Behavior Of Nonlinear Dynamical Systems Research Articles

Wiener–Haar Expansion for the Modeling and Prediction of the Dynamic Behavior of Self-Excited Nonlinear Uncertain Systems

This paper deals with the modeling and the prediction of the dynamic behavior of uncertain nonlinear systems. An efficient method is proposed to treat these problems. It is based on the Wiener–Haar chaos concept resulting from the polynomial chaos theory and it generalizes the use of the multiresolution analysis well known in the signal processing theory. The method provides a powerful tool to describe stochastic processes as series of orthonormal piecewise functions whose weighting coefficients are identified using the Mallat pyramidal algorithm. This paper shows that the Wiener–Haar model allows an efficient description and prediction of the dynamic behavior of nonlinear systems with probabilistic uncertainty in parameters. Its contribution, compared to the representation using the generalized polynomial chaos model, is illustrated by evaluating the two models via their application to the problems of the modeling and the prediction of the dynamic behavior of a self-excited uncertain nonlinear system.

APPROXIMATION OF INVARIANT FOLIATIONS FOR STOCHASTIC DYNAMICAL SYSTEMS

Invariant foliations are geometric structures for describing and understanding the qualitative behaviors of nonlinear dynamical systems. For stochastic dynamical systems, however, these geometric structures themselves are complicated random sets. Thus it is desirable to have some techniques to approximate random invariant foliations. In this paper, invariant foliations are approximated for dynamical systems with small noisy perturbations, via asymptotic analysis. Namely, random invariant foliations are represented as a perturbation of the deterministic invariant foliations, with deviation errors estimated.

Book Review of Non-Traditional Dynamics in Electronics: Theory and Practice

The new book published by Scientific Research Publishers, USA in 2011 is devoted to investigation of properties of dynamic radio physical systems and to development of methodic fundamentals for creation the chaotic oscillation sources and forecasting of nonlinear dynamic system behavior at variation of its parameters. The urgency of such investigation execution in this direction is determined both by the necessity of new ideas development in understanding of nonlinear dynamics of various processes independently upon its affiliation with the specific physics area and by need of the modern radio physics and radio electronics in sources of wide-band chaotic oscillations and signals. This book makes a considerable contribution in this area because it contains theoretical, modeling and experimental results of the analysis of bifurcation phenomena and processes in chaotic systems as well as issues of information transmission using chaotic signals. In this book for the first time the auto-parametric scenario of stochastization is described, conditions are formulated, at which movements in arbitrary dynamic systems are governed by the unified system of spectral-time equations. Phenomena of synchronization of chaotic movements and the intermit synchronization are discovered and investigated in detail. The bifurcation sense of the control parameter of modified logistic map is revealed, which ensures reproduction of strictly chaotic temporal sequences. The aggregate of this book’s results may be classified as the new large scientific achievement. Authenticity of results obtained is determined by its consistent (became classical) laws of dynamic system functioning and coincidence of numerical and physical experiments at testing of offered mathematical models of dynamic systems, and successful functioning of the described radio electronic devices. This book contains preface and six chapters. To my opinion, all materials are reliable and to the right degree are supported by necessary theoretical and experimental substantiation. Authors on the basis of numerical and physical experiments prove earnestly the opportunity of existence of the strange non-chaotic attractor in nonautonomous radio electronic circuits at single-frequency sinusoidal influence and on the basis of result generalization of much number of numerical and physical experiments authors offer and substantiate the feasibility for the wide class oscillator and relaxation generation system of the specific stochastization scenario, which did not meet earlier in the scientific publications on nonlinear dynamics. Theoretical analysis of modified logistic maps (MLM) and appropriate numerical and physical experiments allows authors to discover and investigate conditions of arising of the unknown phenomenon—intermittent synchronization. At last, some results of the book are checked by the functioning radio electronic devices. Scientific and practical importance of investigations

Adaptive Neuro-Fuzzy Control Approach Based on Particle Swarm Optimization

This paper proposes a modified particle swarm optimization algorithm (MPSO) to design adaptive neuro-fuzzy controller parameters for controlling the behavior of non-linear dynamical systems. The modification of the proposed algorithm includes adding adaptive weights to the swarm optimization algorithm, which introduces a new update. The proposed MPSO algorithm uses a minimum velocity threshold to control the velocity of the particles, avoids clustering of the particles, and maintains the diversity of the population in the search space. The mechanism of MPSO has better potential to explore good solutions in new search spaces. The proposed MPSO algorithm is also used to tune and optimize the controller parameters like the scaling factors, the membership functions, and the rule base. To illustrate the adaptation process, the proposed neuro-fuzzy controller based on MPSO algorithm is applied successfully to control the behavior of both non-linear single machine power systems and non-linear inverted pendulum systems. Simulation results demonstrate that the adaptive neuro-fuzzy logic controller application based on MPSO can effectively and robustly enhance the damping of oscillations.

