symmetry and phase locking in a ring of pulse coupled oscillators with distributed delays abstract phase locking in a ring of pulse coupled integrate and fire oscillators with distributed delays is analysed using group theory the period of oscillation of a solution and those related by symmetry is determined self consistently numerical continuation of maximally symmetric solutions in characteristic system length and timescales yields bifurcation diagrams with spontaneous symmetry breaking the stability of phase locked solutions is determined via a linearisation of the oscillator firing map in the weak coupling regime averaging leads to an effective phase coupled model with distributed phase shifts and the analysis of the system is considerably simplified in particular the collective period of a solution is now slaved to the relative phases for odd numbered rings spontaneous symmetry breaking can lead to a change of stability of a travelling wave state via a simple hopf bifurcation the resulting non phase locked solutions are constructed via numerical continuation at these bifurcation points the corresponding behaviour in the integrate and fire system is explored with simulations showing bifurcations to quasiperiodic firing patterns

symmetry and phaselocking in chains of weakly coupled oscillators weakly coupled chains of oscillators with nearest neighbor interactions are analyzed for phaselocked solutions it is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound it is also shown that qualitative behavior of these solutions depends on whether the coupling is diffusive or i synaptic terms defined in the paper the methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two point continuum boundary value problem that is easier to analyze the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law an application to the neurophysiology of motor behavior is discussed briefly

lyapunov function for the kuramoto model of nonlinearly coupled oscillators a lyapunov function for the phase locked state of the kuramoto model of non linearly coupled oscillators is presented it is also valid for finite range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes for the kuramoto model a minimum of the lyapunov function corresponds to a ground state of a system with frustration the interaction between the oscillators xy spins is ferromagnetic whereas the random frequencies induce random fields which try to break the ferromagnetic order i e global phase locking the ensuing arguments imply asymptotic stability of the phase locked state up to degeneracy and hold for any probability distribution of the frequencies special attention is given to discrete distribution functions we argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results but that a partial locking of some but not all sublattices is not to be expected the order parameter of the phase locked state is shown to have a strictly positive lower bound r 1 2 so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded

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

kuramoto model with uniformly spaced frequencies finite n asymptotics of the locking threshold we study phase locking in the kuramoto model of coupled oscillators in the special case where the number of oscillators n is large but finite and the oscillators natural frequencies are evenly spaced on a given interval in this case stable phase locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width called the locking threshold for infinite n the exact value of the locking threshold was calculated 30 years ago however the leading corrections to it for finite n have remained unsolved analytically here we derive an asymptotic formula for the locking threshold when n 1 the leading correction to the infinite n result scales like either n 3 2 or n 1 depending on whether the frequencies are evenly spaced according to a midpoint rule or an end point rule these scaling laws agree with numerical results obtained by pazo d pazo phys rev e 72 046211 2005 pleee81539 375510 1103 physreve 72 046211 moreover our analysis yields the exact prefactors in the scaling laws which also match the numerics

finite dimensional simple lie algebras with a nonsingular derivation abstract we consider the known finite dimensional simple lie algebras of characteristic p 3 and determine all finite dimensional simple lie algebras over an algebraically closed field of characteristic p 7 admitting a nonsingular derivation we also show that the z p n 1 z graded simple lie algebras which admit a nonsingular derivation must be of hamiltonian type

superconformal compactifications in weighted projective space we discuss some aspects of string vacua constructed from orbifolded nonminimal landau ginzburg theories which correspond to calabi yau manifolds in weighted projective space in contrast to previous expectations we find that these theories allow for the construction of numerous stable 2 0 calabi yau vacua most of which are not simply deformations of an underlying 2 2 theory thus indicating that this phenomenologically promising sector of the space of classical vacua is quite robust we briefly discuss methods for extracting the phenomenology of these models and show for example that the full renormalizable superpotential of oursu 5 theories is not corrected by world sheet instantons and is thus given exactly by its tree level value

