asymptotic stability of densities in coupled map lattices abstract sufficient conditions for the asymptotic stability strong convergence of density evolution in finite dimensional piecewise monotone map lattices with both constant and state dependent coupling are given these conditions are quite useful in numerical work since they allow one to precisely define where one expects to see numerical signatures of asymptotic stability asymptotic stability is illustrated with several examples it is also shown that in constantly coupled lattices the density formed by collapsing the higher dimensional density into one dimension can be approximated by the evolution of densities under the action of an appropriately perturbed one dimensional map

a general theory of canonical induction formulae abstract so far there exist three independent constructions of two different canonical versions of brauer s induction theorem for complex characters due to v snaith p symonds and the author canonical in this context means functorial with respect to restrictions along group homomorphisms in this article we axiomatize the situation in which the above canonical induction formulae are constructed mackey functors and related structures arise in this way naturally as a convenient language this approach allows us to construct canonical induction formulae for arbitrary mackey functors in particular we obtain canonical induction formulae for the brauer character ring the group of projective characters the ring of trivial source modules and the ring of linear source modules in most cases it is not difficult to construct such formulae over the rational numbers a much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers we give a sufficient condition in the general framework of mackey functors for a canonical induction formula to be integral as an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings as for example the span of linear characters in the case of the canonical brauer induction formula this will be used in a subsequent article in the case of adams operations and chern classes

transition phenomena in cellular automata rule space abstract we define several qualitative classes of cellular automata ca behavior based on various statistical measures and describe how the space of all cellular automata is organized as a cellular automaton is changed by varying entries in its rule table abrupt changes in qualitative behavior may occur these abrupt changes have the character of bifurcations in smooth dynamical systems or of phase transitions in statistical mechanical systems the most complex ca rules exhibit many of the characteristics of second order transitions suggesting an association between computation complexity and critical phenomena

dynamics of coarse grained variables in spatially extended systems abstract some new ways to probe the dynamics of spatially extended systems giving rise to spatio temporal chaos are explored coarse grained observables suitable for describing large scale behavior are introduced by locally averaging the original variables over a spatial region and their stability spatial correlation function and statistical properties are compared to those of the original variables in general these observables do not obey closed form equations truncation schemes in which high order fourier modes or variables other than the coarse grained ones are discarded are developed and their sensitivity to the parameters is assessed

fibred permutation sets and the idempotents and units of monomial burnside rings we study the units of monomial burnside rings and the idempotents of monomial burnside algebras introducing a tenduction map we realise the unit group and the torsion unit group as a mackey functor

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

dispersion and pollution of meshless solutions for the helmholtz equation it is well known today that the standard finite element method fem is unreliable to compute approximate solutions of the helmholtz equation for high wavenumbers due to the pollution effect consisting mainly of the dispersion i e the numerical wavelength is longer than the exact one unless highly refined meshes are used fem solutions lead to unacceptable solutions in terms of precision while the use of very refined meshed increases the cost in terms of computational times the paper presents an application of the element free galerkin method efgm and focuses on the dispersion analysis in 2d it shows that it is possible to choose the parameters of the method in order to minimize the dispersion and to get extremely good results in comparison with the stabilized fem moreover the present meshless formulation is not restricted to regular distribution of nodes and a simple but real life problem is investigated in order to show the improvement in the accuracy of the numerical results w r fem results

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

chaos and diffusion in deterministic cellular automata abstract it is shown that some of the deterministic one dimensional cellular automata studied recently by wolfram exhibit a kind of spontaneous symmetry breaking the associated kinks bloch walls perform annihilating diffusive walks for infinite lattices and large times in addition close analogies between these automata and strange attractors are exhibited by mapping the automata onto dynamical systems defined by iterations of the unit square

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

spatio temporal chaos and intermittency in a 1 dimensional energy conserving coupled map lattice abstract the klein gordon equation with cubic nonlinearity the 4 equation is considered and an energy conserving difference scheme is proposed for its solution the scheme extended to finite time increments and spacing is then used to define a coupled map lattice for which an energy like functional is conserved the case of linear instability of the vacuum state is considered when this energy is not positive definite and found to lead under certain additional conditions to spatio temporal chaos the statistical properties of this type of solution such as probability densities and correlation functions are calculated strong intermittency whereby the process wanders between two sub manifolds is found and studied in detail

