measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

variable delay feedback control of unstable steady states in retarded time delayed systems we study the stability of unstable steady states in scalar retarded time delayed systems subjected to a variable delay feedback control the important aspect of such a control problem is that time delayed systems are already infinite dimensional before the delayed feedback control is turned on when the frequency of the modulation is large compared to the system s dynamics the analytic approach consists of relating the stability properties of the resulting variable delay system with those of an analogous distributed delay system otherwise the stability domains are obtained by a numerical integration of the linearized variable delay system the analysis shows that the control domains are significantly larger than those in the usual time delayed feedback control and that the complexity of the domain structure depends on the form and the frequency of the delay modulation

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

an analytic approach to practical state space reconstruction we study the three standard methods for reconstructing a state space from a time series delays derivatives and principal components we derive a closed form solution to principal component analysis in the limit of small window widths this solution explains the relationship between delays derivatives and principal components it shows how the singular spectrum scales with dimension and delay time and it explains why the eigenvectors resemble the legendre polynomials most importantly the solution allows us to derive a guideline for choosing a good window width unlike previous suggestions this guideline is based on first principles and simple quantities we argue that discrete legendre polynomials provide a quick and not so dirty substitute for principal component analysis and that they are a good practical method for state space reconstruction

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

a penalty finite element approximation to a signorini two body contact problem in thermoelasticity in this work a numerical approximation by the finite element method of a quasi static frictionless unilateral contact problem between two thermoelastic bodies in two dimensions is considered the approximation scheme is based on a penalty formulation used to replace the signorini contact condition convergence results and error bounds are obtained and some numerical experiments presented

long range order in the scaling behaviour of hyperbolic dynamical systems abstract detailed statistical analyses of chaotic signals have been so far limited to 2 d maps with one expanding and one contracting direction here by studying filtered signals we provide evidence that the existence of a further stable direction can transform self similarity into self affinity and generate phase transitions according to the thermodynamic formalism of dynamical systems in a hyperbolic case the dimension spectrum is computed exactly as a function of the filter cut off frequency moreover the validity of the kaplan yorke formula and of its generalization are rigorously proven finally the scaling function is shown to satisfy a stochastic recursive relation whose lyapunov exponent allows to control the extension of the range of interactions in the associated hamiltonian

estimating dimension from small samples abstract details of a new algorithm to estimate the dimension of a measure from point samples are described the new algorithm emphasizes that dimension need not be the same at all scales that is there need not be a scaling region the algorithm is verified with gaussian measures and is found to be reliable and have less demanding data requirements than conventional estimators using certain convergence properties of gaussian measures it is shown that good estimates can be made from very small samples

multifractal weighted permutation analysis based on renyi entropy for financial time series abstract this paper proposes a new method called multifractal weighted permutation analysis based on renyi entropy mfwpa to calculate generalized dimension of financial time series the generalized dimension obtained by weighted permutation process retains more amplitude information of time series and is closely related to the multifractal properties of the system the advantages of this method are verified by numerical simulations we find mfwpa has a less sensitivity to noise and captures the complexity for different parts of sequences by changing the length of sequences moreover we apply this method to investigate multifractal behaviors of different stock indices and compare it with the classical algorithm called standard multifractal analysis based on partition function sma results show that mfwpa could describe the multifractal behaviors of stock indices in detail and reflect the complexity of time series in addition generalized dimensions of shuffled series are larger than the corresponding original series as a consequence of the removed autocorrelation

complexity and synchronization in chaotic fiber optic systems abstract in this work we investigate experimentally the complexity of chaotic attractors generated by a semiconductor laser subjected to optical feedback and associate their dimensionality with the synchronization efficiency of the corresponding chaotic transmitter receiver configuration the complexity is characterized by calculating the correlation dimension d 2 of experimental chaotic time series for different values of the optical feedback we present the effect of d 2 on the synchronization efficiency and determine the optimal operating condition that leads to the most complex chaotic carrier and simultaneously to the most successful synchronization lastly we associate and explain our experimental results with theoretical predictions in the research literature

