a decoupled monolithic projection method for natural convection problems we propose an efficient monolithic numerical procedure based on a projection method for solving natural convection problems in the present monolithic method the buoyancy linear diffusion and nonlinear convection terms are implicitly advanced by applying the crank nicolson scheme in time to avoid an otherwise inevitable iterative procedure in solving the monolithic discretized system we use a linearization of the nonlinear convection terms and approximate block lower upper lu decompositions along with approximate factorization numerical simulations demonstrate that the proposed method is more stable and computationally efficient than other semi implicit methods preserving temporal second order accuracy

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

chaos synchronization in gap junction coupled neurons depending on temperature the modified hodgkin huxley mhh equations exhibit a variety of dynamical behaviors including intrinsic chaotic firing we analyze synchronization in a large ensemble of mhh neurons that are interconnected with gap junctions by evaluating tangential lyapunov exponents we clarify whether the synchronous state of neurons is chaotic or periodic then we evaluate transversal lyapunov exponents to elucidate if this synchronous state is stable against infinitesimal perturbations our analysis elucidates that with weak gap junctions the stability of the synchronization of mhh neurons shows rather complicated changes with temperature we however find that with strong gap junctions the synchronous state is stable over the wide range of temperature irrespective of whether synchronous state is chaotic or periodic it turns out that strong gap junctions realize the robust synchronization mechanism which well explains synchronization in interneurons in the real nervous system

robust data fusion for multisensor detection systems minimax robust data fusion schemes for multisensor detection systems with discrete time observations characterized by statistical uncertainty are developed and analyzed block sequential and serial fusion rules are considered the performance measures used and made robust with respect to the uncertainties include the error probabilities of the hypothesis testing problem in the block fusion case and the error probabilities and expected numbers of samples or sensors in the sequential and serial fusion cases for different sensor observation statistics the minimax robust fusion rules are derived for two asymptotic cases of interest when the number of sensors is large and when the number of times the fusion center collects the local sensor decisions is large moreover for the case of identical sensor observation statistics and a large number of sensors it is shown that there is no loss in optimality if local tests using likelihood ratios and equal thresholds are used in the sequential fusion rule in all situations the robust decision rules at the sensors and the fusion center are shown to make use of likelihood ratios and thresholds that depend on the least favorable probability distributions of the uncertainty class describing the statistics of sensor observations

effects of the computational time step on numerical solutions of turbulent flow effects of large computational time steps on the computed turbulence were investigated using a fully implicit method in turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined turbulence fluctuations could not be sustained if the computational time step was near or larger than the kolmogorov time scale

information cascade with marginal stability in a network of chaotic elements abstract a newly discovered cascade process of clusters is studied in a network of chaotic elements it is shown that the splitting of clusters and the synchronization of elements are balanced in a class of partially ordered states where marginal stability is sustained over an interval of the bifurcation parameters the partition information creation in bit space shows an avalanche process of information which leads to the anomalous behavior of power spectra roughly fitted by a power law form of the wavenumber lyapunov spectra have accumulation at null exponents analogous with those studied in fluid turbulence models

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

dynamics stability and statistics on lattices and networks these lectures aim at surveying some dynamical models that have been widely explored in the recent scientific literature as case studies of complex dynamical evolution emerging from the spatio temporal organization of several coupled dynamical variables the first message is that a suitable mathematical description of such models needs tools and concepts borrowed from the general theory of dynamical systems and from out of equilibrium statistical mechanics the second message is that the overall scenario is definitely reacher than the standard problems in these fields for instance systems exhibiting complex unpredictable evolution do not necessarily exhibit deterministic chaotic behavior i e lyapunov chaos as it happens for dynamical models made of a few degrees of freedom in fact a very large number of spatially organized dynamical variables may yield unpredictable evolution even in the absence of lyapunov instability such a mechanism may emerge from the combination of spatial extension and non

associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise we study associative memory based on temporal coding in which successful retrieval is realized as an entrainment in a network of simple phase oscillators with distributed natural frequencies under the influence of white noise the memory patterns are assumed to be given by uniformly distributed random numbers on 0 2 ensuremath pi so that the patterns encode the phase differences of the oscillators to derive the macroscopic order parameter equations for the network with an extensive number of stored patterns we introduce an effective transfer function by assuming a fixed point equation of the form of the thouless anderson palmer equation which describes the time averaged output as a function of the effective time averaged local field properties of the networks associated with synchronization phenomena for a discrete symmetric natural frequency distribution with three frequency components are studied based on the order parameter equations and are shown to be in good agreement with the results of numerical simulations two types of retrieval states are found to occur with respect to the degree of synchronization when the size of the width of the natural frequency distribution is changed

