multiparameter control of chaos controlling chaos by using more than one available control parameter is presented as an experimentally feasible way to reduce the transient times that precede stabilization and improve performance in the presence of noise we demonstrate these advantages by applying our method to a numerical example

comparative analysis of formulations for conservative transport in porous media through sensitivity based parameter calibration 1 we apply a general strategy based on global sensitivity analysis gsa and model discrimination criteria to a calibrate the parameters embedded in competing models employed to interpret laboratory scale tracer experiments b rank these models and c estimate the relative degree of likelihood of each model through a posterior probability weight we consider a conservative transport experiment in a uniform porous medium we apply gsa to three transport models based on the classical advection dispersion equation ade a dual porosity dp formulation with mass transfer between mobile and immobile regions and the continuous time random walk ctrw approach gsa is performed through polynomial chaos expansion of the governing equations treating key model parameters as independent random variables we show how this approach allows identification of a the relative importance of model dependent parameters and b the space time locations where the models are most sensitive to these parameters gsa is then employed to assist parameter estimates within a maximum likelihood framework finally formal model identification criteria are employed to a rank the alternative models and b associate each model with a posterior probability weight for the specific case study the gsa based calibration of each model returns an acceptable approximation remarkably accurate in the case of the ctrw model of all available concentration data with calibration being performed using minimum sets of observations corresponding to the most sensitive space time locations

the pim simplex method an extension of the pim triple method to saddles with an arbitrary number of expanding directions abstract the growing interest in non attracting chaotic sets of high dimensional dynamical systems requires the development of numerical techniques for their study the pim triple method h e nusse j a yorke physica d 36 1989 137 is a very good method to obtain trajectories on saddles with one positive lyapunov exponent in this paper we combine the same ideas with an algorithm for finding local extrema of multi variable functions to develop an extension of the method the pim simplex method that is suitable for the study of sets with an arbitrary number of expanding directions

differential embedding of the lorenz attractor ideally an embedding of an n dimensional dynamical system is n dimensional ideally an embedding of a dynamical system with symmetry is symmetric ideally the symmetry of the embedding is the same as the symmetry of the original system this ideal often cannot be achieved differential embeddings of the lorenz system which possesses a twofold rotation symmetry are not ideal while the differential embedding technique happens to yield an embedding of the lorenz attractor in three dimensions it does not yield an embedding of the entire flow an embedding of the flow requires at least four dimensions the four dimensional embedding produces a flow restricted to a twisted three dimensional manifold in r4 this inversion symmetric three manifold cannot be projected into any three dimensional euclidean subspace without singularities

control of chaotic systems using an on line trained linear neural controller abstract we propose a control system including an on line trained linear neural controller to control chaotic systems the control system stabilizes a chaotic orbit onto an unstable fixed point without using the knowledge of the location of the point and the local linearized dynamics at the point furthermore the control system can track the stabilized orbit to the unstable fixed point whose location and local dynamics vary slowly with a variation of the system parameter this paper extends a previous paper konishi and kokame 1995 for more general situations and improves the neural controller proposed in the previous paper both to simplify the training algorithm and to guarantee the convergence of the neural controller the stability analysis of the control system reveals that some unstable fixed points cannot be stabilized in the control system numerical experiments show that the control system works well for controlling high dimensional chaotic systems

control of noisy heteroclinic cycles abstract we develop a control strategy for strongly nonlinear dynamical systems possessing heteroclinic cycles connecting saddle points and subject to small random perturbations in the absence of control attracting cycles lead to solutions which exhibit intermittent behavior consisting of quiescent periods interrupted by rapid bursts we show that suitable feedback control can retain solutions near the saddle points and hence reduce the typical bursting rate the model studied and the form of control are motivated by our interest in reducing turbulence production in the boundary layer

isoperimetric and weingarten surfaces in the schwarzschild manifold we show that any star shaped convex hypersurface with constant weingarten curvature in the desitter schwarzschild manifold is a sphere of symmetry moreover we study an isoperimetric problem for bounded domains in the doubled schwarzschild manifold we prove the existence of an isoperimetric surface for any value of the enclosed volume and we completely describe the isoperimetric surfaces for very large enclosed volume this complements work in h bray s thesis where isoperimetric surfaces homologous to the horizon are studied

model selection and akaike s information criterion aic the general theory and its analytical extensions during the last fifteen years akaike s entropy based information criterion aic has had a fundamental impact in statistical model evaluation problems this paper studies the general theory of the aic procedure and provides its analytical extensions in two ways without violating akaike s main principles these extensions make aic asymptotically consistent and penalize overparameterization more stringently to pick only the simplest of the true models these selection criteria are called caic and caicf asymptotic properties of aic and its extensions are investigated and empirical performances of these criteria are studied in choosing the correct degree of a polynomial model in two different monte carlo experiments under different conditions

sworn testimony of the model evidence gaussian mixture importance game sampling what is the best model the answer to this question lies in part in the eyes of the beholder nevertheless a good model must blend rigorous theory with redeeming qualities such as parsimony and quality of fit model selection is used to make inferences via weighted averaging from a set of k candidate models formula presented and help identify which model is most supported by the observed data formula presented here we introduce a new and robust estimator of the model evidence formula presented which acts as normalizing constant in the denominator of bayes theorem and provides a single quantitative measure of relative support for each hypothesis that integrates model accuracy uncertainty and complexity however formula presented is analytically intractable for most practical modeling problems our method coined gaussian mixture importance game sampling uses bridge sampling of a mixture distribution fitted to samples of the posterior model parameter distribution derived from mcmc simulation we benchmark the accuracy and reliability of game sampling by application to a diverse set of multivariate target distributions up to 100 dimensions with known values of formula presented and to hypothesis testing using numerical modeling of the rainfall runoff transformation of the leaf river watershed in mississippi usa these case studies demonstrate that game sampling provides robust and unbiased estimates of the evidence at a relatively small computational cost outperforming commonly used estimators the game sampler is implemented in the matlab package of dream and simplifies considerably scientific inquiry through hypothesis testing and model selection