information and entropy in strange attractors a technique for analyzing time series data from experiments is presented that provides estimates of four basic characteristics of a system 1 the measure theoretic entropy 2 the accuracy of the measurements 3 the number of measurements necessary to specify a system state and 4 the best delay time t to use in order to construct phase space portraits by the method of delays these characteristics are obtained by separating the entropy of measurements into a part due to noise and parts due to deterministic effects for the technique to work the noise associated with each measurement must be independent of the noise associated with all other measurements an algorithm for implementing the analysis is presented with three examples

on the dynamic properties of axially moving systems abstract the objective of the present paper is a deep analysis of some recent numerical and experimental results regarding the complex dynamics of axially moving systems such important mechanical systems exhibit interesting dynamic behaviors homoclinic orbits sub harmonic responses amplitude modulations and chaos these dynamics have been obtained numerically and in some cases have been experimentally observed using recent techniques of the non linear time series analysis the response of axially moving systems has been studied for a large variety of test cases the correlation dimension of the time series which is deeply related to the minimal dimension of a system able to reproduce the dynamics is estimated lyapunov exponents are evaluated in order to quantify the response regularity the present work gives a contribution towards understanding the complex dynamics observed both in conservative and dissipative systems the dynamical phenomena are analyzed within the unified framework of the non linear time series analysis in the case of experimental data the new non linear filtering techniques based on the embedding techniques have been applied to reduce high noise when classical techniques give bad results

numerical analysis of a frictionless contact problem for elastic viscoplastic materials abstract we consider a mathematical model which describes the unilateral quasistatic contact of two elastic viscoplastic bodies the contact is without friction and it is modeled by the classical signorini boundary conditions the model consists of an evolution equation coupled with a time dependent variational inequality it has been shown that the variational problem of the model has a unique solution here we consider numerical approximations of the problem we use the finite element method to discretize the spatial domain spatially semi discrete and fully discrete schemes are studied for both schemes we show the existence of a unique solution and derive error estimates under appropriate regularity assumptions of the solution we have the optimal order convergence

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

unicite de solutions d equations differentielles sur l espace de wiener soit x l espace de wiener et a un champ de vecteurs sur x on definit la notion de solution de l equation differentielle du dt 1 u t u 0 x on demontre un theoreme d unicite

five turbulent problems abstract five questions which involve strange attractors in one way or another are discussed these are the choice of probability measures to describe turbulence the hausdorff dimension of julia sets for rational maps of the riemann sphere the existence of nonperiodic turbulent crystal structure the mechanism generating intermittency in developed hydrodynamic turbulence the distribution of characteristic exponents in a turbulent fluid

state space reconstruction parameters in the analysis of chaotic time series the role of the time window length abstract the most common state space reconstruction method in the analysis of chaotic time series is the method of delays mod many techniques have been suggested to estimate the parameters of mod i e the time delay and the embedding dimension m we discuss the applicability of these techniques with a critical view as to their validity and point out the necessity of determining the overall time window length w for successful embedding emphasis is put on the relation between w and the dynamics of the underlying chaotic system and we suggest to set w p the mean orbital period p is approximated from the oscillations of the time series the procedure is assessed using the correlation dimension for both synthetic and real data for clean synthetic data values of w larger than p always give good results given enough data and thus p can be considered as a lower limit w p for noisy synthetic data and real data an upper limit is reached for w which approaches p for increasing noise amplitude

estimating correlation dimension from a chaotic time series when does plateau onset occur abstract suppose that a dynamical system has a chaotic attractor a with a correlation dimension d 2 a common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m dimensional space using the delay coordinate technique the estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau on which the dimension estimate is relatively constant for a range of large enough m values the plateaued dimension value is then assumed to be an estimate of d 2 for the attractor in the original full phase space in this paper we first present rigorous results which state that for a long enough data string with low enough noise the plateau onset occurs at m ceil d 2 where ceil d 2 standing for ceiling of d 2 is the smallest integer greater than or equal to d 2 we then numerical examples illustrating the theoretical prediction in addition we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m ceil d 2

