Chaotic Systems

The following sections are included:

• Preface

• Honorary Committee and International Scientific Committee

• Keynote Talks

• Contents

A bifurcation theory is applied to the multimachine power system to investigate the effect of iron saturation on the complex dynamics of the system. The second system of the IEEE second benchmark model of Subsynchronous Resonance (SSR) is considered. The system studied can be mathematically modeled as a set of first order nonlinear ordinary differential equations with (µ = X_c/X_L) as a bifurcation parameter. Hence, bifurcation theory can be applied to nonlinear dynamical systems, which can be written as dx/dt = F(x;µ). The results show that the influence of machine saturation expands the unstable region when the system loses stability at the Hopf bifurcation point at a less value of compensation.

The goal of the paper is chaos examination in multiplicative systems. The paper collects results of numerical simulations as well as presentation of methods applicable in the case of multiplicative systems. Chaos examination concerns one-dimensional multiplicative version of logistic equation and multi-dimensional nonlinear system described with multiplicative derivatives. The classical Lorenz system transformed into multiplicative version was chosen for analysis of stability and chaotic behaviour.

Deterministic chaos machine consisting of the plane triple pendulum as well as of driving and measurement subsystems is presented and studied. The pendulum-driving subsystem consists of two engines of slow alternating currents and optoelectronic commutation. In addition, a mathematical model of the experimental rig is derived as a system of three second order strongly nonlinear ODEs. Mathematical modeling includes details, taking into account some characteristic features (for example, real characteristics of joints built by the use of roller bearings) as well as some imperfections (asymmetry of the forcing) of the real system. Parameters of the model are obtained by a combination of the estimation from experimental data and direct measurements of the system's geometric and physical parameters. A few versions of the model of resistance in the joints are tested in the identification process. Good agreement between both numerical simulation results and experimental measurements have been obtained and presented.

In this paper we examine the links between Ensemble Kalman Filters (EnKF) and Particle Filters (PF). EnKF can be seen as a mean-field process with a PF approximation. We explore the problem of dimensionality on a toy model. To by-pass this difficulty, we suggest using Local Particle Filters (LPF) to catch nonlinearities and feed larger scale EnKF. To go one step forward we conclude with a real application and present the filtering of perturbed measurements of atmospheric wind in the domain of turbulence. This example is the cornerstone of the LPF for the assimilation of atmospheric turbulent wind. These local representation techniques will be used in further works to assimilate singular data of turbulence linked parameters in non-hydrostatic models.

A generic Jacobian is calculated to obtain the Lyapunov exponents Malkus' system. However complete, the Lyapunov exponents obtained from the Jacobian do not appropriately show the distinction between chaos and order. A further explanation for this is required. We show how the waterwheel equations, chaotic as a whole, can be decomposed into a series of convergent equations. Chaos will then come in from the transition between any two of these convergent equations. We finally use a common numerical method, not based on the Jacobian, to obtain Lyapunov exponents that properly make the distinction between chaos and order.

We consider an asset pricing model with wealth dynamics and heterogeneous agents. By assuming that all agents belonging to the same group agree to share their wealth whenever an agent gets in the group (or leaves it), we develop an adaptive model which characterizes the evolution of the wealth distribution when agents switch between different trading strategies. Two groups with heterogeneous beliefs are considered: fundamentalists and chartists. The model results in a nonlinear three-dimensional dynamical system, which is studied in order to investigate complicated dynamics and to explain the effects on wealth distribution among agents in the long run.

A widely debated question in Geosciences is if and how viscous magmas with extreme viscosity contrast mix under natural conditions. Chaotic mixing in magma chambers is thought to play a central role not only in determining the timing and dynamics of volcanic eruptions but may be of equal relevance for the evolutionary history of our planet. To date the dynamics of chaotic mixing have been investigated mostly both in analogue systems and numerical simulations. Here we report the first experimental simulation of chaotic mixing dynamics in molten silicates of geologic relevance and at geologically relevant temperatures (up to 1,700°C). A newly developed device for the in-situ experimental simulation of chaotic dynamics has been successfully developed and employed for this purpose. This device was designed in full awareness of the importance of chaotic dynamics for mixing processes; and earlier studies evidencing that chaotic dynamics equally control magma mixing processes in nature.