energy conserving runge kutta methods for the incompressible navier stokes equations energy conserving methods have recently gained popularity for the spatial discretization of the incompressible navier stokes equations in this paper implicit runge kutta methods are investigated which keep this property when integrating in time firstly a number of energy conserving runge kutta methods based on gauss radau and lobatto quadrature are constructed these methods are suitable for convection dominated problems such as turbulent flows because they do not introduce artificial diffusion and are stable for any time step secondly to obtain robust time integration methods that work also for stiff problems the energy conserving methods are extended to a new class of additive runge kutta methods which combine energy conservation with l stability in this class the radau iia b method has the best properties results for a number of test cases on two stage methods indicate that for pure convection problems the additive radau iia b method is competitive with the gauss methods however for stiff problems such as convection dominated flows with thin boundary layers both the higher order gauss and radau iia b method suffer from order reduction overall the gauss methods are the preferred method for energy conserving time integration of the incompressible navier stokes equations

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

cluster splitting bifurcation in a system of coupled maps abstract we consider cluster splitting bifurcations in a system of globally coupled maps as coupling parameter decreases at these transitions the number of clusters i e groups of elements with identical dynamics increases we demonstrate that different cascades of cluster splitting can occur depending on statistics of redistribution of the oscillators between new born clusters

spatiotemporal chaos in a coupled map lattice with unstable couplings we study the complex spatiotemporal behaviour of a coupled map lattice with a one humped chaotic map and an unstable laplacian coupling bifurcations are numerically investigated and interpreted using low dimensional approximations corresponding to the relevant degrees of freedom of the infinite dimensional system varying the control parameter we find different phases in the chaotic domain such as localized chaos or propagating chaos spatiotemporal intermittency and transient chaos according to our results unstable coupling leads to a number of new features including the effects that i the first bifurcation becomes discontinuous and ii the chaotic regime sets in sooner than for a single map

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

linear stability analysis of retrieval state in associative memory neural networks of spiking neurons we study associative memory neural networks of the hodgkin huxley type of spiking neurons in which multiple periodic spatiotemporal patterns of spike timing are memorized as limit cycle type attractors in encoding the spatiotemporal patterns we assume the spike timing dependent synaptic plasticity with the asymmetric time window analysis for periodic solution of retrieval state reveals that if the area of the negative part of the time window is equivalent to the positive part then crosstalk among encoded patterns vanishes phase transition due to the loss of the stability of periodic solution is observed when we assume fast function for direct interaction among neurons in order to evaluate the critical point of this phase transition we employ floquet theory in which the stability problem of the infinite number of spiking neurons interacting with function is reduced to the eigenvalue problem with the finite size of matrix numerical integration of the single body dynamics yields the explicit value of the matrix which enables us to determine the critical point of the phase transition with a high degree of precision

different types of chaos synchronization in two coupled piecewise linear maps dynamics of a four parameter family of two dimensional piecewise linear endomorphisms which consist of two linearly coupled one dimensional maps is considered we show that under analytically given conditions chaotic behavior in both maps can be synchronized depending on the coupling the parameters chaotic attractor s synchronized state is characterized by different types of stability s1063 651x 96 11409 4

an energy stable and convergent finite difference scheme for the phase field crystal equation we present an unconditionally energy stable finite difference scheme for the phase field crystal equation the method is based on a convex splitting of a discrete energy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present local in time error estimates that ensure the convergence of the scheme while this paper is primarily concerned with the phase field crystal equation most of the theoretical results hold for the related swift hohenberg equation as well

simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems in this paper we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems flows or maps with diagonal diffusive coupling the condition of chaotic synchronization is determined only by two parameters of order i e the largest lyapunov exponent and the coupling coefficient our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular arbitrarily assumed structure of coupling the main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision

features selection and possibility theory abstract a method is presented that uses both cluster analysis and possibility theory to select the most significant variables in multidimensional data the method is applied to a set of biomedical data from electron spin resonance spectroscopy measurements on patients with a brain injury

