networks of theta neurons with time varying excitability macroscopic chaos multistability and final state uncertainty abstract using recently developed analytical techniques we study the macroscopic dynamics of a large heterogeneous network of theta neurons in which the neurons excitability parameter varies in time we demonstrate that such periodic variation can lead to the emergence of macroscopic chaos multistability and final state uncertainty in the collective behavior of the network finite size network effects and rudimentary control via an accessible macroscopic network parameter is also investigated

self organized partially synchronous dynamics in populations of nonlinearly coupled oscillators we analyze a minimal model of a population of identical oscillators with a nonlinear coupling a generalization of the popular kuramoto model in addition to well known for the kuramoto model regimes of full synchrony full asynchrony and integrable neutral quasiperiodic states ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics self organized bunch states and self organized quasiperiodic dynamics the analysis based on the watanabe strogatz ansatz allows us to describe the self organized bunch states in any finite ensemble as a set of equilibria and the self organized quasiperiodicity as a two frequency quasiperiodic regime an analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed

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

phaselocking in a reaction diffusion system with a linear frequency gradient a simple spatially inhomogeneous oscillatory reaction diffusion equation is studied existence of phaselocked waves is proven with a shooting argument stability is determined numerically applications to intestinal motility and a chemical experiment are described

effects of time delayed feedback on chaotic oscillators we study the effects of time delayed feedback on chaotic systems where the delay time is both fixed static case and varying dynamic case in time for the static case typical phase coherent and incoherent chaotic oscillators are investigated detailed phase diagrams are investigated in the parameter space of feedback gain k and delay time tau linear stability analysis by assuming the time delayed perturbation varies as e lambdat where lambda is the eigenvalue gives the boundaries of the stability islands and critical feedback gains k c for both rossler oscillators and lorenz oscillators we also found that the stability island are found when the delay time is about tau n 1 2 t where n is an integer and t is the average period of the chaotic oscillator it is shown that these analytical predictions agree well with the numerical results for the dynamic case we investigate rossler oscillator with periodically modulated delay time stability regimes are found for parameter space of feedback gain and modulation frequency in which it was impossible to be stabilized for a fixed delay time we also trace the detailed routes to the stability near the island boundaries for both cases by investigating bifurcation diagrams

exact results for the kuramoto model with a bimodal frequency distribution we analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed the dynamics of this system has been the subject of long standing interest in 1984 kuramoto proposed several conjectures about its behavior ten years later crawford obtained the first analytical results by means of a local center manifold calculation nevertheless many questions have remained open especially about the possibility of global bifurcations here we derive the system s stability diagram for the special case where the bimodal distribution consists of two equally weighted lorentzians using an ansatz recently discovered by ott and antonsen we show that in this case the infinite dimensional problem reduces exactly to a flow in four dimensions depending on the parameters and initial conditions the long term dynamics evolves to one of three states incoherence where all the oscillators are desynchronized partial synchrony where a macroscopic group of phase locked oscillators coexists with a sea of desynchronized ones and a standing wave state where two counter rotating groups of phase locked oscillators emerge analytical results are presented for the bifurcation boundaries between these states similar results are also obtained for the case in which the bimodal distribution is given by the sum of two gaussians

kac wakimoto characters and universal mock theta functions in recent work bringmann and ono answer a question of kac and show that character formulas for sl r 1 1 modules due to kac and wakimoto are holomorphic parts of nonholomorphic modular functions here we confirm a speculation of ono that these characters are up to a simple q series the universal mock theta functions g 2 w q and g 3 w q of gordon and mcintosh using recent work of bringmann ono kang zwegers and gordon mcintosh we show that g 2 w q and g 3 w q are up to classical theta functions and products the characters of kac and wakimoto as a consequence we include a dictionary that gives a character formula for every classical mock theta function of ramanujan as well as subsequent natural generalizations

