Chaotic Time Series

For various classes of time-delay systems we propose the methods of their model delay-differential equation reconstruction from time series. The methods are based on the characteristic location of extrema in the time series of time-delay systems and the projection of infinite-dimensional phase space of these systems to suitably chosen low-dimensional subspaces. We verify our methods by using them for the recovery of time-delay differential equations from their chaotic solutions and for modelling experimental systems with delay-induced dynamics from their chaotic time series.

This paper is concerned with the automatic induction of parsimonious neural networks. In contrast to other program induction situations, network induction entails parametric learning as well as structural adaptation. We present a novel representation scheme called neural trees that allows efficient learning of both network architectures and parameters by genetic search. A hybrid evolutionary method is developed for neural tree induction that combines genetic programming and the breeder genetic algorithm under the unified framework of the minimum description length principle. The method is successfully applied to the induction of higher order neural trees while still keeping the resulting structures sparse to ensure good generalization performance. Empirical results are provided on two chaotic time series prediction problems of practical interest.

As a powerful paradigm for knowledge representation and a simulation mechanism applicable to numerous research and application fields, Fuzzy Cognitive Maps (FCMs) have attracted a great deal of attention from various research communities. However, the traditional FCMs do not provide efficient methods to determine the states of the investigated system and to quantify causalities which are the very foundation of the FCM theory. Therefore in many cases, constructing FCMs for complex causal systems greatly depends on expert knowledge. The manually developed models have a substantial shortcoming due to model subjectivity and difficulties with accessing its reliability. In this paper, we propose a fuzzy neural network to enhance the learning ability of FCMs so that the automatic determination of membership functions and quantification of causalities can be incorporated with the inference mechanism of conventional FCMs. In this manner, FCM models of the investigated systems can be automatically constructed from data, and therefore are independent of the experts. Furthermore, we employ mutual subsethood to define and describe the causalities in FCMs. It provides more explicit interpretation for causalities in FCMs and makes the inference process easier to understand. To validate the performance, the proposed approach is tested in predicting chaotic time series. The simulation studies show the effectiveness of the proposed approach.

A method (NEMG) proposed in 1992 for diagnosing chaos in noisy time series with 50 or fewer observations entails fitting the time series with an empirical function which predicts an observation in the series from previous observations, and then estimating the rate of divergence or convergence of two nearby trajectories of the function (sensitive dependence on initial conditions). Workers applying NEMG have concluded that chaos exists in some natural biological populations. However, there are insufficient published tests of the reliability of NEMG using time series from biologically realistic models with numerous (>4) state variables. Also, there are unresolved technical issues for NEMG: whether or not the data should be detrended (removal of linear or quadratic trends) before analysis, and how long a sampling interval should be used to define the time series. We addressed these issues by applying NEMG to time series of 50 observations generated by four different chaotic models, ranging from simple models of physical systems to a complex ecosystem model. We also analyzed output from non-chaotic versions of each model. Previously recommended guidelines for specifying parameters affecting NEMG diagnoses of chaos were often ineffective. For the models we examined, NEMG produced high incidences of false determinations of chaos in non-chaotic time series, and of the absence of chaos in chaotic time series. Thus the method appears to be unreliable for the circumstances we examined.

A method (NEMG) proposed in 1992 for diagnosing chaos in noisy time series with 50 or fewer observations entails fitting the time series with an empirical function which predicts an observation in the series from previous observations, and then estimating the rate of divergence or convergence of two nearby trajectories of the function (sensitive dependence on initial conditions). Workers applying NEMG have concluded that chaos exists in some natural biological populations. However, there are insufficient published tests of the reliability of NEMG using time series from biologically realistic models with numerous (>4) state variables. Also, there are unresolved technical issues for NEMG: whether or not the data should be detrended (removal of linear or quadratic trends) before analysis, and how long a sampling interval should be used to define the time series. We addressed these issues by applying NEMG to time series of 50 observations generated by four different chaotic models, ranging from simple models of physical systems to a complex ecosystem model. We also analyzed output from non-chaotic versions of each model. Previously recommended guidelines for specifying parameters affecting NEMG diagnoses of chaos were often ineffective. For the models we examined, NEMG produced high incidences of false determinations of chaos in non-chaotic time series, and of the absence of chaos in chaotic time series. Thus the method appears to be unreliable for the circumstances we examined.

