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

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

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

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

rounding effects of quenched randomness on first order phase transitions frozen in disorder in an otherwise homogeneous system is modeled by interaction terms with random coefficients given by independent random variables with a translation invariant distribution for such systems it is proven that ind 2 dimensions there can be no first order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter discontinuities which would amount to a continuous symmetry breaking in systems which are stochastically invariant under the action of a continuous subgroup ofo n are suppressed by the randomness in dimensionsd 4 specific implications are found in the random field ising model for which we conclude that ind 2 dimensions at all h the gibbs state is unique for almost all field configurations and in the random bond potts model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point the results are explained by the argument of imry and ma 1 the proofs involve the analysis of fluctuations of free energy differences which are shown using martingale techniques to be gaussian on the suitable scale

univalent harmonic mappings and linearly connected domains abstract we investigate the relationship between the univalence of f and of h in the decomposition f h g of a sense preserving harmonic mapping defined in the unit disk d c among other results we determine the holomorphic univalent maps h for which there exists c 0 such that every harmonic mapping of the form f h g with g c h is univalent the notion of a linearly connected domain appears in our study in a relevant way

harmonic univalent mappings into a half plane with nonreal vertical slits a class of harmonic univalent mappings is constructed by applying the method of clunie and sheil small these mappings assuming their values in a half plane with a vertical boundary omit two vertical half lines symmetric w r t the real axis several basic properties are proved

a schwarz pick inequality for harmonic quasiconformal mappings and its applications abstract the main result of this paper is the sharp generalized schwarz pick inequality for euclidean harmonic quasiconformal mappings with convex ranges which generalizes a result given by mateljevic as its applications we obtain the property of quasi isometry with respect to the poincare distance and an analogue of the koebe theorem for this class of mappings

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

estimates on bloch constants for planar harmonic mappings the schwarz lemma and bloch constants for planar bounded harmonic mappings are considered sharper form and better estimates are obtained our results improve the one made by dorff and nowak as well as by chen gauthier and hengartner

a matrix valued kuramoto model the kuramoto model is an important model for studying the onset of phase locking in an ensemble of nonlinearly coupled phase oscillators each oscillator has a natural frequency often taken to be random and interacts with the other oscillators through the phase difference it is known that as the coupling strength is increased there is a bifurcation in which the incoherent state becomes unstable and a stable phase locked solution is born beginning with the work of lohe there have been a number of paper that have generalized the kuramoto model for phase locking to a non commuting setting here we propose and analyze another such model we consider a collection of matrix valued variables that evolve in such a way as to try to align their eigenvector frames the phase locked state is one where the eigenframes all align and thus the matrices all commute we analyze the stability of the equal frequency phase locked state and show that it is stable we also analyze for the case of symmetric matrices a dynamic analog of the twist states arising in the standard kuramoto model and show that these twist states are dynamically unstable

the mathematics of asymptotic stability in the kuramoto model now a standard in nonlinear sciences the kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators while its basic phenomenology has been sketched in early works the corresponding rigorous validation has long remained problematic and was achieved only recently this paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the kuramoto model and provides insights into the principal arguments of proofs this review is complemented with additional original results various examples and possible extensions to some variations of the model in the literature

on the global well posedness of bv weak solutions to the kuramoto sakaguchi equation abstract the kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limit cycle oscillators when the number of oscillators is sufficiently large the dynamics of kuramoto ensemble can be effectively approximated by the corresponding mean field equation namely the kuramoto sakaguchi ks equation this ks equation is a kind of scalar conservation law with a nonlocal flux function due to the mean field interactions among oscillators in this paper we provide a unique global solvability of bounded variation bv weak solutions to the kinetic ks equation for identical oscillators using the method of front tracking in hyperbolic conservation laws moreover we also show that our bv weak solutions satisfy local in time l 1 stability with respect to bv initial data for the ensemble of identical kuramoto oscillators we explicitly construct an exponentially growing bv weak solution generated from bv perturbation of incoherent state for any positive coupling strength this implies the nonlinear instability of incoherent state in a positive coupling strength regime we provide several numerical examples and compare them with our analytical results

bloch constant and landau s theorem for planar p harmonic mappings abstract in this paper our main aim is to introduce the concept of planar p harmonic mappings and investigate the properties of these mappings first we discuss the p harmonic bloch mappings two estimates on the bloch constant are obtained which are generalizations of the main results in colonna 1989 9 as a consequence of these investigations we establish a bloch and landau s theorem for p harmonic mappings