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

complex macroscopic behavior in systems of phase oscillators with adaptive coupling abstract using recent dimensionality reduction techniques in large systems of coupled phase oscillators exhibiting bistability we analyze the complex macroscopic behavior arising when the coupling between oscillators is allowed to evolve slowly as a function of either macroscopic or local system properties for example we observe macroscopic excitability and intermittent synchrony in a system of time delayed kuramoto oscillators with hebbian and anti hebbian learning we demonstrate the robustness of our findings by considering systems with increasing complexity including time delayed oscillators with adaptive network structure and community interaction as well as a system with bimodally distributed frequencies

scaling behaviour at the onset of mutual entrainment in a population of interacting oscillators scaling behaviour of an order parameter and its fluctuations is numerically investigated at the onset of macroscopic mutual entrainment in a population of interacting self oscillators in particular evidence is presented for the power law divergence of the fluctuations with exponents near 1 8 finite size scaling forms are also proposed and verified

travelling waves in chains of pulse coupled integrate and fire oscillators with distributed delays abstract we analyse travelling waves in a chain of pulse coupled integrate and fire oscillators with nearest neighbour coupling and delayed interactions this is achieved by approximating the equations for phase locking in terms of a singularly perturbed two point continuum boundary value problem the latter has a solution provided that a self consistent value for the collective frequency of oscillations can be found we investigate how the qualitative behaviour of travelling waves depends on the distribution of natural frequencies across the chain and the form of delayed interactions a linear stability analysis of phase locked solutions is carried out in terms of perturbations of the firing times of the oscillators it is shown how travelling waves destabilize when the detuning between oscillators or the strength of the coupling becomes too large

calabi yau manifolds motivations and constructions the possible ways of compacitification of thee8 e8 superstring theory to four dimensions are reviewed the phenomenological need forn 1 supersymmetry is argued on quite general grounds to favour the choice of a calabi yau manifold for the compact internal manifold the massless spectrum after compactification is derived in full detail revealing beside the usual particles others that may have great phenomenological impact the technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given a big family of such constructions giving rise to many new calabi yau manifolds is presented and its relevance to the search of a phenomenologically acceptable solution is discussed

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 transition to frequency entrainment in a long chain of pulse coupled oscillators a long chain of pulse coupled oscillators was studied the oscillators interacted via a phase response curve similar to those obtained from pacemaker cells in the heart the natural frequencies were random numbers from a distribution with finite bandwidth stable frequency entrained states were shown always to exist above a critical coupling strength below the critical coupling the probability to have such states was shown to be zero if the number of oscillators is infinite this discontinuity establishes the existence of a phase transition in the thermodynamic limit for weak coupling clusters of frequency entrained oscillators emerged the cluster sizes were exponentially distributed even when the critical coupling was approached at this coupling the mean cluster size diverged to infinity according to a power law the standard deviation of the distribution of mean frequencies in the chain converged to zero also according to a power law

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

almost solubility of lie algebras with almost regular automorphisms the well known theorem of borel mostow kreknin on solubility of lie algebras with regular automorphisms is generalized to the case of almost regular automorphisms it is proved that if a lie algebra l admits an automorphism of finite order n with finite dimensional fixed point subalgebra of dimension dimcl m then l has a soluble ideal of derived length bounded by a function of n whose codimension is bounded by a function of m and n theorem 1 a virtually equivalent formulation is in terms of a n graded lie algebra l whose zero component l0 has finite dimension m the functions of n and of m and n in theorem 1 can be given explicit upper estimates the proof is of combinatorial nature and uses the criterion for solubility of lie rings with an automorphism obtained in e i khukhro siberian math j 42 2001 996 1000 the method of generalized or graded centralizers is developed which was originally created in e i khukhro math ussr sbornik 71 1992 51 63 for almost regular automorphisms of prime order as a corollary we prove a result analogous to theorem 1 on locally nilpotent torsion free groups admitting an automorphism of finite order with the fixed points subgroup of finite rank theorem 3 we also prove an analogous result for lie rings with an automorphism of finite order having finitely many fixed points theorem 2 2003 elsevier inc all rights reserved

network architecture and spatio temporally symmetric dynamics we examine the relation between the structure of a network and the spatio temporally symmetric periodic dynamics it can support for solutions in which no cell is stationary we show that only networks in which all cells interact with each other or which contain a single group of interacting cells which drive the remainder of the network can exhibit such dynamics robustly these characteristics of network architecture are not captured by the typical statistical quantities used to describe network structure we illustrate the existence of spatio temporally periodic solutions through a direct construction using ideas from coupled cell theory and the theory of weakly coupled oscillators and show that these solutions can be stable in a very large region of parameter space while we consider only a special type of network behavior these ideas extend to more general architectures and dynamics

