on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

atoms and regularity for measures in a partially defined free convolution semigroup consider a borel probability measure on the real line and denote by t t 1 the free additive convolution semigroup defined by nica and speicher we show that the singular part of t is purely atomic and the density of t is locally analytic provided that t 1 the main ingredient is a global inversion theorem for analytic functions on a half plane

quantifying spatiotemporal chaos in rayleigh benard convection using large scale parallel numerical simulations we explore spatiotemporal chaos in rayleigh benard convection in a cylindrical domain with experimentally relevant boundary conditions we use the variation of the spectrum of lyapunov exponents and the leading order lyapunov vector with system parameters to quantify states of high dimensional chaos in fluid convection we explore the relationship between the time dynamics of the spectrum of lyapunov exponents and the pattern dynamics for chaotic dynamics we find that all of the lyapunov exponents are positively correlated with the leading order lyapunov exponent and we quantify the details of their response to the dynamics of defects the leading order lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics our results show a transition from boundary dominated dynamics to bulk dominated dynamics as the system size is increased the spectrum of lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high dimensional strange attractor accommodates a range of different chaotic dynamics

a physical interpretation of the he non map abstract the orientation preserving henon map is shown to be equivalent to the period one return map of a periodically kicked harmonic oscillator with a suitable nonlinear coupling to the kicking term the conservative dissipative and extremely dissipative logistic limit cases are each considered all can be derived from a single differential equation describing the kicked oscillator system in each case explicit coordinate and parameter transformations are given relating the return map of the kicked harmonic oscillator to a standard form of the henon map in the dissipative case two studies are presented with periodic kicking replaced by a smooth periodic function each is found to yield a henon like chaotic attractor

two step runge kutta methods and hyperbolic partial differential equations the purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation jameson recently introduced the use of one step runge kutta methods for the numerical solution of the euler equations improvements in efficiency up to 80 may be achieved by using two step runge kutta methods instead of the classical onestep methods these two step runge kutta methods were first introduced by byrne and lambert in 1966 they are designed to have the same number of function evaluations as the equivalent one step schemes and thus they are potentially more efficient by solving a nonlinear programming problem which is specified by stability requirements optimal two step schemes are designed the optimization technique is applicable for stability regions of any shape

fractal dimension and degeneracy of the critical point for iterated maps variations of the three most commonly used definitions of fractal dimension namely the capacity the information dimension and the correlation exponent with the degeneracy of the critical point is studied both numerically and analytically the numerical results agree very well with analytical estimates it is found that these three fractal dimensions show quite different behaviour as the degeneracy of the critical point is varied therefore although they are sometimes used almost interchangeably as measures of fractal dimension they are actually very different concepts

a valued semicircular systems abstract to a von neurnann algebra a and a set of linear maps ij a a i j i such that a ij ij i a a b l2 i is normal and completely positive we associate a von neumann algebra a this von neumann algebra is generated by a and an a valued semicircular system xi i i associated to in many cases there is a faithful conditional expectation e a a if a is tracial then under certain assumptions on a also has a trace one can think of the construction a as an analogue of a crossed product construction we show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction of a particular interest are free krieger algebras which by analogy with crossed products and ordinary krieger factors are defined to be algebras of the form l 0 1 the cores of free araki woods factors are free krieger algebras we study the free krieger algebras and as a result obtain several non isomorphism results for free araki woods factors as another source of classification results for free araki woods factors we compute the invariant of connes for free products of von neumann algebras this computation generalizes earlier work on computation of t s and sd invariants for free product algebras

a new transition for projections of multifractal measures and random maps typical projections of simple multifractal measures with generalized dimensionsdq onto subspaces of dimensiond are considered it is known that fordo d almost all projections have euclidean support here it is shown that if in additiond increases beyondd a typical projection changes from a singular continuous distribution to an absolutely continuous measure with a squareintegrable or even differentiable density and thus from a multifractal to an ordinary distribution with trivial singularity spectrum since projections of strictly self similar measures can be regarded as invariant distributions of iterated function systems such a transition is found also there and is expected to occur in related systems

characterization of sign singular measures abstract situations are considered in which a physical quantity varies rapidly in space from positive values to negative values down to some cutoff scale typically the cutoff scale is determined by some diffusive process e g viscosity and as the cutoff scale decreases e g viscosity decreases the physical quantity as a function of the spatial coordinate varies more and more rapidly approaching a generalized function as the cutoff scale approaches zero as one examines such a generalized function on smaller and smaller scale it is shown that a certain suitably defined quantity sensitive to the intensity of spatial oscillations from positive to negative can exhibit an algebraic divergence the exponent of this algebraic divergence with scale size is one characterization of this situation and is called the cancellation exponent in the spirit of the multifractal formalism we introduce a spectrum of cancellation exponents yielding a more complete characterization partition function and thermodynamic formalisms are employed we also discuss how to associate dq spectra with these signed generalized functions and discuss the relationship of dq to the cancellation exponent spectrum illustrative examples are presented

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

the dimension of affine invariant fractals a generic formula for the hausdorff and box counting dimensions of self affine fractals is presented extensions to the non linear situation and to repellers in discrete dynamical systems are discussed

the hausdorff dimension of some fractals and attractors of overlapping construction we introduce a method to show that the hausdorff dimension of certain fractals of overlapping construction is almost always what would be observed if no overlapping occurred the method is also used to examine the dimension of attractors in some noninjective piecewise linear baker s type transformations

hausdorff dimension of planar self affine sets and measures let x bigcup varphi i x be a strongly separated self affine set in mathbb r 2 or one satisfying the strong open set condition under mild non conformality and irreducibility assumptions on the matrix parts of the varphi i we prove that dim x is equal to the affinity dimension and similarly for self affine measures and the lyapunov dimension the proof is via analysis of the dimension of the orthogonal projections of the measures and relies on additive combinatorics methods

devil staircase behavior of dynamical invariants in chaotic scattering a crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles the fractal dimension of the resulting chaotic saddle increases through the crisis we present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis our analysis indicates that the fractal dimension and other dynamical invariants are a devil staircase type of function of the system parameter our results can also provide insight for similar devil staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos 2000 elsevier science b v all rights reserved

convergence of the k 2 entropy for random noises with power law spectra abstract we discuss a counter example to the traditional expectation that stochastic processes lead to a non convergence of the k 2 entropy in computed or measured time series we consider a simple class of colored random noises with power law spectra which have already been shown to generate finite and predictable values of the correlation dimension we show that the k 2 entropy of these random noises converges as well these results imply that the observation of a finite or null value of the k 2 entropy in the analysis of data is not enough to infer that the system is dominated by a deterministic process such as low dimensional chaos

chaotic attractors of a locally conservative hyperbolic map with overlap abstract we introduce a class of hyperbolic maps which are locally conservative but globally dissipative the dissipation stems not from local contraction of phase space but from the overlap that arises from the noninvertibility of the map the dynamics of these maps are chaotic and the attractors are strange although not necessarily fractal though several examples of this class are provided our analysis concentrates on a single example which is obtained as a modification of the baker s transform we discuss the morphology of the attractors as the map parameter is varied we find that some attractors are thin fractals and others are fat sets of positive measure we are able to characterize the parameter values that lead to these two kinds of attractors and for certain parameter values we can derive analytically exact values for the fractal dimension or for the fat fractal scaling exponent taken together these results lead us to conjecture that the generic attractor of this map has zero measure and full dimension

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

implementation of two step runge kutta methods for ordinary differential equations we investigate the potential for efficient implementation of two step runge kutta methods tsrk a new class of methods introduced recently by jackiewicz and tracogna for numerical integration of ordinary differential equations the implementation issues addressed are the local error estimation changing stepsize using nordsieck technique and construction of interpolants the numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

a lower estimation of the hausdorff dimension for attractors with overlaps we give a lower estimate of the hausdorff dimension for attractors which can be obtained by an overlapping construction

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

operator valued chordal loewner chains and non commutative probability abstract we adapt the theory of chordal loewner chains to the operator valued matricial upper half plane over a c algebra a we define an a valued chordal loewner chain as a subordination chain of analytic self maps of the a valued upper half plane such that each f t is the reciprocal cauchy transform of an a valued law t such that the mean and variance of t are continuous functions of t we relate a valued loewner chains to processes with a valued free or monotone independent increments just as was done in the scalar case by bauer 8 and schleisinger 48 we show that the loewner equation t f t z d f t z v t z when interpreted in a certain distributional sense defines a bijection between lipschitz mean zero loewner chains f t and vector fields v t z of the form v t z g t z where t is a generalized a valued law based on the loewner equation we derive a combinatorial expression for the moments of t in terms of t we also construct non commutative random variables on an operator valued monotone fock space which realize the laws t finally we prove a version of the monotone central limit theorem which describes the behavior of f t as t when t has uniformly bounded support

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

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

two step runge kutta methods implicit two step runge kutta methods are studied it will be shown that these methods require fewer stages to achieve the same order as one step runge kutta methods which means the two step methods are potentially more efficient than one step methods order conditions are derived and examples of two step one stage methods of order 2 and two step two stage methods of order 4 are presented stability properties of these methods with respect to y ay are studied and a stable two step methods of order 2 are characterized two step two stage methods of order 4 which are a stable are found by an extensive computer search semi implicit two stage methods of order 4 were also constructed this is in contrast to the situation encountered in the runge kutta theory where the unique two stage method of order 4 is not semi implicit

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

applications of singular systems analysis to baroclinic chaos abstract the method of singular systems analysis ssa is applied to the phase portrait analysis of high precision time series of measurements of temperature and total heat transport obtained in a high prandtl number pr 26 fluid contained in a rotating cylindrical annulus subject to a horizontal temperature gradient global ssa is found to be a highly effective means of separating weak high frequency oscillatory components from a signal dominated by strong oscillations at low frequency revealing the possible presence of weak inertia gravity oscillations within a large scale baroclinic wave flow ssa embeddings may fail in practice however for signals modulated in amplitude where the carrier and modulation frequencies differ widely unless a sufficiently long window is used a modification of global ssa with short windows consistent with takens method of delays is proposed and verified using data from a chaotic regime of the rotating annulus which may improve ssa embeddings for such cases local ssa is applied to the estimation of topological dimension in quasi periodic and chaotic flows and produces results consistent with more conventional dimension measures

determining the number of integrals of motion by an adapted correlation dimension method an adapted correlation dimension algorithm is used to numerically determine the number of integrals of motion in a variety of conservative classical systems the method is demonstrated on three sample systems that display various degrees of integrable and chaotic behavior the henon heiles hamiltonian an asymmetric top molecule in an electric field and the planetary system hd128311 two additional applications of the method emerge using the adapted correlation dimension algorithm a to study partial barriers and turnstiles in phase space and b to predict the long time stability of planetary systems using short time data

chaos transition of soliton motion on a microstructured lattice abstract soliton motion on a damped driven one dimensional lattice is proposed the lattice model is made of a one dimensional chain equipped at each point with a rotatory molecule the set of coupled difference differential equations for the molecule rotation is obtained for the discrete system using a semi discrete kink solution for which its center of mass is considered as a collective variable we derive the equation of motion for the kink location we show that the kink location is then governed by the equation of the damped sinusoidally driven pendulum when the first order corrections are dropped by means of numerical simulations carried out directly on the lattice model the complex dynamics of the kink location is investigated and compared to that of the well known forced pendulum for some ranges of the control parameters the numerical simulations exhibit a cascade of odd subharmonic components as well as strange attractors which characterizes the chaotic behavior of the kink motion these results agree fairly well with the main features of the transition to chaos for the damped driven pendulum

the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator abstract using recently developed techniques of wiggins 1988 we study the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator we show how homoclinic tori give rise to chaotic dynamics for single degree of freedom quasiperiodically forced oscillators in much the same way as smale horseshoes in the periodically forced case we give a complete description of the bifurcation set in the five dimensional parameter space for the two frequency forced duffing oscillator and then show how the two frequency results can be used to understand the effects of an arbitrary but finite number of frequencies

a nonlinear dynamical systems analysis of fricative consonants acoustic waveforms of the strident fricatives s z sh and yog spoken by two native american english speakers are analyzed using modern chaotic analysis techniques fricative data are extracted from both intervocalic and sustained utterances for comparison acoustic waveforms of the vowels a i and u are also analyzed for 44 of the unvoiced fricative tokens in vcv contexts and 59 of the sustained voiced fricatives indications of low dimensional dynamics could be found with the given limitations of stationarity the low dimensional chaotic behavior is exhibited by a correlation dimension d2 ranging between 3 and 7 2 and by positive maximum lyapunov exponents les for the remaining fricatives results suggest that the dimensional complexity therein is greater than the maximum d2 value that could be reliably estimated from the available data about 7 8 for the intervocalic cases and 9 for the sustained cases intervocalic voiced fricatives are excluded from the analysis due to stat

stability analysis of two step runge kutta methods for delay differential equations abstract we investigate stability properties of two step runge kutta methods with respect to the linear test equation y t ay t by t t 0 y t g t t 0 where a and b are complex parameters it is known that the solution y t to this equation tends to zero as t if b a we will show that under some conditions this property is inherited by any a stable two step runge kutta method applied on a constrained mesh to delay differential equations i e that the corresponding method is p stable

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

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

extreme phase sensitivity in systems with fractal isochrons abstract sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors however even systems with quasi periodic dynamics can exhibit it in this context we report on the fractal properties of the isochrons of some continuous time asymptotically periodic systems we define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons similar results are also obtained with discrete time systems as an illustration of the framework we compute the phase sensitivity coefficient for popular models of bursting neurons suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive unreliable phase response

inverse problems in fractal construction moment method solution abstract the inverse problem for several classes of fractals in one and two dimensions can be profitably solved via the application of the classical theory of moments this new perspective leads to important results on the reconstruction process when only a finite number of moments of the fractal measures are known while discussing the rigorous aspects of the theory we also implement it via efficient numerical algorithms

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

equivalence of nonequilibrium ensembles for simple maps abstract in this paper we discuss the equivalence of different approaches to the same transport problems with particular attention to recent works by breyman et al these suggest a singular equivalence between the stationary states of models from gaussian dynamics and those from chaotic scattering theory although the mentioned equivalence may concern but a limited set of the properties of those systems it is very important from a theoretical standpoint we argue that a special form of dissipativity in the evolution equations of the phase space distributions is to be invoked to understand those facts we discuss one application of periodic orbit theory to illustrate how the stationary states of thermostatted systems can be investigated and some qualitative features obtained

bifurcations of lattice structures equilibrium configurations of a model for a crystal in one spatial dimension with anharmonic and up to third neighbour harmonic interactions are related to orbits in a two or four dimensional space under a nonlinear symplectic mapping in this way a connection is given with dynamical systems with two or three degrees of freedom depending on the parameters the orbits are smooth or stochastic their characterisation in terms of their fixed points and their ljapunov exponents is given the mapping shows a large number of bifurcations of which one can distinguish various types and these are discussed

construction of two step runge kutta methods with large regions of absolute stability we describe the construction of explicit two step runge kutta methods of order p and stage order q p 1 or q p with large regions of absolute stability this process is illustrated for the methods of order p 2 and 3 and leads to new methods which are competitive with explicit runge kutta methods of the same order

η series and a boolean bercovici pata bijection for bounded k tuples let dc k be the space of non commutative distributions of k tuples of selfadjoint elements in a c probability space on dc k one has an operation of free additive convolution and one can consider the subspace dcinf div k of distributions which are infinitely divisible with respect to this operation the linearizing transform for is the r transform one has r r r dc k we prove that the set of r transforms r dcinf div k can also be described as dc k where for dc k we denote m 1 m with m the moment series of the series is the counterpart of r in the theory of boolean convolution as a consequence one can define a bijection b dc k dcinf div k via the formula i rb dc k we show that b is a multi variable analogue of a bijection studied by bercovici and pata for k 1 and we prove a theorem about convergence in moments which parallels the bercovici pata result on the other hand we prove the formula ii b b b with considered in a space dalg k dc k where the operation of free multiplicative convolution always makes sense an equivalent reformulation of ii is that iii dalg k where is an operation on series previously studied by nica and speicher and which describes the multiplication of free k tuples in terms of their r transforms formula iii shows that in a certain sense series behave in the same way as r transforms in connection to the operation of multiplication of free k tuples of non commutative random variables

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

spectral properties of polynomials in independent wigner and deterministic matrices abstract on the one hand we prove that almost surely for large dimension there is no eigenvalue of a hermitian polynomial in independent wigner and deterministic matrices in any interval lying at some distance from the supports of a sequence of deterministic probability measures which is computed with the tools of free probability on the other hand we establish the strong asymptotic freeness of independent wigner matrices and any family of deterministic matrices with strong limiting distribution

spatial chaos in weakly dispersive and viscous media a nonperturbative theory of the driven kdv burgers equation abstract the asymptotic travelling wave solution of the kdv burgers equation driven by the long scale periodic driver is constructed the solution represents a shock train in which the quasi periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions it is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes two types of chaotic behavior are considered the first type is a weak chaos where only a small chaotic deviation from the periodic solution occurs the second type corresponds to the developed chaos where the solution ignores the driver period and represents a random sequence of uncorrelated shocks in the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor while in the case of developed chaos it moves on a repeller both solutions depend on a parameter indicating the reference shock position in the shock train the structure of a one dimensional set to which this parameter belongs is investigated this set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos the capacity dimension of this set is calculated

numerical solutions of fuzzy differential equations by an efficient runge kutta method with generalized differentiability in this paper an extended fourth order runge kutta method is studied to approximate the solutions of first order fuzzy differential equations using a generalized characterization theorem in this method new parameters are utilized in order to enhance the order of accuracy of the solutions using evaluations of both f and f instead of using the evaluations of f only the proposed extended runge kutta method and its error analysis which guarantees pointwise convergence are given in detail furthermore the accuracy and efficiency of the proposed method are demonstrated in a series of numerical experiments

indications of nonlinearity intraindividual specificity and stability of human eeg the unfolding dimension abstract with the help of a parameter which we call the unfolding dimension ww provide a bivariate representation of dynamical eeg analysis applied to human eeg our approach successfully discriminates surrogate data from raw eeg and similarly shows human eeg to be both intraindividually specific and stable over time the heart of our approach is the grassberger procaccia algorithm for the determination of the correlation dimension d 2 within the context of the well known method of time delays to guarantee the reproducibility of results this algorithm as well as an estimate for the k 2 entropy and the determination of f spectra have been integrated into a computer program which encompasses an operator user independent automatic and reproducible specification of both an optimal time delay for calculating the correlation integral as well as an optimal scalar invariant plateau region for the extraction of d 2 and k 2 our embedding protocol applied to mathematical systems is shown to be consistent with findings based on other algorithms in particular with false nearest neighbor considerations and sauer s minimal embedding criteria particular intelligent features of the automation and optimization logic plus the inclusion of estimates of the inherent systematic error make the proposed algorithm especially appropriate for applications to biological psychological data

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

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

full characterization of a strange attractor chaotic dynamics in low dimensional replicator system abstract two chaotic attractors observed in lotka volterra equations of dimension n 3 are shown to represent two different cross sections of one and the same chaotic regime the strange attractor is studied in the equivalent four dimensional catalytic replicator network analytical expressions are derived for the lyapunov exponents of the flow in the centre of the chaotic regime the strange attractor was characterized by the lyapunov dimension dl 2 06 0 02 and the renyi fractal dimensions the set corresponding to the attractor represents a multifractal its singularity spectrum is computed one route in parameter space leading into the chaotic regime and crossing it was studied in detail it leads to a feigenbaum type cascade of bifurcations before the chaos is fully developed the dynamic system passes an internal crisis through a period two tangent bifurcation then a second sequence of period doubling bifurcations leading to another chaotic regime is observed which eventually ends in a crisis and then after a rather complicated periodic regime the attractor finally disappears trajectories in the intermediate periodic regime show transient chaotic bursts of intermittency type a series of one dimensional maps is derived from a properly chosen poincare cross section which illustrates structural changes in the attractor mutations are included in the catalytic replicator network and the changes in the dynamics observed are compared with the predictions of an approach based on perturbation theory the most striking result is the gradual disappearance of complex dynamics with increasing mutation rates the parameter space spanned by the mutation rate and selected parameters of the replication network is split in regions of complex periodic behaviour near the chaotic regime fractal boundaries of these regions are likely to occur but the data available do not allow definitive conclusions yet

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

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

analytic subordination for bi free convolution in this paper we study some analytic properties of bi free additive convolution both scalar and operator valued we show that using properties of voiculescu s subordination functions associated to free additive convolution of operator valued distributions simpler formulas for bi free convolutions can be derived we use these formulas in order to prove a result about atoms of bi free additive convolutions

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

dimension of interaction dynamics a method allowing one to distinguish interacting from noninteracting systems based on available time series is proposed and investigated some facts concerning generalized renyi dimensions that form the basis of our method are proved we show that one can find the dimension of the part of the attractor of the system connected with interaction between its parts we use our method to distinguish interacting from noninteracting systems on the examples of logistic and henon maps a classification of all possible interaction schemes is given

nearest neighbor analysis of a family of fractal distributions we use a central limit theorem for entropy due to ibragimov to obtain limit theorems for linear normalizations of the log minimum distance when observations are sampled from measures belonging to a family of fractal distributions it is shown that in almost all cases the limit distribution is gaussian with parameters determined in part by the hausdorff dimension associated with the underlying measure exceptions to this rule include absolutely continuous measures which obey the classical extreme value limit laws

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

infinite divisibility and a non commutative boolean to free bercovici pata bijection abstract we use the theory of fully matricial or non commutative functions to investigate infinite divisibility and limit theorems in operator valued non commutative probability our main result is an operator valued analogue for the bercovici pata bijection an important tool is voiculescu s subordination property for operator valued free convolution

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

geometrical exponents of contour loops on synthetic multifractal rough surfaces multiplicative hierarchical cascade p model in this paper we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces which are generated by multiplicative hierarchical cascading processes to this end two different classes of multifractal rough surfaces are numerically simulated as the first group singular measure multifractal rough surfaces are generated by using the p model the smoothened multifractal rough surface then is simulated by convolving the first group with a so called hurst exponent h the generalized multifractal dimension of isoheight lines contours d q correlation exponent of contours x l cumulative distributions of areas and perimeters are calculated for both synthetic multifractal rough surfaces our results show that for both mentioned classes hyperscaling relations for contour loops are the same as that of monofractal systems in contrast to singular measure multifractal rough surfaces h plays a leading role in smoothened multifractal rough surfaces all computed geometrical exponents for the first class depend not only on its hurst exponent but also on the set of p values but in spite of multifractal nature of smoothened surfaces second class the corresponding geometrical exponents are controlled by h the same as what happens for monofractal rough surfaces

the periodicity of chaotic impact oscillators in hausdorff phase spaces it is well known that non periodic behavior is one of the most puzzling characteristics of chaotic oscillators so far chaotic dynamical systems have been investigated in euclidean spaces in this paper the concept of non autonomous dynamical systems and that of hausdorff phase spaces are proposed the behavior of chaotic impact oscillators is investigated in hausdorff phase spaces it is discovered that although the non autonomous dynamical systems described by chaotic impact oscillators are non periodic in euclidean phase spaces they are periodic in hausdorff phase spaces this shows that euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena in addition the periodicity of chaotic dynamical systems in hausdorff metric spaces induces a new class of strange invariant sets in euclidean spaces such strange invariant sets may be an ideal symbol of chaotic dynamical systems

on determining the dimension of chaotic flows abstract we describe a method for determining the approximate fractal dimension of an attractor our technique fits linear subspaces of appropriate dimension to sets of points on the attractor the deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques we show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time varying quantity is available for analysis these methods are applied to several dissipative dynamical systems

pseudofractals and the box counting algorithm we show that for sets with the hausdorff besicovitch dimension equal to zero the box counting bc algorithm commonly used to calculate the renyi exponents dq can exhibit perfect scaling suggesting non zero dq the properties of these pathological sets pseudofractals are investigated numerical as well as analytical estimates for dq are obtained a simple indicator is given to distinguish pseudofractals and fractals in practical applications of the bc method histograms made of pseudofractal sets are shown to have pareto tails