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

on the continuity of the srb entropy for endomorphisms we consider classes of dynamical systems admitting markov induced maps under general assumptions which in particular guarantee the existence of srb measures we prove that the entropy of the srb measure varies continuously with the dynamics we apply our result to a vast class of non uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the srb measure in particular we show that the srb entropy of viana maps varies continuously with the map

a note on harmonic quasiconformal mappings abstract in this note we show that a harmonic quasiconformal mapping f u i v with respect to the poincare metric of the upper half plane r 2 onto itself such that v x y v y or u x y u x is a conformal mapping

on the synchronization theory of kuramoto oscillators under the effect of inertia abstract we investigate the synchronized collective behavior of the kuramoto oscillators with inertia effect both the frequency synchronization for nonidentical case and the phase synchronization for identical case are in view as an application of our general theory we show the global frequency synchronization for the three nonidentical oscillator case

noise controlled oscillations and their bifurcations in coupled phase oscillators we derive in gaussian approximation dynamical equations for the first two cumulants of the mean field fluctuations in a system of globally coupled stochastic phase oscillators in these equations the intensity of noise serves as an explicit control parameter its variation generates transitions between three dynamical regimes i stationary ii rotatory and iii locally oscillatory breathing the latter regime has previously not been reported in studies of globally coupled noisy phase oscillators our detailed bifurcation analysis is supported by numerical simulations of an ensemble of coupled stochastic phase oscillators similar regimes are also found in simulations of globally coupled stochastic fitzhugh nagumo elements

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

on the relaxation dynamics of lohe oscillators on some riemannian manifolds we study the collective relaxation dynamics appearing in weakly coupled lohe oscillators in a large coupling regime the lohe models on the unit sphere and unitary group were proposed as a nonabelian generalization of the kuramoto model on the unit circle and their emergent dynamics has been extensively studied in previous literature for some restricted class of initial data based on the lyapunov functional approach and order parameter approach in this paper we extend the previous partial results to cover a generic initial configuration via the detailed analysis on the order parameter measuring the modulus of the centroid in particular we present a detailed relaxation dynamics and structure of the resulting asymptotic states for the lohe sphere model we also present new gradient flow formulations for the lohe matrix models with the same one body hamiltonians on some group manifolds as a direct application of this new formulation we show that every bounded lohe flow which originated from any initial configuration converges asymptotically

complete entrainment of kuramoto oscillators with inertia on networks via gradient like flow abstract we study the asymptotic complete entrainment of kuramoto oscillators with inertia on symmetric and connected network we provide several sufficient conditions for the asymptotic complete entrainment in terms of initial phase frequency configurations strengths of inertia and coupling and natural frequency distributions for this purpose we reinterpret the kuramoto oscillators with inertia as a second order gradient like flow and adopt analytical methods based on several lyapunov functions to apply the convergence estimate studied by haraux and jendoubi 21 our approach does not require any spectral information of the graph associated with the given network structure

landau s theorem for planar harmonic mappings chen gauthier and hengartner obtained two versions of landau s theorem for bounded planar harmonic mappings later dorff and nowak improved their estimates furthermore grigoryan and huang independently improved their results in this note we improve these last results by obtaining better coefficient estimates for bounded and normalized planar harmonic mappings in particular our theorems are sharp when m 1 which are consistent with landau s theorem when m 1

precise coefficient estimates for close to convex harmonic univalent mappings the class sh consists of harmonic univalent and sense preserving functions f in the open unit disk u z z 1 such that f h g where h z z n 2 anzn and g z n 1 a nzn let s0h ch and c0h denote the subclass of sh with a 1 0 the subclass of sh with f being a close to convex mapping and the intersection of s0h and ch respectively in this paper for f c0h and f ch we prove that the harmonic analogue of the bieberbach conjecture and the generalization of the bieberbach conjecture are true

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

non abelian kuramoto models and synchronization we describe non abelian generalizations of the kuramoto model for any classical compact lie group and identify their main properties these models may be defined on any complex network where the variable at each node is an element of the unitary group u n or a subgroup of u n the nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of u n synchronization of trajectories with phase locking occurs as for the kuramoto model for values of the coupling constant larger than a critical value and may be measured by various order and disorder parameters limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables we perform numerical computations for n 2 for which the su 2 group manifold is s3 for a range of natural frequencies and all to all coupling in order to confirm synchronization properties we also describe a second generalization of the kuramoto model which is formulated in terms of real m vectors confined to the m 1 sphere for any positive integer m and investigate trajectories numerically for the s2 model this model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase locked even for large values of the coupling constant

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

landau s theorem for biharmonic mappings abstract in this paper we show the existence of landau constant for biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

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

existence and robustness of phase locking in coupled kuramoto oscillators under mean field feedback motivated by the recent development of deep brain stimulation dbs for neurological diseases we study a network of interconnected oscillators under the influence of mean field feedback and analyze the robustness of its phase locking with respect to general inputs under standard assumptions this system can be reduced to a modified version of the kuramoto model of coupled nonlinear oscillators in the first part of the paper we present an analytical study on the existence of phase locked solutions under generic interconnection and feedback configurations in particular we show that in general no oscillating phase locked solutions can co exist with any non zero proportional mean field feedback in the second part we prove some robustness properties of phase locked solutions namely total stability this general result allows in particular to justify the persistence of practically phase locked states if sufficiently small feedback gains are applied and to give explicit necessary conditions on the intensity of a desynchronizing mean field feedback furthermore the lyapunov function used in the analysis provides a new characterization of the robust phase locked configurations in the kuramoto system with symmetric interconnections