automorphisms and fitting factors of finite groups it is proved that for every prime p there exists a function fp such that if g is a finite solvable group with an automorphism of order p then g contains a nilpotent normal subgroup of index at most fp cg the question is first reduced to a representation theoretic problem which is then analyzed along the lines of the well known theorem b of hall and higman using an elementary argument of brauer and fowler it is shown further that in the case p 2 the assumption of solvability may be omitted thus there is a function f such that for any group g of even order and any involution x in g the fitting factor group gf g has order at most f cg x

phase locking in chains of multiple coupled oscillators phase locking in chains of weakly coupled oscillators with coupling beyond nearest neighbors is studied starting with a piecewise linear coupling function a homotopy method is applied to prove the existence of phase locked solutions numerical examples are provided to illustrate the existence and the properties of the solutions differences between multiple coupling and nearest neighbor coupling are also discussed 2000 elsevier science b v all rights reserved

dynamics of a globally coupled oscillator array abstract we study a set of n globally coupled ordinary differential equations of the form encountered in circuit analysis of superconducting josephson junction arrays particular attention is paid to two kinds of simple time periodic behavior known as in phase and splay phase states some results valid for general n as well as further results for n 2 and n are presented a recurring theme is the appearance of very weak dynamics near the periodic states the implications for josephson junction arrays are discussed

stable rotating waves in two dimensional discrete active media the existence and stability of stable rotating solutions spiral waves in a discrete system of coupled phase models is proven monotone methods are used to obtain existence and qualitative features of the solutions an application of fenichel s results for singular perturbation implies the existence for a one variable model for excitability numerical comparisons with the phase model and a realistic membrane model are made a numerically computed hopf bifurcation shows the existence of the wobbling core observed in spatially continuous models

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

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

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

spontaneous synchronization of coupled oscillator systems with frequency adaptation we study the synchronization of kuramoto oscillators with all to all coupling in the presence of slow noisy frequency adaptation in this paper we develop a new model for oscillators which adapt both their phases and frequencies it is found that this model naturally reproduc es some observed phenomena that are not qualitatively produced by the standard kuramoto model such as long waiting times before the synchronization of clapping audiences by assuming a self consistent steady state solution we find three stability regimes for the coupling constant k separated by critical points k1 and k2 i for k k2 the incoherent state becomes unstable and only the synchronized state exists iii for k1 k2 the average waiting time grows logarithmically with the number of oscillators

synchronization in networks of general weakly nonlinear oscillators we present a general approach to the study of synchrony in networks of weakly nonlinear systems described by singularly perturbed equations of the type x x ef x x 0 by performing a perturbative calculation based on normal form theory we analytically obtain an o approximation to the floquet multipliers that determine the stability of the synchronous solution the technique allows us to prove and generalize recent results obtained using heuristic approaches as well as reveal the structure of the approximating equations we illustrate the results in several examples and discuss extensions to the analysis of stability of multisynchronous states in networks with complex architectures

partial synchronization in populations of pulse coupled oscillators i study the long term behavior of populations of nonlinear oscillators with all to all noninstantaneous pulse coupling with fast enough excitatory coupling both the fully synchronized and the asynchronous state are unstable in this case individual units fire quasiperiodically even though the network as a whole shows a periodic firing pattern the behavior of networks with three or more units is different in this regard from that of two unit networks with inhibitory coupling the network can break up into a variable number of fully synchronized clusters for fast inhibition the number of clusters tends to be large while the number of clusters is smaller for slow inhibition textcopyright 1996 the american physical society

conformists and contrarians in a kuramoto model with identical natural frequencies we consider a variant of the kuramoto model in which all the oscillators are now assumed to have the same natural frequency but some of them are negatively coupled to the mean field these contrarian oscillators tend to align in antiphase with the mean field whereas the positively coupled conformist oscillators favor an in phase relationship the interplay between these effects can lead to rich dynamics in addition to a splitting of the population into two diametrically opposed factions the system can also display traveling waves complete incoherence and a blurred version of the two faction state exact solutions for these states and their bifurcations are obtained by means of the watanabe strogatz transformation and the ott antonsen ansatz curiously this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies

collective dynamics of coupled modulated oscillators with random pinning abstract we present a study of a large pool of coupled oscillators in the presence of a modulated external field random distributed pinning phases introduce a disordering element we find that phase locking of the oscillator community to the harmonics of the frequency of the applied field always is associated with a complete loss of coherence between the oscillators the phase lock regions form islands in parameter space the size of which decreases for increasing coupling strength to vanish completely at a critical value by stability considerations the shape of the phase locked islands is reduced to a two points boundary problem of a first order non autonomous differential equation and an approximation is found for high frequencies the structure of the coherent states is discussed and a first order approximation is found in the limit of strong coherence

synchronization of coupled oscillations an analysis of the comparator model the authors present a model for the study of the synchronization between coupled limit cycle oscillators this model is motivated by the recent observations that revealed the existence of synchronization in the firing patterns of neural cells the study is based on a particular coupling architecture where all the oscillators are coupled to a comparator unit that feeds back to each oscillator the mean value of the total phase the various regimes of this model are analysed in the presence of non uniform external driving

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

nonlinear dynamics of networks the groupoid formalism a formal theory of symmetries of networks of coupled dynamical systems stated in terms of the group of permutations of the nodes that preserve the network topology has existed for some time global network symmetries impose strong constraints on the corresponding dynamical systems which affect equilibria periodic states heteroclinic cycles and even chaotic states in particular the symmetries of the network can lead to synchrony phase relations resonances and synchronous or cycling chaos symmetry is a rather restrictive assumption and a general theory of networks should be more flexible a recent generalization of the group theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network the input sets now the symmetry group becomes a groupoid which is an algebraic structure that resembles a group except that the product of two elements may not be defined the groupoid formalism makes it possible to extend group theoretic methods to more general networks and in particular it leads to a complete classification of robust patterns of synchrony in terms of the combinatorial structure of the network many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology a network of dynamical systems is not just a dynamical system with a high dimensional phase space it is also equipped with a canonical set of observables the states of the individual nodes of the network moreover the form of the underlying ode is constrained by the network topology which variables occur in which component equations and how those equations relate to each other the result is a rich and new range of phenomena only a few of which are yet properly understood

renormalization of oscillator lattices with disorder a real space renormalization transformation is constructed for lattices of nonidentical oscillators with dynamics of the general form dvarphi k dt omega k g summation operator l f lk varphi l varphi k the transformation acts on ensembles of such lattices critical properties corresponding to a second order phase transition toward macroscopic synchronization are deduced the analysis is potentially exact but relies in part on unproven assumptions numerically second order phase transitions with the predicted properties are observed as g increases in two structurally different two dimensional oscillator models one model has smooth coupling f lk varphi l varphi k phi varphi l varphi k where phi x is nonodd the other model is pulse coupled with f lk varphi l varphi k delta varphi l phi varphi k lower bounds for the critical dimensions for different types of coupling are obtained for nonodd coupling macroscopic synchronization cannot be ruled out for any dimension d 1 whereas in the case of odd coupling the well known result that it can be ruled out for d 3 is regained less

lie algebras with almost constant free derivations abstract let l be a finite dimensional lie algebra of characteristic 0 admitting a nilpotent lie algebra of derivations d by a classical result of jacobson if d is constant free that is without non zero constants x 0 for all d only if x 0 then l is nilpotent we prove that if d is almost constant free then l is almost nilpotent in the following precise sense if m is the dimension of the fitting null component with respect to d then l contains a nilpotent subalgebra n of codimension bounded by a function of m and n and the nilpotency class of n is bounded in terms of n only this includes strengthening jacobson s theorem by giving in the case m 0 a bound for the nilpotency class of l in terms of the number of weights of d the main result can also be stated as a theorem about a locally finite lie algebra over an algebraically closed field of characteristic 0 containing a nilpotent subalgebra with finitely many weights such that its fitting null component is of finite dimension then the algebra contains a nilpotent subalgebra of finite codimension with similar bounds for the codimension and the nilpotency class the results on derivations are based on results on z p z graded lie algebras where p is a prime with zero component of dimension m these results in turn generalize the higman kreknin kostrikin theorem on the nilpotency of lie algebras with a regular automorphism of prime order

