INTERNATIONAL  JOURNAL
of
CHAOS THEORY  and  APPLICATIONS
Vol. 4, Issue # 2-3, 1999



Special Issue edited by: Franco Orsucci and Giulio Cassati

Preface: A Historical Perspective
Giulio Casati
University of Milano at Como, ITALY
Keywords: history, statistical mechanics, complexity, quantum mechanics

Regular paper
Signal Analysis and Simulation of the EEG Activity
   M. Balduzzo, T.A. Minelli and L. Turicchia
   Universita di Padova, Italy

Abstract. The classification of brain rhythms and other co-operative phenomena is, at present, based on the analysis of the electroencephalographic (EEG) time series by means of algorithms used in chaos detection and by classical or advanced spectral techniques. Similar methods open the way towards the foundation of a brain mathematical anatomy. The use of more of less realistic neural networks, justified by the low correlation dimension, appears, on the other side, as a promising approach to the elements of a mathematical neurophysiology. We present a review of algorithms of analysis and simulation together to a voltage-driven reset-integrator version of the integrate and fire neural model suitable to explain the EEG chaotic behavior in terms of neuron synchronization and to simulate the rhythm phenomenology.

Keywords: chaos, human electroencephalogram (EEG), brain waves, brain rhythms, synchronization, non-linear models, neurodynamics.

Regular paper
Fractal Dimension Analysis in Human Pathology
G. Bianciardi, M. Del Vecchio, M.M. De Santi, G. Alia, A. Perrone and P. Luzi
University of Siena, Italy

Abstract. Conventional Euclidean morphometry often to analyze biological structure, such as the complex geometry of tumors. Fractal geometry, on the other hand, recently appears to be a useful tool for describing the irregular shapes of many natural objects. Here we present data concerning the fractal analysis of histopathological samples, at light and ultrastructural levels. For the former, 147 samples of Basal cell Carcinoma of the skin, of different diagnostic classes, and 27 samples of urothelial neoplasm were analyzed. For the latter, 13 cases of a benign lesion of the skin, such as chronic dermatitis, and 13 cases of a malignant lesion of the skin, such as early stages of mycosis fungoide were analyzed. Statistical analysis of the data revealed the ability of this approach to distinguish different diagnostic classes, giving importance to fractal analysis to supply useful morphometrical indexes in histopathology.

Keywords: biological structure, classification, fractal analysis, histopathological samples, urothelial neoplasia, dermatitis, mycosis

Regular paper
Special Factors for a Compact Description of Polynucleotides
Alfredo Colosimo and Aldo de Luca
Univ. of Rome “La Sapienza” Roma, Italy

Abstract. A previously proposed approach to the calculation of a Complexity Index based upon the total number of right special factors of a word is systematically used to characterize a natural polynucleotide among other natural or artificial polynucleotides of the same length. A vector related to the distribution of right special factor over the spectrum of their possible lengths has also been used as a new approach to the same task. The two methods were compared inn the analysis of genetic fragments of different origins and about 205000 by long and the latter one shows a higher sensitivity and discrimination power. An interesting dependence of the relative amount of special Factors in native fragments and in their reshuffled and random counterparts is reported and discussed. A software tool implementing the algorithms used in this study is made available.

Keywords: string complexity; biological information; primary structure analysis

Regular paper
A Fractal Scoring System for Quantifying Active Collagen Synthesis During Chronic Liver Disease
F. Grizzi and N. Dioguardi
Researcher of Istituto Clinico Humanitas, Rozzano, Milano, Italy
Scientific Director of Istituto Clinico Humanitas, Rozzano, Milano, Italy

Abstract. In this study, we describe a fractal geometry method, developed to quantify the irregular fibrotic lesions seen in biopsy specimens. The study was conducted on liver biopsy specimens obtained from 26 patients with chronic disease related to hepatitis C virus (HCV). The degree of fibrosis in each specimen was evaluated using quantitative image analysis and a fractal scoring system based on the two parameters of fractal and spectral dimension. The quantitative estimate of collagen irregularity provides numerical data capable of monitoring the active formation of connective tissue and the related changes in architectural tissue occurring during chronic disease. Furthermore, these data can be accurately applied to build mathematical simulations relating to the time of evolution to fibrosis.

Key words: Liver, Fibrosis, Fractals, Fractal dimension, Spectral dimension.

Regular paper
Chaotic dynamics in the discharge of a river
Silvano Bordignon and Francesco Lisi
University of Padua, Padua, Italy

Abstract. In this paper we analyze the discharge time series of an Italian river in order to find some evidence of chaotic behavior it is dynamics. To this purpose we consider a set of different procedures, namely phase portrait of the attractor, correlation dimension, the largest Lyapunov exponent, DVS diagram and non-linear prediction. Their joint application to our data leads to results that enable us not to exclude the presence of a non-linear deterministic dynamics of the chaotic type. These findings provide some interesting insights for better non-linear river flow modeling and prediction.

Keywords: river flow, chaos, non-linear prediction.