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

the winfree model with heterogeneous phase response curves analytical results we acknowledge support by mineco spain under projects no fis2016 74957 p no psi2016 75688 p and no pcin 2015 127 we also acknowledge support by the european union s horizon 2020 research and innovation programme under the marie sklodowskacurie grant agreement no 642563

almost harmonic maass forms and kac wakimoto characters we resolve a question of kac and explain the automorphic properties of char acters due to kac wakimoto pertaining to s mjn highest weight modules for n 1 we prove that the kac wakimoto characters are essentially holomorphic parts of certain general izations of harmonic weak maass forms which we call almost harmonic maass forms using a new approach this generalizes prior work of the first author and ono and the authors both of which treat only the case nd1 we also provide an explicit asymptotic expansion for the characters

dynamical effects of integrative time delay coupling we study coupled dynamical systems wherein the influence of one system on the other is cumulative coupling signals are integrated over a time interval t a major consequence of integrative coupling is that amplitude death occurs over a wider range and in a single region in parameter space for coupled limit cycle oscillators the landau stuart model we obtain an analytic estimate for the boundary of this region while for coupled chaotic lorenz oscillators numerical results are presented for given t we find that there is a critical coupling strength at which the frequency of oscillations changes discontinuously

a framework of rogers ramanujan identities and their arithmetic properties the two rogers ramanujan q series where 0 1 play many roles in mathematics and physics by the rogers ramanujan identities they are essentially modular functions their quotient the rogers ramanujan continued fraction has the special property that its singular values are algebraic integral units we find a framework which extends the rogers ramanujan identities to doubly infinite families of q series identities if a 1 2 and m n 1 then we have infinite product modular function where the p x x q are hall littlewood polynomials these q series are specialized characters of affine kac moody algebras generalizing the rogers ramanujan continued fraction we prove in the case of a that the relevant q series quotients are integral units

characterizing complex networks through statistics of mobius transformations abstract it is well known now that dynamics of large populations of globally all to all coupled oscillators can be reduced to low dimensional submanifolds ws transformation and oa ansatz marvel et al 2009 described an intriguing algebraic structure standing behind this reduction oscillators evolve by the action of the group of mobius transformations of course dynamics in complex networks of coupled oscillators is highly complex and not reducible still closer look unveils that even in complex networks some possibly overlapping groups of oscillators evolve by mobius transformations in this paper we study properties of the network by identifying mobius transformations in the dynamics of oscillators this enables us to introduce some new statistical concepts that characterize the network in particular the notion of coherence of the network or subnetwork is proposed this conceptual approach is meaningful for the broad class of networks including those with time delayed noisy or mixed interactions in this paper several simple random graphs are studied illustrating the meaning of the concepts introduced in the paper

desynchronization transitions in nonlinearly coupled phase oscillators abstract we consider the nonlinear extension of the kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter a bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed we demonstrate that for small ensembles it is typically mediated by stable cluster states that disappear with creation of heteroclinic cycles while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs

the dynamics of chimera states in heterogeneous kuramoto networks abstract we study a variety of mixed synchronous incoherent chimera states in several heterogeneous networks of coupled phase oscillators for each network the recently discovered ott antonsen ansatz is used to reduce the number of variables in the partial differential equation pde governing the evolution of the probability density function by one resulting in a time evolution pde for a variable with as many spatial dimensions as the network bifurcation analysis is performed on the steady states of these pdes the results emphasise the commonality of the dynamics of the different networks and provide stability information that was previously inferred

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

higher dimensional generalizations of the watanabe strogatz transform for vector models of synchronization we generalize the watanabe strogatz ws transform which acts on the kuramoto model in d 2 dimensions to a higher dimensional vector transform which operates on vector oscillator models of synchronization in any dimension for the case of identical frequency matrices these models have conserved quantities constructed from the cross ratios of inner products of the vector variables which are invariant under the vector transform and have trajectories which lie on the unit sphere s d 1 application of the vector transform leads to a partial integration of the equations of motion leaving independent equations to be solved for any number of nodes n we discuss properties of complete synchronization and use the reduced equations to derive a stability condition for completely synchronized trajectories on s d 1 we further generalize the vector transform to a mapping which acts in and in particular preserves the unit ball and leaves invariant the cross ratios constructed from inner products of vectors in this mapping can be used to partially integrate a system of vector oscillators with trajectories in and for d 2 leads to an extension of the kuramoto system to a system of oscillators with time dependent amplitudes and trajectories in the unit disk we find an inequivalent generalization of the mobius map which also preserves but leaves invariant a different set of cross ratios this time constructed from the vector norms this leads to a different extension of the kuramoto model with trajectories in the complex plane that can be partially integrated by means of fractional linear transformations

complex dynamics of an oscillator ensemble with uniformly distributed natural frequencies and global nonlinear coupling we consider large populations of phase oscillators with global nonlinear coupling for identical oscillators such populations are known to demonstrate a transition from completely synchronized state to the state of self organized quasiperiodicity in this state phases of all units differ yet the population is not completely incoherent but produces a nonzero mean field the frequency of the latter differs from the frequency of individual units here we analyze the dynamics of such populations in case of uniformly distributed natural frequencies we demonstrate numerically and describe theoretically i states of complete synchrony ii regimes with coexistence of a synchronous cluster and a drifting subpopulation and iii self organized quasiperiodic states with nonzero mean field and all oscillators drifting with respect to it we analyze transitions between different states with the increase of the coupling strength in particular we show that the mean field arises via a discontinuous transition for a further illustration we compare the results for the nonlinear model with those for the kuramoto sakaguchi model