comparing the locking threshold for rings and chains of oscillators we present a case study of how topology can affect synchronization specifically we consider arrays of phase oscillators coupled in a ring or a chain topology each ring is perfectly matched to a chain with the same initial conditions and the same random natural frequencies the only difference is their boundary conditions periodic for a ring and open for a chain for both topologies stable phase locked states exist if and only if the spread or width of the natural frequencies is smaller than a critical value called the locking threshold which depends on the boundary conditions and the particular realization of the frequencies the central question is whether a ring synchronizes more readily than a chain we show that it usually does but not always rigorous bounds are derived for the ratio between the locking thresholds of a ring and its matched chain for a variant of the kuramoto model that also includes a wider family of models

synchronization transition in scale free networks clusters of synchrony we study the synchronization transition in scale free networks that display power law asymptotic behaviors in their degree distributions the critical coupling strength and the order parameter critical exponent derived by the mean field approach depend on the degree exponent ensuremath lambda which implies a close connection between structural organization and the emergence of dynamical order in complex systems we also derive the finite size scaling behavior of the order parameter finding that the giant cluster of synchronized nodes is formed in different ways between scale free networks with 2l ensuremath lambda l3 and those with ensuremath lambda g3

nonlinear singular perturbation problems and one sided difference schemes we consider the boundary value problem varepsilon y a y y b x y f x 1 leqq x leqq 1 for y 1 and y 1 given and b x 0 equiv 0 b y x y geqq 0 we construct a simple difference scheme approximating varepsilon u xx a u u x b x u f x u t for t geqq 0 1 leqq x leqq 1 with the same boundary conditions and an arbitrary initial guess as the numerical artificial time approaches infinity we prove convergence to a unique steady state solution as the mesh size delta x and the parameter varepsilon approach zero in any order we prove convergence to the solution of the corresponding o d e we use one sided differences to approximate the transport term the method applies equally well to multidimensional analogues we present numerical results verifying the theory

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

clustering and phase resetting in a chain of bistable nonisochronous oscillators we consider a chain with many locally coupled bistable nonisochronous oscillators we show that on the background of a disordered amplitude distribution either phase or frequency clusters form in the chain cluster location varies according to the amplitude distribution in the case of frequency clusters their interaction leads to phase resetting in an isolated oscillator located between two neighboring frequency clusters

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

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

spatiotemporal dynamics of the kuramoto sakaguchi model with time dependent connectivity we study the dynamics of the paradigmatic kuramoto sakaguchi model of identical coupled phase oscillators with various kinds of time dependent connectivity using eulerian discretization we explore the parameter spaces for various types of collective states using the phase plots of the two statistical quantities namely the strength of incoherence and the discontinuity measure in the quasistatic limit of the changing of coupling range we observe how the system relaxes from one state to another and identify a few interesting collective dynamical states along the way under a sinusoidal change of the coupling range the global order parameter characterizing the degree of synchronization in the system is shown to undergo a hysteresis with the coupling range we also study the low dimensional spatiotemporal dynamics of the local order parameter in the continuum limit using the recently developed ott antonsen ansatz and justify some of our numerical results in particular we identify an intrinsic time scale of the kuramoto system and show that the simulations exhibit two distinct kinds of qualitative behavior in two cases when the time scale associated with the switching of the coupling radius is very large compared to the intrinsic time scale and when it is comparable to the intrinsic time scale

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

extended finite size scaling of synchronized coupled oscillators we present a systematic analysis of dynamic scaling in the time evolution of the phase order parameter for coupled oscillators with nonidentical natural frequencies in terms of the kuramoto model this provides a comprehensive view of phase synchronization in particular we extend finite size scaling fss in the steady state to dynamics determine critical exponents and find the critical coupling strength the dynamic scaling approach enables us to measure not only the fss exponent associated with the correlation volume in finite systems but also thermodynamic critical exponents based on the extended fss theory we also discuss how the sampling of natural frequencies and thermal noise affect dynamic scaling which is numerically confirmed

betti numbers of a class of barely g2 manifolds we calculate explicitly the betti numbers of a class of barely g2 manifolds that is g2 manifolds that are realised as a product of a calabi yau manifold and a circle modulo an involution the particular class which we consider are those spaces where the calabi yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces from which they inherit all their 1 1 cohomology and the involutions are free acting