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

sobolev spaces of banach valued functions associated with a markov process we discuss sobolev spaces of banach valued functions they are extensions of sobolev spaces of scalar functions we use a gamma transform of a semigroup associated with a markov process a typical example is the ornstein uhlenbeck process on the wiener space

interior penalty methods for finite element approximations of the signorini problem in elastostatics abstract the use of interior penalty methods as a basis for developing finite element approximations of boundary value problems with constraints is explored particular attention is given to the signorini problem of contact of an elastic body with a rigid foundation error estimates are derived and the results of a numerical experiment are discussed

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

exceptional events as evidence for determinism abstract by focussing attention on close returns of a trajectory to itself the existence of deterministic dynamics underlying a time series can be detected even in very short data sets this provides a practical means of detecting determinism in moderate dimensional e g 7 noisy systems or low dimensional systems with large lyapunov exponents such as computer random number generators

nonlinear electronic circuit part i multiple routes to chaos abstract a nonlinear electronic oscillator suitable for synchronized chaotic communication is studied this circuit is capable of transmitting discrete chaotic signals although the chaotic modes of operation are controlled in an analog way in part i of this review paper the three routes to chaotic operation that appear namely the period doubling intermittency and crisis induced intermittency are thoroughly studied and discussed in all three routes to chaos the appropriate experimental distributions were calculated moreover the chaotic nature of the circuit operation was evaluated by using the grassberger procaccia method calculation of the corresponding minimum embedding dimension the kolmogorov sinai entropy as well as the maximal lyapunov exponent give useful information in order to fully characterize the circuit operation

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

properties of border states of transient chaos transient chaos with absolutely continuous conditionally invariant measure is studied and a general solution of the frobenius perron equation is presented for complete one dimensional maps in particular properties of borderline situations exhibiting phase transitions within the framework of thermodynamical formalism are investigated

bifurcation and chaotic motion of a soliton in a long josephson junction oscillator the fluxon dynamics of a long josephson junction ljj is investigated numerically the effect of the surface resistance of the superconductor on the bifurcation phenomenon is examined in detail we show the occurrence of a period doubling route to chaos and the reappearance of low order periodic motions in the ljj by varying the magnitude of the external force the chaotic attractors of the ljj are characterized using maximal lyapunov exponents and a fractal correlation dimension we show the existence of noise induced oscillations in a force free ljj system finally the influence of noise on the periodic and chaotic dynamics is also presented

duality methods for solving variational inequalities abstract methods of maximal monotone operators are used in order to study from a general point of view duality numerical algorithms for solving variational inequalities with classical algorithms such as uzawa s method for the standard and augmented lagrangian this paper presents some new algorithms which appear to have very good numerical performances

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

difficulties in detecting chaos in a complex system abstract the sequences given by a 7d map are analysed by means of the methods widely used to detect chaos in the real world in order to test their sensitivity to chaotic features of a non linear system determined by comparatively high number of parameters the same diagnostic approaches are applied to the 3d lorenz map for comparison the results show that for some of the sequences yielded from the 7d map the adopted methods are not able to give a straightforward answer to the question if the system is chaotic as in the 3d case since the sequences subject to the analysis are not contaminated by noise and are sufficiently long it could be assumed that such difficulties arise likely due to specific internal features of the more complex system it is found also that an increase from 0 01 to 0 5 of the sampling step determining the sequences obtained from the 7d map masks the chaos in some of them

singular value decomposition in attractor reconstruction pitfalls and precautions applicability of singular value decomposition for reconstructing the strange attractor from one dimensional chaotic time series proposed by broomhead and king is extensively tested and discussed previously published doubts about its reliability are confirmed singular value decomposition by nature a linear method can bring distorted and misleading results when nonlinear structures are studied

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

an attractor in a solar time series a time series analysis of observed solar radio pulsations suggests that there must be a low dimensional attractor the power spectrum cannot be interpreted as a superposition of periodic components estimates of the maximum lyapunov exponent and of the kolmogorov entropy give some indications for a deterministic chaos in order to study the limitations inherent in small data samples we include data from the lorenz model and an artificial noise record consequences for the physical modelling of the pulsation event are discussed

