synthesis of phase locking patterns in networks with cyclic group symmetries a theoretical model for the circular arrays of neurons that are found in cortical barrel structures is presented if the connection stencil is permutation invariant then the network has cyclic group symmetries the properties of cyclic groups are used to formulate an algorithm that synthesizes coupling stencils that are shown using the rotation vector method to guarantee specified patterns of phase locking in both symmetric and asymmetric arrays such patterns are related to stable combinations of firing frequencies and to memory storage in neural networks

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

emergent behaviors of a holonomic particle system on a sphere we study sufficient conditions for the asymptotic emergence of synchronous behaviors in a holonomic particle system on a sphere which was recently introduced by lohe non abelian kuramoto model and synchronization j phys a math theor 42 395101 395126 2009 these conditions depend only on the coupling strength and initial position diameter for identical particles we show that the position diameter approaches zero asymptotically under these sufficient conditions i e all particles approach to the same position for non identical particles the particle positions do not shrink to one point but can be squeezed into some small region whose diameter is inversely proportional to the coupling strength when the coupling strength is large we also provide several numerical results to confirm our analytical findings

emergent dynamics of winfree oscillators on locally coupled networks abstract the winfree model is the first mathematical model for synchronization of weakly coupled oscillators compared to the well known kuramoto model the winfree model does not conserve the total phase this leads to rich dynamic features compared to those produced by other phase models in this paper we study the emergent dynamics of the winfree model on a locally coupled static network a randomly chosen phase configuration undergoes several dynamic phase transitions such as incoherence partial locking complete locking partial oscillator death and complete oscillator death as the coupling strength increases we provide several rigorous analytical results on the emergence of these dynamic features we also provide several numerical simulations and compare their results to the analytical results

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

synchronization of globally coupled phase oscillators singularities and scaling for general couplings abstract the onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling the effect of a stochastic temporal variation in the frequencies is also included the fokker planck equation for the coupled langevin system is reduced to a kinetic equation for the oscillator distribution function instabilities of the phase incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes depending on the coupling the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected a detailed analysis of these singularities to all orders in the normal form expansion is presented physically the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization these predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions

emergent dynamics of a generalized lohe model on some class of lie groups we introduce a lohe group which is a new class of matrix lie groups and present a continuous dynamical system for the synchronization of group elements in a lohe group the lohe group includes classical lie groups such as the orthogonal unitary and symplectic groups and since lohe groups need not be compact global existence of odes may fail the proposed dynamical system generalizes the lohe model lohe in j phys a 43 465301 2010 lohe in j phys a 42 395101 395126 2009 itself a nonabelian generalization of the kuramoto model and alongside we also generalize the analytical framework ha and ryoo in j stat phys 163 411 439 2016 of emergent and unique phase locked states for the construction of the phase locked states we introduce lyapunov functions measuring the ensemble diameter and the dissimilarity between two lohe flows and derive gronwall type differential inequalities for them the global existence of solutions then become a consequence of the boundedness of these lyapunov functions our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states and it leads to the global existence of solutions and the formation and uniqueness of a phase locked asymptotic state as a concrete example we demonstrate how our theory can show emergent phenomenon on the heisenberg group where all initial configurations tend to a unique phase locked state exponentially fast

complete aggregation of the lohe tensor model with the same free flow the lohe tensor model is a first order tensor valued continuous time dynamical system for the aggregation of tensors with the same rank and size it reduces to the well known aggregation models such as the kuramoto model the lohe sphere model and the lohe matrix model as special cases for low rank tensors we present conditions for the solution splitting property asymptotic convergence toward a set of equilibria and analyze two possible asymptotic states completely aggregate state and bi polar state which can emerge from a set of initial data moreover we provide a sufficient framework leading to the aforementioned two asymptotic states in terms of initial data and system parameters