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

the spectrum of the locked state for the kuramoto model of coupled oscillators we analyze the linear stability of the phase locked state in the kuramoto model of coupled oscillators the main result is the first rigorous characterization of the spectrum and its associated eigenvectors for any finite number of oscillators all but two of the eigenvalues are negative and merge into a continuous spectrum as the number of oscillators tends to infinity one eigenvalue is always zero by rotational invariance the final eigenvalue corresponding to a collective mode determines the stability of the locked state

entropy of physical measures for c infty c dynamical systems for a c infty map f on a compact manifold m we prove that for a lebesgue randomly picked point x there is an empirical measure from x with entropy larger than or equal to the top lyapunov exponent of lambda df lambda tm circlearrowleft at x this contrasts with the well known ruelle inequality as a consequence we give some refinement of tsujii s work 23 relating physical and sinai ruelle bowen measures

on the linear combinations of harmonic univalent mappings abstract in this paper we derive several sufficient conditions of the linear combinations of harmonic univalent mappings to be univalent and convex in the direction of the real axis furthermore some illustrative examples and imagine domains of the linear combinations satisfying the desired conditions are enumerated

exponential synchronization of kuramoto oscillators using spatially local coupling abstract we study the generalized kuramoto model of coupled phase oscillators with a finite size and discuss the asymptotic complete phase frequency synchronization the generalized kuramoto model has inherent difficulties in mathematical approaches that this model is governed by nonlinear equations and the kuramoto oscillator is arbitrarily connected with the others to overcome these mathematical barriers many researchers have adopted a linearization of homogeneous solutions and applied a perturbation method however we introduce a new method which just requires some conditions on the smallest and largest positive eigenvalues of the graph laplacian and directly compute the bounds of homogeneous solutions using this method we present analytic results for the generalized kuramoto model more specifically we give a few sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states our sufficient conditions and decay rate depend on the coupling strength the initial phase and natural frequency configurations and the graph laplacian but the conditions are independent of the system size moreover we estimate the time evolution of deviations for the phase and frequency and show that the smallest and largest positive eigenvalues for the graph laplacian affect the stability region and convergence rate for the synchronized states finally we compare our analytic results with numerical simulations using a few examples

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

on the complete synchronization of the kuramoto phase model abstract we discuss the asymptotic complete phase frequency synchronization for the kuramoto phase model with a finite size n we present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations but they are independent of the system size n hence they can be used for the mean field limit for the complete synchronization estimates we estimate the time evolution of the phase and frequency diameters for configurations the initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast in contrast for the non identical oscillators the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations our estimates are based on the monotonicity arguments of extremal phase and frequencies which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states we compare our analytical results with numerical simulations

multistability of phase locking in equal frequency kuramoto models on planar graphs the number n of stable fixed points of locally coupled kuramoto models depends on the topology of the network on which the model is defined it has been shown that cycles in meshed networks play a crucial role in determining n because any two different stable fixed points differ by a collection of loop flows on those cycles since the number of different loop flows increases with the length of the cycle that carries them one expects n to be larger in meshed networks with longer cycles simultaneously the existence of more cycles in a network means more freedom to choose the location of loop flows differentiating between two stable fixed points therefore n should also be larger in networks with more cycles we derive an algebraic upper bound for the number of stable fixed points of the kuramoto model with identical frequencies under the assumption that angle differences between connected nodes do not exceed 2 we obtain n k 1c 2 int nk 4 1 which depends both on the number c of cycles and on the

on exponential synchronization of kuramoto oscillators in this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators the kuramoto model we consider the special case of finite oscillators with distinct bounded natural frequencies our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization this bound improves the one derived by jadbabaie we then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group we also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time

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

on the łojasiewicz exponent of kuramoto model the ojasiewicz inequality and ojasiewicz exponent reveal a fundamental relation between a potential function and its gradient in this paper we explore the ojasiewicz exponent of kuramoto model and prove that the exponent is exactly 12 for equilibria located inside a quarter of circle this implies that the convergence towards such a phase locked state must be exponentially fast in contrast we give an example to see the exponent can be less than 12 for other equilibriums more precisely we prove that the exponent for the bi cluster equilibrium which is located on the boundary of a quarter of circle is 13 this gives an insight for the occurrence of exponential and algebraic convergence of kuramoto model we also present a general theorem for exponential convergence of second order gradient like system by which a criterion for the kuramoto model with inertia is established