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

effects of nonresonant interaction in ensembles of phase oscillators we consider general properties of groups of interacting oscillators for which the natural frequencies are not in resonance such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group we treat the phase dynamics of the groups using the ott antonsen ansatz and reduce it to a system of coupled equations for the order parameters we describe different regimes of cosynchrony in the groups for a large number of groups heteroclinic cycles corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly are possible

complete classification of the macroscopic behavior of a heterogeneous network of theta neurons we design and analyze the dynamics of a large network of theta neurons which are idealized type i neurons the network is heterogeneous in that it includes both inherently spiking and excitable neurons the coupling is global via pulselike synapses of adjustable sharpness using recently developed analytical methods we identify all possible asymptotic states that can be exhibited by a mean field variable that captures the network s macroscopic state these consist of two equilibrium states that reflect partial synchronization in the network and a limit cycle state in which the degree of network synchronization oscillates in time our approach also permits a complete bifurcation analysis which we carry out with respect to parameters that capture the degree of excitability of the neurons the heterogeneity in the population and the coupling strength which can be excitatory or inhibitory we find that the network typically tends toward the two macroscopic equilibrium states when the neuron s intrinsic dynamics and the network interactions reinforce one another in contrast the limit cycle state bifurcations and multistability tend to occur when there is competition among these network features finally we show that our results are exhibited by finite network realizations of reasonable size

slow switching in globally coupled oscillators robustness and occurrence through delayed coupling the phenomenon of slow switching in populations of globally coupled oscillators is discussed this characteristic collective dynamics which was first discovered in a particular class of the phase oscillator model is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population we argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom we also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators although a heteroclinic loop might seem to arise only exceptionally we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony we argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle node bifurcations slow switching arises when a system with a heteroclinic loop is weakly perturbed this will be demonstrated with a vector model by applying weak noises other types of weak symmetry breaking perturbations can also cause slow switching

approximate solution to the stochastic kuramoto model we study kuramoto phase oscillators with temporal fluctuations in the frequencies the infinite dimensional system can be reduced in a gaussian approximation to two first order differential equations this yields a solution for the time dependent order parameter which characterizes the synchronization between the oscillators the known critical coupling strength is exactly recovered by the gaussian theory extensive numerical experiments further show that the analytical results are very accurate below and sufficiently above the critical value we obtain the asymptotic order parameter in closed form which suggests a tighter upper bound for the corresponding scaling as a last point we elaborate the gaussian approximation in complex networks with distributed degrees

experiments on oscillator ensembles with global nonlinear coupling we experimentally analyze collective dynamics of a population of 20 electronic wien bridge limit cycle oscillators with a nonlinear phase shifting unit in the global feedback loop with an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster so that the dependence of the order parameter on the coupling strength is not monotonic after destruction of the cluster the ensemble remains nevertheless coherent i e it exhibits an oscillatory collective mode mean field we show that the system is now in a self organized quasiperiodic state predicted in rosenblum and pikovsky phys rev lett 98 064101 2007 in this state frequencies of all oscillators are smaller than the frequency of the mean field so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic without a nonlinear phase shifting unit the system exhibits a standard kuramoto like transition to a fully synchronous state we demonstrate a good correspondence between the experiment and previously developed theory we also propose a simple measure which characterizes the macroscopic incoherence coherence transition in a finite size ensemble

on the family of ml spectral estimates for mixed spectrum identification a recently developed point spectrum identification procedure based on a family of ar and ml spectral estimates is exploited to arrive at a mixed spectrum identification procedure to this end a variety of properties of the ar and ml estimates as a function of model order are described these properties relate to amplitude convergence resolution and a characterization of the ar spectral artifact which is used to arrive at improved continuous spectral estimates a variety of examples are presented

integrable highest weight modules over affine superalgebras and appell s function we classify integrable irreducible highest weight representations of non twisted affine lie superalgebras we give a free field construction in the level 1 case the analysis of this construction shows in particular that in the simplest case of the sl 2 1 level 1 affine superalgebra the characters are expressed in terms of the appell elliptic function our results demonstrate that the representation theory of affine lie superalgebras is quite different from that of affine lie algebras

