transition between kelvin s equilibria a transition between kelvin s equilibrium states is investigated using nonlinear theory we have shown that the transition of polygonal patterns of the hollow vortex core from mode n 2 through n 4 occurs in two steps quasiperiodicity and frequency locking we have also shown that this transition can be modeled by a one dimensional circle map we extrapolate the present result and hypothesize that the transition between kelvin s equilibria follows the same route and the ratios of locking frequencies form a farey sum and staircase function against the control parameter where the staircase corresponds to the rational frequency ratio n ensuremath 1 n

observation of symmetry breaking state selection and sensitivity in a noisy electronic system abstract measurements of the time evolution of the statistical density as a control parameter is swept through a bifurcation point in the presence of additive noise are presented we have used these densities to study branch selectivity which results from a small additive interaction a universal time scale for all such processes is established the results are in good agreement with a recent theory

filtering and frequency interpretations of singular spectrum analysis new filtering and spectral interpretations of singular spectrum analysis ssa are provided it is shown that the variables reconstructed from diagonal averaging of reduced rank approximations to the trajectory matrix can be obtained from a noncausal convolution filter with zero phase characteristics the reconstructed variables are readily constructed using a two pass filtering algorithm that is well known in the signal processing literature when the number of rows in the trajectory matrix is much larger than number of columns many results reported in the signal processing literature can be used to derive the properties of the resulting filters and their spectra new features of the reconstructed series are revealed using these results two examples are used to illustrate the results derived in this paper

monitoring changing dynamics with correlation integrals case study of an epileptic seizure abstract we describe a procedure and the motivation behind it which rapidly and accurately tracks the onset and progress of an epileptic seizure roughly speaking one monitors changes in the relative dispersion of a re embedded time series the results are robust with respect to variation of adjustable parameters such as embedding dimension lag time and critical distances moreover the general method is virtually unaffected when the data are significantly corrupted by external noise when the information computed for the individual channels is displayed in an appropriate space time plot the progress and geometric location of the seizure are easily seen an interpretation of these results in terms of a cloud of particles moving in an abstract phase space is examined

behavior of sensitivities in the one dimensional advection dispersion equation implications for parameter estimation and sampling design the spatial and temporal variability of sensitivities has a significant impact on parameter estimation and sampling design for studies of solute transport in porous media physical insight into the behavior of sensitivities is offered through an analysis of analytically derived sensitivities for the one dimensional form of the advection dispersion equation when parameters are estimated in regression models of one dimensional transport the spatial and temporal variability in sensitivities influences variance and covariance of parameter estimates several principles account for the observed influence of sensitivities on parameter uncertainty 1 information about a physical parameter may be most accurately gained at points in space and time with a high sensitivity to the parameter 2 as the distance of observation points from the upstream boundary increases maximum sensitivity to velocity during passage of the solute front increases and the consequent estimate of velocity tends to have lower variance 3 the frequency of sampling must be in phase with the s shape of the dispersion sensitivity curve to yield the most information on dispersion 4 the sensitivity to the dispersion coefficient is usually at least an order of magnitude less than the sensitivity to velocity 5 the assumed probability distribution of random error in observations of solute concentration determines the form of the sensitivities 6 if variance in random error in observations is large trends in sensitivities of observation points may be obscured by noise and thus have limited value in predicting variance in parameter estimates among designs 7 designs that minimize the variance of one parameter may not necessarily minimize the variance of other parameters 8 the time and space interval over which an observation point is sensitive to a given parameter depends on the actual values of the parameters in the underlying physical system

on the dimension of complex responses in nonlinear structural vibrations abstract the ability to accurately model engineering systems under extreme dynamic loads would prove a major breakthrough in many aspects of aerospace mechanical and civil engineering extreme loads frequently induce both nonlinearities and coupling which increase the complexity of the response and the computational cost of finite element models dimension reduction has recently gained traction and promises the ability to distill dynamic responses down to a minimal dimension without sacrificing accuracy in this context the dimensionality of a response is related to the number of modes needed in a reduced order model to accurately simulate the response thus an important step is characterizing the dimensionality of complex nonlinear responses of structures in this work the dimensionality of the nonlinear response of a post buckled beam is investigated significant detail is dedicated to carefully introducing the experiment the verification of a finite element model and the dimensionality estimation algorithm as it is hoped that this system may help serve as a benchmark test case it is shown that with minor modifications the method of false nearest neighbors can quantitatively distinguish between the response dimension of various snap through non snap through random and deterministic loads the state space dimension of the nonlinear system in question increased from 2 to 10 as the system response moved from simple low level harmonic to chaotic snap through beyond the problem studied herein the techniques developed will serve as a prescriptive guide in developing fast and accurate dimensionally reduced models of nonlinear systems and eventually as a tool for adaptive dimension reduction in numerical modeling the results are especially relevant in the aerospace industry for the design of thin structures such as beams panels and shells which are all capable of spatio temporally complex dynamic responses that are difficult and computationally expensive to model

