application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

horseshoe and topological entropy estimate of a class of three dimensional cellular neural networks chaos in a class of three dimensional continuous time cellular neural networks is investigated numerical experiments show that this class of cellular neural networks can display chaotic attractors and limit cycles for different parameters by virtue of topological horseshoes theory in dynamical systems a rigorous computer assisted verification for chaotic behavior of the neural network is presented for certain parameter topological entropy estimate of the neural network is also given in terms of the poincare map

stretching based diagnostics and reduction of chemical kinetic models with diffusion a new method for diagnostics and reduction of dynamical systems and chemical kinetic models is proposed the method makes use of the local structure of the normal stretching rates by projecting the dynamics onto the local directions of maximal stretching the approach is computationally very simple as it implies the spectral analysis of a symmetric matrix notwithstanding its simplicity stretching based analysis derives from a geometric basis grounded on the pointwise applications of concepts of normal hyperbolicity theory as a byproduct a simple reduction method is derived equivalent to a local embedding algorithm which is based on the local projection of the dynamics onto the most unstable and or slow modes compared to the time scale dictated by the local tangential dynamics this method provides excellent results in the analysis and reduction of dynamical systems displaying relaxation towards an equilibrium point limit cycles and chaotic attractors several numerical examples deriving from typical models of reaction diffusion kinetics exhibiting complex dynamics are thoroughly addressed the application to typical combustion models is also analyzed

unstable dimension variability a source of nonhyperbolicity in chaotic systems the hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set a familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible respectively noninvertible maps of dimension 3 respectively 2 and higher in particular we investigate a situation first considered by abraham and smale in 1970 for different purposes in which the dimension of the unstable and stable tangent spaces are not constant over the chaotic set we call this unstable dimension variability a simple two dimensional map that displays behavior typical of this phenomenon is presented and analyzed

transmit signal design for optimal estimation of correlated mimo channels we address optimal estimation of correlated multiple input multiple output mimo channels using pilot signals assuming knowledge of the second order channel statistics at the transmitter assuming a block fading channel model and minimum mean square error mmse estimation at the receiver we design the transmitted signal to optimize two criteria mmse and the conditional mutual information between the mimo channel and the received signal our analysis is based on the recently proposed virtual channel representation which corresponds to beamforming in fixed virtual directions and exposes the structure and the true degrees of freedom in the correlated channel however our design framework is applicable to more general channel models which include known channel models such as the transmit and receive correlated model as special cases we show that optimal signaling is in a block form where the block length depends on the signal to noise ratio snr as well as the channel correlation matrix the block signal corresponds to transmitting beams in successive symbol intervals along fixed virtual transmit angles whose powers are determined by nonidentical water filling solutions based on the optimization criteria our analysis shows that these water filling solutions identify exactly which virtual transmit angles are important for channel estimation in particular at low snr the block length reduces to one and all the power is transmitted on the beam corresponding to the strongest transmit angle whereas at high snr the block length has a maximum length equal to the number of active virtual transmit angles and the power is assigned equally to all active transmit angles consequently from a channel estimation viewpoint a faster fading rate can be tolerated at low snrs relative to higher snrs

minimal dynamical systems from pdes using sobolev eigenfunctions abstract a modified karhunen loeve transform is derived using a weighted sobolev norm this approach optimizes the approximation of the higher derivative terms in numerical simulations of partial differential equation and reduces the sizes of the associated system of ordinary differential equations the kuramoto sivashinsky equation is used as an example to demonstrate how a galerkin procedure based on eigenfunctions determined by the sobolev optimality criterion improves the performance of the standard karhunen loeve procedure we present an example of a simulation which required a 19 mode expansion using the k l approach and show how this is reduced to a 7 mode simulation using the sobolev eigenfunctions

a differential algorithm for the lyapunov spectrum abstract we present a new algorithm for computing the lyapunov exponents spectrum based on a matrix differential equation the approach belongs to the so called continuous type where the rate of expansion of perturbations is obtained for all times and the exponents are reached as the limit at infinity it does not involve exponentially divergent quantities so there is no need of rescaling or realigning of the solution we show the algorithm s advantages and drawbacks using mainly the example of a particle moving between two contracting walls

a velocity pressure streamline diffusion finite element method for navier stokes equations a streamline diffusion finite element method is introduced for the time dependent incompressible navier stokes equations in a bounded domain in r squared and r cubed in the case of a flow with a high reynolds number an error estimate is proved and numerical results are given the method is based on a mixed velocity pressure formulation using the same finite element discretization of space time for the velocity and the pressure spaces which consist of piecewise linear functions together with certain least squares modifications of the galerkin variational formulation giving added stability without sacrificing accuracy