chaotic transitions and low frequency fluctuations in semiconductor lasers with optical feedback abstract this paper examines the dynamical origin of low frequency fluctuations lffs in semiconductor lasers subject to time delayed optical feedback in particular we study chaotic transitions leading to the onset of lffs by numerical integration of lang kobayashi equations for a laser pumped near threshold we construct a bifurcation analysis scheme that enables the classification of the different operation regimes of the laser we use the scheme to study the coexistence of the lffs and stable emission on the maximum gain mode mgm which was the subject of recent experiments t heil i fischer w elsaser phys rev a 60 1999 634 our computations suggest that as the feedback level increases the regime of sustained lffs alternates with regions of transient lffs where the laser can achieve stabilization on the mgm exploration of the parameter space reveals strong dependence of the structure of the lff dynamics and the coexistence regime on the value of the linewidth enhancement factor

multi dimensioned intertwined basin boundaries basin structure of the kicked double rotor abstract using numerical computations on a map which describes the time evolution of a particular mechanical system in a four dimensional phase space the kicked double rotor we have found that the boundaries separating basins of attraction can have different properties in different regions and that these different regions can be intertwined on arbitrarily fine scale in particular for the double rotor map if one chooses a restricted region of the phase space and examines the basin boundary in that region then either one observes that the boundary is a smooth three dimensional surface or one observes that the boundary is fractal with dimension d 3 9 and which of these two possibilities applies depends on the particular phase space region chosen for examination furthermore for any region no matter how small for which d 3 9 one can choose subregions within it for which d 3 hence d 3 9 region and d 3 region are intertwined on arbitrarily fine scale other examples will also be presented and analyzed to show how this situation can arise these include one dimensional map cases a map of the plane and the lorenz equations in one of our one dimensional map cases the boundary will be fractal everywhere but the dimension can take on either of two different values both of which lie between 0 and 1 these examples lead us to conjecture that basin boundaries typically can have at most a finite number of possible dimension values more specifically let these values be denoted d 1 d 2 d n choose a volume region of phase space whose interior contains some part of the basin boundary and evaluate the dimension of the boundary in that region then our conjecture is that for all typical volume choices the evaluated dimension within the chosen volume will be one of the values d 1 d 2 d n for example in our double rotor map it appears that n 2 and d 1 3 0 and d 2 3 9

controlling chaotic dynamical systems using time delay coordinates abstract the ott grebogi yorke control method is analyzed in the case that the attractor is reconstructed from a time series using time delay coordinates it turns out that the control formula of ogy should be modified in order to apply to experimental systems if time dekay coordinates are used we reveal that the experimental surface of section map does not only depend on the actual parameter but also on the preceding one in order to meet this dependence two modifications are introduced which lead to a better performance of the control to compare their control abilities they are applied to simulations of a duffing oscillator the issue of measurement noise is considered

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 1 theory this series of three papers describes a cross validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise any number of relevant parameters including nugget effect can be estimated the theory described in this paper is based on a maximum likelihood approach which treats the cross validation errors as gaussian various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases correlation between these errors is weak the log likelihood criterion is optimized through a combination of conjugate gradient algorithms an adjoint state theory is used to efficiently calculate the gradient of the estimation criterion optimize the step size downgradient and compute a lower bound for the covariance matrix of the estimation errors issues related to the identifiability stability and uniqueness of the resulting adjoint state maximum likelihood cross validation asmlcv method are discussed this paper also describes the manner in which asmlcv allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives practical aspects of asmlcv and its application to synthetic data are presented in paper 2 samper and neuman this issue a applications to real hydrogeological data transmissivities and groundwater levels have been presented elsewhere while hydrochemical and isotopic data are analyzed by asmlcv in paper 3 samper and neuman this issue b

complexity in hamiltonian driven dissipative chaotic dynamical systems the existence of symmetry in chaotic dynamical systems often leads to one or several low dimensional invariant subspaces in the phase space we demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is hamiltonian but the full system is dissipative in particular an infinite number of distinct attractors can coexist these attractors can be quasiperiodic strange nonchaotic and chaotic with different positive lyapunov exponents finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors however arbitrarily small perturbations can lead to dynamically distinct chaotic attractors this work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

controlling chaos in high dimensions theory and experiment naval surface warfare center carderock division silver spring maryland 20903 received 27 november 1995 the main contribution of this work is the development of a high dimensional chaos control method that iseffective robust against noise and easy to implement in experiment assuming no knowledge of the modelequations the method achieves control by stabilizing a desired unstable periodic orbit with any number ofunstable directions using small time dependent perturbations of a single system parameter speci cally ourmajor results are as follows first we derive explicit control laws for time series produced by discrete maps second we show how to apply this control law to continuous time problems by introducing straightforwardways to extract from a continuous time series a discrete time series that measures the dynamics of somepoincare map of the original system third we illustrate our approach with two examples of high dimensionalordinary differential equations one autonomous and the other periodically driven fourth we present the resulton our successful control of chaos in a high dimensional experimental system demonstrating the viability ofthe method in practical applications pacs number s 05 45 1bi introduction

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

effects of error covariance structure on estimation of model averaging weights and predictive performance 1 when conducting model averaging for assessing groundwater conceptual model uncertainty the averaging weights are often evaluated using model selection criteria such as aic aicc bic and kic akaike information criterion corrected akaike information criterion bayesian information criterion and kashyap information criterion respectively however this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight close to 100 which cannot be justified by available data and knowledge it was found in this study that this problem was caused by using the covariance matrix c of measurement errors for estimating the negative log likelihood function common to all the model selection criteria this problem can be resolved by using the covariance matrix cek of total errors including model errors and measurement errors to account for the correlation between the total errors an iterative two stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown cek from the residuals during model calibration the inferred cek was then used in the evaluation of model selection criteria and model averaging weights while this method was limited to serial data using time series techniques in this study it can be extended to spatial data using geostatistical techniques the method was first evaluated in a synthetic study and then applied to an experimental study in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport it was found that the total errors of the alternative models were temporally correlated due to the model errors the iterative two stage method using cek resolved the problem that the best model receives 100 model averaging weight and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models using cek obtained from the iterative two stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies

assessment of parametric uncertainty for groundwater reactive transport modeling the validity of using gaussian assumptions for model residuals in uncertainty quantification of a groundwater reactive transport model was evaluated in this study least squares regression methods explicitly assume gaussian residuals and the assumption leads to gaussian likelihood functions model parameters and model predictions while the bayesian methods do not explicitly require the gaussian assumption gaussian residuals are widely used this paper shows that the residuals of the reactive transport model are non gaussian heteroscedastic and correlated in time characterizing them requires using a generalized likelihood function such as the formal generalized likelihood function developed by schoups and vrugt 2010 for the surface complexation model considered in this study for simulating uranium reactive transport in groundwater parametric uncertainty is quantified using the least squares regression methods and bayesian methods with both gaussian and formal generalized likelihood functions while the least squares methods and bayesian methods with gaussian likelihood function produce similar gaussian parameter distributions the parameter distributions of bayesian uncertainty quantification using the formal generalized likelihood function are non gaussian in addition predictive performance of formal generalized likelihood function is superior to that of least squares regression and bayesian methods with gaussian likelihood function the bayesian uncertainty quantification is conducted using the differential evolution adaptive metropolis dream zs algorithm as a markov chain monte carlo mcmc method it is a robust tool for quantifying uncertainty in groundwater reactive transport models for the surface complexation model the regression based local sensitivity analysis and morris and dream zs based global sensitivity analysis yield almost identical ranking of parameter importance the uncertainty analysis may help select appropriate likelihood functions improve model calibration and reduce predictive uncertainty in other groundwater reactive transport and environmental modeling

inverse groundwater modeling with emphasis on model parameterization 1 this study develops an inverse method aiming to circumvent the subjective decision regarding model parameterization and complexity in inverse groundwater modeling the number of parameters is included as a decision variable along with parameter values a parameterization based on b spline surfaces bss is selected to approximate transmissivity and genetic algorithms were selected to perform error minimization a transform based on linear least squares lls is developed so that different parameterizations may be combined by standard genetic algorithm operators first three applications with isotropic anisotropic and zoned aquifer parameters are examined in a single objective optimization problem and the estimated transmissivity is found to be near the true one interestingly in the anisotropic case the algorithm converged to a solution with an anisotropic distribution of control points next a single objective optimization with regularization penalizing complex models is considered and last the problem is expressed in a multiobjective optimization framework moo where the goals are simultaneous minimization of calibration error and model complexity the result of moo is a pareto set of potential solutions where the user can examine the tradeoffs between calibration error and model complexity and select the most suitable model by comparing calibration with prediction errors it appears that the most promising models are the ones near a region where the rate of decrease of calibration error as model complexity increases drops bend of error curve this is a useful result of practical interest in real inverse modeling applications

incorporation of prior information on parameters into nonlinear regression groundwater flow models 1 theory prior information on the parameters of a groundwater flow model can be used to improve parameter estimates obtained from nonlinear regression solution of a modeling problem two scales of prior information can be available 1 prior information having known reliability that is bias and random error structure and 2 prior information consisting of best available estimates of unknown reliability a regression method that incorporates the second scale of prior information assumes the prior information to be fixed for any particular analysis to produce improved although biased parameter estimates approximate optimization of two auxiliary parameters of the formulation is used to help minimize the bias which is almost always much smaller than that resulting from standard ridge regression it is shown that if both scales of prior information are available then a combined regression analysis may be made

an adaptive sparse grid high order stochastic collocation method for bayesian inference in groundwater reactive transport modeling 1 bayesian analysis has become vital to uncertainty quantification in groundwater modeling but its application has been hindered by the computational cost associated with numerous model executions required by exploring the posterior probability density function ppdf of model parameters this is particularly the case when the ppdf is estimated using markov chain monte carlo mcmc sampling in this study a new approach is developed to improve the computational efficiency of bayesian inference by constructing a surrogate of the ppdf using an adaptive sparse grid high order stochastic collocation asg hsc method unlike previous works using first order hierarchical basis this paper utilizes a compactly supported higher order hierarchical basis to construct the surrogate system resulting in a significant reduction in the number of required model executions in addition using the hierarchical surplus as an error indicator allows locally adaptive refinement of sparse grids in the parameter space which further improves computational efficiency to efficiently build the surrogate system for the ppdf with multiple significant modes optimization techniques are used to identify the modes for which high probability regions are defined and components of the asg hsc approximation are constructed after the surrogate is determined the ppdf can be evaluated by sampling the surrogate system directly without model execution resulting in improved efficiency of the surrogate based mcmc compared with conventional mcmc the developed method is evaluated using two synthetic groundwater reactive transport models the first example involves coupled linear reactions and demonstrates the accuracy of our high order hierarchical basis approach in approximating high dimensional posteriori distribution the second example is highly nonlinear because of the reactions of uranium surface complexation and demonstrates how the iterative asg hsc method is able to capture multimodal and non gaussian features of ppdf caused by model nonlinearity both experiments show that asg hsc is an effective and efficient tool for bayesian inference

a statistical concept to assess the uncertainty in bayesian model weights and its impact on model ranking bayesian model averaging bma ranks the plausibility of alternative conceptual models according to bayes theorem a prior belief about each model s adequacy is updated to a posterior model probability based on the skill to reproduce observed data and on the principle of parsimony the posterior model probabilities are then used as model weights for model ranking selection or averaging despite the statistically rigorous bma procedure model weights can become uncertain quantities due to measurement noise in the calibration data set or due to uncertainty in model input uncertain weights may in turn compromise the reliability of bma results we present a new statistical concept to investigate this weighting uncertainty and thus to assess the significance of model weights and the confidence in model ranking our concept is to resample the uncertain input or output data and then to analyze the induced variability in model weights in the special case of weighting uncertainty due to measurement noise in the calibration data set we interpret statistics of bayesian model evidence to assess the distance of a model s performance from the theoretical upper limit to illustrate our suggested approach we investigate the reliability of soil plant model selection following up on a study by wohling et al 2015 results show that the bma routine should be equipped with our suggested upgrade to 1 reveal the significant but otherwise undetected impact of measurement noise on model ranking results and 2 to decide whether the considered set of models should be extended with better performing alternatives

phase space reconstruction for symmetric dynamical systems abstract since the pioneering work of packard et al and takens it has become customary to reconstruct the topology of attractors in phase space from a time series of one dimensional experimental observations by using delay coordinates many practical refinements of the original methods have been developed many experimental systems possess symmetry and bifurcations can cause changes in the symmetry of observed states these changes are quite subtle when the dynamics is chaotic it is therefore important to reconstruct not just the topology of the attractor but its symmetry we indicate how this can be done by extending the packard takens approach to a single equivariant observation taking values not in the real numbers r but in a linear representation v of the symmetry group g in effect a single set of symmetrically related observations is required our central point is that not all plausible choices for such a set can generate embeddings in order for the method to produce an embedding it is necessary that v should be sufficiently complicated more precisely the phase space m must be subordinate to v in a sense introduced by wassermann this concept is technical but unavoidable in this context and it greatly clarifies the issue of embeddability using it we state a symmetric version of the takens embedding theorem and sketch the proof we also discuss the issue of setwise versus pointwise symmetry of an attractor and relate this to the transition from spatial order to spatial disorder in temporally chaotic systems

symmetry and control spatially extended chaotic systems in a recent paper phys rev e 57 1998 1550 it was demonstrated that the symmetries of the evolution equation and the target state have a profound effect on controlling the chaotic behavior in the present paper we extend these results to the cases of time periodic target trajectories and inexact symmetries and apply the developed formalism to the problem of controlling spatiotemporal chaos we use the example of a lattice dynamical system in arbitrary spatial dimension to show that there exists an intimate relationship between the geometry of an extended system and the geometry of feedback control 2000 elsevier science b v all rights reserved

ubiquity of and geostatistics for nonstationary increment random fields 1 nonstationary random fields such as fractional brownian motion and fractional levy motion have been studied extensively in the hydrology literature on the other hand random fields that have nonstationary increments have seen little study a mathematical argument is presented that demonstrates processes with stationary increments are the exception and processes with nonstationary increments are far more abundant the abundance of nonstationary increment processes has important implications e g in kriging where a translation invariant variogram implicitly assumes stationarity of the increments an approach to kriging for processes with nonstationary increments is presented and accompanied by some numerical results

controlling chaos of a dynamical system with discontinuous vector field abstract the paper presents how to control the chaos of dynamical systems with discontinuous vector field through the paradigm of a harmonically forced oscillator having a set up elastic stop it is first shown that the poincare mapping of this oscillator is not smooth near the fixed point corresponding to a periodic motion that grazes the stop thus the current control strategies based on the smooth mapping cannot be directly used to stabilize the chaotic motion near a periodic grazing motion then proposed is a piecewise linear control strategy based on the piecewise linearized poincare mapping reconstructed from sampled data and on the pole assignment in two regions near the fixed point the efficacy of the strategy is finally demonstrated via numerical simulations

on the symmetry breaking bifurcation of chaotic attractors a new type of crisis is shown to exist in a broad class of systems including the lorenz model which leads to an anomalous band splitting or to a symmetry breaking bifurcation of the strange attractor depending on the actual values of the control parameters a piecewise linear model is used to understand the mechanism of this crisis and to obtain exact results

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

optimization of local control of chaos by an evolutionary algorithm an evolutionary algorithm for optimizing local control of chaos is presented based on a lyapunov approach a linear control law and the state space region in which this control law is activated are determined in addition we study a relation between certain adjustable design parameters and a particular measure of the uncontrolled chaotic attractor in the state space region of control ssrc from this relation the objective function to be optimized is derived in that context we assume a linear control law to be given and optimize size and shape of the ssrc using an evolutionary algorithm it is shown by examples how the algorithm can also be applied to higher dimensional systems with possibly more than one positive lyapunov exponent 2000 elsevier science b v all rights reserved

detection of symmetry of attractors from observations i theory abstract barany et al 1993 proposed a method for determining the symmetries of attractors of equivariant systems by averaging certain classes of equivariant maps we use an idea in barany et al to re cast definitions of symmetry detectives assuming that we only have access to equivariant observations from the system detecting from observations allows one to perform averaging in spaces that may have much lower dimension than the phase space this paper generalises and develops their suggestion among the generalisations we consider is the use of nonpolynomial detectives and we show using the notion of prevalence of hunt et al 1992 that our detectives and the detectives of barany et al 1993 dellnitz et al 1994 and golubitsky and nicol 1995 give the correct symmetry of attractors almost certainly in a measure theoretic sense we show that detectives can persistently give incorrect symmetries at isolated points in parametrised systems and discuss how to overcome this we show how one can find the symmetry of an attractor from examination of a poincare section in part ii of this article ashwin and tomes apply these results to find symmetries of attractors in a physical system of four coupled electronic oscillators with s4 symmetry

adaptive control in nonlinear dynamics abstract we extend an adaptive control algorithm recently suggested by huberman and lumer to multi parameter and higher dimensional nonlinear systems this control mechanism is remarkably effective in returning a system to its original dynamics after a sudden perturbation in the system parameters changes the dynamical behaviour we find that in all cases the recovery time is linearly proportional to the inverse of control stiffness for small stiffness in higher dimensions there is an additional optimization problem since increasing stiffness beyond a certain value can retard recovery the control of fixed point dynamics in systems capable of a wide variety of dynamical behaviour is demonstrated we further suggest methods by which periodic motion such as limit cycles can be adaptively controlled and demonstrate the robustness of the procedure in the presence of additive background noise

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

maximum likelihood bayesian averaging of spatial variability models in unsaturated fractured tuff 1 hydrologic analyses typically rely on a single conceptual mathematical model yet hydrologic environments are open and complex rendering them prone to multiple interpretations and mathematical descriptions adopting only one of these may lead to statistical bias and underestimation of uncertainty bayesian model averaging bma hoeting et al 1999 provides an optimal way to combine the predictions of several competing models and to assess their joint predictive uncertainty however it tends to be computationally demanding and relies heavily on prior information about model parameters neuman 2002 2003 proposed a maximum likelihood version mlbma of bma to render it computationally feasible and to allow dealing with cases where reliable prior information is lacking we apply mlbma to seven alternative variogram models of log air permeability data from single hole pneumatic injection tests in six boreholes at the apache leap research site alrs in central arizona unbiased ml estimates of variogram and drift parameters are obtained using adjoint state maximum likelihood cross validation samper and neuman 1989a in conjunction with universal kriging and generalized least squares standard information criteria provide an ambiguous ranking of the models which does not justify selecting one of them and discarding all others as is commonly done in practice instead we eliminate some of the models based on their negligibly small posterior probabilities and use the rest to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes we then average these four projections and associated kriging variances using the posterior probability of each model as weight finally we cross validate the results by eliminating from consideration all data from one borehole at a time repeating the above process and comparing the predictive capability of mlbma with that of each individual model we find that mlbma is superior to any individual geostatistical model of log permeability among those we consider at the alrs

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

model selection on solid ground rigorous comparison of nine ways to evaluate bayesian model evidence bayesian model selection or averaging objectively ranks a number of plausible competing conceptual models based on bayes theorem it implicitly performs an optimal trade off between performance in fitting available data and minimum model complexity the procedure requires determining bayesian model evidence bme which is the likelihood of the observed data integrated over each model s parameter space the computation of this integral is highly challenging because it is as high dimensional as the number of model parameters three classes of techniques to compute bme are available each with its own challenges and limitations 1 exact and fast analytical solutions are limited by strong assumptions 2 numerical evaluation quickly becomes unfeasible for expensive models 3 approximations known as information criteria ics such as the aic bic or kic akaike bayesian or kashyap information criterion respectively yield contradicting results with regard to model ranking our study features a theory based intercomparison of these techniques we further assess their accuracy in a simplistic synthetic example where for some scenarios an exact analytical solution exists in more challenging scenarios we use a brute force monte carlo integration method as reference we continue this analysis with a real world application of hydrological model selection this is a first time benchmarking of the various methods for bme evaluation against true solutions results show that bme values from ics are often heavily biased and that the choice of approximation method substantially influences the accuracy of model ranking for reliable model selection bias free numerical methods should be preferred over ics whenever computationally feasible

sensitivity analysis and assessment of prior model probabilities in mlbma with application to unsaturated fractured tuff 1 previous application of maximum likelihood bayesian model averaging mlbma neuman 2002 2003 to alternative variogram models of log air permeability data in fractured tuff has demonstrated its effectiveness in quantifying conceptual model uncertainty and enhancing predictive capability ye et al 2004 a question remained how best to ascribe prior probabilities to competing models in this paper we examine the extent to which lead statistics of posterior log permeability predictions are sensitive to prior probabilities of seven corresponding variogram models we then explore the feasibility of quantifying prior model probabilities by 1 maximizing shannon s entropy h shannon 1948 subject to constraints reflecting a single analyst s or a group of analysts prior perception about how plausible each alternative model or a group of models is relative to others and 2 selecting a posteriori the most likely among such maxima corresponding to alternative prior perceptions of various analysts or groups of analysts another way to select among alternative prior model probability sets which however is not guaranteed to yield optimum predictive performance though it did so in our example and would therefore not be our preferred option is a minimum maximum approach according to which one selects a priori the set corresponding to the smallest value of maximum entropy whereas maximizing h subject to the prior perception of a single analyst or group maximizes the potential for further information gain through conditioning selecting the smallest among such maxima gives preference to the most informed prior perception among those of several analysts or groups we use the same variogram models and log permeability data as ye et al 2004 to demonstrate that our proposed approach yields the least amount of posterior entropy residual uncertainty after conditioning and enhances predictive model performance as compared to 1 the noninformative neutral case in which all prior model probabilities are set equal to each other and 2 an informed case that nevertheless violates the principle of parsimony

hydrologic consistency as a basis for assessing complexity of monthly water balance models for the continental united states methods to select parsimonious and hydrologically consistent model structures are useful for evaluating dominance of hydrologic processes and representativeness of data while information criteria appropriately constrained to obey underlying statistical assumptions can provide a basis for evaluating appropriate model complexity it is not sufficient to rely upon the principle of maximum likelihood ml alone we suggest that one must also call upon a principle of hydrologic consistency meaning that selected ml structures and parameter estimates must be constrained as well as possible to reproduce desired hydrological characteristics of the processes under investigation this argument is demonstrated in the context of evaluating the suitability of candidate model structures for lumped water balance modeling across the continental united states using data from 307 snow free catchments the models are constrained to satisfy several tests of hydrologic consistency a flow space transformation is used to ensure better consistency with underlying statistical assumptions and information criteria are used to evaluate model complexity relative to the data the results clearly demonstrate that the principle of consistency provides a sensible basis for guiding selection of model structures and indicate strong spatial persistence of certain model structures across the continental united states further work to untangle reasons for model structure predominance can help to relate conceptual model structures to physical characteristics of the catchments facilitating the task of prediction in ungaged basins

combining multiobjective optimization and bayesian model averaging to calibrate forecast ensembles of soil hydraulic models 1 most studies in vadose zone hydrology use a single conceptual model for predictive inference and analysis focusing on the outcome of a single model is prone to statistical bias and underestimation of uncertainty in this study we combine multiobjective optimization and bayesian model averaging bma to generate forecast ensembles of soil hydraulic models to illustrate our method we use observed tensiometric pressure head data at three different depths in a layered vadose zone of volcanic origin in new zealand a set of seven different soil hydraulic models is calibrated using a multiobjective formulation with three different objective functions that each measure the mismatch between observed and predicted soil water pressure head at one specific depth the pareto solution space corresponding to these three objectives is estimated with amalgam and used to generate four different model ensembles these ensembles are postprocessed with bma and used for predictive analysis and uncertainty estimation our most important conclusions for the vadose zone under consideration are 1 the mean bma forecast exhibits similar predictive capabilities as the best individual performing soil hydraulic model 2 the size of the bma uncertainty ranges increase with increasing depth and dryness in the soil profile 3 the best performing ensemble corresponds to the compromise or balanced solution of the three objective pareto surface and 4 the combined multiobjective optimization and bma framework proposed in this paper is very useful to generate forecast ensembles of soil hydraulic models

phase portrait reconstruction using multivariate singular systems analysis abstract the singular systems analysis ssa method of broomhead and king is applied to the reconstruction of phase portraits from experimental time series of temperature measurements taken simultaneously at up to 16 probes within a rotating thermal convection experiment and used to characterize a number of different kinds of flow including the determination of their correlation dimension results are compared with equivalent reconstructions using more conventional method of delays embedding and ssa applied in the time domain only to data obtained from a single thermocouple probe multivariate ssa m ssa is shown to offer significantly improved reconstructions in terms of signal noise ratio and structural uniformity of the attractor over earlier methods this is due mainly to the ability of m ssa to combine simultaneously information sampling the complete range of the spatial cross correlation function in signals with complex spatio temporal structure

