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

probabilistic solution of vibro impact stochastic duffing systems with a unilateral non zero offset barrier abstract this paper proposes a solution procedure for obtaining an approximate probability density function pdf of the response of vibro impact duffing systems with a unilateral non zero offset barrier the impact model is used with classical impacts with instantaneous velocity jumps and the excitation is modelled by a zero mean stationary gaussian white noise first the original vibro impact system is converted into a system without barriers by adopting the zhuravlev non smooth coordinate transformation second the pdf of the converted system is obtained by solving the fokker planck fp equation with the exponential polynomial closure epc method last the pdf of the original system is achieved using the well established methodology on seeking the pdf distribution of a function of a random variable a further parametric study is also conducted to evaluate the effectiveness of the proposed solution procedure comparison with the simulated result shows that when lightly inelastic impacts occur i e the restitution coefficient is very close to 1 0 the technique of the zhuravlev non smooth coordinate transformation is feasible the transformed system can adequately represent the original system consequently the epc method can provide a good approximate pdf solution the tail region of the pdf is also approximated well the analysis further shows that the non zero offset significantly affects the shape of the pdf distribution of displacement

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

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

periodically forced ensemble of nonlinearly coupled oscillators from partial to full synchrony we analyze the dynamics of a periodically forced oscillator ensemble with global nonlinear coupling without forcing the system exhibits complicated collective dynamics even for the simplest case of identical phase oscillators due to nonlinearity the synchronous state becomes unstable for certain values of the coupling parameter and the system settles at the border between synchrony and asynchrony what can be denoted as partial synchrony we find that an external common forcing can result in two synchronous states i a weak forcing entrains only the mean field whereas the individual oscillators remain unlocked to the force and correspondingly to the mean field ii a strong forcing fully synchronizes the system making the phases of all oscillators identical analytical results are confirmed by numerics

riemannian consensus for manifolds with bounded curvature consensus algorithms are popular distributed algorithms for computing aggregate quantities such as averages in ad hoc wireless networks however existing algorithms mostly address the case where the measurements lie in euclidean space in this work we propose riemannian consensus a natural extension of existing averaging consensus algorithms to the case of riemannian manifolds unlike previous generalizations our algorithm is intrinsic and in principle can be applied to any complete riemannian manifold we give sufficient convergence conditions on riemannian manifolds with bounded curvature and we analyze the differences with respect to the euclidean case we test the proposed algorithms on synthetic data sampled from the space of rotations the sphere and the grassmann manifold

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

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

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

new results on the algebraic theory of linear systems the solution of the cover problems abstract we give the complete solution of a large class of problems in linear system theory the so called cover problems these problems are formulated and solved both in the state space and in the input output frameworks the key concept which allows the effective parametrization of all solutions of the cover problems is that of the partial realizations of a sequence of matrices it is shown that the solutions of the state space cover problems can be expressed as state spaces of the partial realizations of appropriately defined sequences of matrices and the solution of the input output cover problems can be expressed as a simple function of the functions of the partial realizations of the sequences of mentioned above

a common framework for complete and incomplete attitude synchronization in networks with switching topology in this paper we study attitude synchronization for elements in the unit sphere in mathbb r scriptscriptstyle scriptscriptstyle 3 and for elements in the three dimensional 3 d rotation group for a network with switching topology the agents angular velocities are assumed to be the control inputs and a switching control law for each agent is devised that guarantees synchronization provided that all elements are initially contained in a region which we identify later in the paper the control law is decentralized and it does not require a common orientation frame among all agents we refer to synchronization of unit vectors in mathbb r scriptscriptstyle 3 as incomplete synchronization and of 3 d rotation matrices as complete synchronization our main contribution lies in showing that these two problems can be analyzed under a common framework where all agents dynamics are transformed into unit vectors dynamics on a sphere of appropriate dimension