a comparative study of computation of lyapunov spectra with different algorithms abstract in this paper we make a detailed numerical comparison between three algorithms for the computation of the full lyapunov spectrum as well as the associated eigenvectors of general dynamical systems they are a the standard method b a differential formulation of the standard method and c a new algorithm which does not require rescaling and reorthogonalization we also bring out the relations among these methods moreover we give a simplified formulation of the new algorithm when the dimensionality of the system is 4 we find that there is reasonable agreement among the lyapunov spectra obtained using the three algorithms in most cases however the standard method seems to be the most efficient followed by the new method and the differential version of the standard method in that order as far as the cpu time for the computation of the lyapunov spectra is concerned the new method is hardly suitable for finding the eigenvectors whereas the other procedures give nearly identical numerical results

modal reduction of pde models by means of snapshot archetypes abstract a new method for constructing low dimensional reduced models of dissipative partial differential equations is proposed the original pde u t f u is projected onto a linear subspace spanned by the so called snapshot archetypes that are selected spatial profiles of u x t the selection rule of the snapshot archetypes characterizes the method two different selection methods are proposed we provide an energetic criterion for the minimum number of archetypes needed for an accurate approximation of the asymptotic dynamics this approach is tested for several pde systems such as the kuramoto sivashinsky equation the arneodo elezgaray reaction diffusion model and the self ignition dynamics of a coal stockpile the latter two systems exhibit a rich bifurcative structure and are suitable for checking the robustness of the snapshot archetype reduced models with respect to parameter variations

the hidden dynamics of the rossler attractor abstract the fast slow dynamics of the rossler model in the chaotic regime are compared to the dynamics of the reduced slow model governing the flow along the slow invariant manifold on which the trajectory is restrained to evolve during the slow part of the attractor it is shown that the dynamics of the reduced model incorporate the slow time scales of the full model however instead of the fast dissipative time scale of the full model that restrains the trajectory on the slow invariant manifold the reduced model generates a new time scale that i relates to the curvature of the manifold and ii becomes faster as higher order corrections are incorporated in the approximation of the manifold and the reduced model it is shown that this new time scale does not characterize the motion of the solution on the manifold but it can be employed as a signal for the strengthening or weakening of the manifold depending on whether it relates to components of the reduced model that tend to lead its solution towards equilibrium or away from it finally it is demonstrated that the new time scale introduced by the reduced model has a significant role in determining the maximum accuracy that can be delivered by the singular perturbation methodology

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

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

back in the saddle again a computer assisted study of the kuramoto sivashinsky equation a numerical and analytical study of the kuramoto sivashinsky partial differential equation pde in one spatial dimension with periodic boundary conditions is presented the structure stability and bifurcation characteristics of steady state and time dependent solutions of the pde for values of the parameter alpha less than 40 are examined the numerically observed primary and secondary bifurcations of steady states as well as bifurcations to constant speed traveling waves limit cycles are analytically verified persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of o 2 their effect on the system dynamics is discussed and several tertiary bifurcations observed numerically are presented

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

a hyperchaos generated from lorenz system abstract this paper presents a four dimension hyperchaotic lorenz system obtained by adding a nonlinear controller to lorenz chaotic system the hyperchaotic lorenz system is studied by bifurcation diagram lyapunov exponents spectrum and phase diagram numerical simulations show that the new system s behavior can be convergent divergent periodic chaotic and hyperchaotic when the parameter varies

joint spatial division and multiplexing the large scale array regime we propose joint spatial division and multiplexing jsdm an approach to multiuser mimo downlink that exploits the structure of the correlation of the channel vectors in order to allow for a large number of antennas at the base station while requiring reduced dimensional channel state information at the transmitter csit jsdm achieves significant savings both in the downlink training and in the csit uplink feedback thus making the use of large antenna arrays at the base station potentially suitable also for frequency division duplexing fdd systems for which uplink downlink channel reciprocity cannot be exploited in the proposed scheme the multiuser mimo downlink precoder is obtained by concatenating a prebeamforming matrix which depends only on the channel second order statistics with a classical multiuser precoder based on the instantaneous knowledge of the resulting reduced dimensional effective channel matrix we prove a simple condition under which jsdm incurs no loss of optimality with respect to the full csit case for linear uniformly spaced arrays we show that such condition is approached in the large number of antennas limit for this case we use szego s asymptotic theory of toeplitz matrices to show that a dft based prebeamforming matrix is near optimal requiring only coarse information about the users angles of arrival and angular spread finally we extend these ideas to the case of a 2 d base station antenna array with 3 d beamforming including multiple beams in the elevation angle direction we provide guidelines for the prebeamforming optimization and calculate the system spectral efficiency under proportional fairness and max min fairness criteria showing extremely attractive performance our numerical results are obtained via asymptotic random matrix theory avoiding lengthy monte carlo simulations and providing accurate results for realistic finite number of antennas and users

