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

bihamiltonian structure of the two component kadomtsev petviashvili hierarchy of type b we employ a lax pair representation of the two component kadomtsev petviashvili hierarchy of type b and construct its bihamiltonian structure with r matrix techniques

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

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

the r matrix theory and the reduction of poisson manifolds the relation between the r matrix approach to the construction of bi hamiltonian structures and the reduction of poisson manifolds is discussed the crucial role of the reduction process is emphasized by showing on the basis of specific examples that the knowledge of a r matrix may not be sufficient the reduction of the poisson structures related with r from the entire algebra to the lax submanifold may become an obstruction in constructing the bi hamiltonian structures of a specific hierarchy of integrable equations

peakons r matrix and toda lattice abstract the integrability of a family of hamiltonian systems describing in a particular case the motion of n peakons special solutions of the so called camassa holm equation is established in the framework of the r matrix approach starting from its lax representation in the general case the r matrix is a dynamical one and has an interesting though complicated structure however for a particular choice of the relevant parameters in the hamiltonian the one corresponding to the pure peakons case the r matrix becomes essentially constant and reduces to the one pertaining to the finite non periodic toda lattice intriguing consequences of this property are discussed and an integrable time discretisation is derived

mastersymmetries angle variables and recursion operator of the relativistic toda lattice conserved quantities bi hamiltonian formulation and recursive structure of the relativistic toda lattice rt are obtained in an algorithmic way without making use of the lax representation furthermore for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries a new hierarchy of completely integrable systems is discovered which turns out to correspond to the negative of the hierarchy of rt thus it is shown that the full algebra of time dependent symmetry group generators for each member of the rt hierarchy is isomorphic to the algebra of first order differential operators with laurent polynomials as coefficients the surprising phenomenon is revealed that the members of the rt hierarchy are connected to their negative counterparts by explicit backlund transformation

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

poisson structures of calogero moser and ruijsenaars schneider models we examine the hamiltonian structures of some calogero moser and ruijsenaars schneider n body integrable models we propose explicit formulations of the bi hamiltonian structures for the discrete models and field theoretical realizations of these structures we discuss the relevance of these realizations as collective field theory for the discrete models

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

on deformations of standard r matrices for integrable infinite dimensional systems simple deformations with a parameter of classical r matrices which follow from decomposition of appropriate lie algebras are considered as a result nonstandard lax representations for some well known integrable systems are presented and integrable evolution equations are constructed

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

technical communique linear dynamic filtering with noisy input and output state estimation problems for linear time invariant systems with noisy inputs and outputs are considered an efficient recursive algorithm for the smoothing problem is presented the equivalence between the optimal filter and an appropriately modified kalman filter is established the optimal estimate of the input signal is derived from the optimal state estimate the result shows that the noisy input output filtering problem is not fundamentally different from the classical kalman filtering problem

the hamiltonian structure of the nonabelian toda hierarchy we show that a subset of the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger operator is endowed with a hamiltonian structure and exhibits an infinite number of integrals of motion in involution we also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem and show that such a relation implies that for the whole class previously introduced there exists an infinite number of conserved quantities

the extended bigraded toda hierarchy we generalize the toda lattice hierarchy by considering n m dependent variables we construct roots and logarithms of the lax operator which are uniquely defined operators with coefficients that are series of differential polynomials in the dependent variables and we use them to provide a lax pair definition of the extended bigraded toda hierarchy generalizing 4 using r matrix theory we give the bi hamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions finally we study the dispersionless limit and its connection with a class of frobenius manifolds on the orbit space of the extended affine weyl groups of the a series defined by dubrovin and zhang 1998 compos math 111 167

integrable systems and loop coproducts we present a generalization of a framework for the construction of classical integrable systems that we call loop coproduct formulation musso 2010 j phys a math theor 43 434026 in this paper the loop coproduct formulation includes systems of gelfand tsetlin type the linear r matrix formulation the sklyanin algebras the reflection algebras the coalgebra symmetry approach and some of its generalizations as particular cases showing that all these apparently different approaches have a common algebraic origin on the other hand all these subcases do not exhaust the domain of applicability of this new technique so that new possible directions of investigation do naturally emerge in this framework