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

consensus on complete riemannian manifolds in finite time abstract in view of the fact that prototypes of riemannian manifolds range from euclidean surfaces to lie groups operator spaces etc the paper explores common consensus schemes for connected complete riemannian manifolds after studying the restriction exponential map we introduce a local consensus protocol with switching communication topologies for the manifolds via the isometric mapping based on one of the euclidean consensus protocols equipped with the compression decompression functions we extend the local consensus protocol and put forth a global consensus algorithm for any such manifolds these consensus schemes can provide common solutions to the consensus problems arising from various nonlinear spaces or abstract spaces belonging to connected complete riemannian manifolds in particular they are simple and concise and can converge in finite time simulation on the unit sphere is provided to verify our consensus schemes and to demonstrate the usage of our proposed techniques

discontinuity induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems a rich variety of dynamical scenarios has been shown to occur when a fixed point of a non smooth map undergoes a border collision this paper concerns a closely related class of discontinuity induced bifurcations those involving equilibria of n dimensional piecewise smooth flows specifically transitions are studied which occur when a boundary equilibrium that is one lying within a discontinuity manifold is perturbed it is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios conditions to classify among the possible simplest scenarios are given for piecewise smooth continuous filippov and impacting systems also we investigate the possible birth of other attractors e g limit cycles at a boundary equilibrium bifurcation

a concept of the mass center of a system of material points in the constant curvature spaces this article demonstrates that in the lobatchevsky space and on a sphere of arbitrary dimensions the concept of the mass center of a system of mass points can be correctly defined presented are a uniform geometric construction for defining the mass center hyperbolic and spheric lever rules the theorem of uniqueness for determining the mass center in these spaces among the compact manifolds only the sphere possesses this property

partial realization of symmetric matrix rational functions abstract the partial realization problem is treated for the special case in which the given data are a sequence of symmetric matrices the induced algebraic structure of the state space for this special case is analyzed and implemented to construct rational matrix functions whose coefficients in the power series expansion match with the given sequence up to a certain finite number the interrelation between the mcmillan degree and the number of coefficients that match is also treated

stationary response of multi degree of freedom vibro impact systems under white noise excitations a multi degree of freedom vibro impact system under white noise excitations is formulated as a stochastically excited and dissipated hamiltonian system the constraints are modelled as non linear springs according to the hertz contact law the exact stationary solution of the system is derived under certain conditions the approximate stationary solutions of the system are also obtained by using the stochastic averaging methods for quasi hamiltonian systems it is shown that the stochastic averaging method for quasi non integrable hamiltonian systems is applicable if the non linear forces according to the hertz contact law take an important role in the response of the system while the stochastic averaging method for quasi integrable hamiltonian systems is applicable if the non linear forces can be neglected an example for stochastically excited two degree of freedom vibro impact system is given to illustrate the application of the procedures in detail

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

on the optimal parameter of the composite laplacian quadratics function abstract recently the composite quadratic lyapunov function has been extended to study of multi agent systems leading to the so called composite laplacian quadratics clqs function compared with quadratic lyapunov functions the clqs function can yield a larger convergence region and is particularly useful in stabilization of multi agent systems with complex dynamics such as differential inclusions in the definition of the clqs function an optimal vector parameter plays a critical role in determining the value of the clqs function and in constructing stabilization laws derived from the clqs function this paper focuses on the properties of the optimal parameter of the clqs function the uniqueness of the optimal parameter is established a distributed computation approach is further proposed which is useful in computing the optimal parameter the robustness issue of the optimal parameter is also investigated for a multi agent system described by linear differential inclusions finally a numerical example is provided to validate the proposed theoretical results

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

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

corner bifurcations in non smoothly forced impact oscillators abstract we consider an impacting mechanical system in which a particle at position u t impacts with a periodically moving obstacle at position z t the motion of which is non smooth in particular we look at corner events when u impacts with z very close to a point where z loses smoothness we show that this leads through a corner bifurcation to complex dynamics in u which can include periodic orbits of arbitrary period and period adding cascades by analysing associated maps close to the corner event we show that this dynamics can be understood in terms of the iterations of a two dimensional piecewise linear discontinuous map we also show some links between this analysis and the difficult problem of understanding the motion of three objects which may have simultaneous impacts