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

on the emergence and orbital stability of phase locked states for the lohe model we study the emergence and orbital stability of phase locked states of the lohe model which was proposed as a non abelian generalization of the kuramoto phase model for synchronization lohe introduced a first order system of matrix valued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phase locked states of the lohe model in this paper we provide an analytical framework to confirm lohe s observations of emergent phase locked states this extends earlier special results on lower dimensions to any finite dimension for the construction and orbital stability of phase locked states we introduce lyapunov functions to measure the ensemble diameter and dissimilarity between two lohe flows and using the time evolution estimates of these lyapunov functions we present an admissible set of initial states and show that an admissible initial state leads to a unique phase locked asymptotic state

some new sequence spaces derived by the domain of generalized difference matrix let l denote any one of the classical spaces c c 0 and p of bounded convergent null and absolutely p summable sequences respectively and l also be the domain of the generalized difference matrix b r s in the sequence space l where 1 p the present paper is devoted to studying on the sequence space l furthermore the b and c duals of the space l are determined and the schauder bases for the spaces c c 0 and p are given and some topological properties of the spaces c 0 1 and p are examined finally the classes l 1 l 2 and l 1 l 2 of infinite matrices are characterized where l 1 c c 0 p 1 and l 2 c c 0 1

emergent behaviors of lohe tensor flocks we present a new aggregation model on the space of rank m tensors with the same size and study emergent dynamics of the proposed model our proposed new aggregation model is general enough to encompass lohe type models such as the kuramoto model the lohe sphere model and the lohe matrix models for the ensemble of real rank 0 rank 1 and rank 2 tensors respectively in this regard we call our proposed model as the lohe tensor model for rank m tensors with the same size for emergent dynamics of the proposed model we employ the ensemble diameter as a lyapunov functional to derive a riccati type differential inequality as a direct application of these differential inequalities we provide two sufficient frameworks leading to the emergent dynamics for homogeneous and heterogeneous ensembles in terms of system parameters and initial data

simulating memory effects with discrete dynamical systems abstract we propose using discrete dynamical systems to model non markovian processes this implies a whole hierarchy of dynamical systems with dimensionality increasing proportional to the memory specific long range memory effects are investigated for a quadratic a cubic and a quartic map markovian processes transform the quadratic map into a normal form of the logistic equation the henon map corresponds to non markovian first generation memory effects higher memories imply new high dimensional systems non markovian processes imply new memory routes to chaos for the three polynomial maps discussed here it is possible to define critical memory lengths above which the systems essentially lose memory

an lmi based controller synthesis for periodic trajectories in a class of nonlinear systems the use of finite dimensional linear time invariant controllers for the stabilization of periodic solutions in sinusoidally forced nonlinear systems is investigated by mixing results concerning absolute stability of nonlinear systems and robustness of linear systems a linear matrix inequality based controller synthesis technique is developed the synthesis algorithm yields the controller maximizing a lower bound of the maximum amplitude of the forcing input for which the corresponding periodic solutions are guaranteed to be stable the duffing oscillator is employed to illustrate the main features of the proposed synthesis technique

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

lmi based lsvf control of a class of nonlinear systems with parametric uncertainty an application to an inverted pendulum system abstractthis work centres around the stabilisation of a nonlinear system containing parametric uncertainty using a new control lyapunov function using lie derivatives which comes up with a linear matrix inequality based design the paper has three major contributions the first one is an extension of a theorem proposed to find the convex concave bounds of nonlinear function towards robustness with some restrictions in the structure of the uncertainty the theory developed here may be applied to find out the bounds of any nonlinear function with uncertainty the next one is the main contribution of this paper in which the form of the control law obtained is linear and has several advantages from a practical point of view over almost all other nonlinear control techniques the third one is the expansion of the proposed control scheme towards underactuated systems to show the effectiveness of the proposed theory the controller design is attempted for both the traditional cart inverted pendulum and the mo

generalization of the sequence space l p derived by weighted mean abstract the sequence space l p was introduced and studied by maddox i j maddox spaces of strongly summable sequences quart j math oxford 2 18 1967 345 355 in the present paper the sequence spaces l u v p of non absolute type which is derived by the generalized weighted mean is defined and proved that the spaces l u v p and l p are linearly isomorphic besides this the and duals of the space l u v p are computed and the basis of that space is constructed further it is established that the sequence space l p u v has ad property and given f dual of the space l p u v finally the matrix mappings from the sequence spaces l u v p to the sequence space and from the sequence space to the sequence space l u v p are characterized