operator theoretic framework for forecasting nonlinear time series with kernel analog techniques abstract kernel analog forecasting kaf alternatively known as kernel principal component regression is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data this paper synthesizes descriptions of kernel methods and koopman operator theory in order to provide a single consistent account of kaf the framework presented here illuminates the property of the kaf method that under measure preserving and ergodic dynamics it consistently approximates the conditional expectation of observables that are acted upon by the koopman operator of the dynamical system and are conditioned on the observed data at forecast initialization more precisely kaf yields optimal predictions in the sense of minimal root mean square error with respect to the invariant measure in the asymptotic limit of large data the presented framework facilitates moreover the analysis of generalization error and quantification of uncertainty extensions of kaf to the construction of conditional variance and conditional probability functions as well as to non symmetric kernels are also shown illustrations of various aspects of kaf are provided with applications to simple examples namely a periodic flow on the circle and the chaotic lorenz 63 system

the analysis of chaotic time series data this paper presents a brief survey of time series analysis methods that are applicable to processes whose behavior can be described as low dimensional chaos the goal of these methods is to allow experimentalists to obtain local estimates of the dynamics directly from a set of data these estimates are often sufficiently accurate to attempt noise reduction prediction and control

a complexity score derived from principal components analysis of nonlinear order measures the generation of a global complexity score for numerical series was derived from a principal components analysis of a group of nonlinear measures of experimental as well simulated series the concept of complexity was demonstrated to be independent from other descriptors of ordered series such as the amount of variance the departure from normality and the relative nonstationarity andto be mainly relatedto the number of ind epend ent elements or operations needed to synthesize the series the possibility of having a univocal ranking of complexity for diverse series opens the way to a wider application of dynamical systems concepts in empirical sciences c 2001 elsevier science b v all rights reserved

characterization of a two parameter mixed mode electrochemical behavior regime using neural networks abstract we use nonlinear signal processing techniques based on artificial neural networks to construct a continuous time model set of ordinary differential equations odes from experimental observations of mixed mode oscillations during the galvanostatic oxidation of hydrogen on platinum in the presence of bismuth and chloride ions the data was in the form of time series of the potential for different values of the applied current and chloride ion concentration we use the model to reconstruct the experimental dynamics and to explore the associated bifurcation structures in phase space using numerical bifurcation techniques we locate stable and unstable periodic solutions calculate eigenvalues and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings for example the fitted model suggests the existence of a sub critical hopf bifurcation near the range of parameters probed in the experiments this might explain the experimental difficulty in locating small amplitude simple oscillations

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

a simple variational approach to a converse of the lagrange dirichlet theorem peter hagedorn 8r jean mawhin communicated by k kigc g ssner 1 introduction the lagrange dirichlet stability theorem states that the equilibrium posi tion of a discrete conservative mechanical sytem is stable in the sense of liapunov if the potential energy u assumes a strict local minimum at this equilibrium we know from painzev s example 22 see also 37 that this sufficient condition stated by lagranoe 13 and proved by lejevne dirich et 17 is not necessary for stability and the history of the inversion of the lagrange dirichlet theorem i e of the search for necessary conditions for stability is rich and interesting we refer to 27 for a discussion and a bibliography and to the more recent references 1 3 6 12 14 16 18 20 21 23 26 28 32 35 36 in 1971 ha6edorn proved for systems of class c 2 that the equilibrium position is unstable if the potential energy u has a strict maximum at this equilibrium 4 5 this result was extended in various directions by ta l fer o 30 bomtin kozlov 2 kozmv 10 liubushin 18 and sofer 29 a common feature of all those proofs initiated in 4 is the application of some method of the calculus of variations namely the theory of geodesics on finsler manifolds to jacobi s form of the principle of least action this leads to technical difficulties which are overcome in the above mentioned papers through a lot of effort and ingenuity the aim of this short paper is to show that the replacement of jacobi s principle of least action by a variant of another variational principle recently introduced by van groesen 33 34 in the study of normal mode vibrations of conservative nonlinear systems allows a simple and elementary treatment in the setting of a hilbert space instead of a finsler manifold of the instabili ty theorem stated in 4 with the minimal regularity assumptions of 18 29 30 in this way the problem is reduced to proving the existence of a minimum for a coercive functional defined on a weakly closed subset of a hilbert space a problem solved at the beginning of any monograph on the direct methods of the calculus of variations see e g 19 the reduction