bistable kinetic model driven by correlated noises steady state analysis a simple rule to obtain the fokker planck equation for a general one dimensional system driven by correlated gaussian white noises is proved by the functional method the fokker planck equation obtained in this paper is applied to the bistable kinetic model we find the following for the steady state 1 in the ensuremath alpha d parameter plane ensuremath alpha is the strength of the additive noise and d is the multiplicative noise strength the critical curve separating the unimodal and bimodal regions of the stationary probability distribution spd of the model is shown to be affected by ensuremath lambda the degree of correlation of the noises as ensuremath lambda is increased the area of the bimodal region in the ensuremath alpha d plane is contracted 2 when we take a point fixed in the ensuremath alpha d plane and increase ensuremath lambda the form of the spd changes from a bimodal to a unimodal structure 3 the positions of the extreme value of the spd of the model sensitively depend on the strength of the multiplicative noise and weakly depend on the additive noise strength 4 for ensuremath lambda 1 the case of perfectly correlated noises when the parameters ensuremath alpha and d take values in the neighborhood of the line ensuremath alpha d in the ensuremath alpha d plane the spd s corresponding to the points ensuremath alpha dg1 and ensuremath alpha d1 exhibit a very different shape of divergence therefore the ratio ensuremath alpha d 1 plays the role of a critical ratio

on parametrizations for the minimal partial realization problem abstract the paper considers parametrizations for minimal partial realizations of a finite sequence of markov parameters first it is shown that all minimal partial realizations have the same set of input and output kronecker indices a parametrization is obtained formulated in terms of a minimal set of extension entries then an i o canonical form is used as a parametrization for the set of minimal partial realizations it is shown that this allows to define the invariants of the problem moreover the i o canonical form is shown to contain a set of parameters that can be chosen such that any minimal partial realization can be represented

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

frequency synchronization and phase agreement in kuramoto oscillator networks with delays we investigate frequency synchronization and phase agreement in networks of non identical kuramoto oscillators with delayed coupling we provide delay dependent lower bounds for the coupling gain so that a network with all to all coupling and constant heterogeneous delays achieves synchronization the given bounds are determined explicitly by the system parameters furthermore we show that frequency synchronization and phase agreement are possible in general directed networks of non identical oscillators which contain a spanning tree if the delays satisfy a self consistency condition networks with delays differ at this point notably from networks without delays because phase agreement is not possible in undelayed non identical kuramoto oscillator networks in addition to the sufficient conditions for synchronization we investigate the connection between phase agreement and the existence of solutions of a necessary self consistency condition

unified cooperative control of multiple agents on a sphere for different spherical patterns in this technical note we consider the cooperative control of multiple agents on a sphere provide the appropriate attraction repulsion interactions between agents with introduction of the state dependent repulsive coefficients and design the unified control laws for agents to achieve three fundamental spherical configurations or patterns rendezvous uniform deployment and formation using both the first and second order models we present the analysis of the stability scaling and geometric patterns of the models on the sphere under different attraction repulsion parameters with illustration of the geometrical patterns and the geometrical isomerism by simulations the closed loop dynamics of agents are invariant with respect to group rotation actions on the sphere

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

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

response probability density functions of duffing van der pol vibro impact system under correlated gaussian white noise excitations abstract this paper aims to investigate the stationary probability density functions pdfs of a duffing van der pol vibro impact system excited by correlated gaussian white noise with the help of non smooth transformation the stationary pdfs are formulated analytically by the stochastic averaging of energy envelope the analytical results are verified by numerical simulation results stochastic bifurcations for different parameters are considered and several special pdf forms are observed in this paper the first form is the shape of the pdf of total energy can be similar to a crater which has a minimum and a maximum the second one is the shape of the joint pdf with three peaks that is to say the section of joint pdf has three maximum and two minimum in addition the influence of the degree of the correlation of gaussian white noises is explored

the matrix partial realization problem an algebraic approach this paper introduces a new approach to the multivariable partial realization problem it formulates the problem in algebraic terminology which sheds new light on the nature of the problem and our existing knowledge of it the structure of the state space is analysed in terms of polynomial models the main idea is analysis of the algebraic structure of the kernels of truncated hankel maps the elements of such kernels are directly related to the partial realization problem in the sense that the columns of the denominator matrix of a partial realization are elements of such kernels the numerator matrix is determined by an appropriate denominator matrix

