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In this contribution, we show that the incorporation of nonlinear dynamical measures into a multivariate discrimination provides a signal classification system that is robust to additive noise. The signal library was composed of nine groups of signals. Four groups were generated computationally from deterministic systems (van der Pol, Lorenz, Rössler and Hénon). Four groups were generated computationally from different stochastic systems. The ninth group contained inter-decay interval sequences from radioactive cobalt. Two classification criteria (minimum Mahalanobis distance and maximum Bayesian likelihood) were tested. In the absence of additive noise, no errors occurred in a within-library classification. Normally distributed random numbers were added to produce signal to noise ratios of 10, 5 and 0 dB. When the minimum Mahalanobis distance was used as the classification criterion, the corresponding error rates were 2.2%, 4.4% and 20% (Expected Error Rate = 89%). When Bayesian maximum likelihood was the criterion, the error rates were 1.1%, 4.4% and 21% respectively. Using nonlinear measures an effective discrimination can be achieved in cases where spectral measures are known to fail.

Most classification errors occurred at low signal to noise ratios when a stochastic signal was misclassified into a different group of stochastic signals. When the within-library classification exercise is limited to the four groups of deterministic signals, no classification errors occurred with clean data, at SNR = 10 dB, or at SNR = 5 dB. A single classification error (Observed Error Rate = 2.5%, Expected Error Rate = 75%) occurred with both classification criteria at SNR = 0 dB.

This tutorial provides a nonlinear dynamics perspective to Wolfram's monumental work on A New Kind of Science. By mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram's empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenomena is that it is based on concepts from nonlinear dynamics and attractors where many fuzzy concepts introduced by Wolfram via brute force observations can be defined and justified via mathematical analysis. The main result of Part I is the introduction of a fundamental concept called linear separability and a complexity index κ for each local Rule which characterizes the intrinsic geometrical structure of an induced "Boolean cube" in three-dimensional Euclidean space. In particular, Wolfram's seductive idea of a "threshold of complexity" is identified with the class of local Rules having a complexity index equal to 2.

Several complexity measures, especially approximate entropy (ApEn) and a new defined complexity measure , of EEG signals or the ones of the mutual information transmission between different channels of EEGs were calculated to distinguish different consciousness levels for different brain functional states. All of the measures decreased with the following order of brain states: rest with eyes open, eyes closed, light sleep and deep sleep. They decreased during epileptic seizures. On the contrary, the averaged mutual information between different channels increased significantly during the epileptic seizure; there is no significant difference among the averaged mutual information for the subject resting with eyes open, closed, being in light sleep and in deep sleep. Thus, the former indexes seem to be promising candidates to characterize different consciousness levels, while the latter seems not.

We consider a simple reentrant model of a manufacturing process, consisting of one machine at which two different types of items have to be processed. The model is completely deterministic: all delivery and processing times are fixed, and are generally incommensurate. Dependent on the arrival and processing times, a queue of waiting items grows, remains constant or disappears. We demonstrate that the dynamics of the system crucially depends on the queue type. Complexity is most observed for the case of growing queue. We characterize this dynamics between order and chaos with the T-entropy and the autocorrelation function.

This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.

We investigated the dynamical characteristics of neuromagnetic responses by recording magnetoencephalographic (MEG) signals to equiluminant flickering stimulus of different color combinations from a group of control subjects, and from a patient with photosensitive epilepsy. By wavelet based time-frequency analysis, we showed that two distinct neuromagentic responses corresponding to stimulus frequency and its time delayed first harmonic were found in control subjects, whereas no harmonic response was obtained for the patient. We applied a battery of methods (sample entropy measuring signal complexity and index of smoothness measuring determinism) based on nonlinear dynamical system theory in conjunction with bootstrapping surrogate analysis. The results suggested that a significant nonlinear structure was evident in the MEG signals for control subjects, whereas nonlinearity was not detected for the patient. In addition, the couplings between distant cortical regions were found to be greater for control subjects. The important role of combinational chromatic sensitivity in sustained cortical excitation was also confirmed. These findings lead to the hypothesis that the healthy human brain is most likely equipped with significantly nonlinear neuronal processing reflecting an inherent mechanism defending against hyper-excitation to chromatic flickering stimulus, and such nonlinear mechanism is likely to be impaired for a patient with photosensitive epilepsy.

