structure preserving approximation of dissipative evolution problems we present a framework for the systematic numerical approximation of nonlinear evolution problems with dissipation the approach is based on rewriting the problem in a canonical form that complies with the underlying energy dissipation structure we show that the corresponding weak formulation then allows for a dissipation preserving approximation by galerkin methods in space and discontinuous galerkin methods in time the proposed methodology is rather general and can be applied to a wide range of applications this is demonstrated by discussion of some typical examples ranging from diffusive partial differential equations to dissipative hamiltonian systems

detection of couplings in ensembles of stochastic oscillators the problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed we suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint moreover we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series the technique applies to weakly coupled nonsynchronized oscillators it is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies efficiency of the technique is demonstrated in numerical experiments

a robust method for detecting interdependences application to intracranially recorded eeg we present a measure for characterizing statistical relationships between two time sequences in contrast to commonly used measures like cross correlations coherence and mutual information the proposed measure is non symmetric and provides information about the direction of interdependence it is closely related to recent attempts to detect generalized synchronization however we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise and to detect also weak interdependences we apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies

detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble a method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail mutual nonlinear prediction also provides information on the directionality of the coupling between systems furthermore the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony numerical examples studied include three classes of unidirectionally coupled systems systems with identical parameters nonidentical parameters and stochastic driving of a nonlinear system this technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool the interrelationships examined include single neuron unit firing the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex and intracellular measurements of integrated excitatory postsynaptic potentials epsp s dynamical interdependence perhaps generalized synchrony was identified in this neuronal network between simultaneous single unit firings between units and the population and between units and intracellular epsp s

synchronization likelihood an unbiased measure of generalized synchronization in multivariate data sets the study of complex systems consisting of many interacting subsystems requires the use of analytical tools which can detect statistical dependencies between time series recorded from these subsystems typical examples are the electroencephalogram eeg and magnetoencephalogram meg which may involve the simultaneous recording of 150 or more time series coherency which is often used to study such data is only sensitive to linear and symmetric interdependencies and cannot deal with non stationarity recently several algorithms based upon the concept of generalized synchronization have been introduced to overcome some of the limitations of coherency estimates e g physica d 134 1999 419 brain res 792 1998 24 however these methods are biased by the degrees of freedom of the interacting subsystems physica d 134 1999 419 physica d 148 2001 147 we propose a novel measure for generalized synchronization in multivariate data sets which avoids this bias and can deal with non stationary dynamics

local predictability and information flow in complex dynamical systems abstract predictability is by observation a local notion in complex dynamical systems its spatiotemporal structure implies a flow or transfer in discrete cases of information that redistributes the local predictability within the state space of concern information flow is a fundamental concept in general physics which has applications in a wide variety of disciplines such as neuroscience material science atmosphere ocean science and turbulence research to name but a few in this study it is rigorously formulated with respect to relative entropy within the framework of a system with many components each signifying a location or a structure given a component the mechanism governing the evolution of its predictability can be classified into two groups one due to the component itself another due to a transfer of information from its peers a measure of the transfer is rigorously derived and an explicit expression obtained this measure possesses a form reminiscent of that we have obtained before with respect to absolute entropy in x s liang and r kleeman a rigorous formalism of information transfer between dynamical system components physica d 227 2007 173 182 in particular when the system is of dimensionality 2 there is no difference between the formalisms with respect to absolute entropy and relative entropy except for a minus sign properties have been explored and discussed particularly discussed is the property of asymmetry or causality which states that information transfer from one component to another carries no hint about the transfer in the other direction in contrast to the transfer of other quantities such as energy this formalism has been applied to the study of the scale scale interaction and information transfer between the first two modes of the truncated burgers equation it is found that all 12 transfers are essentially zero or negligible save for a strong transfer between the sine components from the low frequency mode to the high frequency mode that is to say the predictability of the high frequency mode is controlled by the knowledge of the low frequency mode this result though from a highly idealized system has interesting implications about the dynamical closure problem in turbulence research and atmosphere ocean science i e the subgrid processes may to some extent be parameterized by the large scale dynamics this study can be adopted to investigate the propagation of uncertainties in fluid flows which has important applications in problems such as atmospheric observing platform design and may be utilized to identify the route of information flowing within a complex network

phase shifts of synchronized oscillators and the systolic diastolic blood pressure relation we study the phase synchronization properties of systolic and diastolic arterial pressure in healthy subjects we find that delays in the oscillatory components of the time series depend on the frequency bands that are considered in particular we find a change of sign in the phase shift going from the very low frequency band to the high frequency band this behavior should reflect a collective behavior of a system of nonlinear interacting elementary oscillators we prove that some models describing such systems e g the winfree and the kuramoto models offer a clue to this phenomenon for these theoretical models there is a linear relationship between phase shifts and the difference of natural frequencies of oscillators and a change of sign in the phase shift naturally emerges

weak form of stokes dirac structures and geometric discretization of port hamiltonian systems we present the mixed galerkin discretization of distributed parameter port hamiltonian systems due to the inherent definition of boundary interconnection ports this system representation is particularly useful for the modeling of interconnected multi physics systems and control at the prototypical example of a system of two conservation laws in arbitrary spatial dimension we derive the main contributions i a weak formulation of the underlying geometric stokes dirac structure ii its geometric approximation by a finite dimensional dirac structure using a mixed galerkin approach and power preserving maps on the space of discrete power variables and iii the approximation of the hamiltonian to obtain finite dimensional port hamiltonian state space models the power preserving maps on the discrete bond space offer design degrees of freedom for the discretization which is illustrated at the example whitney finite elements on a 2d simplicial triangulation the resulting schemes can be considered as trade offs between centered approximations and upwinding

higher order asymptotics of toeplitz determinants with symbols in weighted wiener algebras we extend a result of bottcher and silbermann on higher order asymptotics of determinants of block toeplitz matrices with symbols in wiener algebras with power weights to the case of wiener algebras with general weights satisfying natural submultiplicativity monotonicity and regularity conditions

two sided bounds and perturbation results for regularized determinants of infinite order compact operators a compact operator in a separable hilbert space is of infinite order if it does not belong to any schatten von neumann ideal in the paper upper and lower bounds for the regularized determinants of infinite order operators are derived by these bounds perturbations results for the regularized determinants are established

a rigorous formalism of information transfer between dynamical system components i discrete mapping we put the concept of information transfer on a rigorous footing and establish for it a formalism within the framework of discrete maps the resulting transfer measure possesses a property of directionality or transfer asymmetry as emphasized by schreiber t schreiber measuring information transfer phys rev lett 85 2 2000 461 it also verifies the transfer measure for two dimensional systems which was obtained by liang and kleeman x s liang r kleeman information transfer between dynamical system components phys rev lett 95 24 2005 244101 through a different avenue connections to classical formalisms are explored and applications presented we find that in the context of the baker transformation there is always information flowing from the stretching direction to the folding direction while no transfer occurs in the opposite direction we also find that within the henon map system the transfer from the quadratic component to the linear component is of a simple form as expected on physical grounds this latter result is unique to our formalism

phase diagram of a generalized winfree model we study the phase diagram of a generalized winfree model the modification is such that the coupling depends on the fraction of synchronized oscillators a situation which has been noted in some experiments on coupled josephson junctions and mechanical systems we let the global coupling k be a function of the kuramoto order parameter r through an exponent z such that z 1 corresponds to the standard winfree model zl1 strengthens the coupling at low r low amount of synchronization and at zg1 the coupling is weakened for low r using both analytical and numerical approaches we find that z controls the size of the incoherent phase region and that one may make the incoherent behavior less typical by choosing zl1 we also find that the original winfree model is a rather special case indeed the partial locked behavior disappears for zg1 at fixed k and varying ensuremath gamma the stability boundary of the locked phase corresponds to a transition that is continuous for zl1 and first order for zg1 this change in the nature of the transition is in accordance with a previous study of a similarly modified kuramoto model

assessing causality from multivariate time series in this work we propose a general nonparametric test of causality for weakly dependent time series more precisely we study the problem of attribution i e the proper comparison of the relative influence that two or more external dynamics trigger on a given system of interest we illustrate the possible applications of the proposed methodology in very different fields like physiology and climate science

dyson s constant in the asymptotics of the fredholm determinant of the sine kernel we prove that the asymptotics of the fredholm determinant of i k where k is the integral operator with the sine kernel open image in new window on the interval 0 are given by