SPECTRAL ANALYSIS OF EXPANDING ONE-DIMENSIONAL CHAOTIC TRANSFORMATIONS Artur Lopes, S´ılvia Lopes and Rafael Souza Instituto de Matemática Universidade Federal do Rio Grande do Sul Av. Bento Gonçalves, 9500 Porto Alegre – RS – 91540-000 – Brasil ...

Any higher dimensional shift space (X, d) contains many lower dimensional shift spaces obtained by projection onto r-dimensional sublattices L of d where r < d. We show here that any projectional entropy is bounded below by the d entropy and, in the case of certain shifts of finite type satisfying a mixing condition, equality is achieved if and only if the shift of finite type is the infinite product of a lower dimensional projection.

This paper is concerned with the automatic induction of parsimonious neural networks. In contrast to other program induction situations, network induction entails parametric learning as well as structural adaptation. We present a novel representation scheme called neural trees that allows efficient learning of both network architectures and parameters by genetic search. A hybrid evolutionary method is developed for neural tree induction that combines genetic programming and the breeder genetic algorithm under the unified framework of the minimum description length principle. The method is successfully applied to the induction of higher order neural trees while still keeping the resulting structures sparse to ensure good generalization performance. Empirical results are provided on two chaotic time series prediction problems of practical interest.

Time series prediction is a rather difficult problem when the dynamics behind the data originates from a nonlinear system and its functional form is unknown. The hierarchical Bayesian scheme previously proposed by the authors has been shown to be reasonably sound for nontrivial real world applications. A great difficulty implementing the Hierarchical Bayesian scheme lies in the computation of posterior distributions as well as predictive distributions for which Quadratic Approximations have been used so far. This paper attempts to compute predictive mean and error bar for nonlinear time series prediction problems via the Hybrid Monte Carlo scheme, a particular class of Markov Chain Monte Carlo without the Quadratic Approximations. The scheme is tested against two concrete problems; Chaotic time series prediction, and Building airconditioning Load Prediction. The prediction results are compared with those using the Quadratic Approximations which have been used in the previous works of the authors' group. The proposed scheme outperforms the Quadratic Approximations.

A recurrence plot is a two-dimensional visualization technique for sequential data. These plots are useful in that they bring out correlations at all scales in a manner that is obvious to the human eye, but their rich geometric structure can make them hard to interpret. In this paper, we suggest that the unstable periodic orbits embedded in a chaotic attractor are a useful basis set for the geometry of a recurrence plot of those data. This provides not only a simple way to locate unstable periodic orbits in chaotic time-series data, but also a potentially effective way to use a recurrence plot to identify a dynamical system.

A simple, computationally efficient method for forecasting chaotic time series is described which makes implicit use of the Jacobian information in the data. Ifthe embedding dimension of the strange attractor is d, only d+ I linear equations need to be solved,

A simple, computationally efficient method for forecasting chaotic time series is described which makes implicit use of the Jacobian information in the data. Ifthe embedding dimension of the strange attractor is d, only d+ I linear equations need to be solved,

Neural Networks (NN) have been used extensively by researchers and practitioners to forecast financial time series. The forecasting accuracy of NN depends on several design parameters, and fine-tuning them to suit a particular financial time series is essential for attaining lower error levels and minimizing running time. This paper presents the results of a two-level full-factorial Design of Experiment developed to investigate the significant factors that influence the performance of NN in forecasting financial time series. The factors considered in this paper are NN type, number of neurons in the hidden layer, the learning rate of LM algorithm, and the type of output layer transfer function. The methodology is applied to the

In this paper, the idea of a new artificial intelligence based optimization algorithm, which is inspired from the nature of vortex, has been provided briefly. As also a bio-inspired computation algorithm, the idea is generally focused on a typical vortex flow / behavior in nature and inspires from some dynamics that are occurred in the sense of vortex nature. Briefly, the algorithm is also a swarm-oriented evolutional problem solution approach; because it includes many methods related to elimination of weak swarm members and trying to improve the solution process by supporting the solution space via new swarm members. In order have better idea about success of the algorithm; it has been tested via some benchmark functions. At this point, the obtained results show that the algorithm can be an alternative to the literature in terms of single-objective optimization solution ways. Vortex Optimization Algorithm (VOA) is the name suggestion by the authors; for this new idea of intelligent optimization approach.