The standard result of Harrod's growth model is that, because investors react more strongly than savers to a change in income, the long run equilibrium of the economy is unstable. We re-interpret the Harrodian instability puzzle as a local instability problem and integrate his model with a nonlinear investment function. Multiple equilibria and different types of complex behaviour emerge. Moreover, even in the presence of locally unstable equilibria, for a large set of initial conditions the time path of the economy is not diverging, providing a solution to the instability puzzle.

The paper describes the Hick Samuelson Keynes dynamical economic model with discrete time and consumer sentiment. We seek to demonstrate that consumer sentiment may create fluctuations in the economical activities. The model possesses a flip bifurcation and a Neimark-Sacker bifurcation, after which the stable state is replaced by a (quasi-) periodic motion.

Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events or voids among objects. Here we look at the properties of the Shannon entropy function and calculate its corresponding flow curves, relating them to examples of constrained degeneration from ordered processes. We consider univariate and bivariate gamma, and Weibull distributions since these also include exponential distributions.

This paper presents the possibility of designing a linear communication channel by modulating chaotic analog systems. After presenting the general setup, conditions for correct demodulation and linear dynamic input-output behavior are demonstrated. For a linear dynamic relation between the modulating and demodulated signals, channel equalization is used to achieve wider bandwidth transmission. The presented case studies, regarding the Lorenz and Chen systems, highlight the applicability of the proposed method for high speed digital communication. The overall performance of the resulting communication system is analyzed in terms of speed, security and occupied frequency bandwidth. The concluding remarks point towards some directions in further research.

The dynamics of laser models based on the Maxwell Bloch equation is studied. Instances of stability and chaotic behavior are investigated. Special solutions of the system one of which reduces to the Lotka Volterra system are derived. Absence of oscillating solutions in the reduced systems is studied.

One of the principal problems in identifying chaos using nonlinear time-series analysis of real-world data is additive noise. A straightforward procedure in traditional phase-space is introduced that can be used to identify data sets amenable to nonlinear projective noise reduction. This methodology can be used both as a tool for identifying candidates for noise reduction and, with some adaptation, for zeroth order noise quantification. Results for simulation data are compared with that for a quasiperiodic measured time series to illustrate when series can benefit most from nonlinear projective noise reduction.

This work considers Chaos aspects in a setting of arithmetic provided by Observer's Mathematics (see http://www.mathrelativity.com). We prove that the physical speed is a random variable, cannot exceed some constant, and this constant does not depend on an inertial coordinate system. Certain results and communications pertaining to these theorems are provided.

The following sections are included:

• Introduction

• Unified Van-der-Paul/Rayleigh/Duffing Generator

• Unified Poincare Generator

• Conclusion

• References

Our environment, such as natural, social, economics and engineering ones are the world of complex supersystems of various natures. These systems are a collection of various subsystems providing defined functions and interconnected by processes of forced dynamics interaction and exchange of power, matter and information. These supersystems are nonlinear, multidimensional and multilinked. And in these systems are complex transients and have place of critical and chaotic modes. Problems of system synthesis, i.e. finding of common objective laws of control processes in such a dynamics system are much actual, complicated and, in many respects, practically inaccessible for present control theory.

In the report we consider fundamental basis of nonlinear theory of system's synthesis based on synergetics approach in modern control theory as well as its application [1, 2].

The report consists of three parts: Part I General statements; Part II Strategies of synergetics control; Part III Synergetics synthesis of nonlenear systems with state observers.

Traditional algorithms for control power system that were proposed more than half a century ago are applied in our days. We propose principally new synergetics laws for frequency and power for power station units and unit groups. This approach requires development of technique for application. Moreover, directed implementation of synergetics control laws require global rebuilding of existing patterns of power units control. The simplest way of synergetics algorithms implementation is by using hierarchical principal of control system building. So we rate synergetics control laws as dynamical desired values for ordinary algorithms or as correcting signals [1].