subharmonic response of a single degree of freedom nonlinear vibroimpact system to a randomly disordered periodic excitation the subharmonic resonant response of a single degree of freedom nonlinear vibroimpact oscillator with a one sided barrier to narrow band random excitation is investigated the analysis is based on a special zhuravlev transformation which reduces the system to one without impacts or velocity jumps thereby permitting the applications of asymptotic averaging over the period for slowly varying random processes the averaged equations are solved exactly and algebraic equation of the amplitude of the response is obtained in the case without random disorder a perturbation based moment closure scheme is proposed and an iterative calculation equation for the mean square response amplitude is derived in the case with random disorder the effects of damping nonlinear intensity detuning and magnitudes of random excitations are analyzed the theoretical analyses are verified by numerical results theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced at large detuning or large nonlinear stiffness and when intensity of the random disorder increase the steady state solution may change from a limit cycle to a diffused limit cycle and even change to a chaos one

matrix continued fractions are directly related to the maximal a b invariant subspace in ker c the maximal a b invariant subspace in ker c is characterized in terms of polynomial models by its properties pertaining to the hankel map this characterization is directly related to the first stage in a continued fraction representation of the transfer function the atoms of the continued fraction are given explicitly in terms of tocplitz operators we also obtain a complete characterization and parametrization of eligible first atoms based on weiner hopf indexes we show how to construct a first atom which is column proper and determines a feedback equivalent feedback irreducible transfer function

bifurcations in the sakaguchi kuramoto model abstract we analyze the sakaguchi kuramoto model of coupled phase oscillators in a continuum limit given by a frequency dependent version of the ott antonsen system based on a self consistency equation we provide a detailed analysis of partially synchronized states their bifurcation from the completely incoherent state and their stability properties we use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of non universal synchronization transitions

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

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

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

stochastic bifurcations in a bistable duffing van der pol oscillator with colored noise this paper aims to investigate gaussian colored noise induced stochastic bifurcations and the dynamical influence of correlation time and noise intensity in a bistable duffing van der pol oscillator by using the stochastic averaging method theoretically one can obtain the stationary probability density function of amplitude for the duffing van der pol oscillator and can reveal interesting dynamics under the influence of gaussian colored noise stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution which indicates that system parameters noise intensity and noise correlation time respectively can be treated as bifurcation parameters they also imply that the effects of multiplicative noise are rather different from that of additive noise the results of direct numerical simulation verify the effectiveness of the theoretical analysis moreover the largest lyapunov exponent calculations indicate that p and d bifurcations have no direct connection

stochastic responses of vibro impact duffing oscillator excited by additive gaussian noise the stationary responses of vibro impact duffing oscillator excited by additive gaussian white noise are studied by using the quasi conservative averaging method with the help of a non smooth variable transformation and the dirac delta function the response probability density functions pdfs are formulated analytically meanwhile the results are validated numerically using monte carlo simulations

distributed nonlinear control of mobile autonomous multi agents abstract this paper studies the distributed nonlinear control of mobile autonomous agents with variable and directed topology a new distributed nonlinear design scheme is presented for multi agent systems modeled by double integrators with the new design the outputs of the controlled agents asymptotically converge to each other as long as a mild connectivity condition is satisfied moreover the velocity derivative of the output of each agent can be restricted to be within any specified neighborhood of the origin which is of practical interest for systems under such physical constraint the new design is still valid if one of the agents is a leader and the control objective is to achieve leader following as an illustration of the generality and effectiveness of the presented methodology the formation control of a group of unicycle mobile robots with nonholonomic constraints is revisited instead of assuming the point robot model the unicycle model is transformed into two double integrators by dynamic feedback linearization and the proposed distributed nonlinear design method is used to overcome the singularity problem caused by the nonholonomic constraint by properly restricting the velocities simulation results are included to illustrate the theoretical results

synchronization with partial state coupling on so n this paper studies autonomous synchronization of k agent orientations in so n by using links that each convey a priori different partial state information between two agents the latter compare the action of their states on a common link dependent vector of mathbb r n such setting appears when each pair of agents compares the expression in their respective body frames of a common external direction force field heading of moving object etc to synchronize their states agents must combine this information over time or throughout the network a gradient coupling law for synchronization is proposed extensive convergence analysis of the coupled agents is provided both for fixed and time varying reference vectors the results are compared to a similar setting with states in mathbb r n and agents coupled through scalar product of their state with a reference vector

bounds on packings of spheres in the grassmann manifold we derive the gilbert varshamov and hamming bounds for packings of spheres codes in the grassmann manifolds over r and c asymptotic expressions are obtained for the geodesic metric and projection frobenius chordal metric on the manifold