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators a mean field model of nonlinearly coupled oscillators with randomly distributed frequencies and subject to independent external white noises is analyzed in the thermodynamic limit when the frequency distribution isbimodal new results include subcritical spontaneous stationary synchronization of the oscillators supercritical time periodic synchronization bistability and hysteretic phenomena bifurcating synchronized states are asymptotically constructed near bifurcation values of the coupling strength and theirnonlinear stability properties ascertained

phase chaos in coupled oscillators a complex high dimensional chaotic behavior phase chaos is found in the finite dimensional kuramoto model of coupled phase oscillators this type of chaos is characterized by half of the spectrum of lyapunov exponents being positive and the lyapunov dimension equaling almost the total system dimension intriguingly the strongest phase chaos occurs for intermediate size ensembles phase chaos is a common property of networks of oscillators of very different natures such as phase oscillators limit cycle oscillators and chaotic oscillators e g r ossler systems

heterogeneous k core versus bootstrap percolation on complex networks we introduce the heterogeneous k core which generalizes the k core and contrast it with bootstrap percolation vertices have a threshold r i that may be different at each vertex if a vertex has fewer than r i neighbors it is pruned from the network the heterogeneous k core is the subgraph remaining after no further vertices can be pruned if the thresholds r i are 1 with probability f or k ensuremath geqslant 3 with probability 1 ensuremath f the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation which is an activation process we show that there are two types of transitions in this heterogeneous k core process the giant heterogeneous k core may appear with a continuous transition and there may be a second discontinuous hybrid transition we compare critical phenomena critical clusters and avalanches at the heterogeneous k core and bootstrap percolation transitions we also show that the network structure has a crucial effect on these processes with the giant heterogeneous k core appearing immediately at a finite value for any fg0 when the degree distribution tends to a power law p q q ensuremath ensuremath gamma with ensuremath gamma l3

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

multistability in the kuramoto model with synaptic plasticity we present a simplified phase model for neuronal dynamics with spike timing dependent plasticity stdp for asymmetric experimentally observed stdp we find multistability a coexistence of a fully synchronized a fully desynchronized and a variety of cluster states in a wide enough range of the parameter space we show that multistability can occur only for asymmetric stdp and we study how the coexistence of synchronization and desynchronization and clustering depends on the distribution of the eigenfrequencies we test the efficacy of the proposed method on the kuramoto model which is de facto one of the sample models for a description of the phase dynamics in neuronal ensembles

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

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

integrable system constructed out of two interacting superconformal fields we consider an interaction between superconformal fields of the same gradation this entails the construction of a supersymmetric poisson tensor for these fields generating a new class of hamiltonian systems the lax representation is found for one of them by supersymmetrizing the lax operator for the hirota satsuma equation the supersymmetric equation is not reducible to the classical hirota satsuma case we show that our generalized system can be reduced to the the supersymmetric kdv equation a 4 surprisingly the integrals of motion are not reduced to integrals of motion of the supersymmetric kdv equation

separation of variables for quadratic algebras algebras of maillet and reflection equation algebras we consider the problem of variable separation for the classical integrable hamiltonian systems possessing gl n valued lax matrices satisfying quadratic poisson brackets of freidel and maillet phys lett b 262 2 3 278 1991 with spectral parameter dependent a b c d tensors we formulate in terms of the corresponding a b c d tensors a sufficient condition that guarantees that the separating functions a u b u of sklyanin commun math phys 150 1 181 1992 scott e print arxiv hep th 940303 1994 and gekhtman commun math phys 167 593 1995 produce canonical coordinates we consider an important subcase of a b c d algebras namely the case of classical reflection equation algebras e sklyanin j phys a math gen 21 10 2357 1988 and formulate in terms of the corresponding r s matrices the analogous sufficient condition that guarantees the canonicity of the constructed coordinates for the case s 0 we recover our previous results on the variable separation for quadratic sklyanin brackets b dubrovin and t skrypnyk j math phys 60 093506 2019 we consider an example of gl n gl n valued trigonometric a b c d tensors that satisfy the considered condition and find a class of lax matrices for them for which the obtained set of canonical coordinates is complete

