on a random walk with memory and its relation with markovian processes we study a one dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker the feedback is quite efficient and leads to a non diffusive walk the time evolution of the displacement is given by an equivalent markovian dynamical process the probability density for the position of the walker is the same at any time as for a random walk with shrinking steps although the two time correlation functions are quite different

random recursive construction of self similar fractal measures the noncompact case the self similarity of sets measures is often defined in a constructive way in the present paper it will be shown that the random recursive construction model of falconer graf and mauldin williams for statistically self similar sets may be generalized to the noncompact case we define a sequence of random finite measures which converges almost surely to a self similar random limit measure under certain conditions on the generating lipschitz maps we determine the carrying dimension of the limit measure

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

percolation of arbitrary words in 0 1 mathbb n let g be a possibly directed locally finite graph with countably infinite vertex set v let x v be an i i d family of random variables with p x 1 1 p x 0 p finally let 1 2 be a generic element of 0 1 n such a is called a word we say that the word is seen from the vertex if there exists a self avoiding path 1 2 on g starting at and such that x i i for i 1 the traditional problem in site percolation is whether p 1 1 1 is seen from 0 so called ab percolation occurs if p 1 0 1 0 1 0 is seen from 0 here we investigate a whether p all words are seen from 0 for a fixed and b whether p all words are seen from some 1 we show that both answers are positive if g z d or even z d with all edges oriented in the positive direction when d is sufficiently large we show that on the oriented z 3 the answer to a is negative but we do not know the answer to b on z 3 various graphs g are constructed almost all of them trees for which the set of words which can be seen from a given or from some is large even though it is w p 1 not the set of all words

synchronizing sample curves nonparametrically more and more often the outcome of a study is not a random variable but a noisy function for each experimental unit resulting in a sample of curves typically the individual curves vary not only in amplitude or intensity but also with respect to the time axis different subjects experience certain events sooner or later analyzing such data involves finding out the time changes or curve registration among curves following our previous work where modified dynamic time warping is applied to align two curves we formulate a global minimization problem to align all curves in a sample and to compute the aligned average curve algorithms for solving the minimization problem are presented and tested with simulated and real data the test results are promising the method which involves kernel smoothing of regression functions estimates the time changes and the average of the aligned curves from noisy data large sample asymptotics is derived

variable bandwidth kernel estimators of regression curves on montre qu en termes de l erreur en moyenne quadratique integree asymptotique les estimateurs du noyau avec un certain choix de largeur de bande local sont superieurs aux estimateurs ordinaires du noyau avec un choix de largeur de bande global si on utilise des largeurs de bandes optimales

linearized stability of partial differential equations with application to stabilization of the kuramoto sivashinsky equation linearization is a useful tool for analyzing the stability of nonlinear differential equations unfortunately the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations general results giving conditions for when stability or instability of the linearized equation implies the same for the nonlinear equation are given here these results are applied to stability and stabilization of the kuramoto sivashinsky equation a nonlinear partial differential equation that models reaction diffusion systems the stability of the equilibrium solutions depends on the value of a positive parameter nu it is shown that if nu 1 then the set of constant equilibrium solutions is globally asymptotically stable if nu 1 then the equilibria are unstable it is also shown that stabilizing the linearized equation implies local exponential stability of the equation stabilization of the kuramoto sivashinsky equation using a sin

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

tests for separability in nonparametric covariance operators of random surfaces we consider the problem of testing for separability in nonparametric covariance operators of multidimensional functional data is considered we cast the problem in a tensor product of hilbert space framework where the role of the partial trace operator is emphasized and the tests proposed are computationally tractable an applications to acoustic phonetic data is also presented

dynamic retrospective regression for functional data samples of curves or functional data usually present phase variability in addition to amplitude variability existing functional regression methods do not handle phase variability in an efficient way in this article we propose a functional regression method that incorporates phase synchronization as an intrinsic part of the model and then attains better predictive power than ordinary linear regression in a simple and parsimonious way the finite sample properties of the estimators are studied by simulation as an example of application we analyze neuromotor data arising from a study of human lip movement this article has supplementary materials online

self similar random measures a set is called self similar if it is decomposable into parts which are similar to the whole this notion was generalized to random sets in the present paper an alternative axiomatic approach is given which makes precise the following idea using palm distribution theory a random set is statistically self similar if it is statistically scale invariant with respect to any center chosen at random from that set for these sets hausdorff dimension coincides with an intrinsic self similarity index