Modification of measurement theory

To ensure the correctness of measurements in nonlinear dynamic systems in paper was the development the measurement theory. We study the radically new conditions of measurement problems related to the specific behavior of nonlinear dynamic systems. Shown, that between qualitative theory of differential equations and methods of evaluation of measurement uncertainty there is a link that provides an assessment of conditions of measurement.

An impact of noise on invariant manifolds in nonlinear dynamical systems

Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems. However, these nonlinear systems are often subject to random fluctuations or noises. It is thus desirable to quantify the impact of noises on the invariant manifolds. When the noise intensity is small, this impact is estimated via asymptotic analysis in the context of Liapunov–Perron formulation. Namely, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold, with a well-defined bound for the deviation.

The Asian Crisis: The Challenge to Conventional Wisdom

The gap between what conventional wisdom had to say about Asian economic prospects on the dawn of the Asian Crisis and what actually transpired is enormous. After the fact, analysis has focused on the build-up of short-term, unhedged foreign currency debt, which left the region vulnerable to a crisis of confidence that only awaited a trigger, such as an economic slowdown or weakening of exports, or indeed contagion from other economies, to be set in motion. Once underway, various factors that contribute to market volatility, such as behaviour of investors, are adduced to explain the actual course of the crisis, including the overshooting in financial markets. The macroeconomic event of the decade is generally seen and discussed as reflecting idiosyncrasies of Asian business culture, as a crisis of confidence, as a failure of institutional controls or as some combination of the above - factors that do not even appear as arguments in generally accepted core macroeconomic models. Metaphors for the crisis have been drawn from the weather (storms), nuclear physics (meltdown), biology (flu, contagion, virulence), and mass psychology (panic and herd behaviour), all of which are described by non-linear dynamical systems. The paper compares various statements about the Asian Crisis to descriptions of behaviour of non-linear dynamical systems. Preliminary conclusions about the utility of this line of research are drawn.

Searching bifurcations in high-dimensional parameter space via a feedback loop breaking approach

Bifurcations leading to complex dynamical behaviour of non-linear systems are often encountered when the characteristics of feedback circuits in the system are varied. In systems with many unknown or varying parameters, it is an interesting, but difficult problem to find parameter values for which specific bifurcations occur. In this article, we develop a loop breaking approach to evaluate the influence of parameter values on feedback circuit characteristics. This approach allows a theoretical classification of feedback circuit characteristics related to possible bifurcations in the system. Based on the theoretical results, a numerical algorithm for bifurcation search in a possibly high-dimensional parameter space is developed. The application of the proposed algorithm is illustrated by searching for a Hopf bifurcation in a model of the mitogen activated protein kinase cascade, which is a classical example for biochemical signal transduction.

Estabilidad de Sistemas No Lineales Basada en la Teoría de Liapunov

El comportamiento dinámico de los sistemas no lineales es mucho más rico que el de los lineales y su análisis mucho más complicado. Para el análisis de estabilidad, las técnicas basadas en la teoría de Liapunov tienen un lugar destacado. En este articulo se revisa parte de esta teoría incluyendo las técnicas de estimación de la cuenca de atracción. También se repasan los resultados que han aparecido en los últimos años sobre la aplicación a este campo de los métodos numéricos de optimización de suma de cuadrados.