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

generalized phase space projection for nonlinear noise reduction improved phase space projection methods adapted from related work in the linear signal processing field based on subspace decomposition are presented for application to the problem of additive noise reduction in the context of phase space analysis these methods improve upon existing methods such as broomhead king singular spectrum analysis projection by minimizing overall signal distortion subject to constraints on the residual error rather than using a direct least squares fit this results in a range of weighted projections which estimate and compensate for the portion of the principal component s singular values corresponding to noise rather than signal energy and which include least squares ls and least minimum mean square error lmmse as subcases the nature of phase space covariance the key element in construction of the projection matrix is examined across global phase spaces as well as within local neighborhood regions the resulting algorithm illustrated on a noisy henon map as well as on the task of speech enhancement is applicable to a wide variety of nonlinear noise reduction tasks

topology from time series we describe methods for the study of topological properties of the invariant manifolds of experimental dynamical systems we explain how to compute such invariants as the euler characteristic and betti numbers using time series data and suggest a number of potential applications

effects of non gaussian noise near supercritical hopf bifurcation abstract we have studied the effects of non gaussian colored noise in a chemical oscillation system the well known brusselator model in the parameter region close to the supercritical hopf bifurcation with the variation of the parameter q which quantifies the deviation from gaussian character the signal to noise ratio of noise induced oscillation exhibits a bell shaped change indicating the presence of resonant activity the cooperative effects of q and the correlation time on the performance of noise induced oscillation are also investigated interestingly resonance like behavior can be induced by either q or when the other parameter is properly fixed stochastic normal form theory is used to analyze these nontrivial effects and the simulation results are well reproduced this work provides us comprehensive understanding of how non gaussian noise influences the dynamics in chemical oscillation systems

detecting switch dynamics in chaotic time waveform using a parametrized family of nonlinear predictors abstract an algorithm is presented for detecting switch dynamics in chaotic time waveform by the switch dynamics we mean that the chaotic time waveform is measured from a dynamical system whose bifurcation parameters are occasionally switched among a set of slightly different parameter values first the switched chaotic time waveform is divided into windows of short term time waveforms from the set of windowed time waveforms qualitatively similar parametrized family of nonlinear predictors is constructed qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original by characterizing the windows of short term chaotic time waveforms in terms of the qualitative parameters of nonlinear predictors switch dynamics of their associated bifurcation parameters are detected for the lorenz equations the rossler equations and the mackey glass equations efficiency of the algorithm is demonstrated in the experiment chaotic time waveforms contaminated with observational noise is considered

the origins of chaos in a modified van der pol oscillator abstract an experimental and theoretical investigation of a modified van der pol oscillator has led to the discovery of degenerate bifurcations which control the global dynamics of the system a two dimensional parameter space for the oscillator is studied and the measured loci of critical points where a qualitative change of behaviour is observed are given in detail it is shown explicitly by analysis of the experimental data that chaos arises through the silnikov mechanism

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

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

recognizing chaotic time waveforms in terms of a parametrized family of nonlinear predictors abstract consider a chaotic dynamical system which exhibits a variety of chaotic time waveforms with a change in the bifurcation parameters this paper presents an algorithm for estimating the underlying bifurcation parameters of the chaotic time waveforms in experimental situation in which no a priori analytical knowledge of the dynamical system is available first we construct qualitatively similar parametrized family of nonlinear predictors only from several sets of chaotic time waveforms qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original chaotic time waveforms are then characterized in terms of the qualitatively similar bifurcation parameters of the nonlinear predictors we call the characterization of chaotic time waveforms in terms of the underlying bifurcation parameters chaotic time waveform recognition several numerical experiments using the rossler equations show the efficiency of the algorithm the effect of observational noise included in chaotic time waveforms is also considered

