Showing papers in "Chaos Solitons & Fractals in 2000"

Adaptive synchronization of chaotic systems and its application to secure communications

Abstract: This paper addresses the adaptive synchronization problem of the drive–driven type chaotic systems via a scalar transmitted signal. Given certain structural conditions of chaotic systems, an adaptive observer-based driven system is constructed to synchronize the drive system whose dynamics are subjected to the system’s disturbances and/or some unknown parameters. By appropriately selecting the observer gains, the synchronization and stability of the overall systems can be guaranteed by the Lyapunov approach. Two well-known chaotic systems: Rossler-like and Chua’s circuit are considered as illustrative examples to demonstrate the effectiveness of the proposed scheme. Moreover, as an application, the proposed scheme is then applied to a secure communication system whose process consists of two phases: the adaptation phase in which the chaotic transmitter’s disturbances are estimated; and the communication phase in which the information signal is transmitted and then recovered on the basis of the estimated parameters. Simulation results verify the proposed scheme’s success in the communication application.

Sequential synchronization of two Lorenz systems using active control

Abstract: Using techniques from active control theory, we demonstrate that a coupled Lorenz system can be synchronized. The application of the control elements is sequentially applied and the ensuing synchronization is displayed.

On the unification of heterotic strings, M theory and E(∞) theory

Abstract: The work points out the existence of various theoretical and numerical evidences for a possibly deep connection between the high but finite dimensionality of heterotic strings and M theory on the one side and the infinite-dimensional E (∞) Cantorian theory on the other. It is conjectured that various modern strings theories are presumably the best possible description of actual quantum space-time within a finite-dimensional theory. It is further conjectured that the E (∞) theory provides the mathematical framework for a possible stringent proof of this conjecture.

Multistability and cyclic attractors in duopoly games

Abstract: A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T :Ox; yU!O r 1O yU ; r 2O xUU, is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map FOxUaO r 1 r 2UOxU. The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031‐2048] as a model of an economic system, in which the reaction functions r1 and r2 are logistic maps. ” 2000 Elsevier Science Ltd. All rights reserved.

Fractals related to long DNA sequences and complete genomes

Abstract: In visualizing very long DNA sequences, including the complete genomes of several bacteria, yeast and segments of human genes, we encounter fractal-like patterns underlying these biological objects of prominent importance. The method used here to visualize genomes of organisms may well be used as a convenient tool to trace, e.g., evolutionary relatedness of species. We describe the method and explain the origin of the observed fractal-like patterns. ” 2000 Elsevier Science Ltd. All rights reserved.

Modelling supervisory control systems using fuzzy cognitive maps

Abstract: Modelling complex systems and their supervisior has attracted the high interest of many scientists and engineers There has been a need for highly sophisticated Autonomous Intelligent Systems A very promising methodology to model the Supervisor of a plant is the use of Fuzzy Cognitive Maps (FCM) FCM are a combination of Fuzzy Logic and Neural Networks A new mathematical model for FCMs is proposed and its representation is examined in this paper FCM construction is presented through the development of the model for a simple control process problem Then, issues for the application of FCM as the model of the supervisor of a complex system are addressed and a hierarchical two-level structure is proposed

Localization phenomena in structural dynamics

Abstract: A survey of vibration localization phenomena in the context of structural dynamics and vibrations is presented The review covers the more common and relevant cases where mode localization and vibration confinement are likely to occur in engineering structures Examples considered include periodic or nearly periodic multi-span beams and multi-bay trusses, large space structures, space antennas, and almost periodic (ap) structures with circular symmetry, eg, bladed disks in turbomachines Both analytical and numerical methods for analyzing and predicting localization in finite and infinite systems are discussed In this paper, we show how the problem of mode localization and vibration confinement can be formulated as a problem in the theory of stability of differential equations with ap coefficients Using stability theory, new definitions of mode localization can be established for both linear and nonlinear structures The possibility of stabilizing certain nonconservative fluid-structure systems using structural disorder is demonstrated, and stability theorems are given for aeroelastic systems governed by normal operators We also illustrate how the results from localization theory and the associated stability theory can be applied to the vibration control problem, by triggering vibration confinement by active or passive means

Estimation of the largest Lyapunov exponent in systems with impacts

Abstract: The method of estimation of the largest Lyapunov exponent for mechanical systems with impacts using the properties of synchronization phenomenon is demonstrated. The presented method is based on the coupling of two identical dynamical systems and is tested on the classical Duffing oscillator with impacts.