phase locking in weakly heterogeneous neuronal networks abstract we examine analytically the existence and stability of phase locked states in a weakly heterogeneous neuronal network we consider a model of n neurons with all to all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons we consider both inhibitory and excitatory coupling we derive the conditions under which stable phase locking is possible in homogeneous networks many different periodic phase locked states are possible their stability depends on the dynamics of the neuron and the coupling for weak heterogeneity the phase locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable for enough heterogeneity phase locked solutions either lose stability or are destroyed completely we analyze the possible states the network can take when phase locking is broken

geometric singularities and spectra of landau ginzburg models some mathematical and physical aspects of superconformal string compactification in weighted projective space are discussed in particular we recast the path integral argument establishing the connection between landau ginzburg conformal theories and calabi yau string compactification in a geometric framework we then prove that the naive expression for the vanishing of the first chern class for a complete intersection adopted from the smooth case is sufficient to ensure that the resulting variety which is generically singular can be resolved to a smooth calabi yau space this justifies much analysis which has recently been expended on the study of landau ginzburg models furthermore we derive some simple formulae for the determination of the witten index in these theories which are complimentary to those derived using semiclassical reasoning by vafa finally we also comment on the possible geometrical significance ofunorbifolded landau ginzburg theories

stability of a chain of phase oscillators we study a chain of n 1 phase oscillators with asymmetric but uniform coupling this type of chain possesses 2 n ways to synchronize in so called traveling wave states i e states where the phases of the single oscillators are in relative equilibrium we show that the number of unstable dimensions of a traveling wave equals the number of oscillators with relative phase close to ensuremath pi this implies that only the relative equilibrium corresponding to approximate in phase synchronization is locally stable despite the presence of a lyapunov type functional periodic or chaotic phase slipping occurs for chains of lengths 3 and 4 we locate the region in parameter space where rotations corresponding to phase slipping are present

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

computing with arrays of coupled oscillators an application to preattentive texture discrimination recent experimental findings gray et al 1989 eckhorn et al 1988 seem to indicate that rapid oscillations and phase lockings of different populations of cortical neurons play an important role in neural computations in particular global stimulus properties could be reflected in the correlated firing of spatially distant cells here we describe how simple coupled oscillator networks can be used to model the data and to investigate whether useful tasks can be performed by oscillator architectures a specific demonstration is given for the problem of preattentive texture discrimination texture images are convolved with different sets of gabor filters feeding into several corresponding arrays of coupled oscillators after a brief transient the dynamic evolution in the arrays leads to a separation of the textures by a phase labeling mechanism the importance of noise and of long range connections is briefly discussed

phase model analysis of two laser with injected field abstract we study the dynamics of two coupled solid state lasers subject to an injected field a singular perturbation technique is used to reduce the full set of rate equations we present a comprehensive bifurcation analysis of the resulting phase model we find four distinct routes to full entrainment as the injected field amplitude increases bifurcation diagram predicts that the entrainment reflected in the total intensity does not monotonically improve with the injected field strength our numerical simulations suggest that the reduced phase model captures the relevant dynamics well beyond the asymptotic regime in which it is derived

synchronization rates in classes of relaxation oscillators relaxation oscillators arise frequently in physics electronics mathematics and biology their mathematical definitions possess a high degree of flexibility in the sense that through appropriate parameter choices relaxation oscillators can be made to exhibit qualitatively different kinds of oscillations we study numerically four different classes of relaxation oscillators through their synchronization rates in one dimensional chains with a heaviside step function interaction and obtain the following results relaxation oscillators in the sinusoidal and relaxation regime both exhibit an average time to synchrony spl sim n where n is the chain length relaxation oscillators in the singular limit exhibit spl sim n sup p where p is a numerically obtained value less than 0 5 relaxation oscillators in the singular limit with parameters modified so that they resemble spike oscillations exhibit spl sim log n in chains and spl sim log l in two dimensional square networks of length l finally using a sigmoid interaction results in spl sim n sup 2 for relaxation oscillators in the sinusoidal and relaxation regimes indicating that the form of the coupling is a controlling factor in the synchronization rate