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

the multiscale character of evoked cortical activity abstract both the architecture and the dynamics of the brain have characteristic features at different spatial scales however the existence nature and function of dynamical interdependencies between such scales have not been investigated we studied the multiscale properties of functional magnetic resonance imaging fmri data acquired while human subjects viewed a visual image traditional region of interest analysis of this data set revealed evoked activity in primary and extrastriate visual cortex wavelet transform in the spatial domain provides a multiscale representation of this evoked brain activity studying the correlation structure of this representation revealed strong and novel interdependencies in these data within and between different spatial scales we found that such correlations are stronger than those evident in the original data and comparable in magnitude to those obtained after gaussian smoothing however analysis of the data in the wavelet domain revealed additional structure such as positive correlations strong anti correlations and phase lagged interdependencies statistical significance of these effects was inferred through nonparametric bootstrap techniques we conclude that the spatial analysis of functional neuroimaging data in the wavelet domain provides novel information which may reflect complex spatiotemporal neuronal activity and information encoding it also affords a quantitative means of testing hierarchical and multiscale models of cortical activity

phase locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies abstract we study phase locking in a network of coupled nonlinear oscillators with local interactions and random intrinsic frequencies the oscillators are located at the vertices of a graph and interact along the edges they are coupled by sinusoidal functions of the phase differences across the edges and their intrinsic frequencies are independent and identically distributed with finite mean and variance we derive an exact expression for the probability of phase locking in a linear chain of such oscillators and prove that this probability tends to zero as the number of oscillators grows without bound however if the coupling strength increases as the square root of the number of oscillators the probability of phase locking tends to a limiting distribution the kolmogorov smirnov distribution this latter result is obtained by showing that the phase locking problem is equivalent to a discretization of pinned brownian motion the results on chains of oscillators are extended to more general graphs in particular for a hypercubic lattice of any dimension the probability of phase locking tends to zero exponentially fast as the number of oscillators grows without bound we also consider a less stringent type of synchronization characterized by large clusters of oscillators mutually entrained at the same average frequency it is shown that if such clusters exist they necessarily have a sponge like geometry

the convergence of mimetic discretization for rough grids we prove that the mimetic finite difference discretizations of laplace s equation converges on rough logically rectangular grids with convex cells mimetic discretizations for the invariant operators divergence gradient and curl satisfy exact discrete analogs of many of the important theorems of vector calculus the mimetic discretization of the laplacian is given by the composition of the discrete divergence and gradient we first construct a mimetic discretization on a single cell by geometrically constructing inner products for discrete scalar and vector fields then constructing a finite volume discrete divergence and then constructing a discrete gradient that is consistent with the discrete divergence theorem this construction is then extended to the global grid we demonstrate the convergence for the two dimensional laplace equation with dirichlet boundary conditions on grids with a lower bound on the angles in the cell corners and an upper bound on the cell aspect ratios the best convergence rate to be expected is first order which is what we prove the techniques developed apply to far more general initial boundary value problems

statistical mechanics for neural networks with continuous time dynamics the present paper is intended to summariz our rent knowledge about the long time behaviour of nehwrks of graded response neurons with continuous time dynamics we demonstrate the workings of our previously developed statistical mechanical approach to continuous time dynamics by applying it to networks with various forms of synaptic organization leaming rules and neural composition neuron types as encoded in gain functions as well as to networks varying with respect to the ensemble of stored data unbiased and low activity patterns we present phase diagrams and compute distributions of local fields for a variety of examples local field distributions are found to deviate from the gaussian form obtained for stochastic neumns in the wntexi of the replica approach a solution to the low firing rates problem within the framework of nets of analogue neurons is also briefly discussed finally the statisticalmechanical approach to the analysis of continuowtime dynamics is extended to include effects of fast stochastic noise detailed balance solutions are shown to be unique and of canonical form governed by hamiltonians which exhibit a reciprocity relation between potential dynamics and fuing rate dynamics for the potential dynamics the hamiltonian is given by the lyapounw function of the system expressed in terms of the fuing rates and it generates a gibbs distribution over fuing rate space for the noisy firinerate dynamics the same lyapounov function now expressed in terms of neural potentials generates a gibbs distribution over the space of these potentials as a consequence the firing rate dynamics will freeze in configurations saturating the neural input xtput relations whenever such saturation levels exist both types of stationary distribution are shown to exist only under unrealistic assumptions about the noise in the system

approximate factorization as a high order splitting for the implicit incompressible flow equations abstract we apply the method of approximate factorization to derive a second order accurate splitting of the incompressible flow equations stokes or navier stokes equations this is novel because the method of approximate factorization was believed inapplicable to this type of equation system we demonstrate the resulting splitting on a second order crank nicolson discretization and point out its intimate relationship to some existing second order accurate splitting projection methods further we use the approximate factorization method to derive entirely new splittings we first develop a new generalized second order accurate splitting which may be specialized to a variety of applications we indicate its application to finite elements checker board free cell centered discretizations and ocean modeling we then generalized the original splitting to an arbitrarily high order scheme

remarks on the mean field dynamics of networks of chaotic elements abstract fluctuations of the mean field of a globally coupled dynamical systems are discussed the origin of hidden coherence is related with the instability of the fixed point solution of the self consistent perron frobenius equation collective dynamics in globally coupled tent maps are re examined both with the help of direct simulation and the perron frobenius equation collective chaos in a single band state and bifurcation against initial conditions in a two band state are clarified with the return maps of the mean field lyapunov spectra and also the newly introduced lyapunov exponent for the perron frobenius equation future problems on the collective dynamics are discussed

the mean field theory of a q state neural network model the mean field equations of a q state clock neural network model are derived in the replica symmetric approximation using the replica method these equations are studied for the cases q 2 3 4 and q to infinity it is shown that an infinite number of patterns can be stored in the network even in the limit q to infinity the phase diagram and storage capacity of the network are calculated and the information content considered although the overlap achieved with a nominated pattern is shown to decrease as q increases the information stored in the network is shown to increase

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

replica symmetric theory of nonlinear analogue neural networks on the basis of the theory of the naive mean field model of spin glasses analogue neural networks of the hopfield type are investigated in the saturation limit the saddle point equations for the order parameters describing retrieval and spin glass phases of the networks are obtained by means of a statistical mechanical analysis within the framework of the replica symmetric theory and are shown to undergo a modification to that of ags theory due to the absence of the onsager reaction term in the tap equation based on the equations the memory storage capacity of the networks is analysed as a function of the analogue gain beta a small increase in the critical storage capacity is found for finite values of beta compared with that of the ising model networks with corresponding inverse temperature although the qualitative nature of the phase diagram is unchanged

globally coupled circle maps abstract a network of circle maps is investigated successive transitions are found among coherent ordered partially ordered po and turbulent phases in the ordered phase two cluster attractors with similarity have a large basin volume which are reduced to a unique circle map through a scaling transformation with two parameters the po phase is characterized by the increase of partition complexity in the phase chaotic itinerancy is observed whose dynamics consists of quasistationary high dimensional chaotic motion low dimensional attractor ruins and switching among them dependence of the diffusion constant on the size and the nonlinearity clearly deviates from that by a random phase approximation suggesting remaining correlation among elements even in the turbulent phase

cluster synchronization in an ensemble of neurons interacting through chemical synapses in networks of periodically firing spiking neurons that are interconnected with chemical synapses we analyze a cluster state where an ensemble of neurons are subdivided into a few clusters in each of which neurons exhibit perfect synchronization to clarify stability of the cluster state we decompose linear stability of the solution into two types of stabilities stability of a mean state and stabilities of clusters computing floquet matrices for these stabilities we clarify the total stability of the cluster state for any type of neuron and any strength of interaction even if the size of networks is infinitely large first we apply this stability analysis to investigating synchronization in the large ensemble of integrate and fire if neurons in one cluster state we find the change of stability of a cluster which elucidates that in phase synchronization of if neurons occurs with only inhibitory synapses then we investigate entrainment of two clusters of if neurons with different excitability if neurons with fast decaying synapses show low entrainment capability which is explained by a pitchfork bifurcation appearing in the two cluster state with change of synapse decay time constant second we analyze a one cluster state of hodgkin huxley hh neurons and discuss the difference in synchronization properties between if neurons and hh neurons