nonparametric maximum likelihood estimation of the structural mean of a sample of curves a random sample of curves can be usually thought of as noisy realisations of a compound stochastic process x t z w t where z t produces random amplitude variation and w t produces random dynamic or phase variation in most applications it is more important to estimate the so called structural mean t e z t than the crosssectional mean e x t but this estimation problem is difficult because the process z t is not directly observable in this paper we propose a nonparametric maximum likelihood estimator of t this estimator is shown to be surd n consistent and asymptotically normal under the assumed model and robust to model misspecification simulations and a realdata example show that the proposed estimator is competitive with landmark registration often considered the benchmark and has the advantage of avoiding time consuming and often infeasible individual landmark identification

functional convex averaging and synchronization for time warped random curves data that can be best described as a sample of curves are now fairly common in science and engineering when the dynamics of development growth or response over time are at issue subjects or experimental units may experience events at different temporal paces for functional data where trajectories may be individually time transformed it is usually inadequate to use commonly used sample statistics such as the cross sectional mean or median or the cross sectional sample variance if one observes time warped curve data i e random curves or random trajectories that exhibit random transformations of the time scale then the usual l2 norm and metric typically are inadequate one may then consider subjecting each observed curve to a time transformation in an attempt to reverse the warping of the time scale before further statistical analysis dynamic time warping alignment curve registration and landmark based methods have been put forward with the goal of finding adequate empirical time transforma

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

a geometric approach to visualization of variability in functional data abstractwe propose a new method for the construction and visualization of boxplot type displays for functional data we use a recent functional data analysis framework based on a representation of functions called square root slope functions to decompose observed variation in functional data into three main components amplitude phase and vertical translation we then construct separate displays for each component using the geometry and metric of each representation space based on a novel definition of the median the two quartiles and extreme observations the outlyingness of functional data is a very complex concept thus we propose to identify outliers based on any of the three main components after decomposition we provide a variety of visualization tools for the proposed boxplot type displays including surface plots we evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface tempe

unifying amplitude and phase analysis a compositional data approach to functional multivariate mixed effects modeling of mandarin chinese mandarin chinese is characterized by being a tonal language the pitch or f0 of its utterances carries considerable linguistic information however speech samples from different individuals are subject to changes in amplitude and phase which must be accounted for in any analysis that attempts to provide a linguistically meaningful description of the language a joint model for amplitude phase and duration is presented which combines elements from functional data analysis compositional data analysis and linear mixed effects models by decomposing functions via a functional principal component analysis and connecting registration functions to compositional data analysis a joint multivariate mixed effect model can be formulated which gives insights into the relationship between the different modes of variation as well as their dependence on linguistic and nonlinguistic covariates the model is applied to the cospro 1 dataset a comprehensive database of spoken taiwanese mandarin containing appro

nonparametric maximum likelihood estimation for shifted curves the analysis of a sample of curves can be done by self modelling regression methods within this framework we follow the ideas of nonparametric maximum likelihood estimation known from event history analysis and the counting process set up we derive an infinite dimensional score equation and from there we suggest an algorithm to estimate the shape function for a simple shape invariant model the nonparametric maximum likelihood estimator that we find turns out to be a nadaraya watson like estimator but unlike in the usual kernel smoothing situation we do not need to select a bandwidth or even a kernel function since the score equation automatically selects the shape and the smoothing parameter for the estimation we apply the method to a sample of electrophoretic spectra to illustrate how it works

distances and inference for covariance operators a framework is developed for inference concerning the covariance operator of a functional random process where the covariance operator itself is an object of interest for statistical analysis distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable to functional data in particular an infinite dimensional analogue of the procrustes size and shape distance is developed convergence of finite dimensional approximations to the infinite dimensional distance metrics is also shown for inference a frechet estimator of both the covariance operator itself and the average covariance operator is introduced a permutation procedure to test the equality of the covariance operators between two groups is also considered additionally the use of such distances for extrapolation to make predictions is explored as an example of the proposed methodology the use of covariance operators has been suggested in a philological study of cross linguistic dependence as a way to incorporate quantitative phonetic information it is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the romance languages