on stable equal order finite element formulations for incompressible flow problems in this work we discuss stable equal order finite element formulations for incompressible flow problems based on petrov galerkin methods constructed by adding to the classical galerkin formulation least squares of the governing equations continuous and discontinuous pressure interpolations are considered numerical results are presented reinforcing the numerical analysis

chaotic dynamics of coherent structures abstract a variety of chaotic flows evolving in relatively high dimensional spaces are considered it is shown through the use of an optimal choice of basis functions which are consequence of the karhunen loeve procedure that an accurate description can be given in a relatively low dimensional space particular examples of this procedure which are presented are the ginzburg landau equation turbulent convection in an unbounded domain and turbulent convection in a bounded domain

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

the g scheme a framework for multi scale adaptive model reduction the numerical solution of mathematical models for reaction systems in general and reacting flows in particular is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics however the dynamics can develop very slow and very fast time scales separated by a range of active time scales an opportunity to reduce the complexity of the problem arises when the fast active and slow active time scales gaps becomes large we propose a numerical technique consisting of an algorithmic framework named the g scheme to achieve multi scale adaptive model reduction along with the integration of the differential equations des the method is directly applicable to initial value odes and by using the method of lines pdes we assume that the dynamics is decomposed into active slow fast and when applicable invariant subspaces the g scheme introduces locally a curvilinear frame of reference defined by a set of orthonormal basis vectors with corresponding coordinates attached to this decomposition the evolution of the curvilinear coordinates associated with the active subspace is described by non stiff des whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem adjusting the active des dynamically during the time integration is the most significant feature of the g scheme since the numerical integration is accomplished by solving a number of des typically much smaller than the dimension of the original problem with corresponding saving in computational work to demonstrate the effectiveness of the g scheme we present results from illustrative as well as from relevant problems

the reduction of complex dynamical systems using principal interaction patterns abstract a method of constructing low dimensional nonlinear models capturing the main features of complex dynamical systems with many degrees of freedom is described the system is projected onto a linear subspace spanned by only a few characteristic spatial structures called principal interaction patterns pips the expansion coefficients are assumed to be governed by a nonlinear dynamical system the optimal low dimensional model is determined by identifying spatial modes and interaction coefficients describing their time evolution simultaneously according to a nonlinear variational principle the algorithm is applied to a two dimensional geophysical fluid system on the sphere the models based on principal interaction patterns are compared to models using empirical orthogonal functions eofs as basis functions a pip model using 12 patterns is capable of capturing the long term behaviour of the complete system monitored by second order statistics while in the case of eofs 17 modes are necessary

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

on the amount of downlink training in correlated massive mimo channels in this paper we study the effect of downlink dl training on the achievable sum rate of multiuser massive mimo channels with spatial correlations and derive sufficient conditions of the dl channel estimation error covariance matrices that maintain the full multiplexing gain at high data signal to noise ratios snrs given the derived conditions a simple asymptotic upper bound on the average sum rate loss due to channel estimation is obtained we derive in closed form training sequences of limited duration that satisfy these conditions the training duration is variable and increases with the data snr while the sequences lie in a subspace spanned by a variable number of user spatial covariance matrices eigenvectors we additionally study the problem of sequence codebook design and find solutions to this problem for uniform linear and rectangular arrays using asymptotic results for the aforementioned training structure the designed codebooks are observed numerically to be near optimal for a moderate number of base station antennas due to their ability to identify a sufficient limited number of channel directions to train the proposed solutions can substantially reduce dl training overheads while providing achievable rates that are comparable with the rates achieved with perfect channel state information

capacity of a multiple antenna fading channel with a quantized precoding matrix given a multiple input multiple output mimo channel feedback from the receiver can be used to specify a transmit precoding matrix which selectively activates the strongest channel modes here we analyze the performance of random vector quantization rvq in which the precoding matrix is selected from a random codebook containing independent isotropically distributed entries we assume that channel elements are independent and identically distributed i i d and known to the receiver which relays the optimal rate maximizing precoder codebook index to the transmitter using b bits we first derive the large system capacity of beamforming rank one precoding matrix as a function of b where large system refers to the limit as b and the number of transmit and receive antennas all go to infinity with fixed ratios rvq for beamforming is asymptotically optimal i e no other quantization scheme can achieve a larger asymptotic rate we subsequently consider a precoding matrix with arbitrary rank and approximate the asymptotic rvq performance with optimal and linear receivers matched filter and minimum mean squared error mmse numerical examples show that these approximations accurately predict the performance of finite size systems of interest given a target spectral efficiency numerical examples show that the amount of feedback required by the linear mmse receiver is only slightly more than that required by the optimal receiver whereas the matched filter can require significantly more feedback