spatiotemporal chaos in mixed linear nonlinear two dimensional coupled logistic map lattice abstract we investigate a new spatiotemporal dynamics with mixing degrees of nonlinear chaotic maps for spatial coupling connections based on 2dcml here the coupling methods are including with linear neighborhood coupling and the nonlinear chaotic map coupling of lattices and the former 2dcml system is only a special case in the proposed system in this paper the criteria such kolmogorov sinai entropy density and universality bifurcation diagrams space amplitude and snapshot pattern diagrams are provided in order to investigate the chaotic behaviors of the proposed system furthermore we also investigate the parameter ranges of the proposed system which holds those features in comparisons with those of the 2dcml system and the mlncml system theoretical analysis and computer simulation indicate that the proposed system contains features such as the higher percentage of lattices in chaotic behaviors for most of parameters less periodic windows in bifurcation diagrams and the larger range of parameters for chaotic behaviors which is more suitable for cryptography

goal oriented adaptivity using unconventional error representations for the multidimensional helmholtz equation in goal oriented adaptivity goa the error in the quantity of interest qoi is represented using the error functions of the direct and adjoint problems this error representation is subsequently bounded above by element wise error indicators that are used to drive optimal refinements in this work we propose to replace in the error representation the adjoint problem by an alternative operator the main advantage of the proposed approach is that when judiciously selecting such alternative operator the corresponding upper bound of the error representation becomes sharper leading to a more efficient goa while the method can be applied to a variety of problems we focus here on two and threedimensional 2d and 3d helmholtz problems we show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p adaptive process we also provide guidelines for finding operators delivering sharp error representation upper bounds we further extend the results to a convectiondominated diffusion problem as well as to problems with discontinuous material coefficients finally we consider a sonic logging while drilling lwd problem to illustrate the applicability of the proposed method

a simple strategy to assess the error in the numerical wave number of the finite element solution of the helmholtz equation the standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem based on the linear functional output representing the quantity of interest for the assessment of the error in the approximation of the wave number for the helmholtz problem also referred to as dispersion or pollution error this strategy cannot be applied this is because there is no linear extractor producing the wave number from the solution of the acoustic problem moreover in this context the error assessment paradigm is reverted in the sense that the exact value of the wave number is known it is part of the problem data and the effort produced in the error assessment technique aims at obtaining the numerical wave number h as a postprocess of the numerical solution uh the strategy introduced in this paper is based on the ideas used in the a priori analysis a modified equation corresponding to a modified wave number m is introduced then the value of m such that the modified problem better accommodates the numerical solution uh is taken as the estimate of the numerical wave number h thus both global and local versions of the error estimator are proposed the obtained estimates of the dispersion error match the a priori predicted dispersion error and in academical examples the actual values of the error in the wave number

globally coupled maps phase transitions and synchronization abstract bifurcations in a system of coupled maps are investigated using symbolic dynamics it is shown that for coupled shift maps the well known space time mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state for coupled non hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour the anomalous dependence of fluctuations on the system size is attributed to these bifurcations

quantitative characterization of spatiotemporal chaos abstract in this paper one way to characterize spatiotemporal chaos is shown a coupled map lattice is used as a model for the spatiotemporal chaos the local expansion rate is defined for the coupled map lattice in the extension of the low dimensional temporal chaos it is shown that the local expansion rate is closely related to the spatiotemporal pattern of the spatiotemporal chaos the thermodynamic formalism is applied to the local expansion rate the thermodynamic functions describing fluctuations of the local expansion rate are shown the spatiotemporal pattern that fluctuates is shown at the same time it is shown clearly that the thermodynamic functions characterize the spatiotemporal chaos

spatiotemporal chaos in a 2d coupled map lattice with pinning like force abstract spatiotemporal chaos stc has been observed in a coupled map lattice cml which describes magnetic vortex pinning in superconductors for the single site map a transition to chaos is realized by period doubling with feigenbaum scaling in the developed chaos region intermittent laminar and turbulent behaviors of the map take place in this region the natural invariant density and lyapunov exponent are deduced in analytical form analytical expressions of a correlation function and a distribution function of laminar phase durations are represented as well for the cml a state density and correlation functions characterizing stc are calculated

long range order with local chaos in lattices of diffusively coupled odes abstract the existence of non trivial collective behavior in lattices of diffusively coupled differential equations is investigated for a two dimensional square lattice of rossler systems a rotating long range order is observed this case is best described in terms of a complex ginzburg landau cgl equation submitted to the local noise produced by the chaotic rossler units the parameters of this cgl equation are estimated to be in the so called benjamin feir stable region the collective oscillation regime thus corresponds to the linearly stable spatially homogeneous solution of the equivalent cgl equation the possibility of more complex collective behavior in similar systems is discussed

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

wavelike patterns in one dimensional coupled map lattices we investigate the existence of wavelike solution for the logistic coupled map lattices for which the spatiotemporal periodic patterns can be predicted by a simple two dimensional mapping the existence of such wavelike solutions is proved by the implicit function theorem with constraints we also examine the stabilities of these wave solutions under perturbations of uniform small deformation type we show that in some specific cases these perturbations are completely general the technique used in this paper is also applicable to investigate other space time regular patterns

error estimation and adaptivity for the finite element method in acoustics 2d and 3d applications this paper is dedicated to the control of accuracy and to the adaptivity of the finite element simulation of sound propagation assuming time harmonic behaviour the mathematical models are given as boundary value problems for the helmholtz equation two singularities inherent to the operator are demonstrated the k singularity related to the phase shift between the exact and the numerical waves and the singularity corresponding to the singularity at the eigenfrequencies two a posteriori error estimators are developed and the numerical tests show that due to these specific singularities error control cannot in general be accomplished by just transplanting methods that work well in static computations furthermore for low wave numbers it is necessary also to control the influence of the geometric or physical singularities an h adaptive version with refinement is applied to 2d and 3d real life problems

detecting switch dynamics in chaotic time waveform using a parametrized family of nonlinear predictors abstract an algorithm is presented for detecting switch dynamics in chaotic time waveform by the switch dynamics we mean that the chaotic time waveform is measured from a dynamical system whose bifurcation parameters are occasionally switched among a set of slightly different parameter values first the switched chaotic time waveform is divided into windows of short term time waveforms from the set of windowed time waveforms qualitatively similar parametrized family of nonlinear predictors is constructed qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original by characterizing the windows of short term chaotic time waveforms in terms of the qualitative parameters of nonlinear predictors switch dynamics of their associated bifurcation parameters are detected for the lorenz equations the rossler equations and the mackey glass equations efficiency of the algorithm is demonstrated in the experiment chaotic time waveforms contaminated with observational noise is considered

exploring collective behaviors with a multi attractor quartic map we simulate a 2d coupled map lattice formed by individual units consisting of a multi attractor quartic map we show that the interesting recently discovered non trivial collective behaviors where macroscopic quantities show well dened usually regular temporal evolution in spite of the presence of local disorder in space and time also exist over wide parameter domains in the presence of local periodic order in systems having more realistic units allowing coexistence of more than one attractor c 1998 elsevier science b v all rights reserved

spatio temporal intermittency in a 1d convective pattern theoretical model and experiments abstract we describe the occurence of spatio temporal intermittency in a one dimensional convective system that first shows time dependent patterns we recall experimental results and propose a model based on the normal form description of a secondary hopf bifurcation of a stationary periodic structure numerical simulations of this model show spatio temporal intermittent behaviors which we characterize briefly and compare to those given by the experiment

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

nonlinearly driven transverse synchronization in coupled chaotic systems synchronization transitions are investigated in coupled chaotic maps depending on the relative weight of linear versus nonlinear instability mechanisms associated to the single map two different scenarios for the transition may occur when only two maps are considered we always find that the critical coupling el for chaotic synchronization can be predicted within a linear analysis by the vanishing of the transverse lyapunov exponent t however major differences between transitions driven by linear or nonlinear mechanisms are revealed by the dynamics of the transient toward the synchronized state as a representative example of extended systems a one dimensional lattice of chaotic maps with power law coupling is considered in this high dimensional model finite amplitude instabilities may have a dramatic effect on the transition for strong nonlinearities an exponential divergence of the synchronization times with the chain length can be observed above el notwithstanding the transverse dynamics is stable against infinitesimal perturbations at any instant therefore the transition takes place at a coupling enl definitely larger than el and its origin is intrinsically nonlinear the linearly driven transitions are continuous and can be described in terms of mean field results for non equilibrium phase transitions with long range interactions while the transitions dominated by nonlinear mechanisms appear to be discontinuous

dynamics of chaos order interface in coupled map lattices abstract we study a coupled map lattice model with two states a simple fixed point and spatio temporal chaos preparing properly initial conditions we investigate the dynamics of the interface between order and chaos in the one dimensional lattice regimes of irregular and regular front propagation behavior are observed and analyzed by introducing a local front map and a front lyapunov exponent corresponding to these different regimes of front propagation we can characterize different types of transitions from laminar state to chaos using comoving lyapunov exponents in the two dimensional lattice these types of front motion are related to regimes of roughening and flattening of the interface

discrete time dynamical system under observational uncertainty abstract discrete time dynamical systems under observational uncertainty are studied as a result of the uncertainty points on an orbit are surrounded by uncertainty sets the problem of reconstructing the original orbit given the sequence of uncertainty sets is investigated the key property which makes the reconstruction possible is the sensitivity to initial conditions a general reconstructing algorithm is theoretically analyzed and experimentally tested on several low dimensional systems the technique is extended to coupled one dimensional maps with the goal of eventually developing retrospective techniques for partial differential equations exhibiting spatio temporal chaos provided the coupling strength remains small and the coupling term has bounded first derivatives it is conjectured that for dynamical systems with a positive liapunov exponent the observational uncertainty can be reduced exponentially with the length of the orbit used for reconstruction computer experiments with the coupled logistic map are consistent with this conjecture

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

a posteriori error estimation for standard finite element analysis a new residual type estimator based on projections of the error on subspaces of locally supported functions is presented the estimator is defined by a standard element by element refinement first an approximation of the energy norm of the error is obtained solving local problems with homogeneous dirichlet boundary conditions a later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces this estimate is a lower bound of the measure of the actual error the estimator does not need to approximate local boundary conditions for the error equation therefore computation of flux jumps is not necessary moreover the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward the presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error as well as its usefulness in adaptive procedures

the and constructions for biset functors abstract in this article we define the construction and the construction that was crucial in the theory of canonical induction formulas see 3 in the setting of biset functors thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors additionally this provides a unified approach to the study of a class of functors including the burnside ring the monomial burnside ring and global representation ring

pattern dynamics of a coupled map lattice for open flow abstract the pattern dynamics of the one way coupled logistic lattice which can serve as a phenomenological model for open flow is investigated and shown to be extremely rich for medium and large coupling strengths we find spatially periodic quasiperiodic and chaotic patterns with temporal periodicity and analyze their stability with the help of a newly introduced spatial map criteria are established for predicting the down flow bifurcation behavior of the coupled map lattice and for selecting attractors through modulation of the boundary for smaller coupling strengths we find a novel regime in which chaotic defects form a periodic lattice that is shown to be the result of a boundary crisis

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

coupling induced statistical cycling in two diffusively coupled maps abstract this paper discusses the effects of diffusively coupling two identical one dimensional maps attention is focused on situations where the local isolated maps are statistically stable but where the coupled system is not a biologically motivated map and the quadratic map are numerically shown to display this behavior the piecewise linear tent map is then investigated analytically and we give a phase diagram of this system which displays the location of nonequilibrium phase transitions it is conjectured that the diffusive coupling of two chaotic but statistically stable maps i e with asymptotically stable perron frobenius operators can yield a two dimensional system which is not statistically stable whose associated perron frobenius operator is asymptotically periodic

correlations and spectra of an intermittent chaos near its onset point a one parameter family of piecewise linear discontinuous maps which bifurcates from a periodic state of periodm m 2 3 to an intermittent chaos is studied as a new model for the onset of turbulence via intermittency the onset of chaos of this model is due to the excitation of an infinite number of unstable periodic orbits and hence differs from pomeau manneville s mechanism which is a collapse of a pair of stable and unstable periodic orbits the invariant density the time correlation function and the power spectrum are analytically calculated for an infinite sequence of values of the bifurcation parameter which accumulate to the onset point c from the chaos side c 0 the power spectrum near 0 is found to consist of a large number of lorentzian lines with two dominant peaks the highest peak lies around frequency 2 m with the power law envelope l 2 m 4 the second highest peak lies around o 0 with the envelope l 2 the width of each line decreases as and the separation between lines decreases as lg3 1 it is also shown that the liapunov exponent takes the form m and the mean lifetime of the periodic state in the intermittent chaos is given bym 1 ln 1 1

analytical approach for piecewise linear coupled map lattices a simple construction is presented which generalizes piecewise linear one dimensional markov maps to an arbitrary number of dimensions the corresponding coupled map lattice known as a simplicial mapping in the mathematical literature allows for an analytical investigation in particular the spin hamiltonian which is generated by the symbolic dynamics is accessible as an example a formal relation between a globally coupled system and an ising mean field model is established the phase transition in the limit of infinite system size is analyzed and analytical results are compared with numerical simulations

robust space time intermittency and 1 f noise abstract we investigate a type of intermittency that occurs in space as well as time studying a one dimensional lattice of coupled quadratic maps this system naturally forms spatial domains motion of the domain walls causes spatially localized changes from chaotic to almost periodic behavior the almost periodic phases have eigenvalues quite close to one resulting in long lived laminar bursts with a 1 f low frequency spectrum this behavior has aspects of both crutchfield and pomeau manneville intermittency unlike pomeau manneville however the behavior that we observe here is quite robust under changes of parameters

goal oriented adaptivity using unconventional error representations for the 1d helmholtz equation in this work the error of a given output functional is represented using bilinear forms that are different from those given by the adjoint problem these representations can be employed to design novel h p and h p energy norm and goal oriented adaptive algorithms numerical results in 1d show that for wave propagation problems the advantages of this new representation are notorious when selecting the laplace equation as the dual problem specifically the computed upper bounds of the new error representation are sharper than the classical ones used in both energy norm and goal oriented adaptive methods especially when the dispersion pollution error is significant

soliton turbulence in one dimensional cellular automata abstract statistical properties of cellular automata which support simple solitary waves are numerically studied under these rules spatio temporal patterns are sensitively dependent on the collision processes of solitons and other elementary excitations such as breathers kinks and nuclei patterns typically become randomized as collisions proceed turbulent states are realized after many collisions the resulting global patterns can be classified by the types of elementary excitations they contain in many turbulent states the spectra reveal a long range order with a k v anomaly for k 1 the irreversible process leading to turbulent equilibrium is characterized by the transient spectrum together with the allan variance the mean free motion of a soliton in the turbulent states is measured by the mutual information flow and the information loss is shown to obey an inverse power law the transient time to reach an attractor grows exponentially with system size which suggests that in the thermodynamic limit soliton turbulence is not an attractor but rather a transient state

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

a cell dynamical system model of chemical turbulence a cellular automaton like caricature of chemical turbulence on an infinite one dimensional lattice is studied the model exhibits apparently turbulent space time patterns to make this statement precise the following problems or points are discussed 1 the infinite system size limit of such cell dynamical systems and its observability is defined 2 it is proved that the invariant state in the large system size limit of the turbulent phase exhibits spatial patterns governed by a gibbs random field 3 potential characteristics of turbulent space time patterns are critically surveyed and a working definition of weak turbulence is proposed 4 it is proved that the invariant state of the turbulent phase is actually weak turbulent furthermore we conjecture that the turbulent phase of our model is an example of a k system that is not bernoulli

synchronized clusters in coupled map networks i numerical studies we study the synchronization of coupled maps on a variety of networks including regular one and two dimensional networks scale free networks small world networks tree networks and random networks for small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases when nodes show synchronized behavior we observe two interesting phenomena first there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently second we identify two different ways of cluster formation namely self organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings the synchronized clusters may be of dominant self organized type dominant driven type or mixed type depending on the type of network and the parameters of the dynamics we define different states of the coupled dynamics by considering the number and type of synchronized clusters for the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant and the logistic map parameter ensuremath mu for large coupling strengths and nonlinear coupling we find that the scale free networks and the caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity for most of our study we use the number of connections of the order of the number of nodes as the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase

chaotic traveling waves in a coupled map lattice traveling waves triggered by a phase slip are studied in a coupled map lattice a local phase slip affects globally the system which is in strong contrast with kink propagation attractors with different velocities coexist and form quantized bands determined by the number of phase slips if the system size is not far from an integer multiple of the selected wavelength attractors are tori otherwise weak chaos induces modulation of waves or a chaotic itinerancy of traveling states with a long ranged temporal correlation supertransients before the formation of traveling waves are noted in the high nonlinearity regime in the weaker nonlinearity there are fluctuations of domain sizes and brownian like motion of domains propagation of chaotic domains by phase slips is also found

constructive polynomial approximation with a priori error bounds for nonlinear initial value differential problems abstract in this paper we consider an initial value problem y f x y y 0 y 0 where f is a continuous function satisfying a lipschitz condition first the function f x y is approximated by a bernstein polynomial in two variables b n f x y of an appropriate degree according to a prescribed accuracy second by application of the frobenius method to the initial value problem y b n f x y y 0 y 0 an exact series solution of the latter problem is constructed finally the infinite series solution is truncated in order to obtain an explicit polynomial whose error with respect to the exact solution of the original problem is less than the prescribed accuracy in an existence and uniqueness domain which is determined in terms of the initial data

infinite dimensional srb measures we review the basic steps leading to the construction of a sinai ruelle bowen srb measure for an infinite lattice of weakly coupled expanding circle maps and we show that this measure has exponential decay of space time correlations first using the perron frobenius operator one connects the dynamical system of coupled maps on a d dimensional lattice to an equilibrium statistical mechanical model on a lattice of dimension d 1 this lattice model is for weakly coupled maps in a high temperature phase and we use a general but very elementary method to prove exponential decay of correlations at high temperatures

spatio temporal structure in coupled map lattices two point correlations versus mutual information a one dimensional lattice of logistic maps is investigated in the case of strong nonlinearity and strong coupling although the dynamics may be classified as fully developed turbulence spatio temporal structure can be detected by computing time delayed mutual information and two point correlations the correlation is found to be superior in detecting weak structure an improved algorithm for mutual information is described

subdomain based flux free a posteriori error estimators a new residual type flux free error estimator is presented it estimates upper and lower bounds of the error in energy norm the proposed approach precludes the main drawbacks of standard residual type estimators circumvents the need of flux equilibration and results in a simple implementation that uses standard resources available in finite element codes this is specially interesting for 3d applications where the implementation of this technique is as simple as in 2d recall that on the contrary the complexity of the flux equilibration techniques increases drastically in the 3d case the bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs representing quantities of engineering interest the presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest

fibered biset functors abstract the theory of biset functors introduced by serge bouc gives a unified treatment of operations in representation theory that are induced by permutation bimodules in this paper by considering fibered bisets we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules the main result of this paper is the classification of simple fibered biset functors

ks entropy and mean lyapunov exponent for coupled map lattices abstract the statistics of kolmogorov sinai ks entropy for a coupled map lattice model are analyzed from the viewpoint of the thermodynamic formalism it is shown that the fluctuation of ks entropy for a coupled map lattice model satisfies the large deviation statistics also the probability density of lyapunov exponents pdle is studied and it is shown that the pdle gives the measure of the irregularity for the spatio temporal patterns mean lyapunov exponent is introduced and compared with ks entropy

cooperative behavior of a chain of synaptically coupled chaotic neurons abstract a coupled linear chain of hindmarsh rose model neurons with reciprocal inhibition between neighboring neurons exhibited synchronous oscillations in which neighboring neurons burst out of phase and next nearest neighbor neurons burst in phase the bifurcations observed inside this out of phase regime were qualitatively the same for all chains with an even number of neurons and were similar to those observed in a single isolated cell although the organization of the behavior of a chain of coupled neurons was more regular than that of an isolated cell when noise was added to the synaptic coupling strengths there was less hysteresis in the system and many of the bifurcations with smaller basins of attraction were eliminated making the system even more regular these results suggest that in populations of bursting neurons with reciprocal inhibition the chaotic behavior found in single cells is suppressed

the road to chaos by time asymmetric hebbian learning in recurrent neural networks this letter aims at studying the impact of iterative hebbian learning algorithms on the recurrent neural network s underlying dynamics first an iterative supervised learning algorithm is discussed an essential improvement of this algorithm consists of indexing the attractor information items by means of external stimuli rather than by using only initial conditions as hopfield originally proposed modifying the stimuli mainly results in a change of the entire internal dynamics leading to an enlargement of the set of attractors and potential memory bags the impact of the learning on the network s dynamics is the following the more information to be stored as limit cycle attractors of the neural network the more chaos prevails as the background dynamical regime of the network in fact the background chaos spreads widely and adopts a very unstructured shape similar to white noise next we introduce a new form of supervised learning that is more plausible from a biological point of view the network has to learn to react to an external stimulus by cycling through a sequence that is no longer specified a priori based on its spontaneous dynamics the network decides on its own the dynamical patterns to be associated with the stimuli compared with classical supervised learning huge enhancements in storing capacity and computational cost have been observed moreover this new form of supervised learning by being more respectful of the network intrinsic dynamics maintains much more structure in the obtained chaos it is still possible to observe the traces of the learned attractors in the chaotic regime this complex but still very informative regime is referred to as frustrated chaos

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

lyapunov analysis and information flow in coupled map lattices abstract lyapunov analysis for the coupled map lattices is presented the co moving lyapunov exponent is calculated which is related with the propagation of the disturbance in space the propagation speed agrees with the zero crossing point of the co moving lyapunov exponent the co moving mutual information flow is introduced which shows the selective transmission of the information at some speed lyapunov spectra and vectors are calculated spatial structures of the vectors are investigated possible analogy with the anderson localization problem is discussed

front motion metastability and subcritical bifurcations in hydrodynamics metastability occurs whenever under given external conditions as defined by the reynolds number in hydrodynamics more than one linearly stable solution may exist in variational problems the preferred state is usually claimed to be the one with the lowest energy but in hydrodynamics and except for very special conditions no energy functional exists from an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations as turbulent and laminar in pipe flows or boundary layers moves with a constant mean velocity depending on the control parameter the sign of this velocity allows to decide which state is stable and which is metastable possible physical implications of these ideas for various problems of hydrodynamics are discussed

effects of correlation among stored patterns on associative dynamics of chaotic neural network abstract we numerically investigate the effects of correlation among stored patterns on the associative dynamics in a chaotic neural network model in the model there are two kinds of parameters one is a measure of the hopfield like behavior of the retrieval process and another controls the chaotic behavior the parameter dependence of the associative dynamics is also examined the following results are found i two dimensional parameter space is divided into two kinds of associative states by a distinct boundary one is the retrieval state of the association such as the hopfield like retrieval state and another is the wandering state of the associative dynamics where the network retrieves stored patterns and their reverse patterns ii the area of the wandering state becomes larger as the degree of correlation becomes larger iii as the degree of correlation becomes larger both the recall ratio of correlated patterns and the transition frequency between correlated patterns becomes larger in the wandering state iv the whole region of the wandering state in the parameter space is not necessarily chaotic from the view point of the lyapunov dimension but most of the region of the wandering state is chaotic

collective behaviour in one dimensional locally coupled map lattices the authors observe true and quasiperiod 4 collective evolution of average values in one dimensional lattices of logistic maps with variable coupling range the phenomenon appears to be quite prevalent for local coupling beyond nearest neighbours they find that in one dimension it is easy to visualize the structure of the four quasi states these appear to be quite stable to disturbances such as domain reversals

global representation rings abstract in this paper we combine the concepts of burnside ring and character ring in what we call the global representation ring of a finite group we compute all ring homomorphisms from this ring to the complex numbers we compute its spectrum its connected components and determine the primitive idempotents

dynamics at the interface dividing collective chaotic and synchronized periodic states in a cml abstract a study is developed focusing the loss of stability of the interface dividing two regions of different spatial patterns on a coupled map lattice using coupling as the parameter guiding the transition these patterns are constructed over local periodic chaotic attractors generating regions of synchronized collective behavior the discrete feature of the underlying lattice the anisotropy that stems from such discreteness and its possible change to an isotropic system through coupling with large number of neighbors are also investigated

chaotic sub dynamics in coupled logistic maps abstract we study the dynamics of laplacian type coupling induced by logistic family f x x 1 x where 0 4 on a periodic lattice that is the dynamics of maps of the form f x y 1 e f x e f y 1 e f y e f x where e 0 determines strength of coupling our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal in analytical way we detect some regions of parameters for which a horseshoe is present and using simulations global attractors and invariant sets are depicted

multi state coupled map lattices abstract we investigate a two dimensional locally coupled map lattice cml with the local dynamics driven by the multi attractor quartic map in particular we explore a region where two local fixed points exist one being periodic and the other chaotic different sets of initial conditions such as random initial values for each site or arrangements favoring equal weights to the different local attractors were used the system reaches different asymptotic states as the intensity or the topology of the local coupling is varied among the asymptotic states we find either homogeneous collective behavior or mixtures of these with synchronized states these states are characterized and interpreted throughout this work by the distributions of the values of the maps and by the average roughness over the lattice

on quasiperiodic travelling waves in coupled map lattices we investigate quasiperiodic travelling waves qtws in lattices of diffusively coupled logistic maps starting from the assumption that any spatial structure can be broken down into simpler elementary structures a classification scheme for qtws is introduced within this framework the phenomenon of discrete velocities is reviewed and further investigated in addition a new technique is proposed for predicting whether qtws can occur for given parameter values and which they might be