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

self similar measures and intersections of cantor sets it is natural to expect that the arithmetic sum of two cantor sets should have positive lebesgue measure if the sum of their dimensions exceeds 1 but there are many known counterexamples e g when both sets are the middle ca cantor set and ca e 1 1 we show that for any compact set k and for a e ag e 0 1 the arithmetic sum of k and the middle ae cantor set does indeed have positive lebesgue measure when the sum of their hausdorff dimensions exceeds 1 in this case we also determine the essential su premum as the translation parameter t varies of the dimension of the intersection of k t with the middle ca cantor set we also establish a new property of the infinite bernoutlli convolutions lj the distributions of random series e 0 a where the signs are chosen independently with probabilities p 1 p let 1 ql q2 2 for p 2 near 2 and for a e a in some nonempty interval iv is absolutely continuous 2 1 and its density is in lil but not in lq2 we also answer a question of kahane concerning the fourier transform of vl 2

reduced order based feedback control of the kuramoto sivashinsky equation in this paper we consider the kuramoto sivashinsky equation kse which describes the long wave motions of a thin film over a vertical plane solution procedures for the kse often yield a large or infinite dimensional nonlinear system we first discuss two reduced order methods the approximate inertial manifold and the proper orthogonal decomposition and show that these methods can be used to obtain a reduced order system that can accurately describe the dynamics of the kse moreover from this resulting reduced order system the feedback controller can readily be designed and synthesized for our control techniques we use the linear and nonlinear quadratic regulator methods which are the first and second order approximated solutions of the hamilton jacobi bellman equation respectively numerical simulations comparing the performance of the reduced order based linear and nonlinear controllers are presented

fixed points of a destabilized kuramoto sivashinsky equation we consider the family of destabilized kuramoto sivashinsky equations in one spatial dimension u t u x x x x u x x u u x u for 0 and r for certain parameter values shock like stationary solutions have been numerically observed in this work we verify the existence of several such solutions using the framework of self consistent bounds and validated numerics

local models of spatio temporally complex fields abstract we investigate the ability of local models of the one space dimensional kuramoto sivashinsky partial differential equation with periodic boundary conditions obtained by projection on a small set of fourier modes on a short subinterval to reproduce coherent events typical of solutions of the same equation on a much longer interval we find that systems containing as few as two linearly unstable modes can produce realistic local events in the short term but that for more reliable short time tracking and long term statistics three or four interacting modes are required and that the length of the short interval plays a subtle role certain resonant lengths giving superior results

searching for structure in curve samples abstract the shape of a regression curve can to a large extent be characterized by the succession of structural features like extrema inflection points and so on when analyzing a sample of regression curves it is often important to know at an early stage of data analysis which structural features are occurring consistently in each curve of the sample such a definition is usually not easy due to substantial interindividual variation both in the x and the y axis and due to the influence of noise a method is proposed for identifying typical features without relying on an a priori specified functional model for the curves the approach is based on the frequencies of occurrence of structural features as for example maxima in the curve sample along the x axis important tools are nonparametric regression and differentiation and kernel density estimation apart from a theoretical foundation the usefulness of the method is documented by application to two interesting biomedical areas growth and develop

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

wavelet projections of the kuramoto sivashinsky equation i heteroclinic cycles and modulated traveling waves for short systems abstract we study fairly low dimensional projections of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a short interval onto bases spanned by periodic wavelets such projections break the translation reflection symmetry o 2 replacing it by a finite dihedral group dk however we show that for the perrier basdevant wavelets used here the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved in particular we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets we use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and fourier modes which preserve full o 2 symmetry and are also optimal in that fourier truncations maximise the energy l2 norm among all finite dimensional models this study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains

statistical tools to analyze data representing a sample of curves the paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological chemical ldots process such samples of curves often possess the following features there is a typical structural pattern common to all curves of the sample on the other hand individual realizations of the typical shape show different dynamics and intensity in particular typical peaks are shifted from individual to individual differences in dynamics complicate the analysis of samples of curves for example the cross sectional average usually does not reflect an average pattern due to shifts structure is smeared or might even disappear our approach consists in synchronizing the individual curves before determining the average or any further statistics pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity the method requires the introduction of new statistical objects they are defined mathematically their properties are discussed and possible estimators are proposed the asymptotic bias and variance of the estimators are derived an application to visually evoked brain potentials illustrates the approach

alignment of curves by dynamic time warping when studying some process or development in different subjects or units be it biological chemical or physical we usually see a typical pattern common to all curves yet there is variation both in amplitude and dynamics between curves following some ideas of structural analysis introduced by kneip and gasser we study a method dynamic time warping with a proper cost function for estimating the shift or warping function from one curve to another to align the two functions for some models this method can identify the true shift functions if the data are noise free noisy data are smoothed by a nonparametric function estimate such as a kernel estimate it is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity some simulation results are presented to illustrate the performance of this method