On the unification of the fundamental forces and complex time in the E(∞) space

Abstract: Motivated by some recent contributions by various authors in the use of the so-called complex time (0± i t) and Cantorian space ( E (∞) ) in particle physics and cosmology, the present paper gives a preliminary analysis and some possible applications of these relatively new concepts in the quest for a general theory which may unify all the four fundamental forces known at present. In the course of the analysis presented here, we will give a derivation of a fundamental length in conceptually two different but complementary ways. The so-obtained fundamental length turned out to be of the order of magnitude of the Planck length (10 −33 cm ) and we discuss the possible existence of a transcritical fundamental particle with a mass related to the Planck mass (10 −5 g ) and to a mini black hole with a Schwarzschild radius which is of the order of the Planck length.

Modelling of dynamical systems with motion dependent discontinuities

Abstract: The paper introduces two concepts for describing and solving dynamical systems with motion dependent discontinuities such as clearances, impacts, dry friction, or combination of these phenomena. The first approach assumes any dynamic system can be considered as continuous in a finite number of continuous subspaces, which together form so-called global hyperspace. Global solution is obtained by “gluing” local solutions obtained by solving the problem in the continuous subspaces. An efficient numerical algorithm is presented, and then used to solve dynamics of a piecewise oscillator, which has been also verified experimentally. The second approach considers that in reality the system parameters do not change in an abrupt manner. Therefore, a smooth contiunuous function is used to model a transition between the subspaces, in particular the sigmoid function is employed. This allows to control the degree of abruptness on the intersections of the continuous subspaces. An asymmetrical, piecewise linear oscillator has been examined to provide recommendations regarding validity of this approach.

On modifications of Puu's dynamical duopoly

Abstract: Puu's economic dynamical model is extended to the following cases: Bounded rationality, differentiated goods and bounded rationality with delay. It is shown that delay increases stability. Hence firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium. Stability and instability (cycles and chaos) conditions for all these cases are determined.

Fractal dimension and classification of music

Abstract: The fractal aspect of different kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great number of different musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal dimension helps discriminate different categories of music. Then, we used an original statistical technique based on the Bootstrap assumption to find a time window in which fractal dimension reaches a high power of music discrimination. The best discrimination is obtained between 1/44100 and 16/44100 Hertz. We admit that to distinguish some different aspects of music well, the high information quantity is obtained in the high frequency domain. By calculating fractal dimension with the ANAM method, it was statistically proven that fractal dimension could distinguish different kinds of music very well: musics could be classified by their fractal dimensions.

Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation

Abstract: A class of periodic motions of an inverted pendulum with rigid lateral barriers is analysed under the hypothesis that the system is forced by impulsed periodic excitation. Due to the piece-wise linear nature of the problem, the existence and the stability of the cycles are determined analytically. It is found that they depend on both classical (saddle-node and period-doubling) and non-classical bifurcations, the latter involving a `synchronization' between impulses and impacts which leads to the sudden disappearing of the orbits. Attention is paid to the physical interpretation of these bifurcations, and to the determination of analytical criteria for their occurrence. We study how the relative position (with respect to the excitation amplitude) of the local bifurcations determines the system response and the bifurcation scenario. Symmetric and unsymmetric excitations are considered and the regions of stability of the periodic solutions are analytically determined. Finally, a comparison with the case of harmonic excitation is presented showing both analogies and differences, and highlighting how the impulsed excitation allows to obtain stable periodic responses at higher values of the excitation amplitude.

Fractal market hypothesis and two power-laws

Abstract: A fractal approach is used to analyze financial time series by applying different degrees of time resolutions. This leads to the heterogenous market hypothesis (HMH), where different market participants analyze past events and news with different time horizons. A new general model for asset returns is studied in the framework of the fractal market hypothesis (FMH). It concerns capital market systems in which the conditionally exponential dependence (CED) property can be attached to each investor on the market.

The dynamics of a triopoly Cournot game

Abstract: This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for non-invertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three firms are identical is focused.

Nonlinear dynamics of parts in engineering systems

Abstract: By definition, chaotic vibrations arise from nonlinear deterministic physical systems or non-random differential or difference equations. In numerous engineering systems there exist nonlinearities ...

The chaotic universe

Abstract: In this paper we suggest a formulation that would bear out the spirit of Prigogine's “Order Out of Chaos” and Wheeler's “Law Without Law”. In it a typical elementary particle length, namely the pion Compton wavelength arises from the random motion of the N particles in the universe of dimension R. It is then argued in the light of recent work that this is the origin of the laws of physics and leads to a cosmology consistent with observation.

Fractional Fokker–Planck equation

Abstract: By using the definition of the characteristic function and Kramers–Moyal Forward expansion, one can obtain the Fractional Fokker–Planck Equation (FFPE) in the domain of fractal time evolution with a critical exponent α (0 α ⩽1). Two different classes of fractional differential operators, Liouville–Riemann (L–R) and Nishimoto (N) are used to represent the fractal differential operators in time. By applying the technique of eigenfunction expansion to get the solution of FFPE, one finds that the time part of eigenfunction expansion in terms of L–R represents the waiting time density Ψ ( t ), which gives the relation between fractal time evolution and the theory of continuous time random walk (CTRW). From the principle of maximum entropy, the structure of the distribution function can be known.

Parameter estimation in nonlinear stochastic differential equations

Abstract: We discuss the problem of parameter estimation in nonlinear stochastic differential equations (SDEs) based on sampled time series. A central message from the theory of integrating SDEs is that there exist in general two time scales, i.e. that of integrating these equations and that of sampling. We argue that therefore, maximum likelihood estimation is computationally extremely expensive. We discuss the relation between maximum likelihood and quasi maximum likelihood estimation. In a simulation study, we compare the quasi maximum likelihood method with an approach for parameter estimation in nonlinear SDEs that disregards the existence of the two time scales.

Is quantum space-time infinite dimensional

Abstract: The stringy uncertainty relations, and corrections thereof, were explicitly derived recently from the new relativity principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. A simple numerical argument is presented which suggests that quantum space-time may very well be infinite dimensional. A discussion of the repercussions of this new paradigm in Physics is given. A truly remarkably simple and plausible solution of the cosmological constant problem results from the new relativity principle: The cosmological constant is not a constant, in the same vein that energy in Einstein's Special Relativity is observer dependent. Finally, following El Naschie, we argue why the observed D =4 world might just be an average dimension over the infinite possible values of the quantum space-time and why the compactification mechanisms from higher to four dimensions in string theory may not be actually the right way to look at the world at Planck scales.

Effect of noise on the tuning properties of excitable systems

Abstract: We analyze the effect of additive dynamical noise on simple phase-locking patterns in the Fitzhugh–Nagumo (FHN) two-dimensional system in the excitable regime. In the absence of noise, the response amplitude for this system displays a classical resonance as a function of driving frequency. This translates into V-shaped tuning curves, which represent the amplitude threshold for one firing per cycle as a function of forcing frequency. We show that noise opens up these tuning curves at mid-to-low frequencies. We explain this numerical result analytically using a heuristic form for the firing rate that incorporates the frequency dependence of the subthreshold voltage response. We also present stochastic phase-locking curves in the noise intensity-forcing period subspace of parameter space. The relevance of our findings for the tuning of electroreceptors of weakly electric fish and their encoding of amplitude modulations of high frequency carriers are briefly discussed. Our study shows that, in certain contexts, it is essential to take into account the frequency sensitivity of neural responses and their modification by sources of noise.

Hints of a new relativity principle from p-brane quantum mechanics

Abstract: This report is an extension of a previous one hep-th/9812189. Several quantum mechanical wave equations for p-branes are proposed. The most relevant p-brane quantum mechanical wave equations determine the quantum dynamics involving the creation/destruction of p-dimensional loops of topology Sp, moving in a D-dimensional spacetime background, in the quantum state Φ. To implement full covariance we are forced to enlarge the ordinary relativity principle to a new relativity principle, suggested earlier by the author based on the construction of C-space, and also by Pezzaglia's poly-dimensional relativity, where all dimensions and signatures of spacetime should be included on the same footing.