distributed circular formation stabilization for dynamic unicycles this paper investigates the problem of designing distributed control laws making a group of dynamic unicycles converge to a common circle of prespecified radius whose center is stationary but dependent on the initial conditions and travel around the circle in a desired direction the vehicles are required to converge to a formation on the circle expressed by desired separations and ordering of the unicycles the information exchange between unicycles is modelled by a directed graph which is assumed to have a spanning tree a hierarchical approach is proposed which simplifies the control design by decoupling the problem of making the unicycles converge to a common circle from the problem of stabilizing the formation

simultaneous velocity consensus and shape control for a finite number of point agents on the unit circle in this paper we study a second order distributed control system for a finite number of point agents on the unit circle that achieves simultaneously velocity consensus and distance based formation shape control based on tools from riemannian geometry we propose a system of second order differential equations on the n dimensional torus that achieves these two goals we prove convergence of the trajectories to single closed geodesics on a torus and investigate the stability properties of the distributed algorithm

stochastic time delayed systems driven by correlated noises steady state analysis abstract in the present paper a class of stochastic time delayed systems driven by correlated gaussian white noises are considered novikov s theorem is used to derive delay fokker planck equations in the case of small time delays approximate stationary solutions are obtained as a special case the delay fokker planck equation and the approximate stationary probability density function is obtained for a bistable system numerical simulations show that the small delay approximation is a good approximation in the case of small delay furthermore the effects of the correlated noises and the feedback are investigated the critical curve separating the unimodal and bimodal regions of the stationary probability distribution is shown to be affected by the degree of the correlation of the noises and e the strength of the feedback both and e can change the curve of the stationary probability density function from a bimodal to a unimodal structure

synchronization of two interacting populations of oscillators we analyze synchronization between two interacting populations of different phase oscillators for the important case of asymmetric coupling functions we find a much richer dynamical behavior compared to that of symmetrically coupled populations of identical oscillators it includes three types of bistabilities higher order entrainment and the existence of states with unusual stability properties all possible routes to synchronization of the populations are presented and some stability boundaries are obtained analytically the impact of these findings for neuroscience is discussed

a flag representation for finite collections of subspaces of mixed dimensions abstract given a finite set of subspaces of r n perhaps of differing dimensions we describe a flag of vector spaces i e a nested sequence of vector spaces that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation the utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions when the set of subspaces all have the same dimension d the flag mean is related to several commonly used subspace representations for instance the d dimensional subspace in the flag corresponds to the extrinsic manifold mean when the set of subspaces is both well clustered and equidimensional of dimension d then the d dimensional component of the flag provides an approximation to the karcher mean an intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of multiset canonical correlation analysis two examples utilizing the carnegie mellon university pose illumination and expression database cmu pie serve as visual illustrations of the algorithm

almost global consensus on the n sphere this paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multiagent systems that evolve in continuous time on the n dimensional unit sphere or n sphere for any connected undirected graph and all n in mathbb n backslash lbrace 1 rbrace each protocol in said class is shown to yield almost global consensus the feedback laws are negative gradients of lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol this convergence result sheds new light on the general problem of consensus on riemannian manifolds the n sphere for n in mathbb n backslash lbrace 1 rbrace differs from the circle and mathsf so 3 where the corresponding protocols fail to generate almost global consensus moreover we derive a novel consensus protocol on mathsf so 3 by combining two almost globally convergent protocols on the n sphere for n in lbrace 1 2 rbrace theoretical and simulation results suggest that the combined protocol yields almost global consensus on mathsf so 3

stochastic stationary responses of a viscoelastic system with impacts under additive gaussian white noise excitation abstract this paper is devoted to investigate the stochastic stationary responses of a viscoelastic system with impacts under additive gaussian white noise excitation first the viscoelastic force is approximated as equivalent stiffness and damping terms and the original system is replaced by a system without viscoelasticity then the stationary responses of the system are obtained analytically by using non smooth transformation and stochastic averaging method the theoretical analyses are verified by numerical results from monte carlo simulations effects of the noise intensity and the nonlinear damping coefficient on the system responses are investigated at last it is worthy noting that the restitution coefficient the linear damping coefficient and the viscoelastic parameters can induce the occurrence of stochastic p bifurcation