weighted local regression and kernel methods for nonparametric curve fitting abstract weighted local regression a popular technique for smoothing scatterplots is shown to be asymptotically equivalent to certain kernel smoothers since both methods are local weighted averages of the data it is proved that in the fixed design regression model given a weighted local regression procedure with any weight function there is a corresponding kernel method such that the quotients of weights distributed by both methods tend uniformly to 1 as the number of observations increases to infinity it is demonstrated by examples that in some instances the weights are nearly the same for both methods even for small samples the asymptotic equivalence allows the derivation of the leading terms of the mean squared error and of the local limit distribution for weighted local regression further a close correspondence is found between the orders of the polynomial to be locally fitted in weighted local regression and the order number of vanishing moments of the kernel employed in the kernel smoot

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

introduction to archetypal analysis of spatio temporal dynamics abstract a comparison is made between the principal component or karhunen loeve kl decomposition of spatio temporal data and a new procedure called archetypal analysis cutler and breiman 1994 archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values we show that archetypes may be more appropriate than kl when the data do not have elliptical distributions and they are often well suited to studying regimes in which the system spends a long time near a steady state punctuated with quick excursions to outliers in the data set which may represent intermittency we also introduce a variation of archetypal analysis that is designed to track moving structures such as traveling waves or solitions by using this method the traveling part of the motion is separated from the stationary or semi stationary pattern advantages and disadvantages of each method are discussed

hausdorff dimension for randomly perturbed self affine attractors in this paper we shall consider a self affine iterated function system in mathbb r d d 2 where we allow a small random translation at each application of the contractions we compute the dimension of a typical attractor of the resulting random iterated function system complementing a famous deterministic result of falconer which necessarily requires restrictions on the norms of the contraction however our result has the advantage that we do not need to impose any additional assumptions on the norms this is of benefit in practical applications where such perturbations would correspond to the effect of random noise we also give analogous results for the dimension of ergodic measures in terms of their lyapunov dimension finally we apply our method to a problem originating in the theory of fractal image compression

amplitude and phase variation of point processes we develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping known as the problem of separation of amplitude and phase variation the amplitude variation of a real random function y x x is an element of 0 1 corresponds to its random oscillations in the y axis typically encapsulated by its co variation around a mean level in contrast its phase variation refers to fluctuations in the x axis often caused by random time changes we formalise similar notions for a point process and nonparametrically separate them based on realisations of i i d copies pi i of the phase varying point process a key element in our approach is to demonstrate that when the classical phase variation assumptions of functional data analysis fda are applied to the point process case they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure we demonstrate that these induce a natural wasserstein geometry tailored to the warping problem including a formal notion of bias expressing over registration within this framework we construct nonparametric estimators that tend to avoid over registration in finite samples we show that they consistently estimate the warp maps consistently estimate the structural mean and consistently register the warped point processes even in a sparse sampling regime we also establish convergence rates and derivev root n consistency and a central limit theorem in the cox process case under dense sampling showing rate optimality of our structural mean estimator in that case

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

archetypal analysis of spatio temporal dynamics abstract a comparison is made between the principal component or karhunen loeve decomposition of two sets of spatio temporal data one numerical the other experimental and a new procedure called archetypal analysis cutler and breiman 1994 archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values archetypes may be more appropriate than kl when the data do not have elliptical distributions and are often well suited to studying regimes in which the system spends a long time near a steady state punctuated with quick excursions to outliers in the data set which may represent intermittency other advantages and disadvantages of each method are discussed

second order comparison of gaussian random functions and the geometry of dna minicircles given two samples of continuous zero mean iid gaussian processes on 0 1 we consider the problem of testing whether they share the same covariance structure our study is motivated by the problem of determining whether the mechanical properties of short strands of dna are significantly affected by their base pair sequence though expected to be true had so far not been observed in three dimensional electron microscopy data the testing problem is seen to involve aspects of ill posed inverse problems and a test based on a karhunen loeve approximation of the hilbert schmidt distance of the empirical covariance operators is proposed and investigated when applied to a dataset of dna minicircles obtained through the electron microscope our test seems to suggest potential sequence effects on dna shape supplemental material available online