a wavelet method for the characterization of spatiotemporal patterns this paper introduces a wavelet based method for the characterization of spatiotemporal patterns based on the wavelet multiresolution analysis two wavelet indices multiscale accumulative density mad and multiscale accumulative change mac are proposed for the characterization of the dynamics of the spatiotemporal patterns both indices are constructed by using orthogonal wavelet projection operators the mad is a measure of the spatial complexity of a pattern at a given time whereas the mac characterizes the spatial complexity of instantaneous change of the spatiotemporal patterns at a given time the ratio of the mad indices between the lowest and the highest scales reflects the order of coherence in a pattern the time series of both mad and mac provide the dynamical information of morphological pattern evolutions numerical experiments based on the cahn hilliard equation indicate that the proposed method is efficient for quantitatively characterizing the dynamics of the spatiotemporal patterns 2002 published by elsevier science b v

computing lyapunov exponents on a stiefel manifold the problem of numerical computation of a few lyapunov exponents les of finite dimensional dynamical systems is considered from the viewpoint of the differential geometry of stiefel manifolds whether one computes one many or all les of a continuous dynamical system by time integration discrete or continuous orthonormalization is essential for stable numerical integration a differential geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a stiefel manifold however the stiefel manifold is not in general an attracting submanifold of the ambient euclidean space it is a constraint manifold with a weak numerical invariant new numerical algorithms for this problem are then designed which use the fiber bundle characterization of these manifolds this framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew symmetry property numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few les which preserve orthonormality to machine accuracy this idea is also taken a step further by making the stiefel manifold an attracting invariant manifold in which case standard explicit runge kutta algorithms can be used this leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization these class of methods should be particularly effective for computing a few les for large dimension dynamical systems the new schemes are straightforward to implement a test case is presented for illustration and an example from dynamical systems is presented where a few les are computed for an array of coupled oscillators

downlink training sequence design for fdd multiuser massive mimo systems we consider the problem of downlink training sequence design for frequency division duplex multiuser massive multiple input multiple output systems in the general case where users have distinct spatial correlations the training sequences leverage spatial correlations and are designed to minimize the channel estimation weighted sum mean square error mse under the assumption that users employ minimum mse estimators noting that the weighted sum mse function is invariant to unitary rotations of its argument a solution is obtained using a steepest descent method on the grassmannian manifold we extend the proposed design to scenarios with temporal correlations combined with kalman filters at the users and design sequences exploiting the multiuser spatio temporal channel structure finally we consider scenarios where only a limited number of bits b are available at the base station bs to inform the users of each chosen sequence e g sequences are chosen from a set of 2 b vectors known to the bs and users and develop a subspace version of matching pursuit techniques to choose the desired sequences simulation results using realistic channel models show that the proposed solutions improve user fairness with a proper choice of weights lead to accurate channel estimates with training durations that can be much smaller than the number of bs antennas and show substantial gains over randomly chosen sequences for even small values of b

chaos of a new class of hopfield neural networks chaos and bifurcation of a new class of three dimension hopfield neural networks are investigated numerical experiments show that this class of hopfield neural networks can display chaotic attractors and limit cycles for different parameters the lyapunov exponents are calculated a numerical bifurcation analysis with plots is given as well by virtue of horseshoes theory in dynamical systems we give rigorous computer assisted verifications for chaotic behavior of the system for certain parameters quantitative descriptions of the complexity of these neural networks are also given in terms of topological entropy and a brief robustness analysis of this class of hopfield neural networks is also presented

numerical analysis of pattern formation on the surface of transpiring leaves abstract the process of photosynthesis is facilitated by pores on the leaf surface called stomata when a particular stoma is open co 2 is absorbed through its aperture but h 2 o is also lost due to evaporation thus a plant will seek a stomatal aperture that balances its need for co 2 with its aversion to h 2 o loss in order to visualize a particular leaf s stomatal aperture distribution and how it changes with time fluorescence data is collected at regular intervals as digital images resulting in a video sequence it has been observed that stomatal apertures are often synchronized into spatially extended patches in order objectively to analyze this phenomenon we have developed a technique to isolate patches via a three dimensional pde based segmentation method the resulting segmented data is then collapsed to a vector valued time series of much smaller dimension with a hybrid pca archetypal analysis approach this allows for a unique interpretation of the data in terms of statistical measures of the motions of representative patches the technique is illustrated with a data set from a particularly complicated regime collected by the complexity and stomatal behavior research lab at utah state university