a criterion for stability of cluster synchronization in networks with external equitable partitions abstract networks of nonlinear dynamical systems may exhibit a collective state known as cluster synchronization characterized by the formation of groups where the units within each cluster converge to the same behavior that is distinct for each cluster in this brief paper we derive a criterion for local stability in milnor sense of this synchronous state that is expressed as a simple relationship between the maximum lyapunov exponent of the uncoupled dynamics the coupling coefficient and one eigenvalue of the graph laplacian spectrum in the last part of the paper we discuss the application of the criterion to networks design and control

a clustering performance measure based on fuzzy set decomposition clustering is primarily used to uncover the true underlying structure of a given data set and for this purpose it is desirable to subject the same data to several different clustering algorithms this paper attempts to put an order on the various partitions of a data set obtained from different clustering algorithms the goodness of each partition is expressed by means of a performance measure based on a fuzzy set decomposition of the data set under consideration several experiments reported in here show that the proposed performance measure puts an order on different partitions of the same data which is consistent with the error rate of a classifier designed on the basis of the obtained cluster labelings

analysis of velocity components decoupled projection method for the incompressible navier stokes equations we study the temporal accuracy and stability of the velocity components decoupled projection method vdpm for fully discrete incompressible navier stokes equations in particular we investigate the effect of three formulations of the nonlinear convection term which include the advective skew symmetric and divergence forms on the temporal accuracy and stability second order temporal accuracy of the vdpm for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l 2 norm considering the energy evolution we demonstrate that the vdpm is stable when the time step is less than or equal to a constant stability diagrams which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices are obtained using von neumann analysis these diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability and the effects of the formulations of the nonlinear convection term are analyzed overall our results indicate that the vdpm along with an advective discrete convection operator is almost unconditionally stable second order accurate in time and computationally efficient because of the non iterative solution procedure in solving the decoupled momentum equations

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

partition complexity in a network of chaotic elements a network of chaotic elements is investigated with the use of globally coupled maps elements can split into clusters with synchronized oscillation in a partially ordered phase the clustering has an inhomogeneous tree structure like replica symmetry breaking in spin glass the clustering has large variety by attractors and strongly depends on initial conditions the variety of clusterings is characterized by the distribution of partitions originated in spin glass theory the qualitative similarity and quantitative disagreement of the given attractors with a spin glass are clarified

chaotic itinerancy and power law residence time distribution in stochastic dynamical systems chaotic itinerant motion among varieties of ordered states is described by a stochastic model based on the mechanism of chaotic itinerancy the model consists of a random walk on a half line and a markov chain with a transition probability matrix the stability of attractor ruin in the model is investigated by analyzing the residence time distribution of orbits at attractor ruins it is shown that the residence time distribution averaged over all attractor ruins can be described by the superposition of truncated power law distributions if the basin of attraction for each attractor ruin has a zero measure this result is confirmed by simulation of models exhibiting chaotic itinerancy chaotic itinerancy is also shown to be absent in coupled milnor attractor systems if the transition probability among attractor ruins can be represented as a markov chain

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

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

the component consistent pressure correction projection method for the incompressible navier stokes equations abstract in this paper we propose the component consistent pressure correction projection method for the numerical solution of the incompressible navier stokes equations this projection preserves a discrete form of the component consistent condition between components of the solution at every time step we also propose in particular the cnmt2 ccpc method and the rkmt ccpc method both involving one pressure poisson solution per time step we show that they are both of o t 2 for the velocity and o t for the pressure on fixed meshes and finite time intervals numerical tests on flow simulation support our claim that the component consistent pressure correction projection method solves the deviation problem encountered sometimes by the original pressure correction projection method

graph theoretic measures of multivariate association and prediction interpoint distance based graphs can be used to define measures of association that extend kendall s notion of a generalized correlation coefficient the authors present particular statistics that provide distribution free tests of independence sensitive to alternatives involving non monotonic relationships moreover since ordering plays no essential role the ideas that fully applicable in a multivariate setting the authors also define an asymmetric coefficient measuring the extent to which a vector x can be used to make single valued predictions of a vector y the authors discuss various techniques for proving that such statistics are asymptotically normal as an example of the effectiveness of their approach the authors present an application to the examination of residuals from multiple regression 18 references 2 figures 1 table

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

hebbian learning and spiking neurons a correlation based hebbian learning rule at a spike level with millisecond resolution is formulated mathematically analyzed and compared with learning in a firing rate description the relative timing of presynaptic and postsynaptic spikes influences synaptic weights via an asymmetric learning window a differential equation for the learning dynamics is derived under the assumption that the time scales of learning and neuronal spike dynamics can be separated the differential equation is solved for a poissonian neuron model with stochastic spike arrival it is shown that correlations between input and output spikes tend to stabilize structure formation with an appropriate choice of parameters learning leads to an intrinsic normalization of the average weight and the output firing rate noise generates diffusion like spreading of synaptic weights s1063 651x 99 02804 4

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

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

delay induced multistable synchronization of biological oscillators we analyze the dynamics of pulse coupled oscillators depending on strength and delay of the interaction for two oscillators we derive return maps for subsequent phase differences and construct phase diagrams for a broad range of parameters in phase synchronization proves stable for inhibitory coupling and unstable for excitatory coupling if the delay is not zero if the coupling strength is high additional regimes with marginally stable synchronization are found simulations with n 2 oscillators reveal a complex dynamics including spontaneous synchronization and desynchronization with excitatory coupling and multistable phase clustering with inhibitory coupling we simulate a continuous description of the system for n oscillators and demonstrate that these phenomena are independent of the size of the system phase clustering is shown to relate to stability and basins of attraction of fixed points in the return map of two oscillators our findings are generic in the sense that they qualitatively are robust with respect to modeling details we demonstrate this using also pulses of finite rise time and the more realistic model by hodgkin and huxley which exhibits multistable synchronization as predicted from our analysis as well s1063 651x 98 09902 4

clustering in globally coupled inhibitory neurons abstract a model of a large population of identical excitable neurons with a global slowly decaying inhibitory coupling is studied and its patterns of synchrony are examined in addition to converging to a homogeneous fixed point and a homogeneous limit cycle the system exhibits cluster states in which it breaks spontaneously into a few macroscopically big clusters each of which is fully synchronized a method for calculating the stability of cluster states is described and used for investigating the dynamical behavior of the network versus the parameters that describe the neurons and synapses effects of stochastic noise on the network dynamics are discussed at large enough noise the system goes to a globally stationary state low levels of noise preserve the cluster like neuron trajectories in the regime where a noiseless system converges to a fully synchronized periodic state relatively low noise levels cause neurons to burst only every two or more consecutive time periods

controlling a complex dynamical network to attain an inhomogeneous equilibrium abstract many networks are made up of a few groups with nodes in the same group having the same kind of function in this work the problem of controlling a complex dynamical network to attain an inhomogeneous equilibrium point is investigated which means that nodes in the same group achieve the same equilibrium point as an isolated node while different groups correspond to different equilibrium points an open loop constant control approach is first proposed to obtain the inhomogeneous equilibrium point of the network then the feedback pinning control approach is applied to make the inhomogeneous equilibrium point asymptotically stable

unconditionally stable discretizations of the immersed boundary equations the immersed boundary ib method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method many implicit methods have been proposed to try to mitigate this timestep restriction but none are known to be unconditionally stable and the observed instability of even some of the fully implicit methods is not well understood in this paper we prove that particular backward euler and crank nicolson like discretizations of the nonlinear immersed boundary terms of the ib equations in conjunction with unsteady stokes flow can yield unconditionally stable methods we also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space we further demonstrate through computational tests that approximate projection methods which do not provide a discretely divergence free velocity field appear to have a stabilizing influence for these problems and that the implicit methods of this paper when used with the full navier stokes equations are no longer subject to such a strict timestep restriction and can be run up to the cfl constraint of the advection terms

stability conditions for the numerical solution of convection dominated problems with skew symmetric discretizations this paper presents original and close to optimal stability conditions linking the time step and the space step stronger than the cfl criterion delta t leq c delta x alpha with alpha frac 2r 2r 1 r an integer for some numerical schemes we produce when solving convection dominated problems we test this condition numerically and prove that it applies to nonlinear equations under smoothness assumptions

how fast elements can affect slow dynamics a chain of coupled chaotic elements with different time scales is studied in contrast with the adiabatic approximation we find that correlations between elements are transferred from faster to slower elements when the differences in the time scales of the elements lie within a certain range for such correlations to occur three features are essential strong correlations among the elements a bifurcation in the dynamics of the fastest element by changing its control parameter and cascade propagation of the bifurcation by studying coupled rossler equations we demonstrate that fast elements can affect the dynamics of slow elements when these conditions are satisfied the relevance of our results to biological memory is briefly discussed