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

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

bi hamiltonian structures of the coupled akns hierarchy and the coupled yajima oikawa hierarchy the hamiltonian theory for the two component akns hierarchy and yajima oikawa hierarchy is considered from the viewpoint of reduction we show that the second hamiltonian structures of the former is a dirac reduction of the sl 3 current algebra while the latter is related to the classical w 3 4 algebra

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

a two component generalization of the degasperis procesi equation we present two different hamiltonian extensions of the degasperis procesi equation to two component equations the construction is based on the observation that the second hamiltonian operator of the degasperis procesi equation could be considered as the dirac reduced poisson tensor of the second hamiltonian operator of the boussinesq equation the first extension is generated by the hamiltonian operator which is a dirac reduced operator of the generalized but degenerated second hamiltonian operator of the boussinesq equation the second one is obtained by the n 2 supersymmetric extension of the aforementioned method as the byproduct of this procedure we obtain the hamiltonian system of interacting equations which contains the camassa holm and degasperis procesi equations

bi hamiltonian structures for integrable systems on regular time scales a construction of the bi hamiltonian structures for integrable systems on regular time scales is presented the trace functional on an algebra of pseudodifferential operators valid on an arbitrary regular time scale is introduced the linear poisson tensors and the related hamiltonians are derived the quadratic poisson tensors are given by the use of the recursion operators of the lax hierarchies the theory is illustrated by differential counterparts of ablowitz kaup newell segur and kaup broer hierarchies

backlund transformations for the kp and mkp hierarchies with self consistent sources using gauge transformations for the corresponding generating pseudo differential operators ln in terms of eigenfunctions and adjoint eigenfunctions we construct several types of auto backlund transformations for the kp hierarchy with self consistent sources kphscs and mkp hierarchy with self consistent sources mkphscs respectively the backlund transformations from kphscs to mkphscs are also constructed in this way

synchronization and fluctuations coupling a finite number of stochastic units it is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit infinite number of elements in fact since the early work of kuramoto mean field theory has been used to analyze this transition in contrast work that directly deals with finite arrays is relatively scarce in the context of synchronization and yet it is worth noting that finite number effects should be seriously taken into account since in general the limits n where n is the number of units and t where t is time do not commute mean field theory implements the particular choice first n and then t here we analyze an ensemble of three state coupled stochastic units which has been widely studied in the thermodynamic limit we formally address the finite n problem by deducing a fokker planck equation that describes the system we compute the steady state solution of this fokker planck equation that is finite n but t we use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions

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

hybrid phase transition into an absorbing state percolation and avalanches interdependent networks are more fragile under random attacks than simplex networks because interlayer dependencies lead to cascading failures and finally to a sudden collapse this is a hybrid phase transition hpt meaning that at the transition point the order parameter has a jump but there are also critical phenomena related to it here we study these phenomena on the erd s renyi and the two dimensional interdependent networks and show that the hybrid percolation transition exhibits two kinds of critical behaviors divergence of the fluctuations of the order parameter and power law size distribution of finite avalanches at a transition point at the transition point global or infinite avalanches occur while the finite ones have a power law size distribution thus the avalanche statistics also has the nature of a hpt the exponent m of the order parameter is 1 2 under general conditions while the value of the exponent m characterizing the fluctuations of the order parameter depends on the system the critical behavior of the finite avalanches can be described by another set of exponents a and a these two critical behaviors are coupled by a scaling law 1 m a