We address the detection of unstable periodic orbits from experimentally measured transient chaotic time series. In particular, we examine recurrence times of trajectories in the vector space reconstructed from an ensemble of such time series. Numerical experiments demonstrate that this strategy can yield periodic orbits of low periods even when noise is present. We analyze the probability of finding periodic orbits from transient chaotic time series and derive a scaling law for this probability. The scaling law implies that unstable periodic orbits of high periods are practically undetectable from transient chaos.

In this paper the multi step ahead prediction of typical Duffing Chaotic time series and the monthly sunspots real time series are carried out. These two time series are popularized due to their highly chaotic behavior. This paper compares the performance of two neural network configurations namely a Multilayer Perceptron (MLP) and proposed FTLRNN with gamma memory for the duffing time series for 1, 5,10,20,50 and 100-step ahead prediction and for monthly sunspot time series for 1, 6, 12, 18 & 24 month ahead prediction. The standard back propagation algorithm with momentum term has been used for both the models. It is seen that estimated dynamic fully recurrent model clearly outperforms the MLP NN in various performance matrices such as Mean square error (MSE), Normalized mean square error (NMSE) and correlation coefficient (r) on testing as well as training data set for multi step prediction (K=1,5,10,20,50,100) for duffing time series and for the sunspot time series for 1, 6, 12, 18 &24 month ahead prediction. In addition, the output of proposed neural network model closely follows the desired output for all the step ahead prediction. It is observed that suggested recurrent models have the remarkable capability of time series prediction. The major contribution of this paper is that Various parameters like number of processing elements, step size, momentum value in hidden layer, in output layer the various transfer functions like tanh, sigmoid, linear-tan-h and linear sigmoid, different error norms L1, L2 ,Lp to L.

In this paper, we propose a quantization approach, as an alternative of sparsification, to curb the growth of the radial basis function structure in kernel adaptive filtering. The basic idea behind this method is to quantize and hence compress the input (or feature) space. Different from sparsification, the new approach uses the "redundant" data to update the coefficient of the closest center. In particular, a quantized kernel least mean square (QKLMS) algorithm is developed, which is based on a simple online vector quantization method. The analytical study of the mean square convergence has been carried out. The energy conservation relation for QKLMS is established, and on this basis we arrive at a sufficient condition for mean square convergence, and a lower and upper bound on the theoretical value of the steady-state excess mean square error. Static function estimation and short-term chaotic time-series prediction examples are presented to demonstrate the excellent performance.

Time series analysis using nonlinear dynamics systems theory and multilayer neural networks models have been applied to the time sequence of water level data recorded every hour at 'Punta della Salute' from Venice Lagoon during the years 1980-1994. The first method is based on the reconstruction of the state space attractor using time delay embedding vectors and on the characterisation of invariant properties which define its dynamics. The results suggest the existence of a low dimensional chaotic attractor with a Lyapunov dimension, D L , of around 6.6 and a predictability between 8 and 13 hours ahead. Furthermore, once the attractor has been reconstructed it is possible to make predictions by mapping local-neighbourhood to local-neighbourhood in the reconstructed phase space. To compare the prediction results with another nonlinear method, two nonlinear autoregressive models (NAR) based on multilayer feedforward neural networks have been developed. From the study, it can be observed that nonlinear forecasting produces adequate results for the 'normal' dynamic behaviour of the water level of Venice Lagoon, outperforming linear algorithms, however, both methods fail to forecast the 'high water' phenomenon more than 2-3 hours ahead.

This study presents an evolutionary neural fuzzy network, designed using the functional-link-based neural fuzzy network (FLNFN) and a new evolutionary learning algorithm. This new evolutionary learning algorithm is based on a hybrid of cooperative particle swarm optimization and cultural algorithm. It is thus called cultural cooperative particle swarm optimization (CCPSO). The proposed CCPSO method, which uses cooperative behavior among multiple swarms, can increase the global search capacity using the belief space. Cooperative behavior involves a collection of multiple swarms that interact by exchanging information to solve a problem. The belief space is the information repository in which the individuals can store their experiences such that other individuals can learn from them indirectly. The proposed FLNFN model uses functional link neural networks as the consequent part of the fuzzy rules. This study uses orthogonal polynomials and linearly independent functions in a functional expansion of the functional link neural networks. The FLNFN model can generate the consequent part of a nonlinear combination of input variables. Finally, the proposed FLNFN with CCPSO (FLNFN-CCPSO) is adopted in several predictive applications. Experimental results have demonstrated that the proposed CCPSO method performs well in predicting the time series problems.

