate function after appropriate scaling with the interaction parameters the method is considerably simpler than the approach followed in our earlier work which was based on functional analytic arguments applied to variational representations and only worked in a very limited number of situations

periodic orbits for interval maps with sharp cusps abstract we study families of unimodal interval maps having a sharp cusp at a critical point we describe bifurcations and distribution of periodic orbits period adding for attracting periodic orbits is observed that is period k is followed by k 1 etc coexistence of attractors of successive periods as well as of an attracting fixed point and chaos occurs periodic orbits of various periods are grouped into bunches and occupy a relatively small part of the space

analysis of degenerate hopf bifurcations for a nonlinear model of calcium metabolism results of the in viva studies related to calcium metabolism have been generally analysed within the framework of linear mathematical models e g classical compartmental analysis 11 this approach assumes the maintenance of the extracellular calcium concentration at a nearly constant value under physiological conditions plasma calcium homeostasis and the adequacy of a sum of exponential terms the analytical solution of a first order linear differential system to fit kinetic data such as the time course of tracer concentration nonlinear models which exhibit a self oscillatory behaviour have been proposed recently a model of the hormonal control of calcium homeostasis in chicken includes the consideration of a growth function and takes into account complex interrelationships among the calciumregulating hormones and between these hormones and specific metabolic functions 2 another model constructed in our laboratory is a self oscillating model for calcium metabolism in the rat it includes some nonlinear processes involved in bone calcium metabolism and adequately reproduces the kinetic behaviour of radiocalcium in plasma including circadian variations 3 in contrast to linear systems for which the simple dynamic behaviour patterns are predictable and their analytical solutions are commonly applied in calcium metabolism field it is wellestablished that nonlinear systems have potentially numerous and complex spatio temporal expressions excitability periodic oscillations chaos etc therefore the relevance of the proposed nonlinear models and their significance with regard to the control of calcium metabolism require a concomitant knowledge of the set of the dynamic patterns that in theory such nonlinear structures are able to produce the variety of the dynamic behaviours exhibited by the self oscillating model we have established in rat or rather its nonlinear sub structure 4 51 motivated this theoretical work our model of calcium metabolism has been constructed in the general context of compartmental formalism fig 1 briefly this model was based on the explicit consideration the nonlinear properties of the phase transition and nucleation mechanisms associated with bone calcification and their formulation through a single second order autocatalytic process 5 in addition to its ability to generate periodic temporal variations fitting to experimental data fig 2 the model correctly approximates the growth curve of calcium mass as a function of age and proposes a dynamic organization for bone calcium metabolism which distinguishes

scaling for a random polymer letq n be the law of then step random walk on d obtained by weighting simple random walk with a factore for every self intersection domb joyce model of soft polymers it was proved by greven and den hollander 1993 that ind 1 and for every 0 there exist 0 1 and such that under the lawq n asn begin array l i theta beta is the lim it empirical speed of the random walk ii mu beta is the limit empirical distribution of the local times end array a representation was given for and in terms of a largest eigenvalue problem for a certain family of x matrices in the present paper we use this representation to prove the following scaling result as 0 begin array l i beta textstyle 1 over 3 theta beta to b ii beta textstyle 1 over 3 mu beta left left lceil cdot beta textstyle 1 over 3 right rceil right to l 1 eta cdot end array the limitsb 0 and are identified in terms of a sturm liouville problem which turns out to have several interesting properties the techniques that are used in the proof are functional analytic and revolve around the notion of epi convergence of functionals onl 2 our scaling result shows that the speed of soft polymers ind 1 is not right differentiable at 0 which precludes expansion techniques that have been used successfully ind 5 hara and slade 1992a b in simulations the scaling limit is seen for 10 2

static parametric fluctuations give nonstatistical behavior in uncoupled chaotic systems we show that an ensemble of logistic maps with parameters distributed in some range inside a chaotic region cannot be statistical this is so because any parameter range for this map includes periodic windows the effects of this periodicity in the averages of the system can be estimated and are also apparent in the power spectra of average values as a counterexample we show that the behavior for tent maps where chaotic regions do not include periodic windows is statistical miramare trieste october 1993 permanent address departamento de fisica aplicada centra de investigacidn y de estudios avanzados del 1pn unidad merida a p 73 cordemex 97310 merida yucatan mexico

large deviations from the thermodynamic limit in globally coupled maps abstract systems of a large number n of globally coupled maps have become popular as a relatively simple prototype of high dimensional dynamics showing many interesting and typical phenomena like synchronisation cluster formation and multistability and having potential applications in systems like josephson junction arrays or in biophysical models while much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit n there is still need for a sound theory about the asymptotic behaviour of finite n systems as n approaches infinity this article introduces a method by which it is possible to obtain asymptotic estimates for finite size effects that show up as long term deviations from the thermodynamic limit behaviour this method works for systems of globally coupled noisy maps and is based upon the concept of quasipotentials originally developed by freidlin wentzell and others for describing the influence of small random perturbations on the long term behaviour of dynamical systems the problems of explicitly computing quasipotentials in the present context and potential approximation schemes are discussed all the concepts described in this article are illustrated with a simple example