applications of the hausdorff measure of noncompactness in some sequence spaces of weighted means in the present paper we establish some identities or estimates for the operator norms and the hausdorff measures of noncompactness of certain operators on some sequence spaces of weighted means furthermore by using the hausdorff measure of noncompactness we apply our results to characterize some classes of compact operators on those spaces

complete synchronization of kuramoto oscillators with finite inertia abstract we present an approach based on gronwall s inequalities for the asymptotic complete phase frequency synchronization of kuramoto oscillators with finite inertia for given finite inertia and coupling strength we present admissible classes of initial configurations and natural frequency distributions which lead to the complete phase frequency synchronization asymptotically for this we explicitly identify invariant regions for the kuramoto flow and derive second order gronwall s inequalities for the evolution of phase and frequency diameters our detailed time decay estimates for phase and frequency diameters are independent of the number of oscillators we also compare our analytical results with numerical simulations

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

robustness and asymptotic stability for the winfree model on a general network under the effect of time delay we study the robustness and asymptotic stability of the unique equilibrium for the winfree model on a general network under the effect of small time delay time delay is a ubiquitous mechanism in classical and quantum many body systems and it is due to the finite speed of communication in this paper we further investigate the robustness of the equilibrium under the effects of small time delay and general network topology for this we present several sufficient frameworks formulated in terms of the network topology coupling strength time delay and initial configuration in the proposed framework we present a positively invariant arc which contains the unique equilibrium independent of strength of time delay by using energy estimates we show that the unique equilibrium in the invariant arc is asymptotically stable we also provide several numerical simulations and compare them with analytical results we study the robustness and asymptotic stability of the unique equilibrium for the winfree model on a general network under the effect of small time delay time delay is a ubiquitous mechanism in classical and quantum many body systems and it is due to the finite speed of communication in this paper we further investigate the robustness of the equilibrium under the effects of small time delay and general network topology for this we present several sufficient frameworks formulated in terms of the network topology coupling strength time delay and initial configuration in the proposed framework we present a positively invariant arc which contains the unique equilibrium independent of strength of time delay by using energy estimates we show that the unique equilibrium in the invariant arc is asymptotically stable we also provide several numerical simulations and compare them with analytical results

some new paranormed difference sequence spaces and weighted core in this study we define new paranormed sequence spaces by combining a generalized weighted mean and a difference operator furthermore we compute the a b email protected duals and obtain bases for these sequence spaces besides this we characterize the matrix transformations from the new paranormed sequence spaces to the spaces c 0 q c q q and q finally weighted core of a complex valued sequence has been introduced and we prove some inclusion theorems related to this new type of core

certain topological properties and duals of the domain of a triangle matrix in a sequence space the matrix domain of the particular limitation methods cesaro riesz difference summation and euler were studied by several authors in the present paper certain topological properties and and duals of the domain of a triangle matrix in a sequence space have been examined as an application of the characterization of the related matrix classes

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

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

on the schrodinger lohe hierarchy for aggregation and its emergent dynamics the lohe hierarchy is a hierarchy of finite dimensional aggregation models consisting of the kuramoto model the complex lohe sphere model the lohe matrix model and the lohe tensor model in contrast the schrodinger lohe model is the only known infinite dimensional lohe aggregation model in literature in this paper we provide an explicit connection between the schrodinger lohe model and the complex lohe sphere model and then by exploiting this explicit relation we construct infinite dimensional liftings of the lohe matrix and the lohe tensor models in this way we establish the schrodinger lohe hierarchy which corresponds to the infinite dimensional extensions of the lohe hierarchy for the proposed hierarchy we provide sufficient frameworks leading to the complete aggregation in terms of coupling strengths and initial configurations

on the b difference sequence space derived by generalized weighted mean and compact operators abstract in the present paper by using generalized weighted mean and difference matrix b we introduce the paranormed sequence space l u v p b which consist of all sequences whose generalized weighted b difference means are in the linear space l p introduced by i j maddox also we give the basis of this space and compute its and duals further we characterize the classes of matrix mappings from l u v p b to l c and c 0 finally we apply the hausdorff measure of noncompactness to characterize some classes of compact operators given by matrices on the space l p u v b 1 p