partial synchronization and partial amplitude death in mesoscale network motifs we study the interplay between network topology and complex space time patterns and introduce a concept to analytically predict complex patterns in networks of stuart landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology these patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes the underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks in addition it directly links the stability of complete in phase synchronization to only a small subset of topological eigenvalues of a network

superspace conformal field theory conformal sigma models and wess zumino witten wzw models on coset superspaces provide important examples of logarithmic conformal field theories they possess many applications to problems in string and condensed matter theory we review recent results and developments including the general construction of wzw models on type i supergroups the classification of conformal sigma models and their embedding into string theory

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

kac wakimoto characters and non holomorphic jacobi forms in this paper we investigate the automorphic properties of certain characters introduced by kac and wakimoto pertaining to s m n highest weight modules extending previous work of the first author and ono the first author and folsom and the second author we investigate the general case not specializing the jacobi variables we prove that the kac wakimoto characters are essentially holomorphic parts of multivariable mixed h harmonic maass jacobi forms which are certain non holomorphic generalizations of classical holomorphic jacobi forms this also gives extra structure to the previous considered cases

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

analysis of a solvable model of a phase oscillator network on a circle with infinite range mexican hat type interaction interaction with the zeroth and first fourier components we give detailed derivations of the auxiliary equations for the phases and self consistent equations for the amplitudes we solve these equations and characterize the nontrivial solutions in terms of order parameters and the rotation number furthermore we derive the boundaries of the bistable regions and study the bifurcation structures in detail expressions for location dependent resultant frequencies and entrained phases are also derived secondly we treat a different interaction that is composed of mth and nth fourier components where m n and we study its nontrivial solutions in both cases the results of numerical simulations agree quite well with the theoretical results

some statistical methods for random process data from seismology and neurophysiology examples are presented of statistical techniques for the analysis of random process data and of their uses in the substantive fields of seismology and neurophysiology the problems addressed include frequency estimation for decaying cosinusoids signal estimation association measurement causal connection assessment estimation of speed and direction and structural modeling the techniques employed include complex demodulation nonlinear regression probit analysis deconvolution maximum likelihood singular value decomposition fourier analysis and averaging

a maximum likelihood approach to frequency wavenumber analysis this paper is concerned with the likelihood ratio detection and maximum likelihood estimation of plane waves traversing a small array on irregularly placed sensors a new detection statistic is proposed this statistic is related to the high resolution procedure of capon however it has a known null distribution in the case that no signal is present given that a wave is present the asymptotic distribution of the maximum likelihood estimate of the wave parameters is derived the problems of detection and estimation are distinguished they are approached via the fundamental statistical procedures of likelihood ratio testing and maximum likelihood estimation the solutions of both problems are found to be based upon the same quantity the maximum likelihood statistic the development leads to a clarification of the interrelationships and comparative properties of the conventional the least squares the high resolution and the new procedure the approach is via fourier inference

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

the bailey chain and mock theta functions standard applications of the bailey chain preserve mixed mock modularity but not mock modularity after illustrating this with some examples we show how to use a change of base in bailey pairs due to bressoud ismail and stanton to explicitly construct families of q hypergeometric multisums which are mock theta functions we also prove identities involving some of these multisums and certain classical mock theta functions

new infinite families of exact sums of squares formulas jacobi elliptic functions and ramanujan s tau function in this paper we give two infinite families of explicit exact formulas that generalize jacobi s 1829 4 and 8 squares identities to 4n2 or 4n n 1 squares respectively without using cusp forms our 24 squares identity leads to a different formula for ramanujan s tau function n when n is odd these results arise in the setting of jacobi elliptic functions jacobi continued fractions hankel or turanian determinants fourier series lambert series inclusion exclusion laplace expansion formula for determinants and schur functions we have also obtained many additional infinite families of identities in this same setting that are analogous to the function identities in appendix i of macdonald s work macdonald i g 1972 invent math 15 91 143 a special case of our methods yields a proof of the two conjectured kac v g and wakimoto m 1994 in progress in mathematics eds brylinski j l brylinski r guillemin v kac v birkhauser boston boston ma vol 123 pp 415 456 identities involving representing a positive integer by sums of 4n2 or 4n n 1 triangular numbers respectively our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series gustafson s cl nonterminating 6 5 summation theorem and andrews basic hypergeometric series proof of jacobi s 4 and 8 squares identities we have elsewhere applied symmetry and schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n n 1 squares respectively our sums of more than 8 squares identities are not the same as the formulas of mathews 1895 glaisher 1907 ramanujan 1916 mordell 1917 1919 hardy 1918 1920 kac and wakimoto and many others