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

a quasi sure flow property and the equivalence of capacities for differential equations on the wiener space abstract we consider a differential equation on the wiener space we show that the solutions for the differential equation satisfy the flow property quasi everywhere and we obtain the equivalence of capacities under the transformations of the wiener space induced by the solutions by using the quasi flow property

coarse grained embeddings of time series random walks gaussian random processes and deterministic chaos a new method for studying time series is described based on a statistic indicating the degree to which trajectories passing through a small region of an embedding space are parallel the method is particularly suited to time series with a significant correlation time analytic results are presented for brownian motion and gaussian random processes these are generally different from the results for chaotic systems allowing a test for deterministic dynamics in a time series a variety of examples are presented of the application of the method to low and high dimensional systems

effective scaling regime for compution the correlating dimension from chaotic time series abstract in the analysis of chaotic time series a standard technique is to reconstruct an image of the original dynamical system using delay coordinates if the original dynamical system has an attractor then the correlation dimension d2 of its image in the reconstruction can be estimated using the grassberger procaccia algorithm the quality of the reconstruction can be probed by measuring the length of the linear scaling region used in this estimation in this paper we show that the quality is constrained by both the embedding dimension m and more importantly by the delay time for a given embedding dimension and a finite time series there exists a maximum allowed delay time beyond which the size of the scaling region is no longer reliably discernible we derive an upper bound for this maximum delay time numerical experiments on several model chaotic time series support the theoretical argument they also clearly indicate the different roles played by the embedding dimension and the delay time in the reconstruction as the embedding dimension is increased it is necessary to reduce the delay time substantially to guarantee a reliable estimate of d2 our results imply that it is the delay time itself rather than the total observation time m 1 which plays the most critical role in the determination of the correlation dimension

a numerical algorithm for a signorini problem associated with maxwell norton materials by using generalized newton s methods abstract the goal of this work is to present new numerical approaches to obtain computationally efficient adaptive procedures to solve contact problems associated with maxwell norton materials we will describe in detail the numerical methods used to solve such problems and derive iterative algorithms to validate the proposed algorithms we will present some test examples with known analytical solutions finally we will compare the efficiency of the algorithms presented in this paper with others previously proposed by the authors

numerical approximation of the elastic viscoplastic contact problem with non matching meshes this work deals with the approximation of a time dependent variational inequality modelling the unilateral contact problem of elastic viscoplastic bodies in a bidimensional context the problem is approximated in the space variable with nonconforming finite element methods which allow the handling of nonmatching meshes on the contact zone several error estimates are established and the corresponding numerical experiments are achieved

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

numerical solving of frictionless contact problems in perfectly plastic bodies abstract we consider the frictionless unilateral contact problems of a perfectly plastic solid with a rigid foundation and two perfectly plastic solids under an analogous variational formulation for each one we introduce a discrete approximation via the finite element method and in due course prove convergence also we introduce a converging iterative algorithm to the solution of the discrete problems and present some numerical experiences

filtered noise can mimic low dimensional chaotic attractors this contribution presents four results first calculations indicate that when examined by the grassberger procaccia algorithm alone filtered noise can mimic low dimensional chaotic attractors given the ubiquity of signal filtering in experimental investigations this is potentially important second a criterion is derived which provides an estimate of the minimum data accuracy needed to resolve the dimension of an attractor third it is shown that a criterion derived by eckmann and ruelle physica d 56 185 1992 to estimate the minimum number of data points required in a grassberger procaccia calculation can be used to provide a further check on these dimension estimates fourth it is shown that surrogate data techniques recently published by theiler and his colleagues in nonlinear modeling and forecasting edited by m casdagli and s eubanks addison wesley reading ma 1992 can successfully distinguish between linearly correlated noise and nonlinear structure these results and most particularly the first indicate that grassberger procaccia results must be interpreted with far greater circumspection than has previously been the case and that the algorithm should be used in combination with additional procedures such as calculations with surrogate data when filtered signals are examined by this algorithm alone a finite noninteger value of mathit d 2 is consistent with low dimensional chaotic behavior but it is certainly not a definitive diagnostic of chaos