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 euler sequence spaces which include the spaces lp and l ii abstract by e p r we mean the space of all sequences whose euler transforms of order r are in the sequence spaces l p and l see b altay f basar m mursaleen on the euler sequence spaces which include the spaces l p and l i inform sci 2005 in press where 1 p in the present paper we essentially characterize the classes e p r l e 1 r l p and e p r f of infinite matrices for 1 p and give the characterizations of some other matrix mappings from the space e p r to the euler riesz difference etc sequence spaces by means of a given basic lemma we devote the final section of the paper to examining some geometric properties of the space e p r

vulnerability to noise of a phase locked oscillator array abstract the effect of noise on the in phase attractor of a set of n globally coupled oscillators is studied these discrete time maps are associated with the continuous time equations of motion for a series array of josephson junction oscillators we investigate both geometrical properties of the basin of attraction in the large n limit and the implications of this geometry on the average time for the system to escape from the phase locked mode our main results are that the attractor basin maintains a box shaped core of finite radius even as n and that the in phase attractor of a large n array is much less vulnerable to noise than are the out of phase attractors

stability of an n 2 dimensional invariant torus in the kuramoto model at small coupling abstract when the natural frequencies are allocated symmetrically in the kuramoto model there exists an invariant torus of dimension n 2 1 n is the population size a global phase shift invariance allows us to reduce the model to n 1 dimensions using the phase differences and doing so the invariant torus becomes n 2 dimensional by means of perturbative calculations based on the renormalization group technique we show that this torus is asymptotically stable at small coupling if n is odd if n is even the torus can be stable or unstable depending on the natural frequencies and both possibilities persist in the small coupling limit

on generalized means and some related sequence spaces in the present paper we introduce the notion of generalized means and define some related sequence spaces which include the most known sequence spaces as special cases further we study some topological properties of the spaces of generalized means and construct their bases finally we determine the b duals of these spaces and characterize some related matrix classes

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

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

matrix transformations between sequence spaces of generalized weighted means the main object of this paper is to introduce and study some sequence spaces which arise from concept of weighted means determine their b duals and characterize matrix transformations between them

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

almost convergence and generalized difference matrix let f denotes the space of almost convergent sequences introduced by lorentz g g lorentz a contribution to the theory of divergent sequences acta math 80 1948 167 190 and f also be the domain of the generalized difference matrix b r s in the sequence space f in this paper the b and c duals of the spaces f fs and f are determined furthermore two basic results on the space f are proved and the classes f m and m f of infinite matrices are characterized and the characterizations of some other classes are also given as an application of those main results where m is any given sequence space

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

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

systems of matrix riccati equations linear fractional transformations partial integrability and synchronization we partially integrate a system of rectangular matrix riccati equations which describe the synchronization behavior of a nonlinear complex system of n globally connected oscillators the equations take a restricted form in which the time dependent matrix coefficients are independent of the node we use linear fractional transformations to perform the partial integration resulting in a system of reduced size which is independent of n generalizing the well known watanabe strogatz reduction for the kuramoto model for square matrices the resulting constants of motion are related to the eigenvalues of matrix cross ratios which we show satisfy various properties such as symmetry relations for square matrices the variables can be regarded as elements of a classical lie group not necessarily compact satisfying the matrix riccati equations trajectories lie either within or on the boundary of a classical domain and we show by numerical example that complete synchronization can occur even for the mixed case provided that certain unitarity conditions are satisfied we extend the definition of cross ratios to rectangular matrix systems and show that again the eigenvalues are conserved special cases are models with real vector unknowns for which trajectories lie on the unit sphere in higher dimensions with well known synchronization behavior and models with complex vector wavefunctions that describe synchronization in quantum systems possibly infinite dimensional we partially integrate a system of rectangular matrix riccati equations which describe the synchronization behavior of a nonlinear complex system of n globally connected oscillators the equations take a restricted form in which the time dependent matrix coefficients are independent of the node we use linear fractional transformations to perform the partial integration resulting in a system of reduced size which is independent of n generalizing the well known watanabe strogatz reduction for the kuramoto model for square matrices the resulting constants of motion are related to the eigenvalues of matrix cross ratios which we show satisfy various properties such as symmetry relations for square matrices the variables can be regarded as elements of a classical lie group not necessarily compact satisfying the matrix riccati equations trajectories lie either within or on the boundary of a classical domain and we show by numerical example that complete synchronization can occur even for the mixed ca