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

delayed feedback control of collective synchrony an approach to suppression of pathological brain rhythms we suggest a method for suppression of synchrony in a globally coupled oscillator network based on the time delayed feedback via the mean field having in mind possible applications for suppression of pathological rhythms in neural ensembles we present numerical results for different models of coupled bursting neurons a theory is developed based on the consideration of the synchronization transition as a hopf bifurcation

multi branch entrainment and multi peaked order functions in a phase model of limit cycle oscillators with uniform all to all coupling a peculiar type of mutual entrainment is found numerically in a phase model of a large population of limit cycle oscillators with uniform all to all coupling strongly modulated by higher harmonics in which more than one branch of mutually phase locked oscillators exists accompanied by a multi peaked order function for system size n at least en different entrained states are likely to coexist with almost identical order functions also discussed is how to extend the order function theory for dealing with such a phenomenon

one dimensional lattice of oscillators coupled through power law interactions continuum limit and dynamics of spatial fourier modes we study synchronization in a system of phase only oscillators residing on the sites of a one dimensional periodic lattice the oscillators interact with a strength that decays as a power law of the separation along the lattice length and is normalized by a size dependent constant the exponent of the power law is taken in the range 0 1 the oscillator frequency distribution is symmetric about its mean taken to be zero and is nonincreasing on 0 in the continuum limit the local density of oscillators evolves in time following the continuity equation that expresses the conservation of the number of oscillators of each frequency under the dynamics this equation admits as a stationary solution the unsynchronized state uniform both in phase and over the space of the lattice we perform a linear stability analysis of this state to show that when it is unstable different spatial fourier modes of fluctuations have different stability thresholds beyond which they grow exponentially in time with rates that depend on the fourier modes however numerical simulations show that at long times all the nonzero fourier modes decay in time while only the zero fourier mode i e the mean field mode grows in time thereby dominating the instability process and driving the system to a synchronized state our theoretical analysis is supported by extensive numerical simulations

dynamics of heterogeneous oscillator ensembles in terms of collective variables we consider general heterogeneous ensembles of phase oscillators sine coupled to arbitrary external fields starting with the infinitely large ensembles we extend the watanabe strogatz theory valid for identical oscillators to cover the case of an arbitrary parameter distribution the obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion as a particular case of the general setup we consider hierarchically organized ensembles consisting of a finite number of subpopulations whereas the number of elements in a subpopulation can be both finite or infinite next we link the watanabe strogatz and ott antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion the approach is applied to the standard kuramoto sakaguchi model to its extension for the case of nonlinear coupling and to the description of two interacting subpopulations exhibiting a chimera state with these examples we illustrate that although the asymptotic dynamics can be found within the framework of the ott antonsen theory the transients depend on the constants of motion the most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases

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

some characters of kac and wakimoto and nonholomorphic modular functions answering a question of kac we relate the character formulas for certain sl m 1 modules to automorphic forms we show that these q series are the holomorphic parts of nonholomorphic modular functions

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

self emerging and turbulent chimeras in oscillator chains we report on a self emerging chimera state in a homogeneous chain of nonlocally and nonlinearly coupled oscillators this chimera i e a state with coexisting regions of complete and partial synchrony emerges via a supercritical bifurcation from a homogeneous state we develop a theory of chimera based on the ott antonsen equations for the local complex order parameter applying a numerical linear stability analysis we also describe the instability of the chimera and transition to phase turbulence with persistent patches of synchrony

constants of motion for superconducting josephson arrays abstract we show that series arrays of n identical overdamped josephson junctions have extremely degenerate dynamics in particular we prove that such arrays have n 3 constants of motion for all n 3 the analysis is based on a coordinate transformation that reduces the governing equations to an n 3 parameter family of low dimensional systems in the weak coupling limit the reduced equations can be analyzed completely either all solutions approach the synchronous state or they converge to a continuous family of incoherent oscillations depending on a certain parameter value at the transitional value of this parameter the system becomes completely integrable then the phase space is foliated by invariant two dimensional tori for any n 3 the infinite n limit of the system is an integro partial differential equation with rigorously low dimensional dynamics it supports solitons in the integrable case and chaotic waves in the non integrable case