In this work dynamic stability loss of closed cylindrical shells is studied. The hybrid type PDEs governing cylindrical shells dynamics and regarding deflection (Airy's) function and stresses are first derived. Then they are reduced to ODEs and algebraic equations (AE) applying the Bubnov-Galerkin high order approximations method. Dynamic stability loss of the cylindrical shells subject to sign-changeable loading using wavelets is investigated. In addition, the convergence of the Bubnov-Galerkin method and results validation are addressed.

This paper describes the development and use of real-time non-invasive Multivariate Analysis tools for the performance monitoring of atmospheric pressure plasma. The MVA tools (acoustic spectrogram analysis, principal component analysis (PCA) and non-parametric cluster analysis (NPCA) are embedded within a LabVIEW software program. The software program is used to probe a parallel-plate atmospheric pressure process system. It is found that the acoustic frequency spectrum distribution provides a signature of the plasma mode of operation. The signatures are modeled as overtones of the fundamental drive frequency and combination signals (intermodulation distortion). Within these spectrums syncopated patterns are observed. The acoustic signatures are correlated with changing electrical parameters. Using appropriate scaling factors, PCA of the current and voltage waveform are used to generate data set clusters that are deterministic of the acoustic signals. Non-parametric cluster analysis is used to identify and classify the modes.

This paper overviews exact, optimum and approximate decoding and performance results for antipodal chaos shift-keying (CSK) in which a bit is transmitted by modulating a chaotic segment and decoded by use of the corresponding unmodulated reference segment. Both single- and multiple-user versions with both known- and transmitted-reference segments are considered, the so-called coherent and non-coherent cases. There are three main themes to the paper (i) the use of statistical likelihood theory for deriving optimum or improved decoders, (ii) the availability of mathematically exact theory for BER performance of decoders, (iii) qualitative statistical insights provided by simple Gaussian approximations to BER.

The dynamics of a bouncing ball undergoing repeated inelastic impacts with a table oscillating vertically in a sinusoidal fashion is studied using Newtonian mechanics and general relativistic mechanics. An exact mapping describes the bouncing ball dynamics in each theory. We show, contrary to expectation, that the trajectories predicted by Newtonian mechanics and general relativistic mechanics from the same parameters and initial conditions for the ball bouncing at low speed in a weak gravitational field can rapidly disagree completely. The bouncing ball system could be realized experimentally to test which of the two different predicted trajectories is correct.

Starting from the classical Saltzman 2D convection equations, we derive via spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled, generalized Lorenz system. The consideration of this process breaks an important symmetry, couples the dynamics of fast and slow variables, and modifies the structural properties of the attractor. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of O(Ec^-1) is introduced in the system. Moreover, the system is ergodic and hyperbolic, the slow variables feature long term memory with 1/f^3/2 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics.

Sub-critical transitions between two collective regimes with different periods have been observed in a 4-dimensional hypercubic lattice of coupled logistic maps. Statistical findings can be interpreted in terms of macroscopic quantities (measuring global properties) escaping from potential wells in the presence of noise induced by chaos in the microscopic dynamics (local variables). The relevance of these results to a "thermodynamic" understanding of the dynamics of distributed complex systems is briefly discussed.

We develop a statistical theory that describes quantum-mechanical scattering of a particle by a cavity when the geometry is such that the classical dynamics is chaotic. This picture is relevant to a variety of physical systems, ranging from atomic nuclei to mesoscopic systems and microwave cavities; the main application to be discussed in this contribution is the electronic transport through mesoscopic ballistic structures or quantum dots. The theory describes the regime in which there are two distinct time scales, associated with a prompt and an equilibrated response, and is cast in terms of the matrix of scattering amplitudes S. We construct the ensemble of S matrices using a maximum-entropy approach which incorporates the requirements of flux conservation, causality and ergodicity, and the system-specific average of S which quantifies the effect of prompt processes. The resulting ensemble, known as Poisson's kernel, is meant to describe those situations in which any other information is irrelevant. The results of this formulation have been compared with the numerical solution of the Schrödinger equation for cavities in which the assumptions of the theory hold. The model has a remarkable predictive power: it describes statistical properties of the quantum conductance of quantum dots, like its average, its fluctuations, and its full distribution in several cases. We also discuss situations that have been found recently, in which the notion of stationarity and ergodicity is not fulfilled, and yet Poisson's kernel gives a good description of the data. At the present moment we are unable to give an explanation of this fact.

The development of long-wave Marangoni instability under the action of a heat flux modulated in time is studied. The critical Marangoni number for the deformational instability is obtained as a function of frequency.

In this paper, we explore domains in conjunction with the defined parametric space of function , which represents a N-body system in complex form. Interested observations on the computed results reveal limited levels of Herman ring fractals, symmetry broken, and transition to chaos based on phase parameter exp(g_i(z)) and a_i.

Some problems concerning the Markov processes are discussed shortly. Among other things, it is presented an extended Markov property. Also, two variants of the strong Markov property, as they are synthetized by Kiyosi Itô, are discussed.

We consider the classical Heisenberg model (HM) with z-axis anisotropy and external magnetic field. Its phase space is foliated into a family of invariant leaves diffeomorphic to (S²)^Λ The flow on each leaf is Hamiltonian. For the isotropic HM with zero field, the manifold of longitudinal fixed points (LFPs) intersects each leaf orthogonally. In addition, we show that the ferromagnetic (FR) state and the antiferromagnetic (AF) state with non-zero total spin are both stable LFPs. This is a direct implication of a Lemma which extends Lyapunov stability from an invariant subspace to the whole leaf by exploiting the rotational symmetry. The lemma does not apply in the case of zero total spin, and indeed, the AF state on an equal-spins leaf is shown to be unstable.

We extend a previously introduced model for finding eigenvalues and eigenfunctions of PDEs with a certain natural symmetry set based on an analysis of an equivalent transmission line circuit. This was previously applied with success in the case of optical fibers [8], [9] as well as in the case of a linear Schroedinger equation [10], [11] and recently in the case of spherical symmetry (Ball Lightning) [12]. We explore the interpretation of eigenvalues as resonances of the corresponding transmission line model. We use the generic Beltrami problem of non-constant eigen-vorticity in spherical coordinates as a test bed and we locate the bound states and the eigen-vorticity functions.

This paper focuses on the optimization of performance of single-user chaos shift-keying (CSK). More efficient signal transmission is achieved in the coherent case by introducing the class of the so-called deformed circular maps for the generation of spreading. Also, the paired Bernoulli circular spreading (PBCS) is introduced as an optimal choice, which attains the lower bound of bit error rate (BER). As interest shifts to the non-coherent version of the system, attention moves to the receiver end. Maximum likelihood (ML) decoding is utilized serving as an improvement over the correlation decoder. To make the methodology numerically realizable, a Monte Carlo likelihood approach is employed.

In this paper we explore the dynamical behaviour of the q-deformed versions of widely studied 1D nonlinear map-the Gaussian map and another famous 2D nonlinear map-the Henon map. The Gaussian map is perhaps the only 1D nonlinear map which exhibits the co-existing attractors. In this study we particularly, compare the dynamical behaviour of the Gaussian map and q-deformed Gaussian map with a special attention on the regions of the parameter space, where these maps exhibit co-existing attarctors. We also generalize the q-deformation scheme of 1D nonlinear map to the 2D case and apply it to the widely studied 2D quadratic map-the Henon map which is the simplest nonlinear model exhibiting strange attractor.

This paper presents theoretical and experimental results concerning the hypothesis of spatiotemporal chaos in distributed physical systems far from equilibrium. Modern tools of nonlinear time series analysis, such as the correlation dimension and the maximum Lyapunov exponent, were applied to various time series, corresponding to different physical systems such as space plasmas (solar flares, magnetic-electric field components) lithosphere-faults system (earthquakes) brain and cardiac dynamics during or without epileptic episodes. Futhermore, the method of surrogate data was used for the exclusion of 'pseudo chaos' caused by the nonlinear distortion of a purely stochastic process. The results of the nonlinear analysis presented in this study constitute experimental evidence for significant phenomena indicated by the theory of nonequilibrium dynamics such as nonequilibrium phase transition, chaotic synchronization, chaotic intermittency, directed percolation, defect turbulence, spinodal nucleation and clustering.

The Adachi Neural Network (AdNN) [1–5], is a fascinating Neural Network (NN) which has been shown to possess chaotic properties, and to also demonstrate Associative Memory (AM) and Pattern Recognition (PR) characteristics. Variants of the AdNN [6,7] have also been used to obtain other PR phenomena, and even blurring. A significant problem associated with the AdNN and its variants, is that all of them require a quadratic number of computations. This is essentially because all their NNs are completely connected graphs. In this paper we consider how the computations can be significantly reduced by merely using a linear number of computations. To do this, we extract from the original complete graph, one of its spanning trees. We then compute the weights for this spanning tree in such a manner that the modified tree-based NN has approximately the same input-output characteristics, and thus the new weights are themselves calculated using a gradient-based algorithm. By a detailed experimental analysis, we show that the new linear-time AdNN-like network possesses chaotic and PR properties for different settings. As far as we know, such a tree-based AdNN has not been reported, and the results given here are novel.

We study the motion of inertial particles in three-dimensional steady fluid flows that contain a family of two-dimensional invariant manifolds. Using results from Ergodic Theory we derive a condition that predicts if the considered invariant manifold for the flow will persist as an invariant manifold for inertial particles. We illustrate our results for the three-dimensional ABC flow with paremeters corresponding to a non-integrable case.

The development of the last year disaster in the Stock Markets all over the world gave rise to reconsidering the previous models used. It is clear that, even in an organized international or national context, large fluctuations and sudden losses may occur. This paper explores a two populations' model. The populations are conflicting into the same environment (a Stock Market) by following the main rules present, that is mutual interaction between adopters, potential adopters, word-of-mouth communication and of course by taking into consideration the innovation diffusion process. The proposed model has special futures expressed by third order terms providing characteristic stationary points.

Effect of a complicated many-body environment is analyzed on the chaotic motion of a quantum particle in a mesoscopic ballistic structure. The absorption and dephasing phenomena are treated on the same footing in the framework of a schematic microscopic model. The single-particle doorway resonance states excited in the structure via an external channel are damped not only because of the escape onto such channels but also due to ulterior population of the long-lived background states. The transmission through the structure is presented as an incoherent sum of the flow formed by the interfering damped doorway resonances and the retarded flow of the particles re-emitted by the environment.

Music is composed algorithmically from a varying second order recurrence equation which defines sound frequency, x, at each step n knowing sound frequency and amplitude, y, of the previous step, n-1, as: x_n = c - ax_n-1² + by_n-1 where the amplitude is given as y_n-1= f (x_n-1, x_n-2) with f an arbitrary function, and a, b, and c are also arbitrary constants to be chosen by the composer. This compositional model includes as special cases well-known recurrence equations as the logistic map, the Hénon map, and the delayed Julia set. Another feature of our model is that the sound amplitude does not necessarily have the same dependence on previous frequencies in each step. In fact, by the use of discrete form boxcar functions it can change either in consecutive steps or after a finite number of steps. Compositional patterns are also incorporated by use of specific equations for the change of frequency. The effect of initial frequencies chosen is examined as well. Numerical examples are presented to show the chaotic behavior of our model. A final score is presented in musical notation.

We propose a frequency-amplitude triangle whose cosine law is used as a recurrence equation to algorithmically compose music. The triangle is defined by two sides and the angle between them; one of the sides is the sound frequency, x, at a step, the other side is the sound amplitude, y, at the same step, while the angle between them is a free parameter to be chosen by the composer. The resulting third side of the triangle is defined as the square root of the sound frequency, x, of the next step. The sound amplitude, y, at each step depends, in general, on the amplitude and frequency of the same or previous step(s) in any way the composer chooses. To test our proposition for bifurcation and chaos against known solutions, we consider in detail the case where the amplitude depends only on the frequency of the same or some previous step as y_n = b(x_n-m)^λ, with b being a free parameter, m either zero or a positive integer, and λ a real number greater than or equal to zero. If λ is taken as zero, the resulting equation after a change of variables becomes the one parameter logistic recurrence equation. From the known range of the parameter for bifurcation and chaos, we obtain the range of values for sound amplitude and angle θ for periodic and chaotic music. For specific values, the frequencies corresponding to musical notes are obtained from the proposed triangular recurrence equation and musical chaotic scores are composed and presented. Other λ's are also considered and their effect on the periodicity of the frequencies is examined.

Chaos in a fractional-order jerk model using hyperbolic tangent nonlinearity is presented. A fractional integrator in the model is approximated by linear transfer function approximation in the frequency domain. Resulting chaotic attractors are demonstrated with the system orders as low as 2.1.

The direct adaptive regulation of unknown nonlinear dynamical systems in Brunovsky form with modeling error effects, is considered in this chapter. The method is based on a new Neuro-Fuzzy Dynamical System definition, which uses the concept of Fuzzy Adaptive Systems (FAS) operating in conjunction with High Order Neural Network Functions (HONNF's). Since the plant is considered unknown, we propose its approximation by a special form of a Brunovsky type fuzzy dynamical system (FDS) assuming also the existence of disturbance expressed as modeling error terms depending on both input and system states. The development is combined with a sensitivity analysis of the closed loop in the presence of modeling imperfections and provides a comprehensive and rigorous analysis of the stability properties of the closed loop system. Simulations illustrate the potency of the method and its applicability is tested on well known benchmarks, where it is shown that our approach is superior to the case of simple Recurrent High Order Neural Networks (RHONN's).

In this paper, a Lyapunov function is generated to determine the domain of asymptotic stability of a system of three first order nonlinear ordinary differential equations describing the behaviour of a nuclear spin generator (NSG). The generated Lyapunov function, is a simple quadratic form, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations.

Coherent motion of cold atomic wave packets is studied in a one-dimensional optical potential created by a standing laser wave. Their evolution is interpreted in the dressed-state basis where it is the motion in a bipotential. Manifestations of de Broglie-wave chaos, caused by nonadiabatic transitions between components of this bipotential, are found. The probability of those transitions is large in that range of the atom-field detuning and recoil frequency where the classical center-of-mass motion is shown to be chaotic in the sense of exponential sensitivity to small variations in initial conditions or parameters.

In Plasma Physics laboratory of N.C.S.R. "Demokritos" the plasma chemistry method has been used for the restoration and conservation of metallic archaeological objects during the last decades. The obtained experience had led us to conclude that plasma parameters and different status of treated objects are so specific, so as to become unique. In the present paper the theoretical and experimental results of our laboratory are summarized. A treatment table of plasma parameters is given, which claims to be useful for the conservators. It is obvious that this treatment table needs to be completed and extended, so that it meets the uniqueness of each artifact. A theoretical study and the treatment of a variety of iron objects are presented.

The phenomenon of Liesegang banding finds its most striking natural similarity in the band scenery displayed in rock systems. Whereas theoretical modeling studies are extensive in the literature to simulate the reaction-transport dynamics of the medium, experimental simulations in-situ are very scarce. We carry out an empirical observation of the evolution of a deposition pattern embedded in a rock. A sulfuric acid solution is infiltrated into a porous ferruginous limestone rock by means of an infusion pump. This acidization causes the dissolution of the calcite (CaCO₃) mineral and deposition of calcium sulfate in the form of anhydrite (CaSO₄) and mainly gypsum (CaSO₄·2H₂O), behind the advancing dissolution front. We observe the growth of such a deposition pattern and investigate the extent of its resemblance with the well-known Liesegang patterns obtained in the laboratory. The composition of the seemingly banded zones is analyzed by powder X-ray diffraction, both qualitatively and quantitatively. The irregular shape of the band zone boundaries seems to be of fractal nature.

The following sections are included:

• Author Index