Reservoir Computing models are a class of recurrent neural networks that have enjoyed recent attention, in particular, their main family, Echo State Networks (ESNs). These models have a large number of hidden-hidden weights (in the so-called reservoir) forming a recurrent topology. The reservoir is randomly connected with fixed weights during learning: only readout parameters (from reservoir to output neurons) are trained; the reservoir weights are frozen after initialized. Since the reservoir structure is fixed during learning, only its initialization process has an impact on the model's performance. In this work, we introduce an evolutionary method for adjusting the reservoir non-null weights. Moreover, the evolutionary process runs on the frequency space corresponding to a Fourier transformation of the weights. We combine an evolutionary search in the Fourier space with supervised learning for the readout weights. The resulting algorithm, called EvoESN (Evolutionary ESN), obtains promising results modeling two well-known problems of the chaotic time-series area. CCS CONCEPTS • Computing methodologies → Neural networks; Bio-inspired approaches; Genetic algorithms;

Reservoir Computing models are a class of recurrent neural networks that have enjoyed recent attention, in particular, their main family, Echo State Networks (ESNs). These models have a large number of hidden-hidden weights (in the so-called reservoir) forming a recurrent topology. The reservoir is randomly connected with fixed weights during learning: only readout parameters (from reservoir to output neurons) are trained; the reservoir weights are frozen after initialized. Since the reservoir structure is fixed during learning, only its initialization process has an impact on the model's performance. In this work, we introduce an evolutionary method for adjusting the reservoir non-null weights. Moreover, the evolutionary process runs on the frequency space corresponding to a Fourier transformation of the weights. We combine an evolutionary search in the Fourier space with supervised learning for the readout weights. The resulting algorithm, called EvoESN (Evolutionary ESN), obtains promising results modeling two well-known problems of the chaotic time-series area. CCS CONCEPTS • Computing methodologies → Neural networks; Bio-inspired approaches; Genetic algorithms;

We use control of chaos to encode information into the oscillations of the Belousov-Zhabotinsky reaction. An arbitrary binary message is encoded by forcing the chaotic oscillations to follow a specified trajectory. The information manipulating control requires only small perturbations to vary the binary message. In this paper we extend our recent theoretical work ͓Bollt and Dolnik, Phys. Rev. E 64, 1196 ͑1990͔͒ by introducing a new and simplified encoding technique which can be utilized in the presence of experimental noise. We numerically and theoretically study several practical aspects of controlling symbol dynamics including: modeling noisy time-series, learning underlying symbol dynamics, and evaluation of derivatives for control by observing system responses to an intelligent and deliberate sequence of input parameter variations. All of the modeling techniques incorporated here are ultimately designed to learn and control symbol dynamics of experimental data known only as an observed time-series; the simulation assumes no global model. We find that noise affects reliability of encoding information and may cause coding errors. But, if the level of noise is confined to relatively small values, which are achievable in experiments, the control mechanism is robust to the noise. Thus we can still produce a desired symbolic code. However, scarce errors in encoding may occur due to rare but large fluctuations. These errors may be corrected during the decoding process by a variation of the filtering technique suggested by Rosa et al. ͓Phys.

The Grassberger-Procaccia algorithm for estimating the correlation exponent yields estimates biased by systematic errors pronounced mainly for embedding dimension higher than 5. A modification of this algorithm is proposed that removes a part of the bias up to embedding dimension 11.

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.

This paper presents an original method of designing reversible circuits. This method is destined to most popular gate set with three types of gates CNT (Control, NOT and Toffoli). The presented algorithm based on graphical representation of the reversible function is called s-maps. This algorithm allows to find optimal or quasi-optimal reversible circuits. The paper is organized as follows. Section 1 recalls basic concepts of reversible logic. Especially the cascade of the gates as realization of reversible function is presented. In Section 2 there is introduced a classification of minterms distribution. The s-maps are the representation of the reversible functions where the minterms distribution is presented. The choice of the first gate in the cascade depends on possibility of improving the distribution. Section 3 describes the algorithm, namely how to find the optimal or quasi-optimal solutions of the given function.