mean field fluctuation of a network of chaotic elements remaining fluctuation and correlation in the large size limit fluctuation of the mean field is studied for a network of chaotic elements with the use of globally coupled maps as the size n is increased the variance of the mean field decreases up to a crossover size nc but then stays constant where the remaining variance is roughly proportional to 1nc the distribution of the mean field approaches a gaussian form up to nc but keeps some deviation for n nc the breakdown of the law of large numbers implies some hidden coherence among elements confirmed by finite mutual information and lyapunov spectra the breakdown is also supported by the instability of the self consistent solution for the perron frobenius operator the anomalous effect of additional noise to our system is also reported results from other globally coupled maps suggest that the law of large numbers is generally broken in a turbulent phase with local stretching and contraction but is satisfied for a map with stretching only

transverse instability and riddled basins in a system of two coupled logistic maps riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space an example is the synchronized motion of two identical chaotic oscillators the paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps we determine the regions in parameter plane where the transverse lyapunov exponent is negative the bifurcation curves for the transverse destabilization of low periodic orbits embedded in the chaotic attractor are obtained and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations it is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin riddling and that the loss of weak stability when the transverse lyapunov exponent becomes positive does not necessarily destroy the basin of attraction instead the symmetry of the synchronized state may break and the attractor may spread into two dimensional phase space

generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena the integrate and fire if neuron model and the piecewise linear version of fitzhugh nagumo pl neuron model with a time scale parameter mu are frequently being used in the study of synchronization phenomena although the two models are regarded to be different type we show a certain equivalence between them by deriving the coupled if model of an improved version with a firing duration from the recovery variable coupled system of the pl model under taking the limit of mu 0 without a loss of any coupling properties in the coupled if model with the duration time the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model synaptic strength decaying relaxation rate of the synaptic coupling and the parameter exhibiting the firing duration we find some irregularly synchronous behavior with or without constant firing order alternations we show that the duration of an impulse plays an important role in synchronization phenomena

alternating and synchronous rhythms in reciprocally inhibitory model neurons we study pacemaker rhythms generated by two nonoscillatory model cells that are coupled by inhibitory synapses a minimal ionic model that exhibits postinhibitory rebound pir is presented when the post synaptic conductance depends instantaneously on presynaptic potential the classical alternating rhythm is obtained using phase plane analysis we identify two underlying mechanisms release and escape for the out of phase oscillation when the postsynaptic conductance is not instantaneous but decays slowly the two cells can oscillate synchronously with no phase difference in each case different stable activity patterns can coexist over a substantial parameter range

on the strength of attractors in a high dimensional system milnor attractor network robust global attraction and noise induced selection abstract strength of attractor is studied by the return rate to itself after perturbations for a multi attractor state of a globally coupled map it is found that fragile milnor attractors have a large basin volume at the partially ordered phase such dominance of fragile attractors is understood by robustness of global attraction in the phase space change of the attractor strength and basin volume against the parameter and size are studied in the partially ordered phase the dynamics is often described as milnor attractor network which leads to a new interpretation of chaotic itinerancy noise induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude relevance of the observed results to neural dynamics and cell differentiation is also discussed

stability of approximate projection methods on cell centered grids projection methods are a popular class of methods for solving the incompressible navier stokes equations if a cell centered grid is chosen in order to use high resolution methods for the advection terms performing the projection exactly is problematic an attractive alternative is to use an approximate projection in which the velocity is required to be only approximately discretely divergence free the stability of the cell centered approximate projection is highly sensitive to the method used to update the pressure and compute the pressure gradient this is demonstrated by analyzing a model problem and conducting numerical simulations of the navier stokes equations