stochastic response of a vibro impact system with variable mass abstract vibro impact machines are found increasingly broad application in various sectors of the world and the mass disturbance exists in the process of impact which refers to the variable mass system that is a common phenomenon in life with the development of micro electronic and nanotechnology it is essential to investigate the effect of mass disturbance on this random vibro impact system response the stationary system response is the main topic of the manuscript in the light of the discontinuity of vibro impact system the nonsmooth transformation is introduced to overcome the discontinuity then the stochastic averaging technique is applied to the transformed stochastic variable mass system the approximate analytical response can be obtained by solving associated fokker planck kolmogorov equation a van der pol variable mass vibro impact system is given to illustrate the application and availability of this method meanwhile stochastic p bifurcation is studied and a reasonable interpretation is given to explain the procedure of stochastic p bifurcation

double neimark sacker bifurcation and torus bifurcation of a class of vibratory systems with symmetrical rigid stops abstract a multidegree of freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered the system consists of linear components but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints such models play an important role in the studies of mechanical systems with clearances or gaps double neimark sacker bifurcation of the system is analyzed by using the center manifold and normal form method of maps the period one double impact symmetrical motion and homologous disturbed map of the system are derived analytically a center manifold theorem technique is applied to reduce the poincare map to a four dimensional one and the normal form map associated with double neimark sacker bifurcation is obtained the bifurcation sets for the normal form map are illustrated in detail local behavior of the vibratory systems with symmetrical rigid stops near the points of double neimark sacker bifurcations is reported by the presentation of results for a three degree of freedom vibratory system with symmetrical stops the existence and stability of period one double impact symmetrical motion are analyzed explicitly also local bifurcations at the points of change in stability are analyzed thus giving some information on dynamical behavior near the points of double neimark sacker bifurcations near the value of double neimark sacker bifurcation there exist period one double impact symmetrical motion and quasi periodic impact motions the quasi periodic impact motions are represented by the closed circle and tire like attractor in projected poincare sections with change of system parameters the quasi periodic impact motions usually lead to chaos via tire like torus doubling

generalized weiszfeld algorithms for lq optimization in many computer vision applications a desired model of some type is computed by minimizing a cost function based on several measurements typically one may compute the model that minimizes the l 2 cost that is the sum of squares of measurement errors with respect to the model however the lq solution which minimizes the sum of the q th power of errors usually gives more robust results in the presence of outliers for some values of q for example q 1 the weiszfeld algorithm is a classic algorithm for finding the geometric l1 mean of a set of points in euclidean space it is provably optimal and requires neither differentiation nor line search the weiszfeld algorithm has also been generalized to find the l1 mean of a set of points on a riemannian manifold of non negative curvature this paper shows that the weiszfeld approach may be extended to a wide variety of problems to find an lq mean for 1 le q lt 2 while maintaining simplicity and provable convergence we apply this problem to both single rotation averaging under which the algorithm provably finds the global lq optimum and multiple rotation averaging for which no such proof exists experimental results of lq optimization for rotations show the improved reliability and robustness compared to l 2 optimization

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

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

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

control of a driftless bilinear vector field on n sphere in this paper we consider a multi input driftless bilinear system evolving on the n dimensional sphere s n we first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on s n for the general framework we establish the global controllability on s n and propose two linear control laws on s n that achieve asymptotic stabilization of an equilibrium point with an almost global domain of attraction further the asymptotically stable closed loop system trajectories are shown to be arcs on the geodesics of s n for a particular choice of the equilibrium point next we propose two linear time varying control laws to achieve trajectory tracking on s n and show the asymptotic stability of the tracking error a distributed control is designed for the consensus of multiagent bilinear systems on s n with an undirected tree as the communication graph the consensus manifold is shown to have an almost global domain of attraction