Diagnosis of stressed states in the lithosphere of the Baikal Rift System

The results of investigation of the structure and dynamics of stresses at the modern (instrumental) stage of the evolution of the lithosphere in the Baikal Rift System (BRS) are presented. The application of the theoretical methods of nonlinear dynamic systems for diagnosis of the stressed states allowed us to formulate the concept of the complex stressed state of the BRS lithosphere. In this concept, the time distribution of large earthquakes is explained by bifurcations (catastrophes) of stresses. The analysis of the results obtained points to possibilities of a catastrophe of stresses in the BRS lithosphere in the next few years and the consequent realization of large earthquakes with magnitude M ≈ 7 . A model of triple equilibrium bifurcation is suggested for the evolution of stresses in the BRS lithosphere. Rapid processes of stress rearrangement in the BRS lithosphere are consistent with the behavior of complex self-organizing unstable thermodynamic systems [1]. Since they are observed synchronously in three zones of the maxima of structural force inhomogeneities, we classify the BRS as a spatiotemporal open self-organized nonlinear dissipative system [2]. A complex system is considered self-organized if it maintains its instability at a level sufficient for effective resistance to the variations in the active medium, thus saving itself from collapse. We understand self-organization as an antientropic process of buildup, interaction, and maintenance of coherence between the elements of the system with increase of its complexity and formation of the attractor structure. Complex synergetic phenomena are exemplified in the literature as different processes of self-organization in physicochemical and biological systems. Their analysis allows us to establish the general features, which are characteristic of complex phenomena, and formulate the main principles irrespective of the specific nature of the system. Each of such processes has unique features. The role of nonlinearity, fluctuations, oscillations, bifurcations, and the attractor is manifested very clearly in these examples. Therefore, when modeling the behavior of geophysical systems, one should use the advantage of new perspectives discovered by science during the investigation of complex behavior of nonlinear dynamic systems. This is especially important in light of the modern tendencies of forecasting and controlling the seismic process, since forecast and control imply knowledge of the properties and peculiarities of the system under control. They begin with gathering of information about the state of the system, its links, and the logic of its functioning.

Development of bilinear power system representations for small signal stability analysis

In this paper, a new analysis technique for predicting and characterizing nonlinear behavior of stressed power networks is proposed. Making use of an analytical approximation of the system nonlinear model via the use of a truncated Carleman linearization technique, a bilinear state-space model of the power system is developed in which the second and higher order nonlinear terms are explicitly incorporated in the series expansion representation of the system model. The proposed framework enables the study of the dynamic behavior of nonlinear systems, both analytically and numerically, and can be used to represent a wide class of non-linear systems and oscillatory processes. Analytical criteria are developed based on the structural properties of the bilinear state-space model matrices, which predict the existence and stability character of modal interaction in terms of the eigenstructure of the linear system representation. The properties and behavior of the bilinear model are then investigated, and a number of useful results are derived. The present method is quite general and extends readily to higher-dimensional systems. A simplified 2-area, 4-machine system is used to illustrate the proposed procedure. Detailed nonlinear time-domain simulations are conducted to identify the strength of nonlinear behavior arising from interaction of the fundamental modes of oscillation, as well as to check the validity of the analysis.

A model proposal for the chaotic structure of Istanbul stock exchange

Chaos theory is considered a novel way of understanding the behaviour of nonlinear dynamic systems. It is well known that the evaluation of chaotic systems is dependent on initial conditions since exponential growth error is a common characteristic. This present paper evaluates the effects of a nonlinear dynamic system in Istanbul Stock Exchange, based on time series. The reliability of predicting stock behaviour depends on this fact. In other words, the aim is to prove that if ISE daily index return shows chaotic behaviour.

Effect of noise on the bifurcation behavior of nonlinear dynamical systems

Abstract We argue that dynamical noise can dramatically change the dynamics of low dimensional chaotic systems. Moreover, we show that chaos tests are highly sensitive to dynamical noise and this becomes worse when the intensity of the noise increases.

Mechanisms behind the generalized synchronization conditions

A universal mechanism underlying generalized synchronization conditions in unidirectionally coupled stochastic oscillators is considered. The consideration is carried out in the framework of a modified system with additional dissipation. The approach developed is illustrated with model examples. The conclusion is reached that two types of the behavior of nonlinear dynamic systems known as generalized synchronization and noise-induced synchronization, which are viewed as different phenomena, actually represent a unique type of the synchronous behavior of stochastic oscillators and are caused by the same mechanism.

Bifurcation Analysis of Railway Vehicles

To identify the long-time behaviour of nonlinear dynamical systems with respect to the influence of one or more system parameters, numerical bifurcation analysis is an ideal computer-aided method. The objective of the paper is to describe a software environment for such an analysis that is based on the principles of path-following or continuation. A specific viewpoint is the application to ‘realistic’, i.e. detailed and complex simulation models of railway vehicles following a multibody system approach. Stationary as well as periodic behaviour is considered. Three major topics are of primary interest: The integration of a bifurcation software into a software package for the simulation of arbitrary mechanical systems; the direct calculation of periodic solutions (limit cycles); and the handling of differential algebraic equations (DAEs). The algorithms are applied finally to the ‘realistic’ simulation model of a high-speed railway passenger car.