A concept of higher order complexity is proposed in this letter. If a randomness-finding complexity [Rapp & Schmah, 2000] is taken as the complexity measure, the first-order complexity is suggested to be a measure of randomness of the original time series, while the second-order complexity is a measure of its degree of nonstationarity. A different order is associated with each different aspect of complexity. Using logistic mapping repeatedly, some quasi-stationary time series were constructed, the nonstationarity degree of which could be expected theoretically. The estimation of the second-order complexity of these time series shows that the second-order complexities do reflect the degree of nonstationarity and thus can be considered as its indicator. It is also shown that the second-order complexities of the EEG signals from subjects doing mental arithmetic are significantly higher than those from subjects in deep sleep or resting with eyes closed.

In this contribution, eleven different measures of the complexity of multichannel EEGs are described, and their effectiveness in discriminating between two behavioral conditions (eyes open resting versus eyes closed resting) is compared. Ten of the methods were variants of the algorithmic complexity and the covariance complexity. The eleventh measure was a multivariate complexity measure proposed by Tononi and Edelman. The most significant between-condition change was observed with Tononi–Edelman complexity which decreased in the eyes open condition. Of the algorithmic complexity measures tested, the binary Lempel–Ziv complexity and the binary Lempel–Ziv redundancy of the first principal component following mean normalization and normalization against the standard deviation gave the most significant between-group discrimination. A time-dependent generalization of the covariance complexity that can be applied to nonstationary multichannel signals is also described.

The paper stresses the universal role that Cellular Nonlinear Networks (CNNs) are assuming today. It is shown that the dynamical behavior of 3D CNN-based models allows us to approach new emerging problems, to open new research frontiers as the generation of new geometrical forms and to establish some links between art, neuroscience and dynamical systems.

CNN templates for image processing and pattern formation are derived from neural field equations, advection equations and reaction–diffusion equations by discretizing spatial integrals and derivatives. Many useful CNN templates are derived by this approach. Furthermore, self-organization is investigated from the viewpoint of divergence of vector fields.

We study several algorithms to simulate bone mass loss in two-dimensional and three-dimensional computed tomography bone images. The aim is to extrapolate and predict the bone loss, to provide test objects for newly developed structural measures, and to understand the physical mechanisms behind the bone alteration. Our bone model approach differs from those already reported in the literature by two features. First, we work with original bone images, obtained by computed tomography (CT); second, we use structural measures of complexity to evaluate bone resorption and to compare it with the data provided by CT. This gives us the possibility to test algorithms of bone resorption by comparing their results with experimentally found dependencies of structural measures of complexity, as well as to show efficiency of the complexity measures in the analysis of bone models. For two-dimensional images we suggest two algorithms, a threshold algorithm and a virtual slicing algorithm. The threshold algorithm simulates bone resorption on a boundary between bone and marrow, representing an activity of osteoclasts. The virtual slicing algorithm uses a distribution of the bone material between several virtually created slices to achieve statistically correct results, when the bone-marrow transition is not clearly defined. These algorithms have been tested for original CT 10 mm thick vertebral slices and for simulated 10 mm thick slices constructed from ten 1 mm thick slices. For three-dimensional data, we suggest a variation of the threshold algorithm and apply it to bone images. The results of modeling have been compared with CT images using structural measures of complexity in two- and three-dimensions. This comparison has confirmed credibility of a virtual slicing modeling algorithm for two-dimensional data and a threshold algorithm for three-dimensional data.

Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction–diffusion system possess to make the Turing interacting system oscillate.

The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability.

Among other things, a mathematical proof is given which shows none of the complexity-related jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge.

The emergence of complexity is investigated from the viewpoint of the energy balance property and the divergence property of reaction–diffusion cellular neural networks.