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

building systems from simple hyperbolic ones in this article we introduce a technique that derives from the existence and uniqueness of solutions to a simple hyperbolic partial differential equation p d e the existence and uniqueness of solutions to hyperbolic and parabolic p d e s among others we show that starting with an impedance passive system associated to the undamped wave equation we can obtain an impedance passive system associated to the heat conduction equation

determinant formula for the six vertex model with reflecting end using the quantum inverse scattering method for the xxz model with open boundary conditions we obtained the determinant formula for the six vertex model with reflecting end

fractal dimension of spatially extended systems properties of the invariant measure are numerically investigated in 1d chains of diffusively coupled maps the coarse grained fractal dimension is carefully computed in various embedding spaces observing an extremely slow convergence towards the asymptotic value this is in contrast with previous simulations where the analysis of an insufficient number of points led the authors to underestimate the increase of fractal dimension with increasing the dimension of the embedding space orthogonal decomposition is also performed confirming that the slow convergence is intrinsically related to local nonlinear properties of the invariant measure finally the kaplan yorke conjecture is tested for short chains showing that despite the noninvertibility of the dynamical system a good agreement is found between lyapunov dimension and information dimension

construction of multivariate surrogate sets from nonlinear data using the wavelet transform abstract the use of surrogate data has become a crucial first step in the study of nonlinearity in time series data a widely used technique to construct surrogate data is to randomize the phases of the data in the fourier domain in this paper an alternative technique based on the resampling of wavelet coefficients is discussed this approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data it proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties removal of nonlinear structure and computational demands however the wavelet technique may have specific and distinct advantages in the application to complex data sets such as numerical analysis of turbulence and experimental brain imaging data where wavelets give a more parsimonious representation of spatio temporal patterns than fourier modes it is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints

intermittency and regularized fredholm determinants we consider real analytic maps of the interval i 0 1 which are expanding everywhere except for a neutral fixed point at 0 we show that on a certain function space the associated perron frobenius operator has a continuous and residual spectrum contained in the line segment c 0 1 and a point spectrum p which has no points of accumulation outside 0 and 1 furthermore points in p 0 1 are eigenvalues of finite multiplicity we construct a regularized fredholm determinant d which has a holomorphic extension to c and can be analytically continued from each side of c to an open neighborhood of c 0 1 on different riemann sheets in c the zero set of d is in one to one correspondence with the point spectrum of through the conformal transformation the function d z extends to a holomorphic function in a domain which contains the unit disc

detecting generalized synchronization between chaotic signals a kernel based approach a unified framework for analysing generalized synchronization in coupled chaotic systems from data is proposed the key of the proposed approach is the use of the kernel methods recently developed in the field of machine learning several successful applications are presented which show the capability of the kernel based approach for detecting generalized synchronization and dynamical change of the coupling strength between two chaotic systems can be captured by the proposed approach it is also discussed how the kernel parameter is suitably chosen from data

a physically motivated class of scattering passive linear systems we introduce a class of scattering passive linear systems motivated by examples from mathematical physics the state space of the system is x h oplus e where h and e are hilbert spaces we also have a hilbert space e 0 which is dense in e with continuous embedding and e 0 is the dual of e 0 with respect to the pivot space e the input space is the same as the output space and it is denoted by u the semigroup generator has the structure a big begin smallmatrix 0 l l g frac 1 2 k k end smallmatrix big where l in cal l e 0 h and k in cal l e 0 u are such that big begin smallmatrix l k end smallmatrix big with domain e 0 is closed as an unbounded operator from e to h oplus u the operator g in cal l e 0 e 0 is such that rm re langle gw 0 w 0 rangle leq 0 for all w 0 in e 0 the observation operator is c left begin matrix 0 k end matrix right the control operator is b c and the output equatio