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

minute ahead stock price forecasting based on singular spectrum analysis and support vector regression abstract time series modeling and forecasting is an essential and hard task in financial engineering and optimization various models have been proposed in the literature and tested on daily data however a limited attention has been given to intraday data in this regard the current work presents a model for intraday stock price prediction that uses singular spectrum analysis ssa and support vector regression svr coupled with particle swarm optimization pso in particular the ssa decomposes stock price time series into a small number of independent components used as predictors the svr is applied to the task of forecasting and pso is employed to optimize svr parameters the performance of our proposed model is compared to the performance of four models widely used in financial prediction the wavelet transform wt coupled with feedforward neural network ffnn autoregressive moving average arma process polynomial regression polyreg and naive model finally the mean absolute error mae mean absolute percentage error mape and the root mean of squared errors rmse are used as main performance metrics by applying all models to six intraday stock price time series the forecasting results from simulations show that the presented ssa pso svr largely outperforms the conventional wt ffnn arma polynomial regression and naive model in terms of mae mape and rmse therefore our proposed predictive system ssa pso svr shows evident potential for noisy financial time series analysis and forecasting

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

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

some extensions of radial basis functions and their applications in artificial intelligence abstract radial basis functions have recently found interesting applications in artificial intelligence and in particular in the problem of learning to perform a particular task from a set of examples however in many practical cases the radial basis functions method cannot be applied in a straight forward manner because it does not take into account some features that are typical of the problem of learning from examples in this paper we show some extensions of the standard theory introduced in order to deal with a large number of examples and with problems in which different variables play very different roles we present some examples and also point out some open problems

an experimental observation of a new type of intermittency abstract a novel hysteretic form of regular intermittency has been observed experimentally in a variant of the taylor couette problem where the fluid flow is confined between a stationary outer cylinder and a rotating inner cylinder and end plates the results cannot all be explained within the standard pomeau manneville classification of intermittency and an extension to this theory is proposed to account for the observed phenomena a satisfactory agreement is found between experiment and theory both for the reinjection burst frequency scaling law and the influence of noise on the regularity of the bursts

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

noisy precursors of nonlinear instabilities this paper studies the effect of external noise on systems displaying nonlinear instabilities of periodic orbits each class of instability is found to have its own characteristic signature as displayed by the power spectrum results are derived for each of the codimension one instabilities familiar from bifurcation theory

statistical analysis of geopotential height gh timeseries based on tsallis non extensive statistical mechanics abstract in this paper we perform statistical analysis of time series deriving from earth s climate the time series are concerned with geopotential height gh and correspond to temporal and spatial components of the global distribution of month average values during the period 1948 2012 the analysis is based on tsallis non extensive statistical mechanics and in particular on the estimation of tsallis q triplet namely q stat q sens q rel the reconstructed phase space and the estimation of correlation dimension and the hurst exponent of rescaled range analysis r s the deviation of tsallis q triplet from unity indicates non gaussian tsallis q gaussian non extensive character with heavy tails probability density functions pdfs multifractal behavior and long range dependences for all timeseries considered also noticeable differences of the q triplet estimation found in the timeseries at distinct local or temporal regions moreover in the reconstructive phase space revealed a lower dimensional fractal set in the gh dynamical phase space strong self organization and the estimation of hurst exponent indicated multifractality non gaussianity and persistence the analysis is giving significant information identifying and characterizing the dynamical characteristics of the earth s climate

degrees of freedom of a time series we give a formal proof that if f is a smooth dynamics on a d dimensional smooth manifold and is an ergodic and exact dimensional measure with hausdorff dimension dim d 1 then the number d of degrees of freedom of the dynamics can be recovered from the observation of an orbit we implement with this purpose an algorithm based on the analysis of the microstructure of we show how a correct estimation of d permits the computation of the liapunov spectrum with a high accuracy avoiding the issue of the spurious exponents

an adaptive fuzzy system for modeling chaos abstract most chaotic systems do not have even dynamical behavior over the whole phase space in some regions the system stretches and branches more violently than others in the regions where the dynamics are violent finer representation must be given in this paper we make use of this property to model chaos we present an adaptive system based on fuzzy logic it can refine its representation of a region in the phase space if that region requires it it does so by adaptively generating more fuzzy rules to model a region only if that region has very violent dynamics experiments were performed to test the adaptive fuzzy system for capturing the dynamics of a normal dynamical system the van der pol oscillator as well as two chaotic systems the lorenz and rossler attractors results indicate that the fuzzy system can produce an accurate model of the three dynamical systems the adaptive rule generation algorithm allowed the fuzzy system to have an optimal number of rules

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

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

noisy chaos in intraday financial data evidence from the american index the presence of chaos in financial markets was inconclusive due mainly to test misspecification and data type although noisy chaos was investigated in recent studies it was only explored in daily returns which does not necessary mean that continuous intra daily data will exhibit the same dynamics high level noisy chaos is tested in the standard poor s 500 index returns over 4 different frequencies weekly daily 30 min and 5 min basis the dynamics in all frequencies are non chaotic

investigating observability properties from data in nonlinear dynamics investigation of observability properties of nonlinear dynamical systems aims at giving a hint on how much dynamical information can be retrieved from a syste mu sing ac ertain measuring function such an investigation usually requires the knowledge of the system equations this paper addresses the challenging problem of investigating observability properties of a system only from recorded data from previous studies it is known that phase spaces reconstructed from poor observables are characterized by local sharp pleatings local strong squeezing of trajectories and global inhomogeneity a statistic is then proposed to quantify such properties of poor observability such a statistic was computed for a number of bench models for which observability studies had been previously performed it was found that the statistic proposed in this paper estimated exclusively from data correlates generally well with observability results obtained using the system equations it is possible to arrive a tt he same order of observability among the state variables using the proposed statistic even in the presence of noise with standard deviation as high as 10 of the data the paper includes the application of the proposed statistic to sunspot time series

measurement and analysis of complex modulated motions in a weakly nonlinear system abstract the two timing dynamics of a forced weakly nonlinear system are considered experimental results on a roll forced spherical pendulum verify the occurrence of slow time subharmonically and chaotically modulated oscillations in a frequency band near resonance the modulatory behavior follows the occurrence of a hopf bifurcation in the slow time coordinates a phase sensitive detection method for experimentally isolating the slow time behavior is described a lagrangian model for the pendulum is developed and slow time equations are presented a two thirds power scalling law relating roll angle to the appearance of the modulations is derived from the slow time equations and tested with experimental results

sensitivity analysis of distributed parameter systems a method is developed for the calculation of sensitivity coefficients of general distributed parameter systems to the variation of spatially and temporally varying parameters appearing in the system equations and initial and boundary conditions

construction and analysis of differential equations from experimental time series of oscillatory systems abstract we propose a novel method to derive an autonomous system of ordinary differential equations from an experimental time series this model describe quantitatively transient behaviour of a system with limit cycle dynamics and is simplified to an evolution equation for the amplitude and frequency the landau equation in addition the model is used for a qualitative prediction of the influence of periodic excitation on the experimental system using the van der pol transformation and floquet theory in particular the instability of the locking solution including the hopf bifurcation the intermittency scenario and the period doubling cascade is considered the method of constructing a differential equation from an experimental time signal is successfully applied to a velocity fluctuation of the laminar periodic wake behind a circular cylinder

an approximate analytic solution of the lorenz equations multiple time perturbation theory is applied to the lorenz equations to obtain approximate solutions these solutions are used to obtain the form of the mapping relating two successive maxima of one of the variables of the equations good agreement is obtained with the numerical results presented by lorenz

on the instability of equilibrium when the potential has a non strict local minimum instability is studied for a mechanical system with two degrees of freedom whose potential has a non strict local minimum at the origin under suitable conditions of smoothness it is shown that the corresponding equilibrium is unstable when the potential has a local minimum along a curve passing through the origin as a particular case we get instability if the potential is analytic and has a non isolated minimum at the origin these results improve those of g hamel 1 and silla 5 on the same subject