finite mode analysis of the generalized kuramoto sivashinsky equation abstract we present numerical results concerning a five mode truncation of the equation u t uu x u xxx u xx u xxxx 0 subject to periodic boundary conditions we find that for large the system evolves from most initial conditions into a final state consisting of one or two traveling pulses depending on the initial condition and horizontal periodicity this is due to a region of simultaneous stability of the first two branches that bifurcate from the trivial solution an additional two pulse traveling wave which does not bifurcate from u 0 is also present

phase space analysis of bursting behavior in kolmogorov flow abstract pde simulations for the kolmogorov flow are analyzed in terms of phase space concepts the tool used is the proper orthogonal decomposition method which extracts coherent structures and prominent features of a random or turbulent dataset we analyze a quasiperiodic regime and an intermittent regime we derive two eigenfunctions that determine the dynamics and structure of the quasiperiodic case and find a third one associated with the unstable manifold of the bursts of the intermittent regime calculations are performed for streamfunction data and vorticity data which show substantial differences it is argued that the streamfunction data demonstrate the low dimensional phase space dynamics of the large scales whereas the vorticity data show an enstrophy cascade

pairwise curve synchronization for functional data summary data collected by scientists are increasingly in the form of trajectories or curves often these can be viewed as realizations of a composite process driven by both amplitude and time variation we consider the situation in which functional variation is dominated by time variation and develop a curve synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step these initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step a truncated averaging process is used to obtain robust estimation of individual warping functions the method compares well with other available time synchronization approaches and is illustrated with berkeley growth data and gene expression data for multiple sclerosis

weak and strong uniform consistency of kernel regression estimates we study the estimation of a regression function by the kernel method under mild conditions on the window the bandwidth and the underlying distribution of the bivariate observations x i y i we obtain the weak and strong uniform convergence rates on a bounded interval these results parallel those of silverman 1978 on density estimation and extend those of schuster and yakowitz 1979 and collomb 1979 on regression estimation

distribution on warp maps for alignment of open and closed curves alignment of curve data is an integral part of their statistical analysis and can be achieved using model or optimization based approaches the parameter space is usually the set of monotone con

smoothness of projections bernoulli convolutions and the dimension of exceptions erd s 1939 1940 studied the distribution of the random series p 0 n and showed that is singular for infinitely many 1 2 1 and absolutely continuous for a e in a small interval 1 1 solomyak 1995 proved a conjecture made by garsia 1962 that is absolutely continuous for a e 1 2 1 in order to sharpen this result we have developed a general method that can be used to estimate the hausdorff dimension of exceptional parameters in several contexts in particular we prove for any 0 1 2 the set of 0 1 such that is singular has hausdorff dimension strictly less than 1 for any borel set a rd with hausdorff dimension dim a d 1 2 there are points x a such that the pinned distance set x y y a has positive lebesgue measure moreover the set of x where this fails has hausdorff dimension at most d 1 dim a let k denote the middle cantor set for 1 2 and let k r be any compact set peres and solomyak 1998 showed that for a e 0 1 2 such that dim k dimk 1 the sum k k has positive length we show that the set of exceptional in this statement has hausdorff dimension at most 2 dim k dimk 0 for any borel set e rd with dim e 2 almost all orthogonal projections of e onto lines through the origin have nonempty interior and the exceptional set of lines where this fails has dimension at most d 1 dim e if is a borel probability measure on rd with correlation dimension greater than m 2 then for a prevalent set of c1 maps f rd rm in the sense described by hunt sauer and yorke 1992 the image of under f has a density with at least fractional derivatives in l2 rm

suppression of bursting abstract we investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear o 2 symmetric dynamical systems possessing structurally stable heteroclinic cycles the leading order behavior near the equilibria is bilinear and homogeneous in the state variables while nonlinearities representing the symmetry breaking effect of the controller are crucial after a series of simplifying transformations we use ideas from optimal control theory to construct a stabilizing controller this study is motivated by the desire to stabilize the burst sweep cycle in low dimensional models of a turbulent boundary layer in the last two sections we apply the techniques to the 10 dimensional system of aubry et al 1988 kw bilinear systems dynamical systems heteroclinic cycles normal forms optimal control symmetry