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

on edwards model for polymer chains ii the self consistent potential we obtain an existence and uniqueness theorem for the self consistent potential in edwards model for polymer chains and confirm the asymptotic analysis proposed by him on the basis of wkb arguments

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

synchronization and cluster periodic solutions in globally coupled maps abstract the purpose of this work is to investigate mechanisms by which synchronization takes place in networks of elements with global couplings we consider a family of globally coupled nonlinear maps and find for each model sufficient conditions for synchronization we also analyze bifurcations of syncrhonized dynamics to other homogeneous and cluster periodic solutions in terms of corresponding low dimensional maps

collective behavior in coupled chaotic map lattices with random perturbations abstract numerical simulations of coupled map lattices with non local interactions i e the coupling of a given map occurs with all lattice sites often involve a large computer time if the lattice size is too large in order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution this emulates a thermal bath where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude we demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied we considered one dimensional lattices of chaotic logistic maps at outer crisis x 4 x 1 x

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

mean field fluctuation of a network of chaotic elements remaining fluctuation and correlation in the large size limit fluctuation of the mean field is studied for a network of chaotic elements with the use of globally coupled maps as the size n is increased the variance of the mean field decreases up to a crossover size nc but then stays constant where the remaining variance is roughly proportional to 1nc the distribution of the mean field approaches a gaussian form up to nc but keeps some deviation for n nc the breakdown of the law of large numbers implies some hidden coherence among elements confirmed by finite mutual information and lyapunov spectra the breakdown is also supported by the instability of the self consistent solution for the perron frobenius operator the anomalous effect of additional noise to our system is also reported results from other globally coupled maps suggest that the law of large numbers is generally broken in a turbulent phase with local stretching and contraction but is satisfied for a map with stretching only

at the other side of a saddle node we describe phenomena occurring just before a saddle node bifurcation for one parameter families of interval maps in particular as a parameter approaches the bifurcation value attracting periodic orbits of periodsk k 1 k 2 k 3 can appear we make a detailed study of a family of cusp shaped maps where this phenomenon occurs in a pure form

bifurcation analysis of a system of coupled chemical oscillators bromate chlorite iodine abstract we examine the dynamical behavior of two chemical oscillators bromate iodide and chlorite iodide as well as the coupled system bromate chlorite iodide study of the period and amplitude of oscillation at the bifurcation points yields phase portraits for the uncoupled systems these are combined to give insights into the nature of the higher dimensional coupled system which exhibits a wide variety of dynamical phenomena including multiple stationary states birhythmicity and chaos analysis of the topology of the phase portraits and of the experimentally determined return map suggests that the coupled system may possess a lorenz type attractor

a central limit theorem for a one dimensional polymer measure let s n n n 0 be a random walk on the integers having bounded steps the self repellent resp self avoiding walk is a sequence of transformed path measures which discourage resp forbid self intersections this is used as a model for polymers previously we proved a law of large numbers that is we showed the convergence of s n n toward a positive number under the polymer measure the present paper proves a classical central limit theorem for the self repellent and self avoiding walks that is we prove the asymptotic normality of s n n n for large n the proof refines and continues results and techniques developed previously

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

a survey of one dimensional random polymers in the last decade there has been an enormous progress in the mathematical understanding of one dimensional polymer measures which are path measures that suppress self intersections we are currently in the situation that many interesting questions have either been answered or that essential new ideas are needed in this survey paper we discuss the most relevant results open questions and heuristics

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

statistical properties of spatiotemporal dynamical systems in this paper we consider spatiotemporal dynamical systems modeled by coupled or uncoupled but noise driven map lattices in particular we examine reports in the literature where it is found that the distribution of certain mean field quantities violates the law of large numbers hence nonstatistical but not the central limit theorem our results show that the origin of such nonstatistical behavior is due to the statistical dependence between random variables at different lattice sites thus rendering nonapplicable to such situations the law of large numbers and the central limit theorem additional issues explored include the discussion of a special class of systems where nonstatistical behavior is not observed and the physical motivation for considering uncoupled but noise driven map lattices

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

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper