Chaos Theory

The following sections are included:

• Preface

• Honorary Committee and International Scientific Committee

• Keynote Talks

• Contents

We use the Lagrangian approach to study surface transport and mixing of water masses in a selected region of the Japan Sea using velocity fields generated by a numerical MHI multi-level eddy-resolved sea-circulation model. Evolution of patches with a large number of tracers, chosen in different parts of the selected region, is computed. The pictures obtained demonstrate clearly regions of strong mixing and stagnation zones coexisting with each other. Computing finite-time Lyapunov exponents for a long period of time, we plot a Lyapunov synoptic map quantifying surface transport and mixing and revealing Lagrangian coherent structures.

The following sections are included:

• INTRODUCTION

• EXPERIMENTS

• ANALYTICAL TREATMENT

• CONCLUSIONS

• REFERENCES

The approaches to forecasting and control, based on general regularities of instability demonstration in dynamical systems are described. These approaches, developed in the frame of experimental mathematics, make it possible to avoid the attempts of construction, identification, and analysis of approximate models of highly complicated real dynamical objects. In place of that the efforts are made to collect certain experimental material for real model and apply it to the obtaining of forecasting and the construction of control. It is remarked that the appearance of instabilities is a result of general regularities, the account of which leads to some general principles of qualitative control theory.

Much scientific works has been done on the applications of the Brownian motion in such diverse areas as molecular and atomic physics, chemical kinetics, solid-state theory, stability of structures, population genetics, communications, and many other branches of the natural and social sciences and engineering. We shall refer below to some aspects concerning the approximation of Markov chains by a solution of a stochastic differential equation to determine the probability of extinction of a genotype. Thus, the Markovian nature of the problem will be pointed out.

Many-body quantum-mechanical scattering problem is solved asymptotically when the size of the scatterers (inhomogeneities) tends to zero and their number tends to infinity.

A method is given for calculation of the number of small inhomogeneities per unit volume and their intensities such that embedding of these inhomogeneities in a bounded region results in creating a new system, described by a desired potential. The governing equation for this system is a non-relativistic Schrödinger's equation described by a desired potential.

Similar ideas were developed by the author for acoustic and electromagnetic (EM) wave scattering problems.

The previous years' disaster in the stock markets all over the world and the resulting economic crisis lead to serious criticisms of the various models used. It was evident that large fluctuations and sudden losses may occur even in the case of a well organized and supervised context as it looks to be the European Union. In order to explain the economic systems, we explore models of interacting and conflicting populations. The populations are conflicting into the same environment (a Stock Market or a Group of Countries as the EU). Three models where introduced 1) the Lotka-Volterra 2) the Lanchester or the Richardson model and 3) a new model for two conflicting populations. These models assume immediate interaction between the two conflicting populations. This is usually not the case in a stock market or between countries as delays in the information process arise. The main rules present include mutual interaction between adopters, potential adopters, word-of-mouth communication and of course by taking into consideration the innovation diffusion process. In a previous paper (Skiadas, 2010 [9]) we had proposed and analyzed a model including mutual interaction with delays due to the innovation diffusion process. The model characteristics where expressed by third order terms providing four characteristic symmetric stationary points. In this paper we summarize the previous results and we analyze the case of a non-symmetric case where the leading part receives the information immediately while the second part receives the information following a delay mechanism due to the innovation diffusion process (the spread of information) which can be expressed by a third order term. In the later case the non-symmetric process leads to gains of the leading part while the second part oscillates between gains and losses during time.

The extraordinary complexity of classical trajectories of typical nonlinear systems that manifest stochastic behavior is intimately connected with exponential sensitivity to small variations of initial conditions and/or weak external perturbations. In rigorous terms, such classical systems are characterized by positive algorithmic complexity described by the Lyapunov exponent or, alternatively, by the Kolmogorov-Sinai entropy. The said implies that, in spite of the fact that, formally, any however complex trajectory of a perfectly isolated (closed) system is unique and differentiable for any certain initial conditions and the motion is perfectly reversible, it is impractical to treat that sort of classical systems as closed ones. Inevitably, arbitrary weak influence of an environment crucially impacts the dynamics. This influence, that can be considered as a noise, rapidly effaces the memory of initial conditions and turns the motion into an irreversible random process.

In striking contrast, the quantum mechanics of the classically chaotic systems exhibit much weaker sensitivity and strong memory of the initial state. Qualitatively, this crucial difference could be expected in view of a much simpler structure of quantum states as compared to the extraordinary complexity of random and unpredictable classical trajectories. However the very notion of trajectories is absent in quantum mechanics so that the concept of exponential instability seems to be irrelevant in this case. The problem of a quantitative measure of complexity of a quantum state of motion, that is a very important and nontrivial issue of the theory of quantum dynamical chaos, is the one of our concern. With such a measure in hand, we quantitatively analyze the stability and reversibility of quantum dynamics in the presence of external noise.

To solve this problem we point out that individual classical trajectories are of minor interest if the motion is chaotic. Properties of all of them are alike in this case and rather the behavior of their manifolds carries really valuable information. Therefore the phase-space methods and, correspondingly, the Liouville form of the classical mechanics become the most adequate. It is very important that, opposite to the classical trajectories, the classical phase space distribution and the Liouville equation have direct quantum analogs. Hence, the analogy and difference of classical and quantum dynamics can be traced by comparing the classical (W^(c)(I,θ;t)) and quantum (Wigner function W(I,θ;t)) phase space distributions both expressed in identical phase-space variables but ruled by different(!) linear equations.