chaotic saddles at the onset of intermittent spatiotemporal chaos in a recent study rempel and chian phys rev lett 98 014101 2007 it has been shown that nonattracting chaotic sets chaotic saddles are responsible for intermittency in the regularized long wave equation that undergoes a transition to spatiotemporal chaos stc via quasiperiodicity and temporal chaos in the present paper it is demonstrated that a similar mechanism is present in the damped kuramoto sivashinsky equation prior to the onset of stc a spatiotemporally chaotic saddle coexists with a spatially regular attractor after the transition to stc the chaotic saddle merges with the attractor generating intermittent bursts of stc that dominate the post transition dynamics

branching random walk with exponentially decreasing steps and stochastically self similar measures we consider a branching random walk on r whose step size decreases by a fixed factor 0 1 2 the limit measure is almost surely a s absolutely continuous with respect to the lebesgue measure but for pisot 1 it is a s singular 2 for all a 5 1 2 the support of the measure is a s the closure of its interior 3 for pisot 1 the support of the measure is fractured it is a s disconnected and the components of the complement are not isolated on both sides

self modelling warping functions the paper introduces a semiparametric model for functional data the warping functions are assumed to be linear combinations ofqcommon components which are estimated from the data hence the name self modelling even small values ofqprovide remarkable model flexibility comparable with nonparametric methods at the same time this approach avoids overfitting because the common components are estimated combining data across individuals as a convenient by product component scores are often interpretable and can be used for statistical inference an example of classification based on scores is given abstract from author

warped functional regression a characteristic feature of functional data is the presence of phase variability in addition to amplitude variability existing functional regression methods do not handle time variability in an explicit and efficient way in this paper we introduce a functional regression method that incorporates time warping as an intrinsic part of the model the method achieves good predictive power in a parsimonious way and allows unified statistical inference about phase and amplitude components the asymptotic distribution of the estimators is derived and their finite sample properties are studied by simulation an application involving ground level ozone trajectories is presented

order and complexity in the kuramoto sivashinsky model of weakly turbulent interfaces abstract we present a large number of new geometric ergodic and statistical properties of the kuramoto sivashinsky equation modeling interfacial turbulence in various physical contexts in addition this equation has the remarkable property of inertial manifolds where some finite dimensional dynamical system is rigorously equivalent to this infinite dimensional partial differential equation in moderate size domains up to ten periods in length a low dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold the extreme numerical sensitivity of chaos in this dissipative pde requires very high precision methods despite the geometrical complexities of the bifurcation structure some statistical properties remain remarkably simple there is overwhelming evidence that for some parameter values a permanent unsteady state exists an unexpectedly simple diffusive relaxation of the large scale fluctuations is extracted from extensive numerical simulations in these calculations we observe long time tails for the correlation functions of relevant quantities we propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence

clustering multivariate functional data with phase variation when functional data come as multiple curves per subject characterizing the source of variations is not a trivial problem the complexity of the problem goes deeper when there is phase variation in addition to amplitude variation we consider clustering problem with multivariate functional data that have phase variations among the functional variables we propose a conditional subject specific warping framework in order to extract relevant features for clustering using multivariate growth curves of various parts of the body as a motivating example we demonstrate the effectiveness of the proposed approach the found clusters have individuals who show different relative growth patterns among different parts of the body

stability enhancement by boundary control in the kuramoto sivashinsky equation where we refer to 0 as the anti di ussion parameter note that a more general form ut 1uxxxx 2uxx 3uux 0 can always be reduced to 1 1 by appropriate rescaling of t x and u eq 1 1 was derived independently by kuramoto et al 20 22 as a model for phase turbulence in reaction di usion systems and by sivashinsky 33 as a model for plane ame propagation describing the combined in uence of di usion and thermal conduction of the gas on stability of a plane ame front so far it has been well understood that the ks equation can also serve as a mathematical model for cellular instabilities in a variety of situations the ow of thin liquid lms on inclined planes 30 in the limit of large surface tension dendritic fronts in dilute binary alloys 31 and alfven drift waves in plasmas 23 as a nonlinear saturation mechanism of the dissipative trapped ion modes

unstable recurrent patterns in kuramoto sivashinsky dynamics we undertake an exploration of recurrent patterns in the antisymmetric subspace of the one dimensional kuramoto sivashinsky system for a small but already rather turbulent system the long time dynamics takes place on a low dimensional invariant manifold a set of equilibria offers a coarse geometrical partition of this manifold the newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions the attracting set appears surprisingly thin its backbone consists of several smale horseshoe repellers well approximated by intrinsic local one dimensional return maps each with an approximate symbolic dynamics the dynamics appears decomposable into chaotic dynamics within such local repellers interspersed by rapid jumps between them