local predictability and nonhyperbolicity through finite lyapunov exponent distributions in two degrees of freedom hamiltonian systems received 13 september 2007 revised manuscript received 5 november 2008 published 4 december 2008by using nite lyapunov exponent distributions we get insight into both the local and global properties ofa dynamical ow including its nonhyperbolic behavior several distributions of nite lyapunov exponentshave been computed in two prototypical four dimensional phase space hamiltonian systems they have beencomputed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals weanalyze how such distributions serve or not for tracing the orbit nature and local ow properties such as theunstable dimension variability as the axes are allowed or not to tend to the largest stretching direction therelationship between the largest and closest to zero exponent distribution is analyzed it shows a linear depen dency at short intervals related to the number of degrees of freedom of the system finally the hyperbolicityindexes associated to the shadowing times are calculated they provide interesting information at very localscales even when there are no gaussian distributions and the values cannot be regarded as random variables doi 10 1103 physreve 78 066204 pacs number s 05 45 a

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

lyapunov exponents without rescaling and reorthogonalization we present a new method for the computation of lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of m or m m where m is the tangent map this method uses a minimal set of variables does not require renormalization or reorthogonalization can be used to efficiently compute partial lyapunov spectra and does not break down when the lyapunov spectrum is degenerate

natural tangent dynamics with recurrent biorthonormalizations a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic chaotic limit sets abstract this article develops a geometric method referred to as natural tangent dynamics with recurrent biorthonormalizations ntdrb that unifies into a single computational technique the analysis of dynamical systems possessing either a slow manifold associated with an equilibrium point or exhibiting nontrivial limit sets limit cycles and chaotic attractors for the first class of systems the main focus relies on the identification of the slow fast timescales while for the latter dynamics within nontrivial attractors the central issue is the temporal dichotomy between unstable stable components as described by the structure of the lyapunov spectrum the method is based on vector and covector dynamics in the tangent and cotangent bundles induced by the phase space flow and on the periodic application of a biorthonormalization procedure which contrasts the natural tendencies of vectors and covectors to align towards the most unstable slow subspaces the ntdrb method provides the most convenient and natural vector and covector bases for i the characterization of the fast timescales controlling the dynamic relaxation towards a slow manifold ii the identification of the stable and unstable components of the dynamics on a limit set iii the construction of a low dimensional reduced model reproducing the asymptotic behavior of the original systems

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

asymptotic analysis of two reduction methods for systems of chemical reactions abstract this paper concerns two methods for reducing large systems of chemical kinetics equations namely the method of intrinsic low dimensional manifolds ildms due to maas and pope combust flame 88 1992 239 and an iterative method due to fraser j chem phys 88 1988 4732 and further developed by roussel and fraser j chem phys 93 1990 1072 both methods exploit the separation of fast and slow reaction time scales to find low dimensional manifolds in the space of species concentrations where the long term dynamics are played out the asymptotic expansions of these manifolds e 0 where e measures the ratio of the reaction time scales are compared with the asymptotic expansion of m e the slow manifold given by geometric singular perturbation theory it is shown that the expansions of the ildm and m e agree up to and including terms of o e the former has an error at o e 2 that is proportional to the local curvature of m 0 the error vanishes if and only if the curvature is zero everywhere the iterative method generates term by term the asymptotic expansion of m e starting from m 0 the ith application of the algorithm yields the correct expansion coefficient at o e i while leaving the lower order coefficients invariant thus after l applications the expansion is accurate up to and including the terms of o e l the analytical results are illustrated on a planar system from enzyme kinetics michaelis menten henri and a model planar system due to davis and skodje

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

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

a regularized stabilized mixed finite element formulation for viscoplasticity of bingham type abstract in this article we present a new mixed stabilized regularized finite element formulation in primitive variables with continuous velocity and discontinuous pressure interpolations for the steady flow of an incompressible fluid of bingham type this formulation is based on an augmented lagrangian regularization technique and a least squares stabilization method mathematical analyses are performed for the new formulation in terms of stability existence and uniqueness of the solution optimal orders of convergence are obtained mathematically improving on classical methods which also present limitations as regards the values of the yield stress numerical results are presented confirming the theory developed here and they show the robustness of the new method with stability obtained for the velocity and especially for the pressure when the yield stress is very high

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric