a new method to detect nonlinearity in a time series synthesizing surrogate data using a kolmogorov smirnoff tested hidden markov model abstract a way of statistically testing for nonlinearity in a time series is to employ the method of surrogate data this method often makes use of the fourier transform ft in order to generate the surrogate as various authors have shown this can lead to artefacts in the surrogates and spurious detection of nonlinearity can result this paper documents a new method to synthesize surrogate data using a 1st order hidden markov model hmm combined with a kolmogorov smirnoff test ks test to determine the required resolution of the hmm significance test results for a sinewave henon map and gaussian noise time series are presented it is demonstrated that ks tested hmm surrogates can be successfully used to distinguish between a deterministic and stochastic time series then by applying a simple test for linearity using linear and nonlinear predictors it is possible to determine the nature of the deterministic class and hence conclude whether the system is linear deterministic or nonlinear deterministic furthermore it is demonstrated that the method works for periodic functions too where ft surrogates break down

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

evaluation of missing ordinal pattern and its fractional distribution entropy abstract the number of missing ordinal patterns nmp receives extensive attention in this paper we change the perspective and study nmp about time delay and multi scale theory we first highlight the surrogate framework to see if the nmp for a series is statistically different from that of iterative amplitude adjusted fourier transform iaaft surrogates as time delay and scales changes furthermore by means of the fractional calculus we combine nmp with distribution entropy to study the fractional distribution entropy with nmp in order to show the advantages of these methods several simulated data and electrocardiograph ecg signals are chosen to examine the performance of them through experiments we find that nmp about time delay and multi scale theory shows an excellent characteristic in the nonlinearity analysis of different time series especially when applied to ecg signals moreover the results of the fractional distribution entropy show that it can explore the complexity of dynamical systems effectively

a simple homotopy method for determining all isolated solutions to polynomial systems a new homotopy method for solving systems of polynomial equations is presented the homotopy equation is extremely simple it is linear with respect to the homotopy parameter and only one auxiliary parameter is needed to regularize the problem within some limits an arbitrary starting problem can be chosen as long as its solution set is known no restrictions on the polynomial systems are made a few numerical tests are reported which show the influence of the auxiliary parameter resp the starting problem upon the computa tional cost of the method

fractional kinetic equations and universal decoupling of a memory function in mesoscale region it is proved that kinetic equations containing non integer integrals and derivatives appear in the result of reduction of a set of micromotions to some averaged collective motion in the mesoscale region in other words it means that after a proper statistical average the microscopic dynamics is converted into a collective motion in the mesoscopic regime a fractal medium containing weakly and strongly correlated relaxation units has been considered it is shown that in most cases the original of the memory function recovers the riemann liouville integral for a strongly correlated fractal medium a generalization of the riemann liouville integral is obtained for the fractal branched processes one can derive the stretched exponential law of relaxation that is widely used for description of relaxation phenomena in disordered media it is shown that the generalized stretched exponential function describes the averaged collective motion in the fractal branched complex systems it is confirmed that the generalized stretched exponential law can be definitely applicable for description of a set of spontaneous evoked postsynaptic signals in the living matter the application of these fractional kinetic equations for the description of the dielectric relaxation phenomena is also discussed these kinetic equations and their generalizations can be applicable for the description of different relaxation or diffusion processes in the intermediate mesoscale region

applying the method of surrogate data to cyclic time series the surrogate data methodology is used to test a given time series for membership of specific classes of dynamical systems currently there are three algorithms that are widely applied in the literature the most general of these tests the hypothesis of nonlinearly scaled linearly filtered noise however these tests and the many extensions of them that have been suggested are inappropriate for data exhibiting strong cyclic components for such data it is more natural to ask if there exist any long term of period longer than the data cycle length determinism in this paper we discuss existing techniques that attempt to address this hypothesis and introduce a new approach this new approach generates surrogates that are constrained i e they look like the data and for cyclic time series tests the null hypothesis of a periodic orbit with uncorrelated noise we examine various alternative implementations of this algorithm applying it to a variety of known test systems and experimental time series with unknown dynamics 2002 elsevier science b v all rights reserved