parameter uncertainties in models of equivariant dynamical systems abstract we examine the advantages of using equivariant versus nonequivariant phenomenological models for chaotic systems that possess an inversion symmetry numerical experiments used the minimum description length mdl criteria for model selection they indicated that the selection between equivariant versus nonequivariant models should be made on an a priori basis we also examine the relationship between the optimal truncation accuracy of the parameters of the model as predicted by the mdl criteria and the true uncertainty in the parameters of the model numerical experiments indicate that the true uncertainties are poorly approximated by the optimal truncation accuracy they also indicate that the true uncertainties may be useful for determining the presence of an inversion symmetry in reconstructed attractors

a bayesian approach for inverse modeling data assimilation and conditional simulation of spatial random fields 1 this paper addresses the inverse problem in spatially variable fields such as hydraulic conductivity in groundwater aquifers or rainfall intensity in hydrology common to all these problems is the existence of a complex pattern of spatial variability of the target variables and observations the multiple sources of data available for characterizing the fields the complex relations between the observed and target variables and the multiple scales and frequencies of the observations the method of anchored distributions mad that we propose here is a general bayesian method of inverse modeling of spatial random fields that addresses this complexity the central elements of mad are a modular classification of all relevant data and a new concept called anchors data types are classified by the way they relate to the target variable as either local or nonlocal and as either direct or indirect anchors are devices for localization of data they are used to convert nonlocal indirect data into local distributions of the target variables the target of the inversion is the derivation of the joint distribution of the anchors and structural parameters conditional to all measurements regardless of scale or frequency of measurement the structural parameters describe large scale trends of the target variable fields whereas the anchors capture local inhomogeneities following inversion the joint distribution of anchors and structural parameters is used for generating random fields of the target variable s that are conditioned on the nonlocal indirect data through their anchor representation we demonstrate mad through a detailed case study that assimilates point measurements of the conductivity with head measurements from natural gradient flow the resulting statistical distributions of the parameters are non gaussian similarly the moments of the estimates of the hydraulic head are non gaussian we provide an extended discussion of mad vis a vis other inversion methods including maximum likelihood and maximum a posteriori with an emphasis on the differences between mad and the pilot points method

inverse stochastic moment analysis of steady state flow in randomly heterogeneous media 1 nonlocal stochastic moment equations have been used successfully to analyze steady state and transient flow in randomly heterogeneous media conditional on measured values of medium properties we present a nonlinear geostatistical inverse algorithm for steady state flow that makes it possible to further condition such analyses on measured values of state variables our approach accounts for all scales of spatial variability resolvable by the computational grid it is based on recursive finite element approximations of exact nonlocal first and second conditional moment equations hydraulic conductivity is parameterized geostatistically on the basis of measured values at discrete locations and unknown values at discrete pilot points prior estimates of pilot point values are obtained optionally by generalized kriging posterior estimates at pilot points and optionally at measurement points are obtained by calibrating mean flow against measured values of head the parameters are projected onto a computational grid via kriging maximum likelihood calibration allows estimating not only hydraulic but also optionally unknown variogram parameters with or without prior information about the former the approach yields covariance matrices for parameter estimation as well as head and flux prediction errors the latter being obtained a posteriori we implement our inverse approach on highly and mildly heterogeneous media under superimposed mean uniform and convergent flows in a bounded two dimensional domain our examples illustrate how conductivity and head data act separately and jointly to reduce parameter estimation errors and to model predictive uncertainty we also evaluate the functional form of the log conductivity variogram and its parameters using likelihood based model discrimination criteria

evaluating marginal likelihood with thermodynamic integration method and comparison with several other numerical methods evaluating marginal likelihood is the most critical and computationally expensive task when conducting bayesian model averaging to quantify parametric and model uncertainties the evaluation is commonly done by using laplace approximations to evaluate semianalytical expressions of the marginal likelihood or by using monte carlo mc methods to evaluate arithmetic or harmonic mean of a joint likelihood function this study introduces a new mc method i e thermodynamic integration which has not been attempted in environmental modeling instead of using samples only from prior parameter space as in arithmetic mean evaluation or posterior parameter space as in harmonic mean evaluation the thermodynamic integration method uses samples generated gradually from the prior to posterior parameter space this is done through a path sampling that conducts markov chain monte carlo simulation with different power coefficient values applied to the joint likelihood function the thermodynamic integration method is evaluated using three analytical functions by comparing the method with two variants of the laplace approximation method and three mc methods including the nested sampling method that is recently introduced into environmental modeling the thermodynamic integration method outperforms the other methods in terms of their accuracy convergence and consistency the thermodynamic integration method is also applied to a synthetic case of groundwater modeling with four alternative models the application shows that model probabilities obtained using the thermodynamic integration method improves predictive performance of bayesian model averaging the thermodynamic integration method is mathematically rigorous and its mc implementation is computationally general for a wide range of environmental problems

on the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3 manifolds in this note we study constant mean curvature surfaces in asymptotically flat 3 manifolds we prove that outside a given compact subset in an asymptotically flat 3 manifold with positive mass stable spheres of given constant mean curvature are unique therefore we are able to conclude that the foliation of stable spheres of constant mean curvature in an asymptotically flat 3 manifold with positive mass outside a given compact subset is unique

low dimensional dynamics for the complex ginzburg landau equation abstract a method is presented which results in low dimensional dynamical systems for situations in which low dimensional attractors are known to exist the method is based on the use of the karhunen loeve procedure for the determination of an optimal basis and the subsequent use of the galerkin procedure to generate the dynamical system the method is applied to two problems for the ginzburg landau equation for which large databases have been obtained in each instance a dynamical system is generated which has roughly twice the number of degrees of freedom as the hausdorff dimension of the exact case it is also demonstrated that the approximations are robust in that they are accurate over a wide range of parameter space

generalised information criteria in model selection summary the problem of evaluating the goodness of statistical models is investigated from an information theoretic point of view information criteria are proposed for evaluating models constructed by various estimation procedures when the specified family of probability distributions does not contain the distribution generating the data the proposed criteria are applied to the evaluation of models estimated by maximum likelihood robust penalised likelihood bayes procedures etc we also discuss the use of the bootstrap in model evaluation problems and present a variance reduction technique in the bootstrap simulation

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

on hydrological model complexity its geometrical interpretations and prediction uncertainty knowledge of hydrological model complexity can aid selection of an optimal prediction model out of a set of available models optimal model selection is formalized as selection of the least complex model out of a subset of models that have lower empirical risk this may be considered equivalent to minimizing an upper bound on prediction error defined here as the mathematical expectation of empirical risk in this paper we derive an upper bound that is free from assumptions on data and underlying process distribution as well as on independence of model predictions over time we demonstrate that hydrological model complexity as defined in the presented theoretical framework plays an important role in determining the upper bound the model complexity also acts as a stabilizer to a hydrological model selection problem if it is deemed ill posed we provide an algorithm for computing complexity of any arbitrary hydrological model we also demonstrate that hydrological model complexity has a geometric interpretation as the size of model output space the presented theory is applied to quantify complexities of two hydrological model structures sac sma and sixpar it detects that sac sma is indeed more complex than sixpar we also develop an algorithm to estimate the upper bound on prediction error which is applied on five different rainfall runoff model structures that vary in complexity we show that a model selection problem is stabilized by regularizing it with model complexity complexity regularized model selection yields models that are robust in predicting future but yet unseen data

statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability severe obstruction to shadowing of computer generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability that is when the dimension of the unstable eigenspace changes along a trajectory in the invariant set no true trajectory of reasonable length can be found to exist near any numerically generated trajectory an important quantity characterizing the shadowability of numerical trajectories is the shadowing time which measures for how long a trajectory remains valid this time depends sensitively on initial condition here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors an algebraic scaling for short times and an exponential scaling for long times the exponential behavior depends on system details but the small time algebraic behavior appears to be universal we describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

inverse modeling of dynamic nonequilibrium in water flow with an effective approach 1 observations of water flow in unsaturated soils often show dynamic effects indicated by nonequilibrium between water contents and water potential a phenomenon that cannot be modeled with the richards equation the objective of this article is to formulate an effective process description of dynamic nonequilibrium flow in variably saturated soil which is both flexible enough to match experimental observations and as parsimonious as possible to allow unique parameter estimation by inverse modeling in the conceptual model water content is partitioned into two fractions water in one fraction is in equilibrium with the pressure head whereas water in the second fraction is in nonequilibrium described by the kinetic equilibration approach of ross and smettem 2000 between the two fractions an instantaneous equilibration of the pressure head is assumed the new model termed the dual fraction nonequilibrium model requires only one additional parameter compared to the nonequilibrium approach of ross and smettem we tested the model with experimental data from multistep outflow experiments conducted on two soils and compared it to the richards equation the nonequilibrium model of ross and smettem and the dual porosity model of philip 1968 the experimental data were evaluated by inverse modeling using a robust markov chain monte carlo sampler the results show that the proposed model is superior to the richards equation and the ross and smettem model in describing dynamic nonequilibrium effects occurring in multistep outflow experiments the three popular model selection criteria akaike information criterion bayesian information criterion and deviance information criterion all favored the new model because of its smaller number of parameters

quantification of uncertainty in pedotransfer function based parameter estimation for unsaturated flow modeling 1 while pedotransfer functions ptfs have long been applied to estimate soil hydraulic parameters for unsaturated flow and solute transport modeling the uncertainty associated with the estimates is often ignored the objective of this study is to evaluate uncertainty of the ptf estimated soil hydraulic parameters and its effect on numerical simulation of moisture flow contributing to the parameter estimation uncertainty are 1 the ptf intrinsic uncertainty caused by limited data used for ptf training and 2 the ptf input uncertainty in pedotransfer variables i e ptf inputs the ptf intrinsic uncertainty is assessed using the bootstrap method by generating multiple bootstrap realizations of the soil hydraulic parameters the realizations follow normal or lognormal distributions the ptf input variables i e bulk density and soil texture are obtained using the cokriging technique the ptf input uncertainty is quantified by assuming that the cokriging estimates follow a normal distribution our results show that the ptf input uncertainty dominates over the ptf intrinsic uncertainty and determines the spatial distribution of the ptf parameter estimation uncertainty when the parameter estimation uncertainty is included the spatial variability of the measured soil hydraulic parameters is better captured this is also the case for the observed moisture contents whose spatial variability is well bracketed by the prediction intervals however this is only possible after the ptf input uncertainty is considered these results suggest that additional sample acquisition for the ptf input variables would have a more favorable impact on reduction of the parameter estimation uncertainty than collecting additional soil hydraulic parameter measurements for ptf development

application of stochastic control techniques to chaotic nonlinear systems a control algorithm based on stochastic control techniques is devised for chaotic nonlinear systems the algorithm uses a state estimator based on the kalman filter and yields performance improvements in at least some regions of state space with respect to that obtainable by use of a controller utilizing only the conditional mean of the system state vector the method is applied to two typical chaotic nonlinear systems the henon heiles system and the lorenz system and their behavior with control is explored numerically

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

route to high dimensional chaos we present a route to high dimensional chaos that is chaos with more than one positive lyapunov exponent in this route as a system parameter changes a subsystem becomes chaotic through say a cascade of period doubling bifurcations after which the complementary subsystem becomes chaotic leading to an additional positive lyapunov exponent for the whole system a characteristic feature of this route as suggested by numerical evidence is that the second largest lyapunov exponent passes through zero continuously three examples are presented a discrete time map a continuous time flow and a population model for species dispersal in evolutionary ecology