Dynamical behavior of nonlinear viscoelastic Timoshenko beams with damage on a viscoelastic foundation

Based on convolution-type constitutive equations for linear viscoelastic materials with damage and the hypotheses of Timoshenko beams with large deflections, the nonlinear equations governing dynamical behavior of Timoshenko beams with damage on viscoelastic foundation were firstly derived. By using the Galerkin method in spatial domain, the nonlinear integro-partial differential equations were transformed into a set of integro-ordinary differential equations. The numerical methods in nonlinear dynamical systems, such as the phase-trajectory diagram, Poincare section and bifurcation figure, were used to solve the simplified systems of equations. It could be seen that simplified dynamical systems possess the plenty of nonlinear dynamical properties. The influence of load and material parameters on the dynamic behavior of nonlinear system were investigated in detail.

Methods to reduce non-linear mechanical systems for instability computation

Non-linear dynamical structures depending on control parameters are encountered in many areas of science and engineering. In the study of non-linear dynamical systems depending on a given control parameter, the stability analysis and the associated non-linear behaviour in a near-critical steady-state equilibrium point are two of the most important points; they make it possible to validate and characterize the non-linear structures. Stability is investigated by determining eigenvalues of the linearized perturbation equations about each steady-state operating point, or by calculating the Jacobian of the system at the equilibrium points. While the conditions and the values of the parameters which cause instability can be investigated by using linearized equations of motion studies of the non-linear behaviour of vibration problems, on the other hand, require the complete non-linear expressions of systems. Due to the complexity of non-linear systems and to save time, simplifications and reductions in the mathematical complexity of the non-linear equations are usually required. The principal idea for these non-linear methods is to reduce the order of the system and eliminate as many non-linearities as possible in the system of equations. In this paper, a study devoted to evaluating the instability phenomena in non-linear models is presented. It outlines stability analysis and gives a non-linear strategy by constructing a reduced order model and simplifying the non-linearities, based on three non-linear methods: the centre manifold concept, the rational approximants and the Alternating Frequency/Time domain method. The computational procedures to determine the reduced and simplified system via the centre manifold approach and the fractional approximants, as well as the approximation of the responses as a Fourier series via the harmonic balance method, are presented and discussed. These non-linear methods for calculating the dynamical behaviour of non-linear systems with several degrees-of-freedom and non-linearities are tested in the case of mechanical systems with many degrees-of-freedom possessing polynomial non-linearities. Results obtained are compared with those estimated by a classical Runge-Kutta integration procedure. Moreover, an extension of the centre manifold approach using rational approximants is proposed and used to explore the dynamics of non-linear systems, by extending the domain of convergence of the non-linear reduced system and evaluating its performance and suitability.

Multiple time-scales in nonlinear flight mechanics: diagnosis and modeling

There are often disparate time-scales in the dynamics of flight, creating the potential for reduced-order modeling to simplify simulation, analysis and design. There have been notable successes in developing reduced-order models; however, in the case of nonlinear dynamics, which one must typically deal with in guidance problems, there has not been a systematic, reliable means of diagnosing disparate time-scales and developing reduced-order models. Focusing on two time-scale behavior in nonlinear dynamical systems, we recall Fenichel’s characterization of the geometric structure of the flow and his theorem establishing the existence and properties of coordinates adapted to this structure. Adapted coordinates are difficult to construct directly, without an appropriate singularly perturbed model of the dynamics. We discuss the use of Lyapunov exponents and vectors to diagnose two time-scale behavior and to determine the corresponding tangent space structure for the linearized dynamics. The structure of the linearized flow can then be translated into the manifold structure of the nonlinear flow. We briefly mention the use of Lyapunov vectors to locate a slow manifold and contrast this approach with two existing approaches. The minimum time to climb problem provides an example of two time-scale behavior and motivates the discussion.

Analytical properties of the Sprott’s chaotic flows

A method is developed to describe the asymptotic (final, at t→∞) behaviour of nonlinear dynamical systems. A systematic examination of all the nineteen chaotic systems proposed by Sprott was preformed by this method. It was found that thirteen of them can be reformulated into oscillatory type second order ordinary differential equations with memory term as forcing one. The rest of the systems cannot be recast in this form, but two of them admit existence of nondifferential relationships between the phase variables at t→∞ The possibility for statistical treatment of the systems in chaotic regime is also discussed. The results could be used for further studies, e.g. numerical analysis of the transient phenomenon.

Universal Algorithm of the Nonequilibrium Dynamics of Chaotic Systems

Scenarios of transition of nonlinear dynamical systems to chaos are considered based on the study of the behavior of nonlinear dynamical systems represented as feedback systems and analysis of universality and scale invariance (fractal) properties of threshold dynamical structures at points of unstable equilibrium of these systems.