The primary purpose of this paper is to show that simple dissipation can bring about oscillations in certain kinds of asymptotically stable nonlinear dynamical systems; namely when the system is locally active where the dissipation is introduced. Furthermore, if these nonlinear dynamical systems are coupled with appropriate choice of diffusion coefficients, then the coupled system can exhibit spatio-temporal oscillations. The secondary purpose of this paper is to show that spatio-temporal oscillations can occur in spatially discrete reaction diffusion equations operating on the edge of chaos, provided the array size is sufficiently large.

We investigate the applicability of the permutation entropy H and a synchronization index γ that is based on the changing tendency of temporal permutation entropies to analyze noisy time series from nonstationary dynamical systems with poorly understood properties. Using model systems, we first study the interdependencies of parameters involved in the calculation of both measures. Having identified appropriate parameter settings we then analyze long-lasting EEG time series recorded from an epilepsy patient. Our findings indicate that γ could be of interest for studies on the predictability of epileptic seizures.

The generation and progression of epileptiform activity, especially that associated with ictal paroxysmal neuronal discharges (seizures), is usually studied in terms of its temporal evolution rather than its spatial organization. The characterization of the spatio-temporal dynamics of epileptiform activity represents a major challenge in neuroscience, due to the very intricate nature of brain structure and function. Our study is an initial attempt to reveal the structure hidden under the spatial organization of the synchronization patterns in neuronal activity associated with epilepsy. Analysis of the phase synchronization patterns from magnetoencephalographic recordings in an epileptic patient revealed a decrease in complexity during seizures. Distinct patterns of synchronized activity were observed during interictal and ictal (seizure) activity, and new tools to quantify and visualize the information contained in a synchrony pattern are proposed. The results reported here support previous observations on the high local synchronization in seizures.

Throughout this paper I will argue that dynamical and structural instabilities are the sources of complexity and pattern formation. The argument can be accomplished by defending two theses. First, a demarcation thesis: two different approaches are predominant in mathematical sciences today — the symmetry-approach and the complexity-approach. Second, a synthesis thesis: although the two approaches differ, they can be connected and, further, to some degree integrated: instabilities are core concepts connecting the two approaches. However, in a specific sense we can say: evolution leads from symmetry to complexity by transitions across borders of instabilities. This paper will provide further arguments in favor of a structural unity in phenomenological diversity [Mainzer, 2005].

The complex dynamics of Alfvén waves described by the derivative nonlinear Schrödinger equation is investigated. In a region of the parameters space where multistability is observed, this complex system is driven towards an intermittent regime by the addition of noise. The effects of Gaussian and non-Gaussian noise are compared. In the intermittent regime, the Alfvén wave exhibits random qualitative changes in its dynamics as the result of a competition between three attractors and a chaotic saddle embedded in the fractal basin boundary.

We present an algorithm, completely random in nature, that, without invoking a fitness function or purposeful design, produces symmetric replicas in a population of so-called cellicules. Cellicules consist of "cells" arranged in a structure with geometric symmetry S. Each cell has one of two possible states, thus defining a state-configuration pattern on a cellicule. The algorithm acts recurrently on a population of cellicules, possibly randomly initialized, through a random "copying interaction" between two randomly selected cellicules that first undergo a random reorientation in accordance with the symmetry S. The dynamics of the algorithm is analyzed in detail for several symmetries. This shows that it is a random walk with absorbing states which correspond to a population in which all cellicules have an identical S-symmetric configuration pattern. We discuss some aspects concerning the evolution of cellicule-populations under mixing and mutation, and some variations on the basic algorithm.

A mutualism is an interaction where the involved species benefit from each other. We study a two-dimensional hexagonal three-state cellular automaton model of a two-species mutualistic system. The simple model is characterized by four parameters of propagation and survival dependencies between the species. We map the parametric set onto the basic types of space-time structures that emerge in the mutualistic population dynamic. The structures discovered include propagating quasi-one-dimensional patterns, very slowly growing clusters, still and oscillatory stationary localizations. Although we hardly find such idealized patterns in nature, due to inreased complexity of interaction phenomena, we recognize our findings as basic spatial patterns of mutualistic systems, which can be used as baseline to build up more complex models.