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

integrated fuzzy clustering abstract cluster analysis is a valuable tool for exploratory pattern analysis especially when the information available is not complete and or the data model is affected by uncertainty and imprecision here a new integrated clustering method based on a fuzzy approach is proposed its implementation consists of the combination of hierarchical fuzzy algorithms the performance and accuracy of the methodology are tested on biomedical images an application to the segmentation of magnetic resonance images is discussed

study of a high dimensional chaotic attractor the nature of a very high dimensional chaotic attractor in an infinite dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high dimensional attractor in the phase space we introduce two complementary bases from which the attractor is observed one the lyapunov basis composed of the lyapunov vectors and the another the fourier basis composed of the fourier modes we introduce the exterior subspaces on the basis of the lyapunov vectors and observe the chaotic motion projected onto these exteriors it is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace goes out of the attractor the origin of such a phenomenon is attributed to more fundamental features of our attractor which become manifest when the attractor is observed from the lyapunov basis a counterpart of the phenomenon can be observed also on the fourier basis because there is a statistical one to one correspondence between the lyapunov vectors and the fourier modes in particular a statistical property of the high pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

synchronizing multiple chaotic maps with a randomized scalar coupling abstract synchronization between chaotic systems has recently been a topic of great interest among physicists and engineers in addition to theoretical results a number of applications in communications and control have also been proposed we have previously shown that identical chaotic maps can under certain conditions be synchronized by a common noise like input this raises the question whether the chaotic output from a map can synchronize other maps to itself in this paper we present results on the synchronization of two identical maps where the output of one map is used to drive both maps we then apply this method to synchronizing identical but internally inhomogeneous populations of chaotic oscillators using a randomly constructed scalar coupling signal which tends to white noise as the number of oscillators increases such a system may have applications in secure communication

asynchronous states in networks of pulse coupled oscillators we use a mean field approach to analyze the stability of the asynchronous state in a population of all to all pulse coupled nonlinear oscillators we determine the conditions that must be satisfied by the time constants and phase dependence characterizing the coupling between the oscillators in order for the asynchronous state to be stable we also consider the effects of noise this work complements results on synchronous states in similar models and allows us to study the validity of firing rate models commonly used for neural networks

what matters in neuronal locking exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons under standard conditions a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire if the postsynaptic potential is decreasing oscillations are bound to be unstable this is a kind of locking theorem and boils down to a subtle interplay of axonal delays postsynaptic potentials and refractory behavior the theorem also allows for mixtures of excitatory and inhibitory interactions on the basis of the locking theorem we present a simple geometric method to verify the existence and local stability of a coherent oscillation

collective synchronization of pulse coupled oscillators and excitable units abstract a phase model is proposed and studied for a population of integrate and fire type oscillators and excitable units with pulsatile coupling the units are subject to external noise and their mutual connection is all to all an evolution equation is derived for the phase distribution in a closed form its steady state analysis in the limit of weak noise shows the possibility of collective bistability the evolution equation is simplified drastically when the coupling as well as noise is sufficiently weak this reduction of equation enables analytical calculation of various quantities among others the stability of the steady distribution is analyzed explicity and the onset of collective oscillations is described in terms of the hopf bifurcation it is also shown that the present model is improved in some respects by introducing absolute refractoriness

complex spatiotemporal patterns in two lattice models with instability abstract in this paper we consider two lattice models i the discretized version of a simple nonlinear partial differential equation pde with a singular term and ii a coupled map lattice cml with unstable coupling both systems are defined by simple iteration rules involving the nearest neighbour grid points of the corresponding one dimensional lattice we show numerically that these models result in an apparently stochastic spatiotemporal behaviour and exhibit interesting regimes as the function of the parameters of the models in particular we find that by varying the relative weight b of the singular term in the pde the total width of the solution scales with b the cml leads to an intermittent behaviour between structured and turbulent regimes

riddling bifurcations in coupled piecewise linear maps abstract a mechanism for riddling bifurcations in the system of coupled piecewise linear maps is described we give sufficient conditions for the occurrence of locally and globally riddled basins based on the properties of absorbing areas of the chaotic attractors on the invariant manifold it is also shown that riddled basins are preserved upon bifurcation of the chaotic attractors as long as the attractor after bifurcation is located in the absorbing area of the attractor before bifurcation

collective synchronisation in lattices of nonlinear oscillators with randomness the authors study mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies it is shown rigorously that the model exhibits no long range order in one dimension and that in higher dimensional lattices large clusters of synchronised oscillators necessarily have a sponge like structure surprisingly the phase locking behaviour of the mean field model is completely different from that of any finite dimensional lattice indicating that d infinity is the upper critical dimension for phase locking