from collective oscillations to collective chaos in a globally coupled oscillator system abstract various types of collective behaviors are discovered in globally coupled ginzburg landau oscillators when the coupling is sufficiently weak the oscillators are either in complete synchrony or their phases are scattered completely randomly for finite coupling new collective behaviors emerge such as splitting of the population into a small number of clusters or their fusion into a continuous stringlike distribution in the phase plane low dimensional chaotic dynamics arises from the coupled motion of 3 point clusters chaotic motion is also exhibited by fused clusters which is of extremely high dimension possibly proportional to the system size as is implied from its lyapunov analysis in the latter type of chaos the motion of the string is in some cases characterized by repeated stretching and foldings it is argued how this kind of coherent behavior seen on a collective level does not contradict the high dimensionality of the corresponding chaotic attractor

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

on the euler sequence spaces which include the spaces l p and l i in the present paper we introduce the euler sequence space e p r consisting of all sequences whose euler transforms of order r are in the space p of non absolute type which is the bk space including the space p and prove that the spaces e p r and p are linearly isomorphic for 1

on the relaxation dynamics of the kuramoto oscillators with small inertia for the kuramoto oscillators with small inertia we present several quantitative estimates on the relaxation dynamics and formational structure of a phase locked state pls for some classes of initial configurations in a super critical regime where the coupling strength is strictly larger than the diameter of natural frequencies we present quantitative relaxation dynamics on the collision numbers and the structure of pls in a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies we provide a sufficient condition for an asymptotically pls solution in particular we show the existence of slow relaxation to a pls when there are exactly two natural frequencies this generalizes the earlier results of choi et al asymptotic formation and orbital stability of phase locked states for the kuramoto model physica d 241 735 754 2012 10 1016 j physd 2011 11 011 choi et al complete synchronization of kuramoto oscillators with finite inertia physica d 240

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

contractivity of transport distances for the kinetic kuramoto equation we present synchronization and contractivity estimates for the kinetic kuramoto model obtained from the kuramoto phase model in the mean field limit for identical kuramoto oscillators we present an admissible class of initial data leading to time asymptotic complete synchronization that is all measure valued solutions converge to the traveling dirac measure concentrated on the initial averaged phase in the case of non identical oscillators we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration if two initial radon measures have the same natural frequency density function and strength of coupling we show that the wasserstein p distance between corresponding measure valued solutions is exponentially decreasing in time this contraction principle is more general than previous l 1 contraction properties of the kuramoto phase model

uncertainty in phase frequency synchronization of large populations of globally coupled nonlinear oscillators abstract numerical evidence as well as an analytical description is given for the existence of a kind of uncertainty principle in the mean field model of the kuramoto type of large populations of nonlinearly coupled oscillators here the noise plays the role of the planck constant this means that there is a limit to the possibility of synchronizing both phases and frequencies the collective phase synchronization competes in fact within such a model with the frequency synchronization the explanation rests on the effect of the noise which is responsible for loss of accuracy comparison is made with a more sophisticated model where such uncertainty is absent when the coupling parameter is sufficiently large

emergent behaviors of the schrodinger lohe model on cooperative competitive networks abstract we present several sufficient frameworks leading to the emergent behaviors of the coupled schrodinger lohe s l model under the same one body external potential on cooperative competitive networks the s l model was first introduced as a possible phenomenological model exhibiting quantum synchronization and its emergent dynamics on all to all cooperative networks has been treated via two distinct approaches lyapunov functional approach and the finite dimensional reduction based on pairwise correlations in this paper we further generalize the finite dimensional dynamical systems approach for pairwise correlation functions on cooperative competitive networks and provide several sufficient frameworks leading to the collective exponential synchronization for small systems consisting of three and four quantum subsystem we also show that the system for pairwise correlations can be reduced to the lotka volterra model with cooperative and competitive interactions in which lots of interesting dynamical patterns appear e g existence of closed orbits and limit cycles

asymptotic description of transients and synchronized states of globally coupled oscillators abstract a two timescale asymptotic method has been introduced to analyze the multimodal mean field kuramoto model of oscillator synchronization in the high frequency limit the method allows to uncouple the probability density in different components corresponding to the different peaks of the oscillator frequency distribution each component evolves towards a stationary state in a comoving frame and the overall order parameter can be reconstructed by combining them synchronized phases are a combination of traveling waves and incoherent solutions depending on parameter values our results agree very well with direct numerical simulations of the nonlinear fokker planck equation for the probability density numerical results have been obtained by finite differences and a spectral method in the particular case of bimodal symmetric and asymmetric frequency distribution with or without external field we also recover in a very easy and intuitive way the only other known analytical results those corresponding to reflection symmetric bimodal frequency distributions near bifurcation points

robust h control of a nonlinear uncertain system via a stable nonlinear output feedback controller a new approach to solving a nonlinear robust h control problem using a stable nonlinear output feedback controller is presented in this article the class of nonlinear uncertain systems being considered is characterised in terms of integral quadratic constraints and global lipschitz conditions describing the admissible uncertainties and nonlinearities respectively the nonlinear controller is able to exploit the plant nonlinearities through the inclusion of a copy of the known plant nonlinearities in the controller the h control objective is to obtain an absolutely stable closed loop system with a specified disturbance attenuation level the solution to this control problem involves stabilising solutions to parametrised algebraic riccati equations we apply a differential evolution algorithm to solve a non convex nonlinear optimisation problem arising in the controller synthesis

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

robust input output feedback linearization to make input output feedback linearization a practical and systematic design methodology for single input nonlinear systems two problems need to be addressed one is to handle systematically the difficulties associated with the internal dynamics or zero dynamics when the relative degree is less than the system order the other is to account for the effect of model uncertainties in the successive differentiations of the output of interest while the first problem has recently received considerable attention the second has been largely unexplored this paper represents a preliminary study of a systematic methodology to account robustly for parametric uncertainties in the original system model the approach is based on combining sliding control ideas with the recursive construction of a closed loop lyapunov function and is illustrated with a simple example

matrix transformations on some sequence spaces related to strong cesaro summability and boundedness the spaces a 0 r d a c r d and a r d introduced by aydin and basar c aydin f basar some new difference sequence spaces appl math comput 157 3 2004 677 693 can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero summable and bounded by the cesaro method of order 1 here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable summable to zero or bounded by the strong cesaro method of order 1 with index p 1 we determine the b duals of the new spaces and characterize matrix transformations on them into the sets of bounded convergent and null sequences

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

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

practical quantum synchronization for the schrodinger lohe system we present a practical synchronization method for the schrodinger lohe s l system distinct potentials the s l model describes the spatial temporal evolution of the wave functions of quantum lohe oscillators on a network with lohe couplings when the potential effects are ignored complete wave function synchronization cwfs can emerge in the sense that the l 2 distance between wave functions exponentially approaches zero for a class of initial wave functions in contrast when the lohe oscillators are under the effect of potential forces cwfs cannot occur in this study we employ a weaker concept of quantum synchronization for discussing the asymptotic collective behavior of the s l model this concept leads to practical synchronization of the s l model in practical synchronization the l 2 distance between wave functions can be upper bounded by the inverse power of the square root of the coupling strength as the coupling strength increases thus the l 2 discrepancy between wave functions arbitrarily decreases as the coupling strength increases we present a sufficient analytical framework for this practical synchronization which is a generalization of the earlier result in choi s h and ha s y 2014 j phys a math theor 47 355104

brief paper robust adaptive motion force control for wheeled inverted pendulums previous works for wheeled inverted pendulums usually eliminate nonholonomic constraint force in order to make the control design easier under the assumption that the friction force from the ground is as large as needed nevertheless such an assumption is unfeasible in practical applications in this paper adaptive robust motion force control for wheeled inverted pendulums is investigated with parametric and functional uncertainties the proposed robust adaptive controls based on physical properties of wheeled inverted pendulums make use of online adaptation mechanism to cancel the unmodelled dynamics based on lyapunov synthesis the proposed controls ensure that the system outputs track the given bounded reference signals within a small neighborhood of zero and guarantee the semi global uniform boundedness of all closed loop signals the effectiveness of the proposed controls is verified through extensive simulations

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

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

averaging of globally coupled oscillators abstract we study a specific system of symmetrically coupled oscillators using the method of averaging the equations describe a series array of josephson junctions we concentrate on the dynamics near the splay phase state also known as the antiphase state ponies on a merry go round or rotating wave we calculate the floquet exponents of the splay phase periodic orbit in the weak coupling limit and find that all of the floquet exponents are purely imaginary in fact all the floquet exponents are zero except for a single complex conjugate pair thus nested two tori of doubly periodic solutions surround the splay phase state in the linearized averaged equations we numerically integrate the original system and find startling agreement with the averaging results on two counts the observed ratio of frequencies is very close to the prediction and the solutions of the full equations appear to be either periodic or doubly periodic as they are in the averaged equations such behavior is quite surprising from the point of view of generic dynamical systems theory one expects higher dimensional tori and chaotic solutions we show that the functional form of the equations and not just their symmetry is responsible for this nongeneric behavior

partial entrainment in the finite kuramoto sakaguchi model abstract although modifications of the kuramoto model have been the subject of extensive research the model itself is not yet fully understood we offer several results and observations some analytic others through simulations we derive a sufficient condition for the existence of a solution exhibiting partial entrainment with respect to a given subset of oscillators the result also implies persistence of the entrainment behavior under perturbations the critical values of the coupling strength defining the transitions between different forms of partial entrainment are predicted by an analytical approximation based on the fact that oscillators with large differences in their natural frequencies have little influence on each other s entrainment behavior the predictions agree with the actual values obtained by simulations we indicate by simulations that entrainment can disappear with increasing coupling strength and that in arrays of josephson junctions a similar phenomenon can be observed where it is also possible that a junction leaving one entrained subset joins another entrained subset

some paranormed sequence spaces of non absolute type derived by weighted mean abstract the sequence spaces l p c p and c 0 p were introduced and studied by maddox i j maddox paranormed sequence spaces generated by infinite matrices proc cambridge philos soc 64 1968 335 340 in the present paper the sequence spaces u v p of non absolute type which are derived by the generalized weighted mean are defined and proved that the spaces u v p and p are linearly isomorphic where denotes the one of the sequence spaces l c or c 0 besides this the and duals of the spaces u v p are computed and the basis of the spaces c 0 u v p and c u v p is constructed additionally it is established that the sequence space c 0 u v has ad property and given the f dual of the space c 0 u v p finally the matrix mappings from the sequence spaces u v p to the sequence space and from the sequence space to the sequence spaces u v p are characterized

adaptive fuzzy logic control of dynamic balance and motion for wheeled inverted pendulums in this paper adaptive fuzzy logic control of dynamic balance and motion is investigated for wheeled inverted pendulums with parametric and functional uncertainties the proposed adaptive fuzzy logic control based on physical properties of wheeled inverted pendulums makes use of a fuzzy logic engine and a systematic online adaptation mechanism to approximate the unknown dynamics based on lyapunov synthesis the fuzzy control ensures that the system outputs track the given bounded reference signals to within a small neighborhood of zero and guarantees semi global uniform boundedness of all closed loop signals the effectiveness of the proposed control is verified through extensive simulations

population dynamics of spiking neurons fast transients asynchronous states and locking an integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate and fire type is discussed it is analytically shown that transients from a state of incoherent firing can be immediate the stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived the response of a population of noisy integrate and fire neurons to an input current of small amplitude is calculated and characterized by a linear filter l the stability of perfectly synchronized locked solutions is analyzed

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

some new sequence spaces derived by the domain of the triple band matrix let l denote any of the classical spaces c c 0 and p of bounded convergent null and absolutely p summable sequences respectively and let l b also be the domain of the triple band matrix b r s t in the sequence space l where 1