The universe of chaos and quanta

Abstract: In recent papers it was shown that stochastic processes in the universe as a whole lead to discrete space time at Compton scales as also non-relativistic Quantum Mechanics. In this paper, we deduce the Dirac equation and thence a unified formulation of quarks and leptons. In the process several puzzling empirical results and coincidences are shown to be a consequence of the theory. These include the discreteness of the charge, handedness of quarks, their fractional charge, confinement and masses and the handedness of neutrinos, the so-called accidental relation that the classical Kerr–Newman metric describes, the field of an electron including the purely quantum mechanical gyromagnetic ratio g=2 , as also the many large number coincidences made famous by Dirac and Weinberg's mysterious empirical formula that relates the pion mass to the Hubble Constant. A cosmology based on fluctuations related to the above stochastic space-time discretization, consistent with latest observations is also seen to follow.

Relative multifractal analysis

Abstract: We introduce a general formalism for the multifractal analysis of one probability measure with respect to another. As an example, we analyse the multifractal structure of one graph directed self-conformal measure with respect to another.

Binary nonlinearization of AKNS spectral problem under higher-order symmetry constraints

Abstract: Binary nonlinearization of AKNS spectral problem is extended to the cases of higher-order symmetry constraints. The Hamiltonian structures, Lax representations, r -matrices and integrals of motion in involution are explicitly proposed for the resulting constrained systems in the cases of the first four orders.The obtained integrals of motion are proved to be functionally independent and thus the constrained systems are completely integrable in the Liouville sense.

Chaos in periodically forced discrete-time ecosystem models

Abstract: Natural populations whose generations are non-overlapping can be modelled by difference equations that describe how the populations evolve in discrete time-steps. These ecosystem models are, in general, nonlinear and contain system parameters that relate to such properties as the intrinsic growth-rate of a species. Typically, the parameters are kept constant. In this study, in order to simulate cyclic effects due to changes in environmental conditions, periodic forcing is applied to system parameters in four specific models, comprising three well-known, single-species models due to May, Moran–Ricker, and Hassell, and also a Maynard Smith predator–prey model. It is found that, in each case, a system that has simple (e.g., periodic) behavior in its unforced state can take on extremely complicated behavior, including chaos, when periodic forcing is applied, dependent on the values of the forcing amplitudes and frequencies. For each model, the application of forcing is found to produce an effective increase in the parameter space over which the system can behave chaotically. Bifurcation diagrams are constructed with the forcing amplitude as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, and attractor crises.

Application of chaos degree to some dynamical systems

Abstract: Chaos degree defined through two complexities in information dynamics is applied to some deterministic dynamical models. It is shown that this degree well describes the chaotic feature of the models.

Signal-to-noise ratio gain by stochastic resonance in a bistable system

Abstract: It was shown recently that the signal-to-noise ratio (SNR) could be improved by stochastic resonance (SR) in certain monostable systems and certain systems with monotonous nonlinearity working in the nonlinear response (NLR) regime. Here we demonstrate th

On the synchronization of different chaotic oscillators

Abstract: The synchronization of two different chaotic oscillators is studied, based on an open-loop control – the entrainment control. We consider two types of synchronization: complete synchronization and effectively complete synchronization. The sufficient conditions that two different systems can be synchronized by this method is discussed. Furthermore, a hierarchical idea to synchronize multiple chaotic subsystems is proposed.

A symplectic mapping for the ergodic magnetic limiter and its dynamical analysis

Abstract: A model for a new bidimensional sympletic mapping describing magnetic field line trajectories in a tokamak perturbed by ergodic magnetic limiter coils is presented. Numerical examples of these trajectories, computed for plasma described by large aspect-ratio equilibria, simulate the main characteristics of trajectories in the toroidal geometry. Also the importance of the symplecticity of the new mapping regarding certain features of non-linear dynamical analysis, for which a large number of iterations is necessary, is shown. Thus, some standard algorithms, such as the Lyapunov exponents and the rotational transforms, are applied with precision in order to characterize regular and chaotic regions in the parameter space, improving the study of bifurcations, routes to chaos, and diffusion in this system.