correlation dimension and spectrum for the almost periodic bouncing ball abstract we analyze the dependence of the autocorrelation functions and correlation dimensions on the degree of almost periodicity of the classical bouncing ball model

rational dynamical zeta functions for birational transformations we propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables depending on two parameters this conjectured function is a simple rational expression with integer coecients this yields an algebraic value for the topological entropy furthermore the generating function for the arnold complexity is also conjectured to be a rational expression with integer coecients with the same singularities as for the dynamical zeta function this leads at least in this example to an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy we also give a semi numerical method to eectively compute the arnold complexity c 1999 published by elsevier science b v all rights reserved

finite element methods for constrained problems in elasticity several different variational formulations of boundary value problems with constraints are discussed with particular reference to constrained problems in elasticity special attention is given to exterior penalty methods a discussion of the conditions necessary for penalty methods to provide a basis for stable and convergent finite element methods is given in particular the use of reduced integration is discussed and criteria on the order of reduced integration rules sufficient to produce stable and convergent schemes are described applications of reduced integration penalty methods to incompressible elasticity problems and contact problems are described

chaos in segments from korean traditional singing and western singing the time series of the segments from a korean traditional song gwansanyungma and a western song la mamma morta were investigated using chaotic analysis techniques it is found that the phase portrait in the reconstructed state space of the time series of the segment from the korean traditional song has a more complex structure in comparison with the segment from the western songs the segment from the korean traditional song has the correlation dimension 4 4 and two positive lyapunov exponents which show that the dynamic related to the korean traditional song is a high dimensional hyperchaotic process on the other hand the segment from the western song with only one positive lyapunov exponent and the correlation dimension 2 5 exhibits low dimensional chaotic behavior

reconstruction expansion as a geometry based framework for choosing proper delay times abstract the quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter here we develop a new computationally efficient approach to choosing that quantifies reconstruction expansion from the identity line of the embedding space we show that reconstruction expansion is related to the concept of reconstruction signal strength and that increased expansion corresponds to diminished effects of measurement error thus reconstruction expansion represents a simple geometrical framework for choosing furthermore we describe the role of dynamical error in attractor expansion and argue that algorithms for determining should be considered as attempts at estimating an upper bound to the optimal delay

on the control of complex dynamic systems a method is described for the limited control of the dynamics of systems which generally have several dynamic attractors associated either with maps or first order ordinary differential equations ode in rn the control is based on the existence of convergent regions cc k 1 2 in the phase space of such systems where there is local convergence of all nearby orbits the character of the control involves the entrainment and subsequent possible migration of the experimental system from one attractor to another entrainment means that limt x t g t 0 where x t rn is the system s controlled dynamics and the goal dynamics g t gk has any topological form but is limited dynamically and to regions of phase space gk contained in some ck gk ck the control process is initiated only when the system enters a basin of entrainment bek gk associated with the goal region gk aside from this macroscopic initial state information about the system no further feedback of dynamic information concerning the response of the system is required the experimental reliability of the control requires that the regions bek be convex regions in the phase space which can apparently be assured if gk ck simple illustrations of these concepts are given using a general linear and a piecewise linear ode in rn in addition to these entrainment goals migration goal dynamics is introduced which intersects two convergent regions g cj g cj i j and permits transferring the dynamics of a system from one attractor to another or from one convergent region to another in the present study these concepts are illustrated with various one dimensional maps involving one or more attractors and convergent regions several theorems concerning entrainment are derived for very general continuous one dimensional maps sufficient conditions are also established which ensure near entrainment for a system when the dynamic model of the system is not exactly known the applications of these concepts to higher dimensional maps and flows will be presented in subsequent studies

on the metric properties of the feigenbaum attractor the two standard literature definitions of the function associated with the feigenbaum attractor are not equivalent the method due to vulet al and feigenbaum is used to calculate the haussdorff dimension of the feigenbaum attractor using as input the trajectory scaling functions the two calculations yield the same hausdorff dimensiond 0 5380451435 to within the accuracy of the computation