The paramount property of the classical dynamical chaos is the exponentially fast structuring of the system's phase space on finer and finer scales. On the contrary, degree of structuring of the corresponding Wigner function is restricted by the quantization of the phase space. This makes Wigner function more coarse and relatively "simple" as compared to its classical counterpart. Fourier analysis affords quite suitable ground for analyzing complexity of a phase space distribution, that is equally valid in classical and quantum cases. We demonstrate that the typical number of Fourier harmonics is indeed a relevant measure of complexity of states of motion in both classical as well as quantum cases. This allowed us to investigate in detail and introduce a quantitative measure of sensitivity to an external noisy environment and formulate the conditions under which the quantum motion remains reversible. It turns out that while the mean number of harmonics of the classical phase-space distribution of a non-integrable system grows with time exponentially during the whole time of the motion, the time of exponential upgrowth of this number in the case of the corresponding quantum Wigner function is restricted only to the Ehrenfest interval 0 < t < t_E - just the interval within which the Wigner function still satisfies the classical Liouville equation. We showed that the number of harmonics increases beyond this interval algebraically. This fact gains a crucial importance when the Ehrenfest time is so short that the exponential regime has no time to show up. Under this condition the quantum motion turns out to be quite stable and reversible.

Logistic-like first order iterative maps, defined here as X_n+1 = r X_n^λ(1-X_n)^μ, are examined. The parameters r, λ and μ are positive real numbers, while the variable x and its map range from 0 to 1, the latter yielding the upper value of r for which full chaos is obtained. Depending on the values of λ and μ, the resulting x's can have a totally different behavior from those of the logistic map, given by λ = μ =1. The focus here is on fixed points since their existence, for given values of λ and μ, is necessary for obtaining chaotic x's. The purpose of the paper is four-fold: first, to define regions of existence for the fixed point(s) in terms of the parameters r, λ and μ; second, to determine the nature of the fixed points, whether they are attractors (stable), repellors (unstable) or super-stable, according to the values of the parameters; third, to define those maps for which the fixed points can be written in explicit algebraic form; and fourth, for iterative nearby maps, to obtain their fixed points in an approximate algebraic form in terms of the exact fixed points. The approximation is based on Newton's method, one step from the nearest iterative map whose fixed points can be obtained exactly, in explicit form. The validity of the fixed point approximation depends on the stability of the fixed points and is, subsequently, established in respect of well defined surfaces of the parameters r, λ and μ.

The ε expansion in the theory of developed d-dimensional turbulence is amended by renormalization of random forcing in the stochastic Navier-Stokes equation, which takes into account additional divergences in two-dimensional space. The Kolmogorov constant and skewness factor calculated in the one-loop approximation of the improved perturbation theory are in reasonable agreement with their recommended experimental values.

The non-linear behaviour of rubbing cylindrical rotors have been studied in several papers. In such systems rich dynamics have been found for frequencies above the natural frequency. Below natural frequency the solution was found to be stationary. In this paper the influence of blades is studied. A Jeffcott rotor with three blades is used and the contacts are described by large displacement beam theory. The model shows that no stationary point will exist and complex behaviour will occur below the natural frequency. For the studied rotor, failure due to high stresses will occur at driving frequencies below 50% of the natural frequency and instability at 80% of the natural frequency. The paper shows that the dynamics of bladed rotors differs from the dynamics of rubbing circular rotors. If a bladed rotor is used it is essential to study a model with blades. Otherwise the general conclusions on the dynamics can be wrong.

This paper presents numerical methods of modelling composites, which allow observation of fibre breaking during static loads without using invasive experimental methods. Our simulations are based on cellular automata, which is an alternative method to study behaviour of dynamical systems. We assume defects evolution in composite as a dynamical system depending on external and internal forces and properties of fibres.

We discuss recent work with E.Floratos (JHEP 1004:036,2010) on Nambu Dynamics of Intersecting Surfaces underlying Dissipative Chaos in R³. We present our argument for the well studied Lorenz and Rössler strange attractors. We implement a flow decomposition to their equations of motion. Their volume preserving part preserves in time a family of two intersecting surfaces, the so called Nambu Hamiltonians. For dynamical systems with linear dissipative sector such as the Lorenz system, they are specified in terms of Intersecting Quadratic Surfaces. For the case of the Rössler system, with nonlinear dissipative part, they are given in terms of a Helicoid intersected by a Cylinder. In each case they foliate the entire phase space and get deformed by Dissipation, the irrotational component to their flow. It is given by the gradient of a surface in R³ specified in terms of a scalar function. All three intersecting surfaces reproduce completely the dynamics of each strange attractor.

We propose a unified methodology, based on renormalization group theory, for finding out existence of periodic solutions in a plethora of nonlinear dynamical systems appearing across disciplines. The technique will be shown to have the non-trivial ability of classifying the solutions into limit cycles and periodic orbits surrounding a center.

Tools from chaos theory that have found recent use in analysing financial markets have been applied to the US Dollar and Euro buying and selling rates against the Turkish currency. The reason for choosing the foreign exchange rate in this analysis is the fact that foreign currency is an indicator of not only the globalization of economy but also savings and investment. In order to test the globality assumption and to ascertain the degree of involvement of local conditions in Turkey, the Euro and US dollar exchange rates have been subjected to the same analysis.

Chaotic communications schemes based on synchronization aim to provide security over the conventional communication schemes. Symbolic dynamics based on synchronization methods has provided high quality synchronization [5]. Symbolic dynamics is a rigorous way to investigate chaotic behavior with finite precision and can be used combined with information theory [13]. In previous works we have studied the kneading theory analysis of the Duffing equation [3] and the symbolic dynamics and chaotic synchronization in coupled Duffing oscillators [2] and [4]. In this work we consider the complete synchronization of two identical coupled unimodal and bimodal maps. We relate the synchronization with the symbolic dynamics, namely, defining a distance between the kneading sequences generated by the map iterates in its critical points and defining n-symbolic synchronization. We establish the synchronization in terms of the topological entropy of two unidirectional or bidirectional coupled piecewise linear unimodal and bimodal maps. We also give numerical simulations with coupled Duffing oscillators that exhibit numerical evidence of the n-symbolic synchronization.

The paper belongs to the field of chaotic based cryptography. It relies on some ideas from classical fundamentals as mixing functions suggested by C.E. Shannon for secrecy systems, and on the known publication of M.S. Baptista, a paper essentially implying the ergodicity assumption of the chaotic signal in cipher design. In this paper, the first step was to choose a chaotic system of a higher dimension than Baptista used, aiming to obtain a more complex system having a hyperchaotic behavior. The proposed algorithm is based on Generalized Hénon Map (3D), stated in literature as hyperchaotic for a number of bifurcation parameters. The paper advances a new mixing enciphering scheme based on the Generalized Hénon Map (GHM), which may be used as an inner element in a cipher, providing a good practical diffusion and confusion. A random variable transform is applied on the state of the chaotic system at each iteration in order to obtain a new random variable of a quasi uniform law. This new random variable is further transformed, through a series of other functions containing elements of the secret key, into a discrete random variable. The discrete values – which are ASCII numbers - are combined by a simple relation with the plain message, also in ASCII format. It is obtained a first mask of the original message, involving the GHM. On this result (in its binary representation form) other simple transformations that depend on the state of the GHM are applied. That finally allows getting a transformed version of the message that can be included in one of the states of the GHM without disturbing its chaotic behavior. The results, including a perception of the diffusion and the confusion involved, are illustrated on natural text and jpeg image.

This paper explores the effect of the noise in the reinjection probability densities (RPD) for type-II and type-III intermittencies by using the temporal series of iterative maps. The RPD are calculated by means of a new method proposed in Refs. [1] and [2] and the results are compared with both, numerical simulations and analytical calculations. In addition, we provide an explanation for the gap observed in early experiments around the unstable point in the Poincaré map. We show that and added white noise approaches the RPD to the case of uniform reinjection for small distances of iterations to the unstable point. For large distances the RPD should be incremented with respect to the noiseless case. These numerical results suggest the existence of a noise induced reinjection mechanism.

The aim of the study is to develop quantitative parameters of human electroencephalographic (EEG) recordings with epileptic seizures. We used long-lasting recordings from subjects with epilepsy obtained as part of their clinical investigation. The continuous wavelet transform of the EEG segments and the wavelet-transform modulus maxima method enable us to evaluate the energy spectra of the segments, to fin lines of local maximums, to gain the scaling exponents and to construct the singularity spectra. We have shown that the significant increase of the global energy with respect to background and the redistribution of the energy over the frequency range are observed in the patterns involving the epileptic activity. The singularity spectra expand so that the degree of inhomogenety and multifractality of the patterns enhances. Comparing the results gained for the patterns during different functional probes such as open and closed eyes or hyperventilation we demonstrate the high sensitivity of the analyzed parameters (the maximal global energy, the width and asymmetry of the singularity spectrum) for detecting the epileptic patterns.

In this article we examine fractal curves and synthesis algorithms in musical composition and research. First we trace the evolution of different approaches for the use of fractals in music since the 80's by a literature review. Furthermore, we review representative fractal algorithms and platforms that implement them. Properties such as self-similarity (pink noise), correlation, memory (related to the notion of Brownian motion) or non correlation at multiple levels (white noise), can be used to develop hierarchy of criteria for analyzing different layers of musical structure. L-systems can be applied in the modelling of melody in different musical cultures as well as in the investigation of musical perception principles. Finally, we propose a critical investigation approach for the use of artificial or natural fractal curves in systematic musicology.

Chaos synchronization was extensively studied the last two decades leading to several synchronization and modulation methods. The main aim of the research was to develop wide bandwidth, spread spectrum like and secure communication systems. Taking into account the high sensitivity of all synchronization and modulation methods to channel noise and parameter mismatch, digital implementations aiming at chaos encryption were preferred lately. Several authors developed, in their studies, different encryption approaches, most of them being successfully cryptanalized, due to the direct influence of the constant system parameters onto their nonlinear dynamics.

The present contribution proposes a time variant approach to chaotic encryption. The proposed method is based on chaos synchronization and plaintext modulation onto the chaotic dynamics of the emitter. In order to improve communication confidentiality, one or several parameters of the emitter system are modulated with pseudo-random digital sequences, thus drastically increasing the length of the encryption key. At the decryption end, the corresponding receiver parameters are also time variant. Exact knowledge of the emitter pseudo-random digital sequence shape and timing are necessary for correct decryption of the cipher-text. It is also worth noting the importance of the modulating sequence amplitude or dynamic range due to its influence on the instantaneous value of the emitter sensitivity and, by consequence, its capacity in hiding the transmitted plaintext. Thus a parametric analysis is presented in order to find a valid parameter range for possible modulation. For the digital implementation of the proposed encryption/decryption systems, although input and output are fixed point, the internal structure of both emitter and receiver must be implemented in floating point to obtain the closest behavior with the chaotic prototype that has analog valued state variables.

Both analog and discrete examples for the encryption/decryption system are analyzed. Presented simulations confirm the theoretical results and highlight the great improvement of the communication security by the proposed approach. The concluding remarks point towards some directions in further research.

Models of chance operations, random equations, stochastic processes, and chaos systems have inspired composers as historical as Wolfgang Amadeus Mozart. As these models advance and new processes are discovered or defined, composers continue to find new inspirations for musical composition. Yet, the relative artistic merits of some of these works are limited. This paper explores the application of extra-musical processes to the sonic arts and proposes aesthetic considerations from the point of view of the artist. Musical examples demonstrate possibilities for working successfully with algorithmic and generative processes in sound, from formal decisions to synthesis.

In the search for extrasolar planets in mean-motion resonance using the transit method, the delay in time-delay phase space reconstruction may be chosen to either lower noise at maximum depth or, to lower noise where one looks for planets in resonance, as the transits begin and end. The trade-offs in the selection of the delay will be discussed for the case of a simulated light curve.

The perturbations of chirped dissipative solitons are analyzed in the spectral domain. It is shown, that the structure of the perturbed chirped dissipative soliton is highly nontrivial and has a tendency to an enhancement of the spectral perturbations especially at the spectrum edges, where the irregularities develop. Even spectrally localized perturbations spread over a whole soliton spectrum. As a result of spectral irregularity, the chaotic dynamics develops due to the spectral loss action. In particular, the dissipative soliton can become fragmented though remains localized.

In this paper, a modified chaotic shift keying method is proposed to transmit digital bits securely over a communication channel. The scheme is based upon encrypting the digital bits 0 and 1 into infinite levels by applying the keystream such that there is no recognisable pattern in the encoded transmitted signal. The encoded transmitting signal generated is shown to resist popular attack method therefore realizing a secure and trustworthy digital communication system.

This work considers the solution of Cauchy problem (initial value problem) in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics (see www.mathrelativity.com) and applies this solution to free wave equation, the linear (time-dependent) Schrodinger equation, the (time-dependent) Airy equation, the Korteweg-de Vries (KdV) equation, and quantum theory of two-slit interference. Certain results and communications pertaining to these problems are provided.

This paper presents some of the author's experiences with computer aided composition (CAC): the modeling of physical movements is used to obtain plausible musical gestures in interaction with constraint programming (rule based expert systems) in order to achieve precisely structured, consistent musical material with strong inner logic and syntax in pitch material. The "Constraints Engine" by Michael Laurson implemented in OpenMusic (IRCAM) or PWGL (Sibelius Academy) can be used to set up an interactive framework for composition, which offers a balance of freedom (allowing chance operations and arbitrary decisions of the composer) and necessity (through strict rules as well as through criteria for optimization). Computer Aided Composition is moving far beyond being "algorithmic" or "mechanical". This paper proposes an approach based on evolutionary epistemology (by the Austrian biologist and philosopher Rupert Riedl). The aim is a holistic synthesis of artistic freedom and coherent structures similar to the grown order of nature.

A class of predator–prey models suggested by the continuous form of the following two dimensional map is studied

Biological systems are dynamic and possess properties that depend on two key elements: initial conditions and the response of the system over time. Conceptualizing this on tumor models will influence conclusions drawn with regard to disease initiation and progression. Alterations in initial conditions dynamically reshape the properties of proliferating tumor cells. The present work aims to test the hypothesis of Wolfrom et al., that proliferation shows evidence for deterministic chaos in a manner such that subtle differences in the initial conditions give rise to non-linear response behavior of the system. Their hypothesis, tested on adherent Fao rat hepatoma cells, provides evidence that these cells manifest aperiodic oscillations in their proliferation rate. We have tested this hypothesis with some modifications to the proposed experimental setup. We have used the acute lymphoblastic leukemia cell line CCRF-CEM, as it provides an excellent substrate for modeling proliferation dynamics. Measurements were taken at time points varying from 24h to 48h, extending the assayed populations beyond that of previous published reports that dealt with the complex dynamic behavior of animal cell populations. We conducted flow cytometry studies to examine the apoptotic and necrotic rate of the system, as well as DNA content changes of the cells over time. The cells exhibited a proliferation rate of nonlinear nature, as this rate presented oscillatory behavior. The obtained data have been fit in known models of growth, such as logistic and Gompertzian growth.

Important aspects of chaotic behavior appear in systems of low dimension, as illustrated by the Map Module 1. It is indeed a remarkable fact that all systems tha make a transition from order to disorder display common properties, irrespective of their exacta functional form. We discuss evidence for 1/f power spectra in the chaotic time series associated in classical and quantum examples, the one-dimensional map module 1 and the spectrum of ⁴⁸Ca.

A Detrended Fluctuation Analysis (DFA) method is applied to investigate the scaling properties of the energy fluctuations in the spectrum of ⁴⁸Ca obtained with a large realistic shell model calculation (ANTOINE code) and with a random shell model (TBRE) calculation also in the time series obtained with the map mod 1. We compare the scale invariant properties of the ⁴⁸Ca nuclear spectrum sith similar analyses applied to the RMT ensambles GOE and GDE. A comparison with the corresponding power spectra is made in both cases. The possible consequences of the results are discussed.

This paper describes the thermal gas effluent and ion acoustic pressure wave interaction between the fundamental drive frequency and its harmonics within an atmospheric helium discharge. Deconvolution of the acoustic signal and the electrical signals reveal that the plasma jet undergoes a change in operational mode from chaotic (where the plasma is spatially and temporally inhomogeneous at the electrode surface) to stable (periodic in nature) when the plasma expands away from the electrodes and into the reactor cylinder. This effect is strongly influence by the helium flow and input power. In addition the generated acoustic signals is found to have a frequency response to that of a closed-end cylinder column which supports antinodes of n = 1 and 3. Decoding of the acoustic signal allows the helium thermal gas temperature to be obtained: T_gas ~ 290 K. The signal allows the axial gap distance between the jet nozzle and work surface to be estimated which has technology importance in terms of plasma metrology and in the basic understanding of atmospheric pressure plasma jet physics.

Quantum graphs are widely used to investigate properties of quantum chaos. Experimentally, quantum graphs are simulated by microwave graphs (networks) consisting of joints, microwave cables and other microwave components such as attenuators and circulators. This is possible due to an equivalency of the one-dimensional Schrödinger equation describing a quantum system and the telegraph equation describing an ideal microwave network. We present the results of the experimental study of the enhancement factor W_s,β for irregular fully connected hexagon undirected and directed microwave graphs in the presence of absorption. We measured the two-port scattering matrix Ŝ for the undirected microwave graphs which statistical properties of eigenfrequencies, typical for the systems with time reversal symmetry (TRS), can be described by Gaussian Orthogonal Ensemble (GOE) in Random Matrix Theory (RMT), and for the directed graphs possessing the properties of the systems with broken time reversal symmetry, which can be described by Gaussian Unitary Ensemble (GUE) in RMT. The measurements were performed as a function of absorption, which was varied by microwave attenuators.

In this paper we introduce a new approach to algorithmic sound composition using a bespoke technique combining coupled Cellular Automata (CA) and Histogram Mapping Synthesis. Two CA are used: a hodge podge machine and a growth model. The latter serves as control of the former. The hodge podge machine can exhibit different kinds of behaviour depending on the values of a set of rule parameters. Our method explores the fact that different simultaneous behaviours can be evolved within the same automaton if we bring into play different sets of parameter values. However, we restrict the number of parameter sets to two. Therefore, the CA growth model will have only two states and will delimit two dynamic zones in the hodge podge machine, each of which governed by a different set of parameter values. The predictable evolution of the two zones will produce a controlled dynamic sound spectrum. Among all the possibilities that this process affords for the composition of a variety of sounds algorithmically, we highlight its application to the attack portion of a sound, making it dynamically more complex than the rest of the sound.

This study proposes a new forcing scheme suitable for massively-parallel finite-difference simulations of steady isotropic turbulence. The proposed forcing scheme, named reduced-communication forcing (RCF), is based on the idea of the conventional large-scale forcing, but requires much less data communication, leading to a high parallel efficiency. It has been confirmed that the RCF works intrinsically in the same manner as the conventional large-scale forcing. Comparisons have revealed that the fourth-order finite-difference model run in combination with the RCF (FDM-RCF) is as good as the spectral model, while requiring less computational costs.

Information causality measures, i.e. transfer entropy and symbolic transfer entropy, are modified using the concept of surrogate data in order to identify correctly the presence and direction of causal effects. The measures are evaluated on multiple bivariate time series of known coupled systems of varying complexity and on a range of embedding dimensions. The proposed modifications of the causality measures are found to reduce the bias in the estimation of the measures and preserve the zero level in the absence of coupling.

During the last two decades, low dimensional chaotic or self-organized criticality (SOC) processes have been observed by our group in many different physical systems such as space plasmas, the solar or the magnetospheric dynamics, the atmosphere, earthquakes, the brain activity as well as in informational systems. All these systems are complex systems living far from equilibrium with strong self-organization and phase transition character. The theoretical interpretation of these natural phenomena needs a deeper insight into the fundamentals of complexity theory. In this study, we try to give a synoptic description of complexity theory both at the microscopic and at the macroscopic level of the physical reality. Also, we propose that the self-organization observed macroscopically is a phenomenon that reveals the strong unifying character of the complex dynamics which includes thermodynamical and dynamical characteristics in all levels of the physical reality. From this point of view, macroscopical deterministic and stochastic processes are closely related to the microscopical chaos and self-organization. In this study the scientific work of scientists such as Wilson, Nicolis, Prigogine, Hooft, Nottale, El Naschie, Castro, Tsallis, Chang and others is used for the development of a unified physical comprehension of complex dynamics from the microscopic to the macroscopic level.

Classical central limit theorems, culminating in the theory of infinite divisibility, accurately describe the behaviour of stochastic phenomena with asymptotically negligible components. The classical theory fails when a single component may assume an extreme protagonism. The early developments of the speculation theory didn't incorporate the pioneer work of Pareto on heavy tailed models, and the proper setup to conciliate regularity and abrupt changes, in a wide range of natural phenomena, is Karamata's concept of regular variation and the role it plays in the theory of domains of attraction, [8], and Resnick's tail equivalence leading to the importance of generalized Pareto distribution is the scope of extreme value theory, [13]. Waliszewski and Konarski discussed the applicability of the Gompertz curve and its fractal behaviour for instance in modeling healthy and neoplasic cells tissue growth, [15]. Gompertz function is the Gumbel extreme value model, whose broad domain of attraction contains intermediate tail weight laws with a wide range of behaviour.

Aleixo et al. investigated fractality associated with Beta (p,q) models, [1], [2], [10] and [11]. In this work, we introduce a new family of probability density functions tied to the classical beta family, the Beta*(p,q) models, some of which are generalized Pareto, that span the possible regular variation of tails. We extend the investigation to other extreme stable models, namely Fréchet's and Weibull's types in the General Extreme Value (GEV) model.

In this paper the study of complex phenomenon in buck converter under voltage mode control, operating in discontinious current mode, within Matlab/Simulink simulation environments is provided. To perform simulations different types of models are used: based on discrete-time maps, differential equations and real elements (including different nonidealities). The main goal of this paper is to detect the ability of various Matlab/Simulink models to identify and to explore different types of complex behaviour, such as chaos and bifurcataions, in switch-mode DC-DC converters, as system parameters are changed, as well as to estalish the possibilities of each model in this kind of investigation. Simulations are carried out by means of Matlab/Simulink simulation environment that provides wide range of blocks and elements for complete investigation procedure, including the implementation of all types of mentioned models and appropriate result data postprocessing and visualization. The verification of accuracy of developed models is based on the detection of Feigenbaum numbers. All models with definite level of precision are able to reveal that under certain circuit parameters period doubling route to chaos is observed.

Composition is a combination of determined combinations of notes, durations and timbres usually decided upon in advance by a composer who plans carefully the sounds she desires. There is also always an element of chance present in acoustic music due to the 'human' element of the performance in that the performers will add their own interpretation of the dynamics and errors in terms of precise durations and pitches. Some composers have exploited this chance element more than others, allowing more space within the composition for the performers to make choices during the course of the piece. Composers such as Cage and Bussotti offer varying degrees of freedom within pieces resulting in unpredictability of the resulting sound of the composition. Other composers attempt to control as far as possible every parameter of the music as seen in serialist composers such as Webern and Boulez.

This paper is delivered from the point of view of a composer who is intrigued by the relationship between the notation and the resultant sound, specifically, in terms of the relationship between the written elements determined by the composer and the unpredictability that arises due to those elements which cannot or are deliberately not written. These elements are then left to the interpretation and/or choice of the performer during the performance resulting in a composition which differs sonically from performance to performance. Chaos offers this combination of determination and the appearance of disorder: a clear structure within which are a number of elaborate chaotic-appearing options. The paper will focus on a composition-in-progress for voices which will offer the performers some choices based on the idea of sensitivity on initial conditions. Each singer will be provided with a set of headphones through which they will be fed a choice of pitches, the choices made for the first few pitches will determine the choices provided to the singer later on in the composition.

The paper will also outline the concepts behind the piece and how the use of chaos will provide compositional parameters with the overall development of the piece being determined by choices made by the performers. The paper will also briefly explain the unusual tuning systems set in place for the piece which is a system proposed by Dr Rob Sturman based on non-linear dynamics.

Usually randomness appears as a sophisticated extension of deterministic models, that are then presented as expectation of some class of random models (this approach is exceedingly well managed in the classical Barucha-Reid's treatise on random functions and stochastic processes). The works [1], [2], [3] and [5] summarize previous studies by the authors, using stochastic definitions of extensions of Cantor's fractal to put forward appropriate deterministic models, that in a precise sense are the expectation of a structured class of models, and investigated bifurcations, Allee' effect, and the Hausdorff dimension. Beta(p,q) models, with either p = 1 or q = 1, or the classical Verhulsts model (p = q = 2), proportionate interesting computable models for which computations both of Hausdorff dimension and probabilities can be explicitly evaluated, either analytically or using the Monte Carlo method.

The present extension, axed on arbitrary symbolic dynamical systems, further develops new fundamental classes of geometric constructions, and exploits the interplay of determinism and randomness on the richness of the limit fractal set, in a recursive construction. This sheds new light on the concept of Hausdorff dimensionality. We show that the dependence of the random order statistics is at the core of the apparent anomaly of consistently smaller Hausdorff dimensions of the random sets, when compared with the corresponding "expected" deterministic counterparts. We also recover Falconner's, Pesin's and Weiss' (among others) ideas on recursive geometric constructions as a straightforward approach to important issues in fractality and chaos.

We overview several analytic methods of predicting the emergence of chaotic motion in nonlinear oscillatory systems. A special attention is given to the second method of Lyapunov, a technique that has been widely used in the analysis of stability of motion in the theory of dynamical systems but received little attention in the context of chaotic systems analysis. We show that the method allows formulating a necessary condition for the appearance of chaos in nonlinear systems. In other terms, it provides an analytic estimate of an area in the space of control parameters where the largest Lyapunov exponent is strictly negative. A complementary area thus comprises the values of controls, where the exponent can take positive values, and hence the motion can become chaotic. Contrary to other commonly used methods based on perturbation analysis, such as e.g., Melnikov criterion, harmonic balance, or averaging, our approach demonstrates superior performance at large values of the parameters of dissipation and nonlinearity. Several classical examples including mathematical pendulum, Duffing oscillator, and a system of two coupled oscillators, are analyzed in detail demonstrating advantages of the proposed method compared to other existing techniques.

We study Melnikov conditions predicting appearance of chaos in Duffing oscillator with hardening type of non-linearity under two-frequency excitation acting in the vicinity of the principal resonance. Since Hamiltonian part of the system contains no saddle points, Melnikov method cannot be applied directly. After separating the external force into two parts, we use a perturbation analysis that allows recasting the original system to the form suitable for Melnikov analysis. At the initial step, we perform averaging at one of the frequencies of the external force. The averaged equations are then analyzed by traditional Melnikov approach, considering the second frequency component of the external force and the dissipation term as perturbations. The numerical study of the conditions for homoclinic bifurcation found by Melnikov theory is performed by varying the control parameters of amplitudes and frequencies of the harmonic components of the external force. The predictions from Melnikov theory have been further verified numerically by integrating the governing differential equations and finding areas of chaotic behavior. Mismatch between the results of theoretical analysis and numerical experiment is discussed.

In this paper we explore the life expectancy limits by based on the stochastic modeling of mortality and applying the first exit or hitting time theory of a stochastic process. The main assumption is that the health state or the "vitality", according to Strehler and Mildvan, of an individual is a stochastic variable and thus it was introduced and applied a first exit time density function to mortality data. The model is used to estimate the development of mortality rates in the late stages of the human life span, to make better fitting to population mortality data including the infant mortality, to compare it with the classical Gompertz curve, and to make comparisons between the Carey medfly data and the population mortality data estimating the health state or "vitality" functions. Furthermore, we apply the model to the life table data of Italy, France, USA, Canada, Sweden, Norway and Japan, and we analyze the characteristic parameters of the model and make forecasts. The case of female mortality in Sweden is extensively studied and forecasts to 2025 and 2050 are presented.

Chaotic music is composed from a proposed iterative map depicting the letter m, relating the pitch, duration and loudness of successive steps. Each of the two curves of the letter m is based on the classical logistic map. Thus, the generating map is x_n+1 = r x_n(1/2 - x_n) for x_n between 0 and 1/2 defining the first curve, and x_n+1 = r (x_n - 1/2)(1 - x_n) for x_n between 1/2 and 1 representing the second curve. The parameter r which determines the height(s) of the letter m varies from 2 to 16, the latter value ensuring fully developed chaotic solutions for the whole letter m; r = 8 yielding full chaotic solutions only for its first curve. The m-model yields fixed points, bifurcation points and chaotic regions for each separate curve, as well as values of the parameter r greater than 8 which produce inter-fixed points, inter-bifurcation points and inter-chaotic regions from the interplay of the two curves. Based on this, music is composed from mapping the m- recurrence model solutions onto actual notes. The resulting musical score strongly depends on the sequence of notes chosen by the composer to define the musical range corresponding to the range of the chaotic mathematical solutions x from 0 to 1. Here, two musical ranges are used; one is the middle chromatic scale and the other is the seven- octaves range. At the composer's will and, for aesthetics, within the same composition, notes can be the outcome of different values of r and/or shifted in any octave. Compositions with endings of non-repeating note patterns result from values of r in the m-model that do not produce bifurcations. Scores of chaotic music composed from the m-model and the classical logistic model are presented.

Chaotic sound waveforms generated algorithmically are considered to study their timbre characteristics of harmonic and inharmonic overtones, loudness and onset time. Algorithms employed in the present work come from different first order iterative maps with parameters that generate chaotic sound waveforms. The generated chaotic sounds are compared with each other in respect of their waveforms' energy over the same time interval. Interest is focused in the logistic, double logistic and elliptic iterative maps. For these maps, the energy of the algorithmically synthesized sounds is obtained numerically in the chaotic region. The results show that for a specific parameter value in the chaotic region for each one of the first two maps, the calculated sound energy is the same. The energy, though, produced by the elliptic iterative map is higher than that of the other two maps everywhere in the chaotic region. Under the criterion of equal energy, the discrete Fourier transform is employed to compute for the logistic and double logistic iterative maps, a) the generated chaotic sound's power spectral density over frequency revealing the location (frequency) and relative loudness of the overtones which can be associated with fundamental frequencies of musical notes, and b) the generated chaotic sound's frequency dependent phase, which together with the overtones' frequency, yields the overtones' onset time. It is found that the synthesized overtones' loudness, frequency and onset time are totally different for the two generating algorithms (iterative maps) even though the sound's total generated power is equal. It is also demonstrated that, within each one of the iterative maps considered, the overtone characteristics are strongly affected by the choice of initial loudness.

The effect of rainbow color sequence on composing chaotic algorithmic music is examined. The mathematical range of the chaotic algorithm is mapped onto musical notes whose sequence follows the sequence of the seven main rainbow colors and their in-between five auxiliary colors. Each musical note is identified with the frequency of a color by a frequency shift. As a result, for a single rainbow, the scale of the chaotic music comprises an ascending chromatic F major scale without the thirteenth note, followed by its corresponding descending chromatic scale, for a total of twenty four notes. For aesthetic purposes, a note can be placed in any octave at the composer's will. The effect of a double rainbow on composing chaotic music is also studied. It is known from nature that the outer bow has its color sequence reversed. Thus, in this case, the double rainbow musical scale comprises forty eight notes on a repeated reversed full chromatic F major scale without the thirteenth note in the ascent or the first note in the descent, resembling in shape the letter w. Colorless regions in the rainbow or dark (Alexander's bands) regions in a supernumerary rainbow are included in the musical range as rests. With the musical scale based on the described rainbow mapping, chaotic music is composed from an algorithm defined by a semi-elliptical first order iterative map. The minor axis of the ellipse is defined by the range of the mathematical pitch from 0 to 1 while the semi-major axis by that of the succeeding pitch from 0 to r/2; r is a free parameter that varies from 1 to 2 to be chosen by the composer. The lower limiting value of the free parameter r corresponds to a circle of radius 1/2 yielding steady state music whereas all the other values of r correspond to ellipses. Chaotic compositions result from r values between 1.95 and 2, the latter value yielding full chaos from an ellipse with its major axis double its minor axis. Fixed notes are obtained for all r's, i.e., notes to be avoided for a prematurely ending composition. Chaotic musical scores are composed from the semi-elliptical iterative map with different initial pitches, on a single or a double rainbow musical scale.

A one-parameter highly chaotic attractor is presented and its application to a dual-channel, single-attractor, private communication system is demonstrated based on self-synchronization and chaotic masking techniques. Only a single attractor is required for a dual-channel transmitter or receiver, and can be either the well-known Lorenz attractor, the Lorenz-like attractor, or the one-parameter highly chaotic attractor developed in this paper. The latter is particularly well suited for an application to private communications due to the relatively high values of both the maximum Lyapunov exponent of 2.6148 and the maximum Kaplan-Yorke dimension of 2.1921. An advantage of the dual channel is the possibly twice increase in higher speed.

We present results of an extensive analysis of classical and quantum signatures of chaos in the geometric collective model (GCM) and the interacting boson model (IBM) of nuclei. Apart from comparing the regular fraction of the classical phase space and the Brody parameter for the nearest neighbor spacing distribution in the quantum case, we also adopt (i) the Peres lattices allowing one to distinguish ordered and disordered parts of spectra and to reveal main ordering principles of quantum states, (ii) the geometrical method to determine the position where the transition from order to chaos occurs, and (iii) we look for the 1/f^α power law in the power spectrum of energy level fluctuations. The Peres method demonstrates the adiabatic separation of collective rotations in the IBM.

Today, computers are based on automata model of computations i.e. Turing Machine and are designed to be deterministic; chaotic behavior is undesirable when considering the stability of algorithms for example. However, when attempting to build the theory of the phenomena of Collective Intelligence (CI), it appears that molecular models of computations relying on chaotic behavior of its components must be used as the computational model. Moreover, Chaos emerges as the essential component for Collective Intelligence (CI) computational processes, providing some required computational mechanisms and computational properties. The paper attempts to define Collective Intelligence and describe relations between CI and Chaos.

We briefly analyze and demonstrate several nonlinear signal generators, according to the principles proposed in [1]. These systems apparently should not oscillate, because they include only capacitors in the positive feedback loops. However, the design attempted to make use of the well-known parasitic elements of the capacitors to create a selective feedback loop with physical capacitors only. The use of the resonant (piezoelectric) load adds a new potential mode of oscillation. The design aimed to make achievable as many as possible frequency points where oscillation modes ay exist, against the rather common belief that a single oscillation mode can be supported at one time in electronic oscillators. For this purpose, variable, independent gains have been provided to the positive feedback loops that correspond to the various modes in the circuit. We describe the schemes, briefly analyze their behavior, show simulation and experimental results, and discuss potential uses. The paper is largely based on [1].

In a cylindrical cold rf plasma, a variety of drifts and other sorts of waves are usually observed; when a turbulence is created, the state becomes chaotic and then the plasma turns out to be more unstable. In the present work, an external signal is enforced on the plasma's waves (or turbulence), which strongly affects the physical magnitudes of the plasma instabilities. The final result is that plasma stabilization occurs when plasma waves are synchronized with the external signal. Moreover, nonlinear phenomena occur, such as a vigorous coupling among the waves' frequencies, which affect the Hall conductivity. Another significant observation is the influence of boundaries on the interaction waves.

The paper presents a novel method for evaluating the different natural processes. The method is based on the chaos game technique, and evaluation of the results is based on the measurements of the fractal dimension of the obtained Sierpinski triangles. Pathways of the natural rivers channels case study has been used for the demonstration of the methodology, despite the method can be applied to many natural phenomena. Authors would like to encourage others to make similar analysis, because more case studies are needed to draw more solid conclusions.

This paper explores the cutting force oscillations. Forces have been measured during the stainless steel turning. We provide the results of standard statistical analysis of the corresponding time series together with their recurrence properties. We claim that the system, which is initially in a regular vibration region for some fairly larger cutting depth, is unstable to chaotic oscillation appearance. This could have the important implication to the process control procedure.

The following sections are included:

• Author Index