role of model selection criteria in geostatistical inverse estimation of statistical data and model parameters 1 we analyze theoretically the ability of model quality sometimes termed information or discrimination criteria such as the negative log likelihood nll bayesian criteria bic and kic and information theoretic criteria aic aicc and hic to estimate 1 the parameter vector of the variogram of hydraulic log conductivity y ln k and 2 statistical parameters and proportional to head and log conductivity measurement error variances respectively in the context of geostatistical groundwater flow inversion our analysis extends the work of hernandez et al 2003 2006 and riva et al 2009 who developed nonlinear stochastic inverse algorithms that allow conditioning estimates of steady state and transient hydraulic heads fluxes and their associated uncertainty on information about conductivity and head data collected in a randomly heterogeneous confined aquifer their algorithms are based on recursive numerical approximations of exact nonlocal conditional equations describing the mean and co variance of groundwater flow log conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete pilot points optionally the maximum likelihood function on which the inverse estimation of y at pilot points is based may include a regularization term reflecting prior information about y the relative weight assigned to this term and its components and as well as are evaluated separately from other model parameters to avoid bias and instability this evaluation is done on the basis of criteria such as nll kic bic hic aic and aicc we demonstrate theoretically that whereas all these six criteria make it possible to estimate kic alone allows one to estimate validly and and thus we illustrate this discriminatory power of kic numerically by using a differential evolution genetic search algorithm to minimize it in the context of a two dimensional steady state groundwater flow problem we find that whereas and the integral scale of y can be estimated on the basis of a zero order mean flow equation the sill of the y variogram is estimated more accurately by a second order approximation of flow this notwithstanding kic prefers the simpler zero order moment over the more complex second order version

incorporation of prior information on parameters into nonlinear regression groundwater flow models 2 applications this paper investigates factors influencing the degree of improvement in estimates of parameters of a nonlinear regression groundwater flow model by incorporating prior information of unknown reliability consideration of expected behavior of the regression solutions and results of a hypothetical modeling problem lead to several general conclusions first if the parameters are properly scaled linearized expressions for the mean square error mse in parameter estimates of a nonlinear model will often behave very nearly as if the model were linear second by using prior information the mse in properly scaled parameters can be reduced greatly over the mse of ordinary least squares estimates of parameters third plots of estimated mse and the estimated standard deviation of mse versus an auxiliary parameter the ridge parameter specifying the degree of influence of the prior information on regression results can help determine the potential for improvement of parameter estimates fourth proposed criteria can be used to make appropriate choices for the ridge parameter and another parameter expressing degree of overall bias in the prior information results of a case study of truckee meadows reno sparks area washoe county nevada conform closely to the results of the hypothetical problem in the truckee meadows case incorporation of prior information did not greatly change the parameter estimates from those obtained by ordinary least squares however the analysis showed that both sets of estimates are more reliable than suggested by the standard errors from ordinary least squares

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

large outlying stable constant mean curvature spheres in initial data sets we give examples of asymptotically flat three manifolds m g which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold this resolves a question raised by huisken and yau invent math 124 281 311 1996 on the other hand we show that such surfaces cannot exist when m g has nonnegative scalar curvature this result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces

controlling chaos in systems of coupled oscillators abstract we study the dynamical behavior of a one dimensional network of phase oscillators displaying a variety of cluster or traveling wave solutions either stable or unstable according to the type of interaction function for some range of parameters these solutions coexist with a chaotic attractor we analyze a mechanism for controlling chaos and stabilizing the periodic solutions we also analyze the minimal number of controllers needed to stabilize the system and the relation between the location of the controllers and the symmetry of the reference state to be stabilized

structure selection techniques applied to continous time nonlinear models this paper addresses the problem of choosing the multinomials that should compose a polynomial mathematical model starting from data the mathematical representation used is a nonlinear differential equation of the polynomial type some approaches that have been used in the context of discrete time models are adapted and applied to continuous time models two examples are included to illustrate the main ideas models obtained with and without structure selection are compared using topological analysis the main differences between structure selected models and complete structure models are i the former are more parsimonious than the latter ii a predefined fixed point configuration can be guaranteed for the former and iii the former set of models produce attractors that are topologically closer to the original attractor than those produced by the complete structure models 2001 elsevier science b v all rights reserved

on large volume preserving stable cmc surfaces in initial data sets let m g be a complete 3 dimensional asymptotically flat manifold with everywhere positive scalar curvature we prove that given a compact subset k subset m all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid k this complements the results of g huisken and s t yau and of j qing and g tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to schwarzschild with mass m gt 0 the analysis in g huisken and s t yau and in j qing and g tian takes place in the asymptotic regime of m here we adapt ideas from the minimal surface proof of the positive mass theorem by r schoen and s t yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of m that is far from euclidean

highly parameterized inverse estimation of hydraulic conductivity and porosity in a three dimensional heterogeneous transport experiment 1 assessing the impact of parameter estimation accuracy in models of heterogeneous three dimensional 3 d groundwater systems is critical for predictions of solute transport a unique experimental data set provides concentration breakthrough curves btcs measured at a 0 253 cm3 scale over the 13 8 8 cm3 domain 53 000 measurement locations advective transport is used to match the first temporal moments of btcs or mean arrival times m1 averaged at 0 253 and 1 0 cm3 scales through simultaneous inversion of highly parameterized heterogeneous hydraulic conductivity k and porosity fields pilot points parameterize the fields within eight layers of the 3 d medium and estimations are completed with six different models of the k relationship parameter estimation through advective transport shows accurate estimation of the observed m1 values results across the six different k relationships have statistically similar fits to the observed m1 values and similar spatial estimates of m1 along the main flow direction the resulting fields provide the basis for forward transport modeling of the advection dispersion equation ade using the estimated k and fields demonstrates that advective transport coupled with inversion using dense spatial field parameterization provides an efficient surrogate for the ade these results indicate that there is not a single set of model parameters or a single k relationship that leads to a best representation of the actual experimental sand packing pattern i e nonuniqueness additionally knowledge of the individual sand k and values along with their arrangement in the 3 d experiment does not reproduce the observed transport results at small scales small scale variation in the packing and mixing of the sands causes large deviations from the expected transport results as highlighted in forward ade simulations highly parameterized inverse estimation is able to identify those regions where variations in mixing and packing alter the expected property values and significantly improve results relative to the naive application of the experimentally derived property values impacts of the observation scale the scale over which results are averaged and the number of observations and parameters on the final estimations are also examined results indicate existence of a representative element volume rev at 0 25 cm3 the existence of subgrid scale heterogeneity that impacts transport and the accuracy of highly parameterized models with even relatively small amounts of observations finally this work suggests that local heterogeneity features below the rev scale are difficult to incorporate into parameterized models highlighting the importance of addressing prediction uncertainty for small scale variability i e uncaptured variability in modeling practice