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

numerical analysis of a viscoplastic problem with contact condition taking place in an aluminium casting we consider a viscoplastic problem with contact condition modelling the thermomechanical deformations during casting of aluminium alloy slabs we introduce a fictitious domain method to impose the metallostatic pressure on the thermal free boundary finally we present some numerical results for a particular casting process during both the start and stationary stages

estimation of the dimension of a noisy attractor a simple method is proposed to estimate the correlation dimension of a noisy chaotic attractor the method is based on the observation that the noise induces a bias in the observed distances of trajectories which tend to appear farther apart than they are under the assumption of noise being strictly bounded in amplitude this leads to a rescaling of interpoint distances on the attractor a correlation integral function is obtained that accounts for this effect of noise the applicability of the method is illustrated with two examples viz the lorenz attractor with additive noise and experimental time series of pressure fluctuation data measured in gas solid fluidized beds

evaluation of probabilistic and dynamical invariants from finite symbolic substrings comparison between two approaches abstract while evaluating scaling functions of fractal dimensions and lyapunov exponents from time series in the traditional way assumptions are made concerning the grammar of the underlying dynamical system it is implicitly assumed that the length of the substrings considered is sufficient to capture the grammatical properties of the system in this contribution we show where this assumption becomes relevant we give an example of a simple grammatical rule which leads to badly behaving convergence properties of the associated scaling functions as another consequence of our investigations we conclude that whenever a finite grammar is encountered the cycle expansion approach of cvitanovic using periodic orbits should preferably be used

numerical analysis of a frictionless viscoelastic contact problem with normal damped response abstract we consider a mathematical model which describes the frictionless contact between a viscoelastic body and a deformable foundation we model the material s behavior with a nonlinear kelvin voigt constitutive law the process is assumed to be quasistatic and the contact is modeled with a general normal damped response condition we present the variational formulation of the problem including the existence of a unique weak solution to the model we then study the numerical approach to the problem using a fully discrete finite elements scheme with an explicit discretization in time we state the existence of the unique solution for the scheme and derive an error estimate on the approximate solutions finally we present some numerical results involving examples in one and two dimensions

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

optimal embedding parameters a modelling paradigm the reconstruction of a dynamical system from a time series requires the selection of two parameters the embedding dimension de and the embedding lag many competing criteria to select these parameters exist and all are heuristic within the context of modelling the evolution operator of the underlying dynamical system we show that one only need be concerned with the product de we introduce an information theoretic criterion for the optimal selection of the embedding window dw de for infinitely long time series this method is equivalent to selecting the embedding lag that minimises the nonlinear model prediction error for short and noisy time series we find that the results of this new algorithm are data dependent and are superior to estimation of embedding parameters with the standard techniques

reconstructing attractors from scalar time series a comparison of singular system and redundancy criteria abstract a delay vector phase space reconstruction in which the delay time satisfiesa minimum redundancy criterion is compared with a reconstruction obtained using a singular system approach minimum redundancy produces the better reconstruction the reconstructions are compared using a distortion functional d which measures how well the location of a point in the original phase space can be determined on the basis of its image under the reconstruction process the superiority of the redundancy analysis over the singular system analysis is found to arise from the former s foundation on the notion of general independence as opposed to the latter s foundation on the notion of linear independence

numerical analysis and simulations of a dynamic frictionless contact problem with damage the process of dynamic frictionless contact between a viscoelastic body and a reactive foundation which includes material damage is modelled numerically analyzed and simulated contact is modelled with the normal compliance condition the damage of the material resulting from tension or compression is described by the damage field which measures the pointwise fractional decrease in the load carrying capacity of the material the evolution of the damage is governed by a parabolic differential inclusion a fully discrete scheme for the problem is proposed and error estimates on the numerical solutions derived the scheme has been implemented and numerical simulations of the evolution of the mechanical state of the system and its material damage are presented