dynamics of the fractional oscillator abstract the integral equation of motion of a simple harmonic oscillator is generalized by taking the integral to be of arbitrary order according to the methods of fractional calculus to yield the equation of motion of a fractional oscillator the solution is obtained in terms of mittag leffler functions using laplace transforms the expressions for the generalized momentum and the total energy of the fractional oscillator are also obtained numerical application and the phase plane representation of the dynamics are discussed

learning chaotic attractors by neural networks an algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series during training the algorithm learns to short term predict the time series at the same time a criterion developed by diks van zwet takens and de goede 1996 is monitored that tests the hypothesis that the reconstructed attractors of model generated and measured data are the same training is stopped when the prediction error is low and the model passes this test two other features of the algorithm are 1 the way the state of the system consisting of delays from the time series has its dimension reduced by weighted principal component analysis data reduction and 2 the user adjustable prediction horizon obtained by error propagation partially propagating prediction errors to the next time step the algorithm is first applied to data from an experimental driven chaotic pendulum of which two of the three state variables are known this is a comprehensive example that shows how well the diks test can distinguish between slightly different attractors second the algorithm is applied to the same problem but now one of the two known state variables is ignored finally we present a model for the laser data from the santa fe time series competition set a it is the first model for these data that is not only useful for short term predictions but also generates time series with similar chaotic characteristics as the measured data

pseudozeros of multivariate polynomials the pseudozero set of a system f of polynomials in n complex variables is the subset of cn which is the union of the zero sets of all polynomial systems g that are near to f in a suitable sense this concept is made precise and general properties of pseudozero sets are established in particular it is shown that in many cases of natural interest the pseudozero set is a semialgebraic set also estimates are given for the size of the projections of pseudozero sets in coordinate directions several examples are presented illustrating some of the general theory developed here finally algorithmic ideas are proposed for solving multivariate polynomials

maximum likelihood estimation of the entropy of an attractor in this paper a maximum likelihood estimate of the kolmogorov entropy of an attractor is proposed that can be obtained directly from a time series also the relative standard deviation of the entropy estimate is derived it is dependent on the entropy and on the number of samples used in the estimation

table errata higher transcendental functions vol iii mcgraw hill new york 1955 mr 16 586 by a erdelyi w magnus f oberhettinger and f g tricomi

kolmogorov smirnov test distinguishes attractors with similar dimensions recent advances in nonlinear dynamics have led to more informative characterizations of complex signals making it possible to probe correlations in data to which traditional linear statistical and spectral analyses were not sensitive many of these new tools require detailed knowledge of small scale structures of the attractor knowledge that can be acquired only from relatively large amounts of precise data that are not contaminated by noise not the kind of data one usually obtains from experiments there is a need for tools that can take advantage of coarse grained information but which nevertheless remain sensitive to higher order correlations in the data we propose that the correlation integral now much used as an intermediate step in the calculation of dimensions and entropies can be used as such a tool and that the kolmogorov smirnov test is a convenient and reliable way of comparing correlation integrals quantitatively this procedure makes it possible to distinguish between attractors with similar dimensions for example it can unambiguously distinguish p 10 mathrm ensuremath 8 the lorenz r ossler and mackey glass delay 17 attractors whose correlation dimensions are within 1 of each other we also show that the kolmogorov smirnov test is a convenient way of comparing a data set with its surrogates

long range cohesive interactions of non local continuum faced by fractional calculus abstract a non local continuum model including long range forces between non adjacent volume elements has been studied in this paper the proposed continuum model has been obtained as limit case of two fully equivalent mechanical models i a volume element model including contact forces between adjacent volumes as well as long range interactions distance decaying between non adjacent elements ii a discrete point spring model with local springs between adjacent points and non local springs with distance decaying stiffness connecting non adjacent points under the assumption of fractional distance decaying interactions between non adjacent elements a fractional differential equation involving marchaud type fractional derivatives has been obtained for unbounded domains it is shown that for unbounded domains the two mechanical models revert to lazopoulos and eringen model with fractional distance decaying functions it has also been shown that for a confined bar the stress strain relation is substantially different from that obtained simply using the truncated marchaud derivatives since a double integral instead of convolution integral appears moreover in the analysis of bounded domains the governing equations turn out to an integro differential equation including only the integral part of marchaud fractional derivatives on finite interval the mechanical boundary condition for the proposed model has been introduced consistently on the basis of mechanical considerations and the constitutive law of the proposed continuum model has been reported by mathematical induction several numerical applications have been reported to show verify and assess the concepts listed in this paper

on the hausdorff dimension of fractal attractors we consider such mappingsx n 1 f xn of an interval into itself for which the attractor is a cantor set for the same class of mappings for which the feigenbaum scaling laws hold we show that the hausdorff dimension of the attractor is universally equal tod 0 538

fractional oscillator driven by a gaussian noise abstract the time dependent response is studied of a fractional oscillator driven by a gaussian white noise applying the method of characteristic functionals combined with the technique of integral transforms we derive explicit formulas for the variances of position and velocity and the asymptotic power spectrum of the oscillator the effects of the order of fractional derivative on statistical characteristics of the oscillator are analyzed numerically

the multiscale character of evoked cortical activity abstract both the architecture and the dynamics of the brain have characteristic features at different spatial scales however the existence nature and function of dynamical interdependencies between such scales have not been investigated we studied the multiscale properties of functional magnetic resonance imaging fmri data acquired while human subjects viewed a visual image traditional region of interest analysis of this data set revealed evoked activity in primary and extrastriate visual cortex wavelet transform in the spatial domain provides a multiscale representation of this evoked brain activity studying the correlation structure of this representation revealed strong and novel interdependencies in these data within and between different spatial scales we found that such correlations are stronger than those evident in the original data and comparable in magnitude to those obtained after gaussian smoothing however analysis of the data in the wavelet domain revealed additional structure such as positive correlations strong anti correlations and phase lagged interdependencies statistical significance of these effects was inferred through nonparametric bootstrap techniques we conclude that the spatial analysis of functional neuroimaging data in the wavelet domain provides novel information which may reflect complex spatiotemporal neuronal activity and information encoding it also affords a quantitative means of testing hierarchical and multiscale models of cortical activity

reliable detection of nonlinearity in experimental time series with strong periodic components abstract testing with phase randomised surrogate signals has been used extensively to search for interesting nonlinear dynamical structure in experimental time series in this paper we argue that in the case of experimental time series with strong periodic components the method of phase randomised surrogate data may not be particularly suitable to test for nonlinearity since construction of such surrogates by fft requires a time series whose length is a power of 2 we demonstrate that in the case of nearly periodic signals this approach will almost always produce spurious detection of nonlinearity this error can be fixed by adjusting the length of the time series such that it becomes an integer multiple of the dominant periodicity the resulting time series will not be a power of 2 and requires the use of a dft to generate surrogate data dft based surrogates no longer detect spurious nonlinearity but cannot be used to detect periodic nonlinearity we propose a new test nonlinear cross prediction nlcp which avoids some of the problems associated with phase randomised surrogate data and which allows reliable detection of both periodic and aperiodic nonlinearity in the test the original data are used to construct a nonlinear model to predict the original data set as well as amplitude inverted and time reversed versions of the original data lower predictability of the amplitude inverted or time reversed copies reflect respectively an asymmetric amplitude distribution and time irreversibility both of these indicate nonlinearity in the data set

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

on the detection of determinism in a time series the nonlinearly scaled distributions of the strengths of the orthogonal modes in the data of a time series are compared with that derived from its surrogate counterpart to assess its chaoticity or the stochastic nature chaoticity manifests in the relatively decreasing strengths of the weaker modes with increasing dimension of the orthogonal spaces mapping the process the distributions of the strengths of the modes for the data and the surrogate are compared statistically singular value decomposition is used for the implementation the ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated series including linear and nonlinear stochastic noise and processes described by lorenz rossler and mackey glass equations henon map and ecological map and some real life examples

using surrogate data to detect nonlinearity in time series we address the issue of reliably discriminating between chaos and noise from a time series in particular we are interested in avoiding claims of chaos when simpler models such as linearly correlated noise can explain the data we take a statistical approach and use a form of bootstrapping to detect nonlinearity by showing that a given linear model is unlikely to have produced the data our method requires the careful statement of a null hypothesis which characterizes a candidate linear process the generation of an ensemble of surrogate data sets which are similar to the original time series but consistent with the null hypothesis and the computation of a discriminating statistic for the original and for each of the surrogate data sets the idea is to test the original time series against the null hypothesis by checking whether the discriminating statistic computed for the original time series differs significantly from the statistics computed for each of the surrogate sets we present algorithms for generating surrogate data under various null hypotheses and we show the results of numerical experiments on artificial data using correlation dimension lyapunov exponent and forecasting error as discriminating statistics finally we consider a number of experimental timemore series including sunspots electroencephalogram eeg signals and fluid convection and evaluate the statistical significance of the evidence for nonlinear structure in each case 52 refs 10 figs less

response characteristics of a fractional oscillator abstract the integral equation of motion of a driven fractional oscillator is obtained by generalizing the corresponding equation of motion of a driven harmonic oscillator to include integrals of arbitrary order according to the methods of fractional calculus the green s function solution for the fractional oscillator is obtained in terms of mittag leffler functions using laplace transforms the response and resonance characteristics of the fractional oscillator are studied for several cases of forcing function

error covariance matrix estimation of noisy and dynamically coupled time series we estimate the covariance matrix of the errors in several dynamically coupled time series corrupted by measurement errors we say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system the estimation of the covariance matrix of the errors is made using a noise reduction algorithm that efficiently exploits the information contained jointly in the dynamically coupled noisy time series the method is particularly powerful for short length time series with high uncertainties

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

nonholonomic constraints with fractional derivatives we consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples the corresponding equations of motion are derived using variational principle we prove that fractional constraints can be used to describe the evolution of dynamical systems in which some coordinates and velocities are related to velocities through a power law memory function

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

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

testing for nonlinearity in irregular fluctuations with long term trends we describe a method for investigating nonlinearity in irregular fluctuations short term variability of time series even if the data exhibit long term trends periodicities such situations are theoretically incompatible with the assumption of previously proposed methods the null hypothesis addressed by our algorithm is that irregular fluctuations are generated by a stationary linear system the method is demonstrated for numerical data generated by known systems and applied to several actual time series

pseudozeros of polynomials and pseudospectra of companion matrices it is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix a in this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding the epsilon pseudozero set z epsilon p is the set of zeros of all polynomials hat p obtained by coefficientwise perturbations of p of size leq epsilon this is a subset of the complex plane considered earlier by mosier and is bounded by a certain generalized lemniscate the epsilon pseudospectrum lambda epsilon a is another subset of bbb c defined as the set of eigenvalues of matrices hat a a e with vert e vert leq epsilon it is bounded by a level curve of the resolvent of a we find that if a is first balanced in the usual eispack sense then z epsilon vert p vert p and lambda epsilon vert a vert a are usually quite close to one another it follows that the matlab roots algorithm of balancing the companion matrix then computing its eigenvalues is a stable algorithm for polynomial zerofinding experimental comparisons with the jenkins traub imsl and madsen reid harwell fortran codes confirm that these three algorithms have roughly similar stability properties

absence of long range order in one dimensional spin systems for a one dimensional ising model with interaction energy e left mu right sum limits 1 leqslant i j leqslant n j j i mu iota mu j left j k geqslant 0 mu iota pm 1 right it is proved that there is no long range order at any temperature when s n sum limits k 1 n kj left k right o left log n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right the same result is shown to hold for the corresponding plane rotator model when s n o left left log n mathord left vphantom log n log log n right kern nulldelimiterspace log log n right right

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

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

root neighborhoods of a polynomial the root neighborhoods of p z a polynomial over the complex field are the sets of complex numbers that are the roots of polynomials which are near to p z the term near means that the coefficients of the polynomials are within some fixed magnitude of the coefficients of p z a necessary and sufficient condition for a complex number to be in the root neighborhoods is given and it is proved that each root neighborhood contains at least one root of p z and the same number of roots of each near polynomial finally a necessary and sufficient condition is given for a root neighborhood to contain more than one root of p z and consequently more than one root of any of the near polynomials

conditional entropies phase synchronization and changes in the directionality of information flow in neural systems we devised a novel measure that dynamically evaluates temporal interdependences between two coupled units based on the properties of the distributions of their relative interevent intervals we investigate its properties on the system of two coupled non identical r ossler oscillators and a system of non identical hindmarsh rose models of thalamocortical neurons and show that the measure highlights the properties of phase synchronization observed in those two systems we postulate that the observed properties of the phase lag in conjunction with the experimentally observed activity dependent synaptic modification in the neural systems may drive the changes of the direction of information flow in a neural network and thus the measure can play an important role in assessing those changes

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

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

adaptation of nonlinear systems through dynamic entropy estimation we show that adaptation and control of the target system can be achieved by linking the modification of its dynamical properties to the estimated difference of the distribution of the dynamical characteristics of the control and target signals subsequently the target system which has initially different dynamical properties adjusts its dynamics via changes of its control parameters and synchronizes with the control one the differences in the evolving probability distributions are evaluated through entropy estimation causing the adaptation to be based solely on the statistical properties of the control and target signals without explicit knowledge of the underlying equations of the system

simulation of fluidized beds with lattice gas cellular automata this paper introduces an approach for the simulation of the hydrodynamic behaviour of gas solid fluidized beds via the use of lattice gas cellular automata this approach is based on a two speed model developed by u frisch b hasslacher and y pomeau simulation runs for different configurations of the automaton produce results that can be compared to actual data the simulations show when and how bubbling will occur values for the bubble diameters as a function of bed height as well as bed porosities in two horizontal planes have been obtained from the simulations from these results correlation diagrams and the kolmogorov entropy are calculated generally the results of the simulations are qualitatively in good agreement with experimental observations showing that this new approach could provide a useful tool in predicting the fluid dynamic behaviour of fluidized beds

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

damping characteristics of a fractional oscillator abstract a study of sinusoidally forced oscillations of a fractional oscillator shows that the system exhibits a rich variety of damping characteristics while some aspects of the damping mimic the characteristic features of a damped harmonic oscillator there are others which do not find any parallel in the damped harmonic oscillator system it is clearly demonstrated that the free and forced oscillations of a fractional oscillator are characterized by different damping parameters while both depend on the fractional index the free oscillation damping depends on the natural frequency 0 of the oscillator the forced oscillation damping depends in addition on the driving frequency furthermore there is a different power law tail associated with each of these cases

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

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

lorenz like systems and classical dynamical equations with memory forcing an alternate point of view for singling out the origin of chaos an alternate view for the emergence of chaos in lorenz like systems is presented in this paper for such purpose the lorenz problem is reformulated in a classical mechanical form and it turns out to be equivalent to the problem of a damped and forced one dimensional motion of a particle in a two well potential with a forcing term depending on the memory of the particle past motion the dynamics of the original lorenz system in the proposed particle phase space can then be rewritten in terms of a one dimensional first exit time problem the emergence of chaos turns out to be due to the discontinuous solutions of the transcendental equation ruling the time for the particle to cross the intermediate potential wall the whole problem is tackled analytically deriving a piecewise linearized lorenz like system that preserves all the essential properties of the original model