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

a shocking display of synchrony abstract this article explores the kuramoto model describing the synchronization of a population of coupled oscillators two versions of this model are considered a discrete version suitable for a population with a finite number of oscillators and a continuum model found in the limit of an infinite population when the strength of the coupling between the oscillators exceeds a threshold the oscillators partially synchronize we explore the transition in the continuum model which takes the form of a bifurcation of a discrete mode from a continuous spectrum we use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition and compare with the synchronization predicted by the discrete model there are similarities with instabilities in ideal plasmas and inviscid fluids but these are superficial

explosive synchronization with partial degree frequency correlation networks of kuramoto oscillators with a positive correlation between the oscillators frequencies and the degree of their corresponding vertices exhibit so called explosive synchronization behavior which is now under intensive investigation here we study and discuss explosive synchronization in a situation that has not yet been considered namely when only a part typically a small part of the vertices is subjected to a degree frequency correlation our results show that in order to have explosive synchronization it suffices to have degree frequency correlations only for the hubs the vertices with the highest degrees moreover we show that a partial degree frequency correlation does not only promotes but also allows explosive synchronization to happen in networks for which a full degree frequency correlation would not allow it we perform a mean field analysis and our conclusions were corroborated by exhaustive numerical experiments for synthetic networks and also for the undirected and unweighed version of a typical benchmark biological network namely the neural network of the worm caenorhabditis elegans the latter is an explicit example where partial degree frequency correlation leads to explosive synchronization with hysteresis in contrast with the fully correlated case for which no explosive synchronization is observed

critical behavior of k core percolation numerical studies k core percolation has served as a paradigmatic model of discontinuous percolation for a long time recently it was revealed that the order parameter of k core percolation of random networks additionally exhibits critical behavior thus k core percolation exhibits a hybrid phase transition unlike the critical behaviors of ordinary percolation that are well understood those of hybrid percolation transitions have not been thoroughly understood yet here we investigate the critical behavior of k core percolation of erd ifmmode mbox h o else h o fi s r enyi networks we find numerically that the fluctuations of the order parameter and the mean avalanche size diverge in different ways thus we classify the critical exponents into two types those associated with the order parameter and those with finite avalanches the conventional scaling relations hold within each set however these two critical exponents are coupled finally we discuss some universal features of the critical behaviors of k core percolation and the cascade failure model on multiplex networks

local and non local multiplicative poisson vertex algebras and differential difference equations we develop the notions of multiplicative lie conformal and poisson vertex algebras local and non local and their connections to the theory of integrable differential difference hamiltonian equations we establish relations of these notions to q deformed w algebras and lattice poisson algebras we introduce the notion of adler type pseudodifference operators and apply them to integrability of differential difference hamiltonian equations

multi hamiltonian structures for a class of degenerate completely integrable systems in this paper a class of degenerate i e associated to a degenerate poisson structure completely integrable systems is studied which generalizes the so called odd and even master systems introduced and studied by mumford and by vanhaecke it is shown that all these completely integrable systems called the generalized master systems admit a multi hamiltonian formulation and a systematic construction of this multi hamiltonian structure is described

lax nijenhuis operators for integrable systems the relationship between lax and bi hamiltonian formulations of dynamical systems on finite or infinite dimensional phase spaces is investigated the lax nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the lenard recursion relations while the converse holds for an operator with a simple spectrum explicit higher order hamiltonian structures for the toda system a second hamiltonian structure of the euler equation for a rigid body in n dimensional space and the quadratic adler gelfand dickey structure for the kdv hierarchy are derived using the lax nijenhuis equation

super toda lattices the lax formalism as described by oevel et al and in an earlier and more fundamental form by semenov kostant symes and adler can easily be generalized to the case where anticommuting variables are involved the so called supercase in this article this super lax formalism is applied to the well known associative superalgebra g mat m n subspaces of g to which the super poisson structures can be restricted arise in a natural way taking l in one of these subspaces formally leads to superextensions of the hierarchy of nonrelativistic toda lattices in the simplest case where only nearest neighbor interaction is involved the equations are explicitly solved furthermore the relevant two super hamiltonian structures are explicitly calculated finally a superextension of the relativistic toda lattice with a super hamiltonian structure is described herein