faint phase synchronization detection through structured orthomax rotations in singular spectrum analysis multivariate singular spectrum analysis m ssa with a structured varimax rotation is a method that allows a deep characterization of phase synchronization ps phenomena in an almost automatic fashion it has been increasingly used in the study of ps in networks of nonlinear real world and numeric systems this paper investigates the impact of the other recently developed structured orthomax rotations on the m ssa ability to characterize ps the results show that by using the structured quartimax rotation a very faint and intermittent ps regime can be detected in contrast with the structured varimax which demands a stronger more consolidated ps regime this is due to the fact that the different rotations do not have the same efficiency in achieving a simple structure of the m ssa eigenvectors nevertheless for well established ps regimes the same robustness of the original m ssa approach against high levels of additive gaussian noise was found for the structured quartimax and biquartimax rotations however for all approaches we found an overshoot of the qualitative range for the ps onset due to noise

simultaneous modeling of nonlinear deterministic and stochastic dynamics abstract we present a method which can simultaneously model the nonlinear deterministic and stochastic dynamics underlying an observed time series it is formulated to treat markov processes in continuous and discrete time the procedure which we call sequin stochastic equation inference is a generalization of that which we have developed previously for continuous time systems borland and haken 1992a b 1993a b haken 1988 the theory behind the method as well as its numerical implementation is discussed simulations using sequin to analyze finite size time series stemming from nonlinear processes with additive and multiplicative dynamical noise in continuous time are shown furthermore we show that sequin can extract the underlying dynamics of discrete time noisy chaotic processes such as the logistic map with additive dynamical noise the success of the method in being able to characerize a system consisting partly of a deterministic chaotic element and partly of a stochastic element indicates its possible application to many interesting real world problems

does sunspot numbers cause global temperatures a reconsideration using non parametric causality tests abstract in a recent paper gupta et al 2015 analyzed whether sunspot numbers cause global temperatures based on monthly data covering the period 1880 1 2013 9 the authors find that standard time domain granger causality test fails to reject the null hypothesis that sunspot numbers do not cause global temperatures for both full and sub samples namely 1880 1 1936 2 1936 3 1986 11 and 1986 12 2013 9 identified based on tests of structural breaks however frequency domain causality test detects predictability for the full sample at short 2 2 6 months cycle lengths but not the sub samples but since full sample causality cannot be relied upon due to structural breaks gupta et al 2015 conclude that the evidence of causality running from sunspot numbers to global temperatures is weak and inconclusive given the importance of the issue of global warming our current paper aims to revisit this issue of whether sunspot numbers cause global temperatures using the same data set and sub samples used by gupta et al 2015 based on an nonparametric singular spectrum analysis ssa based causality test based on this test we however show that sunspot numbers have predictive ability for global temperatures for the three sub samples over and above the full sample thus generally speaking our non parametric ssa based causality test outperformed both time domain and frequency domain causality tests and highlighted that sunspot numbers have always been important in predicting global temperatures

distinguishing between low dimensional dynamics and randomness in measured time series the success of current attempts to distinguish between low dimensional chaos and random behavior in a time series of observations is considered first we discuss stationary stochastic processes which produce finite numerical estimates of the correlation dimension and k2 entropy under naive application of correlation integral methods we then consider several straightforward tests to evaluate whether correlation integral methods reflect the global geometry or the local fractal structure of the trajectory this determines whether the methods are applicable to a given series if they are we evaluate the significance of a particular result for example by considering the results of the analysis of stochastic signals with statistical properties similar to those of observed series from the examples considered it is clear that the correlation integral should not be used in isolation but as one of a collection of tools to distinguish chaos from stochasticity

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

using renyi parameter to improve the predictive power of singular value decomposition entropy on stock market abstract this paper generalizes the method of traditional singular value decomposition entropy by incorporating orders q of renyi entropy we analyze the predictive power of the entropy based on trajectory matrix using shanghai composite index and dow jones index data in both static test and dynamic test in the static test on sci results of global granger causality tests all turn out to be significant regardless of orders selected but this entropy fails to show much predictability in american stock market in the dynamic test we find that the predictive power can be significantly improved in sci by our generalized method but not in dji this suggests that noises and errors affect sci more frequently than dji in the end results obtained using different length of sliding window also corroborate this finding

the effect of noise on pitchfork and hopf bifurcations we present the results of an experimental and numerical investigation into the effects of noise on pitchfork and hopf bifurcations good quantitative agreement is found between calculations and experiment in the case of the pitchfork we find that natural imperfections override the effects of the noise however novel noise amplification effects have been uncovered in the study of the hopf bifurcation these destroy the critical event found in the noise free case and could be of considerable practical importance in systems containing dynamic bifurcations

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

flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus we present evidence for a sequence of bifurcations in simulations of the differentially heated baroclinic rotating annulus similar to bifurcations of the logistic map the met office oxford rotating annulus laboratory simulation morals code is used to construct a detailed numerical regime diagram for the annulus and the distribution of regimes in parameter space is discussed the bifurcations are observed in a sequence of runs at high temperature forcing identified by poincare sections of the dominant temperature mode amplitude time series higher order return maps and predictions using quadratic fits to the data are used to verify this result and lyapunov exponents are calculated to identify and quantify the chaotic parts of the sequence

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

correlation of local stretchings as a way of characterising chaotic dynamics amid noise abstract a positive lyapunov exponent can result from either chaotic dynamics or noise as a possible means of assessing the degree to which the exponent is noise affected and as a possible index in its own right the localised divergence of two neighbouring trajectories from a fiducial trajectory is compared in the expectation of finding strong correlation when the source of stretching is deterministic chaos and little or no correlation when the source is coloured or uncoloured noise the method is applied to various equation systems and to a periodic time series from the self excited oscillation of flexible tube collapsed by external pressure and conveying a flow

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

reconstructing bifurcation diagrams only from time waveforms abstract a simple algorithm is proposed for reconstructing bifurcation diagrams only from time waveforms of a dynamical system without knowing an explicit form of the dynamics and its dependence on the parameters reconstruction means that the obtained bifurcation diagram is qualitatively similar to that of the original system the algorithm consists of two steps first a nonlinear predictor is sought using a particular class of functions with many parameters second the karhunen loeve transform is used to find only significant parameters contributing the bifurcations the algorithm is tested against two families of dynamical system the henon family and the coupled logistic delayed logistic family

quasicycles in the stochastic hybrid morris lecar neural model intrinsic noise arising from the stochastic opening and closing of voltage gated ion channels has been shown experimentally and mathematically to have important effects on a neuron s function study of classical neuron models with stochastic ion channels is becoming increasingly important especially in understanding a cell s ability to produce subthreshold oscillations and to respond to weak periodic stimuli while it is known that stochastic models can produce oscillations quasicycles in parameter regimes where the corresponding deterministic model has only a stable fixed point little analytical work has been done to explore these connections within the context of channel noise using a stochastic hybrid morris lecar ml model we combine a system size expansion in k and a quasi steady state qss approximation in persistent na in order to derive an effective langevin equation that preserves the low dimensional planar structure of the underlying deterministic ml model the qss analysis exploits the fact that persistent na channels are fast by calculating the corresponding power spectrum we determine analytically how noise significantly extends the parameter regime in which subthreshold oscillations occur

analytical mini max methods for hamiltonian brake orbits of prescribed energy on considere des caracterisations explicites de solutions periodiques de systemes hamiltoniens autonomes d ordre 2 de la forme q v q q t r n ou le potentiel v satisfait les conditions suivantes v c 2 r n v x v o 0 v est strictement convexe sur r n

reconstructing attractors from filtered time series abstract in this paper we investigate the criteria necessary to reconstruct an attractor from a time series that has been passed through an iir filter to guarantee that the whole of the attractor is reconstructed requires uniform bounds on the contraction rate of the dynamics however we show that if we use a condition that is only based on the average contraction rates the lyapunov exponents then a set of high probability can still be reconstructed we argue that in practice this is all that is necessary

optimal reconstruction of dynamical systems a noise amplification approach in this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements these statistics can be evaluated on any reconstructed attractor thereby allowing a direct comparison among different approaches uniform or nonuniform delay vectors pca legendre coordinates etc it can also be used to select the most appropriate parameters of a reconstruction strategy in the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function its definition is based on theoretical arguments on noise amplification the complexity of the reconstructed attractor and a direct measure of local stretch which constitutes an irrelevance measure the proposed method is demonstrated on synthetic and experimental time series