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Multilayer perceptrons are successfully used in an increasing number of nonlinear signal processing applications. The backpropagation learning algorithm, or variations hereof, is the standard method applied to the nonlinear optimization problem of adjusting the weights in the network in order to minimize a given cost function. However, backpropagation as a steepest descent approach is too slow for many applications. In this paper a new learning procedure is presented which is based on a linearization of the nonlinear processing elements and the optimization of the multilayer perceptron layer by layer. In order to limit the introduced linearization error a penalty term is added to the cost function. The new learning algorithm is applied to the problem of nonlinear prediction of chaotic time series. The proposed algorithm yields results in both accuracy and convergence rates which are orders of magnitude superior compared to conventional backpropagation learning.

We introduce a numerical approximation method for estimating an unknown parameter of a ͑primary͒ chaotic system which is partially observed through a scalar time series. Specifically, we show that the recursive minimization of a suitably designed cost function that involves the dynamic state of a fully observed ͑second-ary͒ system and the observed time series can lead to the identical synchronization of the two systems and the accurate estimation of the unknown parameter. The salient feature of the proposed technique is that the only external input to the secondary system is the unknown parameter which needs to be adjusted. We present numerical examples for the Lorenz system which show how our algorithm can be considerably faster than some previously proposed methods.

We present a technique for building deterministic models of the nonlinear dynamics underlying observed time series. It is formulated from maximum entropy principle within the framework of information theory. Two numerical examples of chaotic time series illustrate the method. Furthermore, the theory on which the method is based, provides an entropy-like quantity that characterizes the suitability of the model. It is deÿned over the space of parameters of the model. We illustrate, with two applications, how this entropy can be a useful tool for exploring the phase space, and establishing a criterion for choosing a convenient parametrized functional form.

A hierarchical approximation of a generic chaotic attractor can be formulated in terms of unstable periodic orbits. We demonstrate the possibility of extracting the most dominant unstable periodic orbits from a measurement of a time-continuous Aow. Since unstable periodic orbits not only represent the static properties of the system but also dominate the dynamics, they can be used to fit models of the Aow and to predict the orbit. We show that predictions for the Roessler system using unstable periodic orbits extracted from a time series of moderate length are significantly better than those from other approaches.

Time series prediction is a remarkable research interest, which is widely followed by scientists / researchers. Because many fields include analyzing processes over such time series, different kinds of approaches, methods, and techniques are often employed in order to achieve alternative prediction ways. It seems that Artificial Intelligence oriented solutions have strong potential on providing effective and accurate prediction approaches in even most complicated time series structures. In the sense of the explanations, this study aims to

Optimization is one of the most remarkable research interests of the Artificial Intelligence field. In time, many different kinds of techniques regarding to 'intelligent optimization' have been developed and introduced to the associated literature. In this context, even a sub-research area called as Swarm Intelligence has taken part under the literature of Artificial Intelligence. At this point, objective of this paper is to introduce an alternative Swarm Intelligence based optimization algorithm, which is inspired from algorithmic thinking-reasoning. In detail, solution approach of the algorithms is based on graphs as similar to some already known algorithms like Ant Colony Optimization, and Intelligent Water Drops Algorithms. As different, the algorithm introduced here tries to get the optimum solution on graph-based solution space by employing some logical decision loops that are similar to basic if-then-else structures of algorithms. Because of that the algorithm is called as 'Algorithmic Reasoning Optimization (ARO)'. The paper briefly introduces background and solution approach of the ARO and provides some results regarding to its evaluation.

We propose a method that allows one to estimate the parameters of model scalar time-delay differential equations from time series. The method is based on a statistical analysis of time intervals between extrema in the time series. We verify our method by using it for the reconstruction of time-delay differential equations from their chaotic solutions and for modeling experimental systems with delay-induced dynamics from their chaotic time series.

In this paper, two issues are addressed: (1) the applicability of the delay-coordinate embedding method to transient chaotic time series analysis, and (2) the Hilbert transform methodology for chaotic signal processing.A common practice in chaotic time series analysis has been to reconstruct the phase space by utilizing the delay-coordinate embedding technique, and then to compute dynamical invariant quantities of interest such as unstable periodic orbits, the fractal dimension of the underlying chaotic set, and its Lyapunov spectrum. As a large body of literature exists on applying the technique to time series from chaotic attractors, a relatively unexplored issue is its applicability to dynamical systems that exhibit transient chaos. Our focus will be on the analysis of transient chaotic time series. We will argue and provide numerical support that the current delay-coordinate embedding techniques for extracting unstable periodic orbits, for estimating the fractal dimension, and f...