the algebra of q pseudodifferential symbols and the q wkp n algebra in this paper we continue with the program to explore the topography of the space of w type algebras in the present case the starting point is the work of khesin lyubashenko and roger on the algebra of q deformed pseudodifferential symbols and their associated integrable hierarchies the analysis goes on by studying the associated hamiltonian structures for which compact expressions are found the fundamental poisson brackets yield q deformations of wkp and related w type algebras which in specific cases coincide with the ones constructed by frenkel and reshetikhin the construction underlies a continuous correspondence between the hamiltonian structures of the toda lattice and the kp hierarchies

r structures yang baxter equations and related involution theorems there are several ways to construct functions in involution on a lie bi algebra a lie algebra equipped with a second lie bracket for the solvable systems associated to the casimir functions a second hamiltonian formulation can be constructed and a class of bi hamiltonian korteweg de vries like evolution equations with explicit space dependence is derived translating the casimir functions with the flow of a special vector field yields another set of functions in involution lenard relations are found for the corresponding hamiltonian systems finally solutions of the classical yang baxter equation lead to an analog of compatible hamiltonian pairs the invariants of the resulting hereditary operators are in involution

central limit behavior in the kuramoto model at the edge of chaos abstract we study the relationship between chaotic behavior and the central limit theorem clt in the kuramoto model we calculate sums of angles at equidistant times along deterministic trajectories of single oscillators and we show that when chaos is sufficiently strong the pdfs of the sums tend to a gaussian consistently with the standard clt on the other hand when the system is at the edge of chaos i e in a regime with vanishing lyapunov exponents robust q gaussian like limit distributions naturally emerge consistently with recently proved generalizations of the clt

nonlocal quadratic poisson algebras monodromy map and bogoyavlensky lattices a new lax representation for the bogoyavlensky lattice is found and its r matrix interpretation is elaborated the r matrix structure turns out to be related to a highly nonlocal quadratic poisson structure on a direct sum of associative algebras the theory of such nonlocal structures is developed and the poisson property of the monodromy map is worked out in the most general situation some problems concerning the duality of lax representations are raised

what is the relativistic volterra lattice we develop a systematic procedure of finding integrable 6ldquo relativistic regular one parameter deformations for integrable lattice systems our procedure is based on the integrable time discretizations and consists of three steps first for a given system one finds a local discretization living in the same hierarchy second one considers this discretization as a particular cauchy problem for a certain 2 dimensional lattice equation and then looks for another meaningful cauchy problem which can be in turn interpreted as a new discrete time system third one has to identify integrable hierarchies to which these new discrete time systems belong these novel hierarchies are called then relativistic the small time step h playing the role of inverse speed of light we apply this procedure to the toda lattice and recover the well known relativistic toda lattice as well as to the volterra lattice and a certain bogoyavlensky lattice for which the relativistic deformations were not known previously

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

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

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

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

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

r matrix theory formal casimirs and the periodic toda lattice the nonunitary r matrix theory and the associated bi and trihamiltonian schemes are considered the language of poisson pencils and of their formal casimirs is applied in this framework to characterize the bihamiltonian chains of integrals of motion pointing out the role of the schur polynomials in these constructions this formalism is subsequently applied to the periodic toda lattice some different algebraic settings and lax formulations proposed in the literature for this system are analyzed in detail and their full equivalence is exploited in particular the equivalence between the loop algebra approach and the method of differential difference operators is illustrated moreover two alternative lax formulations are considered and appropriate reduction algorithms are found in both cases allowing us to derive the multihamiltonian formalism from r matrix theory the systems of integrals for the periodic toda lattice known after flaschka and henon and their functional relations are recovered through systematic application of the previously outlined schemes