Regular paper
Fractal Patterns of the Complex Cell Structures
Gabriele A. Losa
Instituto di Patologia and Instituto di Ricerca in Fisica e Matematica, Locarno, Switzerland

Abstract. The irregularity and self-similarity under scale changes are the main attributes of the morphological complexity of cells and tissues, either normal or pathological. In other words, the shape of a self-similar object does not change when the measuring scale changes, because any part of it may look similar as the original object. However, size and geometrical parameters of an irregular object differ when inspected at increasing resolution, which reveals more details. The application of the principle of the fractal geometry, unlike to the conventional Euclidean geometry developed for describing regular and ideal geometrical shapes practically unknown in nature, enables to measure the fractal dimension, contour length, surface area and other dimensional parameters of almost all irregular and complex biological tissues. Over the last decades, a large amount of experimental evidence has been accumulated showing that for the biological tissues, fractal patterns or self-similar structures could be observed only within a “scaling window.” The measure of this window must be experimentally established. Within the window, experimental data set are following a straight line with slope (1-D), whereby the fractal dimension D is invariant with respect to changes of magnification. Through selected examples, we demonstrate the application of the fractal approach to measuring irregular and complex membrane structures of normal blood lymphocytes and of breast cancer cells.

Keywords: biological tissues, lymphocytes, neoplastic transformation, leukemic cells, membrane contours
 

Regular paper
Neural Modeling of Non-Linear Process: Relevance of the Takens-Mane Theorem
Francesco Masulli, Riccardo Parenti  and Leonard Studer
Universita di Genova, Genova, Italy,

Abstract. In this paper, we test a constructive methodology for shaping neural networks models of non-linear dynamic systems. The method is supported by results and prescriptions related to the Takens-Mane theorem and it is based on the measurement of the first minimum of the mutual information of the output signal, and in the application of the method of global false nearest neighbors to determine the embedding dimension. We present a numerical experiment to assess this constructive approach to the identification of a non-linear dynamic system and the application to the design of a neural network to forecasting a time series generated by an accelerometer coupled to a 150 MW steam turbine.

Keywords: neural network, prediction, predictor design, vibrations, steam engine

Regular paper
On the Fibonacci’s Attractor and the Long Orbits in the 3n + 1 Problem
Danilo Merlini and Nicoletta Sala
Cerfim (Rechearch Center of Mathematics and Physics), Locarno, Switzerland, Università di Economia e Finanza, Verbania, Italy, Largo Argentina Università  della Svizzera Italiana, Mendrisio, Switzerland

Abstract. In this paper, we consider some aspects of the 3n + 1 problem, namely we have divided the study of the 3n+ 1 problem in three parts: (1) the growth of the chalice; (2) the Fibonacci’s attractor; (3) the long orbits. We find the inverse orbits of the process and prove that the special sequence of numbers, which govern their structure, has a suggested property. Our numerical computation of the orbits of length up to 32 will be used to compute our value of the Collatz’s constant (the growth constant associated to the above map). We have found:

  •  a new model to study and to simplify this problem (using only the odd numbers);
  • a numerical sequence that is in excellent agreement with our theoretical values, obtained by mean of a Fibonacci’s attractor;
  •  some simple models to analyze the data using C++ programs on personal computer (IBM compatible).
    • Keywords: Iterative processes, inverse orbits, Collatz’s constant, stochastic Fibonacci sequence, Fibonacci’s attractor.
     

    Regular paper
    Orthographic Structuring of Human Speech and Texts: Linguistic Application of Recurrence Quantification Analysis
    Franco Orsucci, Kimberly Walter, Alessandro Giuliani, Charles L. Webber Jr., Josep P. Zbilut
    International University, Italy, American University, Rome, Italy, Instituto Superiore di Sanita, Rome, Italy, Loyola University, Maywood, USA, Rush University, Chicago, USA

    Abstract. A methodology based upon recurrence quantification analysis is proposed for the study of orthographic structure of written texts. Five different orthographic data sets (20th century Italian poems, 20th century American poems, contemporary Swedish poems with their corresponding Italian translations, Italian speech samples, and American speech samples) were subjected to recurrence quantification analysis. This procedure has been found useful in the quantitative assessment of ordered series in several fields. Recurrence quantification was developed from recurrence plots as applied to the analysis of non-linear systems in the physical sciences, and is based on the computation of a distance matrix of the elements of an ordered series (in this case the letters constituting selected speech and poetic texts). The results show the possibility of demonstrating invariance between different language exemplars, despite the apparent low-level of coding (orthography). Comparison with the actual texts confirms the ability of the method to reveal recurrent structures and their complexity. Using poems as a reference standard for judging speech complexity, the technique exhibits language independence, order dependence and freedom from pure statistical characteristics of the studied sequences, as well as consistency with easily identifiable texts. Such studies may provide phenomenological markers of hidden structure as coded by the purely orthographic level.

    Keywords: natural language, nonlinear analysis, poems, written texts, recurrent behavior, speaker discrimination

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