This article proposes a runtime model that relates server energy consumption to its overall thermal envelope, using hardware performance counters and experimental measurements. While previous studies have attempted system-wide modeling of server power consumption through subsystem models, our approach is different in that it links system energy input to subsystem energy consumption based on a small set of tightly correlated parameters. The proposed model takes into account processor power, bus activities, and system ambient temperature for real-time prediction on the power consumption of long running jobs. Using the HyperTransport and QuickPath Link structures as case studies and through electrical measurements on example server subsystems, we develop a chaotic time-series approximation for runtime power consumption, arriving at the Chaotic Attractor Predictor (CAP). With polynomial time complexity, CAP exhibits high prediction accuracy, having the prediction errors within 1.6% (or ...

We present an estimation procedure and analyse spectral properties of stochastic processes of the kind Zt = Xt + ξt = ϕ(Tt(ψ)) + ξt, for t ∊ Z, where T is a deterministic map, ϕ is a given function and ξt is a noise process. The examples considered in this paper generalize the classical harmonic model Zt = Acos(ω0t + ψ) + ξt, for t ∊ Z. Two examples are developed at length. In the first one, the spectral measure is discrete and in the second it is continuous. In the second example, the time series is obtained from a chaotic map. These two examples exhibit the extremal cases of different possibilities for the spectral measure of time series and they are both associated with ergodic deterministic transformations with noise. We present a method for obtaining explicitly the spectral density function (second example) and the autocorrelation coefficients (first example). In the first example the rotation number plays an important role. We also consider large deviation properties of the es...

Padua and Libao, is a versatile method used in solving the Inverse Frobenius-Perron problem (IFPP) for chaotic systems based on time series data. While their initial work focused on the logistic map and prime gap data, this article highlights the broader significance of LP-Conjugation as a powerful tool for the IFPP. Any chaotic map can be employed with established chaotic time series data to reconstruct unknown maps. LP-Conjugation utilizes a known chaotic map ψ(•) and invariant distribution F(•) to construct an unknown chaotic map θ(•) with known invariant distribution H(•). The article presents a detailed proof, illustrating the method's robustness and adaptability. Further illustrations on how to construct a chaotic map from this logistic-based chaotic map sample is also provided. Real-world examples demonstrate LP-Conjugation's efficacy in diverse applications, solidifying its role as a valuable approach for solving the IFPP and understanding chaotic dynamics.

The nonlinear electronic converter used by Rulkov and collaborators [Rulkov et al., 2001], which is the core of their chaotic oscillator, is modeled and simulated numerically by means of an appropriate direct relationship between the experimental values of the electronic components of the system and the mathematical model. This relationship allows us to analyze the chaotic behavior of the model in terms of a particular bifurcation parameter k. Varying the parameter k, quantitative results of the dynamics of the numerical system are presented, which are found to be in good agreement with the experimental measurements that we performed as well. Moreover, we show that this nonlinear converter belongs to a class of 3-D systems that can be mapped to the unfolded Chua&#39;s circuit. We also report a wavelet transform analysis of the experimental and numerical chaotic time series data of this chaotic system. The wavelet analysis provides us with information on such systems in terms of the ...

We investigate the scaling of the average time r between intermittent bursts for a chaotic system that undergoes a homoclinic tangency crisis, which causes a sudden expansion in the attractor. The system studied is a periodically driven (frequency f), nonlinear, magnetoelastic ribbon. The observed behavior of~is well fit by a power-law scaling r-If-fe I~, where f=fãt the crisis. We identify the unstable periodic orbit mediating the crisis, and determine its linearized eigenvalues from experimental data. The critical exponent y found from the scaling of i is shown to agree with that theoretically predicted fora two-dimensional map on the basis ofthe eigenvalues ofthe mediating periodic orbit.

Inspired by natural biochemicals that perform complex information processing within living cells, we design and simulate a chemically implemented feedforward neural network, which learns by a novel chemical-reaction-based analogue of backpropagation. Our network is implemented in a simulated chemical system, where individual neurons are separated from each other by semipermeable cell-like membranes. Our compartmentalized, modular design allows a variety of network topologies to be constructed from the same building blocks. This brings us towards general-purpose, adaptive learning in chemico: wet machine learning in an embodied dynamical system.