the closed form stationary probability distribution of the stochastically excited vibro impact oscillators abstract the response of vibro impact oscillators under random excitations has been studied using various techniques in the last three decades however the results available were smooth approximations of the intrinsically non smooth response of vibroimpact oscillators this paper proposes a new procedure for the closed form stationary probability density function pdf of the response of single degree of freedom sdof vibro impact system under gaussian white noise excitation first the zhuravlev ivanov transformation is adopted to convert a vibro impact oscillator with one side barrier into an oscillator without barrier the probabilistic description of the system is subsequently defined through the corresponding fokker planck kolmogorov fpk equation the closed form stationary pdf of the response is obtained by solving the reduced fpk equation using the iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow two examples are examined to illustrate the effectiveness of the proposed procedure good agreements are found between the analytical solutions and the simulated results of the pdfs and logarithmic pdfs of the examples the reported studies also show that the proposed procedure can deal with case of small restitution factor in particular the closed form solution for the case of small restitution factor is firstly obtained in literature and can be used as the benchmark problem to examine the other method in random vibration furthermore the approximate solutions obtained with the proposed procedure are piece wise form reflecting the true nature of the intrinsically discontinuous vibro impact oscillators

synchronization of a josephson junction array in terms of global variables we consider an array of josephson junctions with a common lcr load application of the watanabe strogatz approach physica d 74 197 1994 allows us to formulate the dynamics of the array via the global variables only for identical junctions this is a finite set of equations analysis of which reveals the regions of bistability of the synchronous and asynchronous states for disordered arrays with distributed parameters of the junctions the problem is formulated as an integro differential equation for the global variables here stability of the asynchronous states and the properties of the transition synchrony asynchrony are established numerically

synchronization of diffusively coupled systems on compact riemannian manifolds in the presence of drift abstract recently it has been shown that the synchronization manifold is an asymptotically stable invariant set of diffusively coupled systems on riemannian manifolds we regionally investigate the stability properties of the synchronization manifold when the systems are subject to drift when the drift vector field is quad i e satisfies a certain quadratic inequality and the underlying riemannian manifold is compact we prove that a sufficiently large algebraic connectivity of the underlying graph is sufficient for the synchronization manifold to remain asymptotically stable for drift vector fields which are quad or contracting we explicitly characterize the rate at which the solution converges to the synchronization manifold our main result is that the synchronization manifold is asymptotically stable even for drift vector fields which are only locally lipschitz continuous as long as the algebraic connectivity of the underlying graph is sufficiently large

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

high dimensional kuramoto models on stiefel manifolds synchronize complex networks almost globally abstract the kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature some applications e g quantum synchronization and rigid body attitude synchronization involve high dimensional kuramoto models where each oscillator lives on the n sphere or so n these manifolds are special cases of the compact real stiefel manifold st p n using tools from optimization and control theory we prove that the generalized kuramoto model on st p n converges to a synchronized state for any connected graph and from almost all initial conditions provided p n satisfies p 2 3 n 1 and all oscillator frequencies are equal this result could not have been predicted based on knowledge of the kuramoto model in complex networks over the circle in that case almost global synchronization is graph dependent it applies if the network is acyclic or sufficiently dense this paper hence identifies a property that distinguishes many high dimensional generalizations of the kuramoto models from the original model

consensus optimization on manifolds the present paper considers distributed consensus algorithms that involve n agents evolving on a connected compact homogeneous manifold the agents track no external reference and communicate their relative state according to a communication graph the consensus problem is formulated in terms of the extrema of a cost function this leads to efficient gradient algorithms to synchronize i e maximizing the consensus or balance i e minimizing the consensus the agents a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time varying the cost function is linked to a specific centroid definition on manifolds introduced here as the induced arithmetic mean that is easily computable in closed form and may be of independent interest for a number of manifolds the special orthogonal group so n and the grassmann manifold text it grass p n are treated as original examples a link is also drawn with the many existing results on the circle

synchronization and balancing on the n torus in this paper we study the behavior of a network of n agents each evolving on the circle we propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the possibly time varying and unidirectional communication graphs the global convergence analysis on the n torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the euclidean space

asymptotic stabilization of submanifolds embedded in riemannian manifolds we study control problems in which systems whose states spaces are riemannian manifolds must be steered towards embedded submanifolds of their state spaces in a stable fashion i e feedback laws rendering the given submanifold asymptotically stable are sought bringing the closed loop to the form of a gradient system with drift we find that the gradient part can be scaled in a fashion so as to guarantee asymptotic stability of the desired submanifold under this circumstance we show that solutions of the system can be brought to an arbitrarily small neighborhood of the submanifold in arbitrary time thereafter we derive an algebraic condition under which the control vector fields can be brought to the desired form lastly we recast our algebraic condition in terms of vertical and horizontal spaces for the case that the submanifold is an equivalence class

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

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper