self similar set identification in the time scale domain we study the wavelet transform of deterministic self similar signals and derive their properties as well as a new algorithm for identification of the self similarity parameter we also include such applications as characterization and analysis of real chaotic signals in the presence of additive noise

nonlinear structures in electroencephalogram signals abstract we apply a nonlinear prediction algorithm to investigate the presence of nonlinear structure in electroencephalogram eeg recordings the eeg signal could be modeled as a realization of a nonlinear model plus a residual noise uncorrelated gaussian noise using linear and nonlinear models we analyze the statistical nature of these residual noises in the case of epileptic patients and normal subjects we found that the residual noise presents gaussian distribution for epileptic patients if a nonlinear model is used whereas in the case of normal subjects the residual noise will exhibit a gaussian distribution only if a linear model autoregressive is used these results provide another evidence of the nonlinear character of the epileptic seizure recordings while the normal eeg seems to be better described as linearly correlated noise

dimension spectra of fractal measures from uniform partitions and correlation integrals the spectra of generalized dimensions dq and of local exponents f for fractal measures are evaluated by using the uniform partitions to compute the free energy the numerical results obtained from optimal algorithms are compared with the analytical results obtained from the free energy evaluated with dynamical partitions in the case of ifs measures it is proved that the spectra dq obtained from correlation integrals and dynamical partitions are the same even for q 1 and for q 1 only if the support of the measure is not fractal or if the dynamical partitions are a subset of the uniform partitions the spectra obtained from a numerical approximation of the correlation integrals provide the correct result for any value of q the algorithms based on the uniform partitions are fast and can be used for real time analysis of digitized images

uniformity of the phase space and fluctuations in thermal equilibrium abstract general relations are found between the measure of the uniformity of distributions on the phase space and the first moments and correlations of extensive variables for systems close to thermal equilibrium the role played by the parameter of the renyi entropy for the analysis of their fluctuations and correlations is studied analytical results are verified and illustrated by direct simulations of quantum systems of ideal fermions and bosons problems of finite statistics usual in experiments and simulations are addressed and discussed and solved by finding unbiased estimators for renyi entropies and uniformities

nonlinear dynamics in driven damped sine gordon systems abstract many quasi one and two dimensional systems can be modeled by sine gordon equations illustrations are given for driven and damped sine gordon equations of behaviors that are typical of ways that complexity in time and space can be manifested examples include an ac response to a dc driver and spontaneous pattern formation and low dimensional chaos under the influence of ac or ac dc drivers the importance of the coexistence of coherence and chaos for transport phenomena is emphasized

nonlinear analysis of river flow time sequences within the field of chaos theory several methods for the analysis of complex dynamical systems have recently been proposed in light of these ideas we study the dynamics which control the behavior over time of river flow investigating the existence of a low dimension deterministic component the present article follows the research undertaken in the work of porporato and ridolfi 1996a in which some clues as to the existence of chaos were collected particular emphasis is given here to the problem of noise and to nonlinear prediction with regard to the latter the benefits obtainable by means of the interpolation of the available time series are reported and the remarkable predictive results attained with this nonlinear method are shown

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

dynamical complexity detection in short term physiological series using base scale entropy physiological systems generate complex fluctuations in their output signals that reflect the underlying dynamics the base scale entropy method was proposed as a complexity measure to investigate the complexity of time series the advantages of this method are simplicity and extremely fast calculation for very short data sets this method enables analyzing very short nonstationary and noisy data sets we employed this method for short term physiological time series for analysis of heart rate variability signals the results show that the simple and easily calculated measure can effectively detect the complexity dissimilarity of physiological time series in different physiological or pathological states which is convenient for clinical applications

a technique for measuring fractal dimensions from time series on a real time scale abstract a dedicated electronic instrument for measuring fractal dimension of chaotic systems called dimensiometer is described the pointwise correlation dimension can be obtained from experimental time series on a real time scale this technique employs a discrete sequence of the time intervals between the successive intersections of a single variable x t and an arbitrary level c the dimensiometer is tested by means of a simple electronic chaos oscillator the experimental results are compared with the numerical calculations carried out for the corresponding dynamical model

a comparison of correlation and lyapunov dimensions this paper investigates the relation between the correlation d2 and the kaplan yorke dimension dky of three dimensional chaotic flows besides the kaplan yorke dimension a new lyapunov dimension d derived using a polynomial interpolation instead of a linear one is compared with dky and d2 various systems from the literature are used in this analysis together with some special cases that span a range of dimension 2 dky 3 a linear regression to the data produces a new fitted lyapunov dimension of the form dfit 1 3 where 1 and 3 are the largest and smallest lyapunov exponents respectively this form correlates better with the correlation dimension d2 than do either dky or d additional forms of the fitted dimension are investigated to improve the fit to d2 and the results are discussed and interpreted with respect to the kaplan yorke conjecture 2004 elsevier b v all rights reserved

a scaling law how an attractor s volume depends on noise level abstract we investigate the meaning of the dimension of a strange attractor for systems with noise more specifically we investigate the effect of adding noise of magnitude e to a deterministic system with d degrees of freedom if the attractor has dimension d and d d then its volume is zero the addition of noise may be an important physical probe for experimental situations useful for determining how much of the observed phenomena in a system is due to noise already present when the noise is added the attractor a e has positive volume we conjecture that the generalized volume of a e is proportional to e d d for e near 0 and show this relationship is valid in several cases for chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly

pricing american asian options with higher moments in the underlying distribution we develop a modified edgeworth binomial model with higher moment consideration for pricing american asian options with lognormal underlying distribution for benchmark comparison our algorithm is as precise as that of chalasani et al p chalasani s jha f egriboyun a varikooty a refined binomial lattice for pricing american asian options rev derivatives res 3 1 1999 85 105 if the number of the time steps increases if the underlying distribution displays negative skewness and leptokurtosis as often observed for stock index returns our estimates can work better than those in chalasani et al p chalasani s jha f egriboyun a varikooty a refined binomial lattice for pricing american asian options rev derivatives res 3 1 1999 85 105 and are very similar to the benchmarks in hull and white j hull a white efficient procedures for valuing european and american path dependent options j derivatives 1 fall 1993 21 31 the numerical analysis shows that our modified edgeworth binomial model can value american asian options with greater accuracy and speed given higher moments in their underlying distribution

the hilbert phenomenon in chaotic motions abstract it is proved that chaotic dynamical systems have d 1 singularity due to such singularity and its periodicity in hausdorff metric spaces a chaotic dynamical system induces a hilbert iterated system which was discovered at the end of the 19th century this implies that the most puzzling discovery in nature in the 20th century can be put into deep correspondence with that in thoughts in the 19th century all this is verified by the numerical experiments of chaotic vibrations of a beam and a tension slack oscillator

multifractal analysis and topological properties of a new family of weighted koch networks abstract weighted complex networks especially scale free networks which characterize real life systems better than non weighted networks have attracted considerable interest in recent years studies on the multifractality of weighted complex networks are still to be undertaken in this paper inspired by the concepts of koch networks and koch island we propose a new family of weighted koch networks and investigate their multifractal behavior and topological properties we find some key topological properties of the new networks their vertex cumulative strength has a power law distribution there is a power law relationship between their topological degree and weight strength the networks have a high weighted clustering coefficient of 0 41004 which is independent of the scaling factor c in the limit of large generation t the second smallest eigenvalue 2 and the maximum eigenvalue n are approximated by quartic polynomials of the scaling factor c for the general laplacian operator while 2 is approximately a quartic polynomial of c and n 1 5 for the normalized laplacian operator then we find that weighted koch networks are both fractal and multifractal their fractal dimension is influenced by the scaling factor c we also apply these analyses to six real world networks and find that the multifractality in three of them are strong

computing connectedness disconnectedness and discreteness we consider finite point set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set we use the minimal spanning tree of the finite set of points to compute the number and size of its connected components by extrapolating the limiting behavior of these quantities as 0 we can say whether the underlying set appears to be connected totally disconnected or perfect we demonstrate the effectiveness of our techniques for a number of examples including a family of fractals related to the sierpinski triangle cantor subsets of the plane the henon attractor and cantori from four dimensional symplectic sawtooth maps for zero measure cantor sets we conjecture that the growth rate of the number of components as 0 is equivalent to the box counting dimension 2000 elsevier science b v all rights reserved

dimension measurement on high dimensional systems abstract using experimental data from a variety of laboratory electrical systems the upper limits on dimension measurement are examined and it is shown that contrary to standard scaling estimates it can be possible to reliably measure correlation dimensions larger than 10 the practical implications of the constraints on such a measurement are examined including the role of statistical errors geometrical artifacts dimension algorithm metric choice and the process power spectrum these ideas are applied to electrical 1 f noise from both macroscopic and microscopic sources to show that it cannot be explained by a deterministic model with fewer than roughly 20 degrees of freedom

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

characterization of the spatial complex behavior and transition to chaos in flow systems abstract we introduce a spatial lyapunov exponent to characterize the complex behavior of non chaotic but convectively unstable flow sytems this complexity is of spatial type and is due to sensitivity to the boundary conditions we show that there exists a relation between the spatial complexity index we define and the comoving lyapunov exponents in such systems the transition to chaos i e the occurrence of a positive lyapunov exponent can manifest itself in two different ways in the first case from neither chaotic nor spatially complex behavior to chaos one observes the typical scenario i e as the system size grows up the spectrum of the lyapunov exponents gives rise to a density in the second case when the chaos develops from a convectively unstable situation one observes only a finite number of positive lyapunov exponents

on weighted u statistics for stationary processes a weighted u statistic based on a random sample x1 xn has the form un p 1 i jn wiijk xi xj where k is fixed symmetric measurable function and the wi are symmetric weights a large class of statistics can be expressed as weighted u statistics or variations thereof this paper establishes the asymptotic normality of un when the sample observations come from a non linear time series and linear processes msc 2000 subject classifications primary 60f05 secondary 60g10

theoremes de dualite pour les corps de fonctions sur des corps locaux superieurs let k be the function field of a smooth projective curve x over a higher dimensional local field k we define tate shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of k coming from a closed point of x we establish duality theorems between tate shafarevich groups for finite groups schemes and for tori

finite size lyapunov exponent review on applications in dynamical systems the growth of infinitesimal perturbations is well characterized by the lyapunov exponents in many situations of interest however important phenomena involve finite amplitude perturbations which are ruled by nonlinear dynamics out of tangent space and thus cannot be captured by the standard lyapunov exponents we review the application of the finite size lyapunov exponent fsle for the characterization of non infinitesimal perturbations in a variety of systems in particular we illustrate their usage in the context of predictability of systems with multiple spatio temporal scales of geophysical relevance in the characterization of nonlinear instabilities and in some aspects of transport in fluid flows we also discuss the application of the fsle to more general aspects such as chaos noise detection and coarse grained descriptions of signals this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

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

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

learning time series evolution by unsupervised extraction of correlations as a consequence we are able to model chaotic and nonchaotic time series furthermore one critical point in modeling time series is the determination of the dimension of the embedding vector used i e the number of components of the past that are needed to predict the future with this method we can detect the embedding dimension by extracting the influence of the past on the future i e the correlation of remote past and future optimal embedding dimensions are obtained for the henon map and the mackey glass series when noisy data corrupted by colored noise are used a model is still possible the noise will then be decorrelated by the network in the case of modeling a chemical reaction the most natural architecture that conserves the volume is a symplectic network which describes a system that conserves the entropy and therefore the transmitted information

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

relaxation closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks attractors in asymmetric neural networks with deterministic parallel dynamics present a chaotic regime at symmetry where the average length of the cycles increases exponentially with system size and an oscillatory regime at high symmetry where the average length of the cycles is 2 we show both with analytic arguments and numerically that there is a sharp transition at a critical symmetry between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size and the weights of their attraction basins are distributed as in a random map with reversal symmetry the timescale after which cycles are reached grows exponentially with system size n and the exponent vanishes in the symmetric limit where the transition can be related to the dynamics of the infinite system where cycles are never reached using the closing probabilities as a tool we also study the relaxation of the function where is the local field experienced by the neuron i in the symmetric system it plays the role of a ljapunov function which drives the system towards its minima through steepest descent this interpretation survives even if only on the average also for small asymmetry this acts like an effective temperature the larger the asymmetry the faster the relaxation and the higher the asymptotic value reached e reaches very deep minima in the fixed points of the dynamics which are reached with vanishing probability and attains a larger value on the typical attractors which are cycles of length 2

chaos in a variable mass relaxation oscillator model for the leaky tap abstract a variable mass relaxation oscillator model for the behaviour of the leaky tap is numerically investigated different regions of its three dimensional parameter space have been searched and interesting dynamical behaviour has been found the dynamics of the system is thoroughly characterized using projections of phase space trajectories 2 and 3 dimensional time delay plots power spectra lyapunov exponents hausdorff and correlation dimensions and bifurcation diagrams dynamical variables obtained from the model like drop masses velocities and drip intervals are used for reconstructing attractors we found several periodic intermittent and chaotic attractors we conclude that the rich dynamics of the model investigated represents an improvement in describing the experimentally observed behaviour of the actual dripping faucet system

a semi lagrangian approach for american asian options under jump diffusion a semi lagrangian method is presented to price continuously observed fixed strike asian options at each timestep a set of one dimensional partial integrodifferential equations pides is solved and the solution of each pide is updated using semi lagrangian timestepping crank nicolson and second order backward differencing timestepping schemes are studied monotonicity and stability results are derived with low volatility values it is observed that the nonsmoothness at the strike in the payoff affects the convergence rate a subquadratic convergence rate is observed

coherence and chaos in the driven damped sine gordon equation measurement of the soliton spectrum abstract a numerical procedure is developed which measures the sine gordon soliton and radiation content of any field t which is periodic in space the procedure is applied to the field generated by a damped driven sine gordon equation this field can be either temporally periodic locked to the driver or chaotic in either case the numerical measurement shows that the spatial structure can be described by only a few spatially localized soliton wave train modes the numerical procedure quantitatively identifies the presence number and properties of these soliton wave trains for example an increase of spatial symmetry is accompanied by the injection of additional solitons into the field

strong laws for l and u statistics strong laws of large numbers are given for l statistics linear combinations of order statistics and for u statistics averages of kernels of random samples for ergodic stationary processes extending classical theorems of hoeffding and of helmers for lid sequences examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied and an application is given to estimation of correlation dimension of invariant measures

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

nonlinear analysis of the fracture surface of a single crystal brittle solid abstract the dynamical behavior of a spatial fracture mechanism in a rapid crack in a single crystal brittle solid was under investigation wide thin strips like specimens were fractured under three point bending the spatial mechanism was photographed and digitized and the results were then studied using nonlinear analysis the nonlinear tools used were correlation dimension false nearest neighbor null hypothesis surrogate data and the dominant lyapunov exponent the governing equation of motion in the analyzed region is a differential equations with a minimum of seven independent dynamical variables in contrast to a single independent variable used in analytical calculations

fractal analysis of measure representation of large proteins based on the detailed hp model abstract the notion of measure representation of protein sequences is introduced based on the detailed hp model multifractal analysis and detrended fluctuation analysis are then performed on the measure representations of a large number of long protein sequences it is concluded that these protein sequences are not completely random sequences through the measure representations and the values of the d q spectra and related c q curves the values of the exponent from the detrended fluctuation analysis show that the k strings with the ordering in the measure representation exhibit strong long range correlation for substrings with length k 5 the d q spectra of all proteins studied are multifractal like and sufficiently smooth for the c q curves to be meaningful the c q curves of all proteins resemble a classical phase transition at a critical point an ifs model is found to simulate the measure representation of protein sequences very well from the estimated values of parameters in the ifs model we think the non polar residues and uncharged polar residues play a more important role than other kinds of residues in the protein folding process

an approach for characterizing coupling in dynamical systems abstract the study of coupling in dynamical systems dates back to christian hyugens who in 1665 discovered that pendulum clocks with the same length pendulum synchronize when they are near to each other in that case the observed synchronous motion was out of phase in this paper we propose a new approach for measuring the degree of coupling and synchronization of a dynamical system consisting of interacting subsystems the measure is based on quantifying the active degrees of freedom e g correlation dimension of the coupled system and the constituent subsystems the time delay embedding scheme is extended to coupled systems and used for attractor reconstruction of the coupled dynamical system we use the coupled lorenz rossler and henon model systems with a coupling strength variable for evaluation of the proposed approach results show that we can measure the active degrees of freedom of the coupled dynamical systems and can quantify and distinguish the degree of synchronization or coupling in each of the dynamical systems studied furthermore using this approach the direction of coupling can be determined

limits on inferring the past here we define and study the properties of retrodictive inference we derive equations relating retrodiction entropy and thermodynamic entropy and as a special case show that under equilibrium conditions the two are identical we demonstrate relations involving the kullback leibler divergence and retrodiction probability and bound the time rate of change of retrodiction entropy as a specific case we invert various langevin processes inferring the initial condition of n particles given their final positions at some later time we evaluate the retrodiction entropy for langevin dynamics exactly for special cases and find that one s ability to infer the initial state of a system can exhibit two possible qualitative behaviors depending on the potential energy landscape either decreasing indefinitely or asymptotically approaching a fixed value we also study how well we can retrodict points that evolve based on the logistic map we find singular changes in the retrodictivity near bifurcations counterintuitively the transition to chaos is accompanied by maximal retrodictability

a scaling exponent based detector of chaos in oscillatory circuits abstract in this paper a scaling exponent based approach is proposed to determine the state of chaotic circuits and the scaling exponent is calculated using detrended fluctuation analysis dfa the corresponding detector is designed using the fact that the scaling exponent changes for various states of chaotic circuits simulation examples in this paper are performed for chua s circuit and other chaotic systems and compared with the state of the art in the field the proposed detector outperforms existing techniques in ability to distinguish the chaotic and periodic states in the circuits for relatively high noise

phase portrait reconstruction using multivariate singular systems analysis abstract the singular systems analysis ssa method of broomhead and king is applied to the reconstruction of phase portraits from experimental time series of temperature measurements taken simultaneously at up to 16 probes within a rotating thermal convection experiment and used to characterize a number of different kinds of flow including the determination of their correlation dimension results are compared with equivalent reconstructions using more conventional method of delays embedding and ssa applied in the time domain only to data obtained from a single thermocouple probe multivariate ssa m ssa is shown to offer significantly improved reconstructions in terms of signal noise ratio and structural uniformity of the attractor over earlier methods this is due mainly to the ability of m ssa to combine simultaneously information sampling the complete range of the spatial cross correlation function in signals with complex spatio temporal structure

nonlinearity in high frequency stock returns evidence from the athens stock exchange abstract this study investigates empirically the presence of nonlinearities in the athens composite share price index high frequency returns a preliminary analysis indicates that volatility exhibits a periodic intraday inverse j shaped pattern associated with the opening and closing of the market periodicity is then removed employing a flexible fourier form subsequently an arma figarch model over several frequencies yields that return volatility is long memory and self similar nonlinear analysis with the use of the embedding dimension suggests that the filtered return process does not exhibit deterministic or higher order stochastic nonlinearity rather it is reminiscent of a random process we conclude that the acspi data are nonlinear however nonlinearity is attributed to persistent arch effects

reshape scale method a novel multi scale entropic analysis approach abstract the reshape scale rs method was introduced in this article as a novel approach to perform multi scale transition of sample entropy this method was able to quantify the orderliness in the signal by determining the distance over which the subsequent data points can remain affiliated to one another entropic half life enhl was introduced to characterize such an affiliation the method was tested for 1 f processes for different values furthermore the dependency of the multi scale entropy analysis developed by costa et al 2002 6 to the probability density function and the standard deviation of autoregressive signals was studied and discussed

on the exit times approach for e entropy and turbulent signals abstract we review a recently proposed approach to the computation of the e entropy of a given signal based on the exit time statistics i e one codes the signal by looking at the instants when the fluctuations are larger than a given threshold e moreover we show how the exit times statistics when applied to experimental turbulent data is able to highlight the intermediate dissipative range of turbulent fluctuations

a note on the behaviour of the correlation integral in the presence of a time series summary the behaviour of the correlation integral is well known when applied to simple chaotic regimes and to pure random noise the asymptotic behaviour of a chaotic system is represented by an infinite set of points called an attractor the dimension of which can be estimated by the correlation integral essentially the correlation integral of a time series is a normalized count of the number of close pairs of points of the series lying in phase space we examine the expected behaviour of the integral for some stochastic process models which place a weak structure on the noise such as simple autoregressive and moving average formulations

uniform partitions and a dimensions spectrum for lacunar measures a multifractal analysis has been proposed to investigate complex systems which exhibit a nonuniform local structure multifractal measures are normalized mass distributions inhomogeneous at every scale to every point p is associated the exponent of the power law r describing the asymptotic behaviour as r 0 of the mass of a sphere of radius r and centre p the measure properties are specified by the dimension spectrum f of the set of points p with the same scaling exponent this spectrum is nontrivial even when the mass distribution has no holes measure with euclidean support and the box counting dimension is an integer in this case it can be evaluated by considering a sequence of uniform grids whose spacing decreases as 2 n known as uniform partitions we consider here a family of iterated function system fractal measures with support on the unit square with random defects specified by three parameters a scale a probability p and an attenuation factor given a partition with a diameter less than any cell is chosen with probability p and its mass is multiplied by the corresponding pre measure is obtained after normalization in all the examples examined the f spectrum appears to widen monotonically as decreases this result suggests the possibility of using the method to detect the presence of defects in sponge like structures such as cancellous bone sponge like by analysing their radiographs since the grey intensity is related to the mass density of the structure a widening of the f spectrum is indeed observed in osteoporotic bone radiographs compared to normal bone radiographs this may be attributed to the rarefaction of the cancellous bone

generalized synchronization of chaos in electronic circuit experiments abstract two examples of one way coupled electronic circuits displaying generalized synchronization of chaos are considered in one of them the dynamics of both the response and the driving systems represent a double scroll chaos oscillator in another example the double scroll oscillator is driven by an electronic analog of the mackey glass system to detect the generalized synchronization an auxiliary response system that is a replica of the original one is used in these systems we have discovered two types of generalized synchronization namely a strong and a weak synchronization which correspond to the existence of a smooth and a nonsmooth map from the trajectories of the driving attractor to the trajectories of the response system respectively

slow and fast dynamics in coupled systems a time series analysis view abstract we study the dynamics of systems with different timescales when access only to the slow variables is allowed we use the concept of finite size lyapunov exponent fsle and consider both the cases when the equations of motion for the slow components are known and the situation when a scalar time series of one of the slow variables has been measured a discussion on the effects of parametrizing the fast dynamics is given we show that although the computation of the largest lyapunov exponent can be practically infeasible in complex dynamical systems the computation of the fsle allows to extract information on the characteristic time and on the predictability of the large scale slow time dynamics even with moderate statistics and unresolved small scales

measure representation and multifractal analysis of complete genomes this paper introduces the notion of measure representation of dna sequences spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes the main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria from the measure representations and the values of the dq spectra and related cq curves it is concluded that these complete genomes are not random sequences in fact spectral analyses performed indicate that these measure representations considered as time series exhibit strong long range correlation here the long range correlation is for the k strings with dictionary ordering and it is different from the base pair correlations introduced by other people for substrings with length k58 the dq spectra of all organisms studied are multifractal like and sufficiently smooth for the cq curves to be meaningful with the decreasing value of k the multifractality lessens the cq curves of all bacteria resemble a classical phase transition at a critical point but the analogous phase transitions of chromosomes of nonbacteria organisms are different apart from chromosome 1 of c elegans they exhibit the shape of double peaked specific heat function a classification of genomes of bacteria by assigning to each sequence a point in two dimensional space d21 d1 and in three dimensional space d21 d1 d22 was given bacteria that are close phylogenetically are almost close in the spaces d21 d1 and d21 d1 d22

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

correlation dimension of financial market abstract in this paper correlation dimension is applied to financial data analysis we calculate the correlation dimensions of some real market data and find that the dimensions are significantly smaller than those of the simulation data based on geometric brownian motion based on the analysis of the chinese and us stock market data the main results are as follows first by calculating three data sets for the chinese and us market we find that large market volatility leads to a significant decrease in the dimensions second based on 5 min stock price data we find that the chinese market dimension is significantly larger than the us market this shows a significant difference between the two markets for high frequency data third we randomly extract stocks from a stock set and calculate the correlation dimensions and find that the average value of these dimensions is close to the dimension of the original set in addition we analyse the intuitional meaning of the relevant dimensions used in this paper which are directly related to the average degree of the financial threshold network the dimension measures the speed of the average degree that varies with the threshold value a smaller dimension means that the rate of change is slower

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

cohomology of tori over p adic curves we establish a duality in the cohomology of arbitrary tori over smooth but not necessarily projective curves over a p adic field this generalises lichtenbaum tate duality between the picard group and the brauer group of a smooth projective curve

limit behaviour and scaling relations of two kinds of noisy logistic map in the vicinity of chaos threshold and their robustness abstract in this paper we numerically investigate the distribution of the sums of the iterates of the logistic map and the relationships among the important properties of the nonlinear dynamics in the vicinity of the chaos threshold by adding two kinds of contributions with different densities the first one is the well known white noise whereas the second is a newly defined one named as quartic term which makes contributions from the own structure of the map as the chaos threshold is approached the iterates of the standard logistic map i e noise free have strong correlations and the standard central limit theorem is not valid anymore in a recent work tirnakli 2009 it has been shown that the limit distribution seems to converge to q gaussian distribution which maximizes the nonadditive entropy s q 1 i p i q q 1 under appropriate conditions in this work we investigate the effect of these contributions i e white noise and quartic term on the limit distribution and on the range of the obtained q gaussian distribution as a result of these findings under the existence of white noise and also the quartic term we analyse the validity of the scaling relations among correlation fractality the lyapunov divergence and q gaussian distributions which have recently been observed in afsar 2014 the results obtained here strengthen the argument that the central limit behaviour is given by a q gaussian as the chaos threshold is approached and indicate that the scaling relations obtained for the standard logistic map among the range of the q gaussian the correlation dimension the correlation length the lyapunov exponent fractality and the distance from the chaos threshold are robust under the existence of white noise and the quartic term

computation of lyapunov spectra effect of interactive noise and application to a chemical oscillator abstract a refined version of the sano sawada algorithm to compute lyapunov spectra has been implemented and tested with several models which include periodic chaotic and hyperchaotic attractors under conditions designed to mimic experimental situations it was found that reliable lyapunov exponents can be obtained for time series of moderate length and point density also in the presence of interactive noise as long as its amplitude does not exceed 1 of the oscillation amplitude the algorithm was then applied to experimental data from the co oxidation on pt 110 clearly showing the occurrence of chaos as well as hyperchaos

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

spatiotemporal chaos and noise low dimensional chaotic dynamical systems can exhibit many characteristic properties of stochastic systems such as broad fourier spectra they are distinguishable from stochastic processes through finite values for their dimension lyapunov exponents and kolmogorov sinai entropy we discuss how these characteristic observables are modified in spatiotemporal chaotic systems like coupled map lattices we analyze with the help of lyapunov concepts how the stochastic limit is approached and how these properties can be observed directly through local dimension measurements from reconstructed time series finally we discuss the interaction of spatiotemporal attractors with external noise and possible connections to problems of pattern selection and stability

small scale turbulence in the plankton low order deterministic chaos or high order stochasticity abstract almost all regions of the oceans are heavily influenced by the effects of physical processes such as turbulence which in turn influence the distribution and ecology of organisms that occupy these regions there is a real need for additional focus placed on the precise knowledge of both physical and biological processes which is often difficult using basic time series analysis in that way we applied nonlinear analysis techniques to high frequency time series of temperature salinity and phytoplankton concentrations recorded in different hydrodynamical regimes related to tidal forcing in a tidally mixed coastal ecosystem techniques devoted to the identification of low order deterministic chaos cannot find evidence of chaos while a lower dimensionality was identified in low hydrodynamic conditions the results rather suggest stochastic time series with many degrees of freedom no obvious attractor in phase space trajectory lle indistinguishable from zero absence of convergence of the correlation integral we then applied to these data specific multifractal analysis techniques and showed that these time series clearly exhibit high order stochasticity in addition the stochastic structure of purely passive scalars i e temperature and salinity remained invariant while the one of phytoplankton biomass must be regarded as highly structured in time by both hydrodynamic and advective processes

limit theorems for von mises statistics of a measure preserving transformation for a measure preserving transformation t of a probability space x mathcal f mu and some d ge 1 we investigate almost sure and distributional convergence of random variables of the form begin aligned x rightarrow frac 1 c n sum 0 le i 1 ldots i d n f t i 1 x ldots t i d x n 1 2 ldots end aligned where c 1 c 2 ldots are normalizing constants and the kernel f belongs to an appropriate subspace in some l p x d mathcal f otimes d mu d we establish a form of the individual ergodic theorem for such sequences using a filtration compatible with t and the martingale approximation we prove a central limit theorem in the non degenerate case for a class of canonical totally degenerate kernels and d 2 we also show that the convergence holds in distribution towards a quadratic form sum m 1 infty lambda m eta 2 m in independent standard gaussian variables eta 1 eta 2 ldots

on the convergence rates of ipa and fdc derivative estimators we show that under the sufficient conditions usually given for infinitesimal perturbation analysis ipa to apply for derivative estimation a finite difference scheme with common random numbers fdc has the same order of convergence namely o n 1 2 provided that the size of the finite difference interval converges to zero fast enough this holds for both one and two sided fdc this also holds for different variants of ipa such as some versions of smoothed perturbation analysis spa which is based on conditional expectation finally this also holds for the estimation of steady state performance measures by truncated horizon estimators under some ergodicity assumptions our developments do not involve monotonicity but are based on continuity and smoothness we give examples and numerical illustrations which show that the actual difference in mean square error mse between ipa and fdc is typically negligible we also obtain the order of convergence of that difference which is faster than the c

invariant trajectory classification of dynamical systems with a case study on ecg an invariant pattern recognition framework for classification of phase space trajectories of nonlinear dynamical systems is presented using statistical shape theory known external influences can be discriminated from true changes of the system the external effects are modeled as a transformation group acting on the phase space and variation of the trajectories not explained by the transformations is accounted for using principal component analysis the approach suggested is highly adaptable to a wide range of situations and individual differences the methodology presented is applied to detect abnormalities in electrocardiograms results based on measured data indicate that the model developed is resistant to the effects of respiration and body position changes which are abundant in ambulatory conditions and cause significant morphological artifacts in the signal the results also show that the detection of an artificially induced acute myocardial infarction is achieved with high performance due to its low computational complexity the method developed can be implemented in real time the method developed also adapts to morphological changes caused by various heart conditions

exploring dynamic property of traffic flow time series in multi states based on complex networks phase space reconstruction versus visibility graph abstract a new method based on complex network theory is proposed to analyze traffic flow time series in different states we use the data collected from loop detectors on freeway to establish traffic flow model and classify the flow into three states based on k means method we then introduced two widely used methods to convert time series into networks phase space reconstruction and visibility graph furthermore in phase space reconstruction we discuss how to determine delay time constant and embedding dimension and how to select optimal critical threshold in terms of cumulative degree distribution in the visibility graph we design a method to construct network from multi variables time series based on logical or finally we study and compare the statistic features of the networks converted from original traffic time series in three states based on phase space and visibility by using the degree distribution network structure correlation of the cluster coefficient to betweenness and degree degree correlation

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

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

lichtenbaum tate duality for varieties over p adic fields s lichtenbaum has proved in l1 that there is a nondegenerate pairing pic c x br c br k q z 1 between the picard group and the brauer group of a nonsingular projective curve c over a p adic field k a finite extension of the p adic numbers qp his proof consists of a reduction via explicit cocycle calculations in galois cohomology to a combination of tate duality for group schemes over p adic fields and the autoduality of the jacobian of a smooth curve in this paper we will reconstruct the above duality as a purely formal combination of a generalized form of tate duality over p adic fields and a form of poincar e duality for curves over arbitrary fields of characteristic zero this gives a more conceptual proof of lichtenbaum s result and an analogue in higher dimensions

nonlinear determinism in time series measurements of two dimensional turbulence experiments on two dimensional and three dimensional turbulent flows in a rotating annulus are analyzed using linear and nonlinear time series predictors the models are used to predict the time series a time s ahead and to calculate the velocity incrementu s u sn s u sn between the current value of the time series sn and a point time s apart for two dimensional flow the nonlinear model provides superior predictions to the linear model foru s positive and large in contrast for three dimensional turbulence the prediction of the nonlinear model is no better than the linear model for two dimensional turbulence the probability density functions for the predictedu s from the linear model have an exponential tail while for the nonlinear model the tail exhibits power law decay the scaling exponent of this power law can be explained using arguments of the kolmogorov 1941 theory our findings contradict the common assumption that two dimensional turbulence shares the unpredictability properties of three dimensional turbulent flows 2002 elsevier science b v all rights reserved