This paper presents a new numerical approach to the study of nonperiodicity in signals, which can complement the maximal Lyapunov exponent method for determining chaos transitions of a given dynamical system. The proposed technique is based on the continuous wavelet transform and the wavelet multiresolution analysis. A new parameter, the scale index, is introduced and interpreted as a measure of the degree of the signal's non-periodicity. This methodology is successfully applied to three classical dynamical systems: the Bonhoeffer-van der Pol oscillator, the logistic map, and the Henon map.

Cardiac diseases are one of the main reasons of mortality in modern, industrialized societies, and they cause high expenses in public health systems. Therefore, it is important to develop analytical methods to improve cardiac diagnostics. Electric activity of heart was first modeled by using a set of nonlinear differential equations. Latter, variations of cardiac spectra originated from deterministic dynamics are investigated. Analyzing the power spectra of a normal human heart presents His-Purkinje network, possessing a fractal like structure. Phase space trajectories are extracted from the time series graph of ECG. Lower values of fractal dimension, D indicate dynamics that are more coherent. If D has non-integer values greater than two when the system becomes chaotic or strange attractor. Recently, the development of a fast and robust method, which can be applied to multichannel physiologic signals, was reported. This manuscript investigates two different ECG systems produced from normal and abnormal human hearts to introduce an auxiliary phase space method in conjunction with ECG signals for diagnoses of heart diseases. Here, the data for each person includes two signals based on V 4 and modified lead III (MLIII) respectively. Fractal analysis method is employed on the trajectories constructed in phase space, from which the fractal dimension D is obtained using the box counting method. It is observed that, MLIII signals have larger D values than the first signals (V 4), predicting more randomness yet more information. The lowest value of D (1.708) indicates the perfect oscillation of the normal heart and the highest value of D (1.863) presents the randomness of the abnormal heart. Our significant finding is that the phase space picture presents the distribution of the peak heights from the ECG spectra, giving valuable information about heart activities in conjunction with ECG.

This paper focuses on the study of a one dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example-the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.

In this paper, we review our work on a time series forecasting methodology based on the combination of unsupervised clustering and artificial neural networks. To address noise and non-stationarity, a common approach is to combine a method for the partitioning of the input space into a number of subspaces with a local approximation scheme for each subspace. Unsupervised clustering algorithms have the desirable property of deciding on the number of partitions required to accurately segment the input space during the clustering process, thus relieving the user from making this ad hoc choice. Artificial neural networks, on the other hand, are powerful computational models that have proved their capabilities on numerous hard real-world problems. The time series that we consider are all daily spot foreign exchange rates of major currencies. The experimental results reported suggest that predictability varies across different regions of the input space, irrespective of clustering algorithm. In all cases, there are regions that are associated with a particularly high forecasting performance. Evaluating the performance of the proposed methodology with respect to its profit generating capability indicates that it compares favorably with that of two other established approaches. Moving from the task of one-step-ahead to multiple-step-ahead prediction, performance deteriorates rapidly.

The nonlinear dynamics of in-line bubbles rising with coalescence in non-Newtonian Carboxymethylcellulose sodium (CMC) fluids was studied through the techniques such as the multiresolution signal decomposition and the chaotic time series analysis. The temporary signals of bubble passages collected by an optical sensing device at different heights were investigated by a 12-level wavelet decomposition and the scalewise characteristics of bubble motion were extracted and analyzed. The chaotic time series analysis distinguished the periodicity or the deterministic chaos of bubble motion successsfully. The calculation of Kolmogorov entropy proves that in the ranges of experimental heights and gas flowrates, the bubble rising dynamics becomes more chaotic with the increase of height, and reaches the maximum chaotic extent in a certain height, while with the further increase of height, the chaotic extent decreases slowly. With the increase of gas flowrate, at the lower height, the bubble rising dynamics changes from periodicity to deterministic chaos, and at the higher heights it reaches the maximum chaotic extent in a certain gas flowrate; however, for both cases, it has little change in the higher gas flowrates. Moreover, with the increase of CMC concentration, the bubble rising dynamics becomes less chaotic when the height is beyond a certain value.