Nonlinear Systems With Stochastic Fluctuations

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Nonlinear Systems with Stochastic Fluctuations

Author : Wei Xu
Publisher : Walter de Gruyter
ISBN 13 : 9783110313055
Total Pages : 200 pages
Book Rating : 4.3/5 (13 download)

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Book Synopsis Nonlinear Systems with Stochastic Fluctuations by : Wei Xu

Download or read book Nonlinear Systems with Stochastic Fluctuations written by Wei Xu and published by Walter de Gruyter. This book was released on 2016-07-15 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book considers some typical nonlinear dynamical systems with various random noises. It presents the reviews about nonlinear and stochastic dynamics, with some historic background and recent developments. Theoretical and numerical approaches will be carried out for some typical nonlinear dynamical systems. It is written with a wide audience in mind and presents basic techniques for readers to learn and investigate stochastic dynamics. This book serves as a reference book or textbook for students and researchers in mathematics, engineering, physics, chemistry, biology and social sciences, and other professionals who are interested in the subject and help them quickly get into the field and ready for studies.

Stochastic Nonlinear Systems in Physics, Chemistry, and Biology

Author : L. Arnold
Publisher : Springer Science & Business Media
ISBN 13 : 3642680380
Total Pages : 239 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis Stochastic Nonlinear Systems in Physics, Chemistry, and Biology by : L. Arnold

Download or read book Stochastic Nonlinear Systems in Physics, Chemistry, and Biology written by L. Arnold and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the invited papers of the interdisciplinary workshop on "Stochastic Nonlinear Systems in Physics, Chemistry and Biology" held at the Center for Interdisciplinary Research (ZIF), University of Bielefeld, West Germany, October 5-11, 1980. The workshop brought some 25 physicists, chemists, and biologists - who deal with stochastic phenomena - and about an equal number of mathematicians - who are experts in the theory of stochastic processes - together. The Scientific Commitee consisted of L. Arnold (Bremen), A. Dress (Bielefeld), W. Horsthemke (Brussels), T. Kurtz (Madison), R. Lefever (Brussels), G. Nicolis (Brussels), and V. Wihstutz (Bremen). The main topics of the workshop were the transition from deterministic to stoch astic behavior, external noise and noise induced transitions, internal fluctuations, phase transitions, and irreversible thermodynamics, and on the mathematical side, approximation of stochastic processes, qualitative theory of stochastic systems, and space-time processes. The workshop was sponsored by ZIF, Bielefeld, and by the Universities of Bremen and Brussels. We would like to thank the staff of ZIF and H. Crauel and M. Ehrhardt (Bremen) for the perfect organization and their assistance. In addition, our thanks go to Professor H. Haken for having these Proceedings included in the Se ries in Synergetics. Bremen and Brussels L. Arnold and R. Lefever December 1980 v Contents Part I. Introduction: From Deterministic to Stochastic Behavior On the Foundations of Kinetic Theory By B. Misr~ and I. Prigogine (With 1 Figure) ............................. .

Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems

Author : M. Reza Rahimi Tabar
Publisher : Springer
ISBN 13 : 3030184722
Total Pages : 280 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems by : M. Reza Rahimi Tabar

Download or read book Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems written by M. Reza Rahimi Tabar and published by Springer. This book was released on 2019-07-04 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on a central question in the field of complex systems: Given a fluctuating (in time or space), uni- or multi-variant sequentially measured set of experimental data (even noisy data), how should one analyse non-parametrically the data, assess underlying trends, uncover characteristics of the fluctuations (including diffusion and jump contributions), and construct a stochastic evolution equation? Here, the term "non-parametrically" exemplifies that all the functions and parameters of the constructed stochastic evolution equation can be determined directly from the measured data. The book provides an overview of methods that have been developed for the analysis of fluctuating time series and of spatially disordered structures. Thanks to its feasibility and simplicity, it has been successfully applied to fluctuating time series and spatially disordered structures of complex systems studied in scientific fields such as physics, astrophysics, meteorology, earth science, engineering, finance, medicine and the neurosciences, and has led to a number of important results. The book also includes the numerical and analytical approaches to the analyses of complex time series that are most common in the physical and natural sciences. Further, it is self-contained and readily accessible to students, scientists, and researchers who are familiar with traditional methods of mathematics, such as ordinary, and partial differential equations. The codes for analysing continuous time series are available in an R package developed by the research group Turbulence, Wind energy and Stochastic (TWiSt) at the Carl von Ossietzky University of Oldenburg under the supervision of Prof. Dr. Joachim Peinke. This package makes it possible to extract the (stochastic) evolution equation underlying a set of data or measurements.

Formation of Spatio-Temporal Patterns in Stochastic Nonlinear Systems

Author : Felix M Ller
Publisher : Sudwestdeutscher Verlag Fur Hochschulschriften AG
ISBN 13 : 9783838134208
Total Pages : 164 pages
Book Rating : 4.1/5 (342 download)

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Book Synopsis Formation of Spatio-Temporal Patterns in Stochastic Nonlinear Systems by : Felix M Ller

Download or read book Formation of Spatio-Temporal Patterns in Stochastic Nonlinear Systems written by Felix M Ller and published by Sudwestdeutscher Verlag Fur Hochschulschriften AG. This book was released on 2012-08 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: The physical essence of complex structures goes beyond Newton's dynamical laws. In this book systems are investigated which exhibit excitable, oscillatory and bistable behavior as basic modes of nonlinear dynamics. Addition of stochastic fluctuations contribute to the appearance of complex behavior. Bifurcations of local behavior as well as nucleation of spatially extended structures are studied. Typical fronts and spirals appear but also unusual patterns such as moving clusters and inverted waves. Methods are presented to find expressions for front velocities in the presence of system boundaries. An abstract two-state model with two waiting time distributions is considered representing excitable dynamics. Instantaneous and delayed response of the ensemble's response is analyzed. Furthermore transport and diffusion of Brownian particles in a spatio-temporal oscillating potential is discussed. As a cause of nearly dispersion-less transport synchronization mechanisms can be identified.

Non-Linear Transformations of Stochastic Processes

Author : P. I. Kuznetsov
Publisher : Elsevier
ISBN 13 : 1483282686
Total Pages : 515 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Non-Linear Transformations of Stochastic Processes by : P. I. Kuznetsov

Download or read book Non-Linear Transformations of Stochastic Processes written by P. I. Kuznetsov and published by Elsevier. This book was released on 2014-05-12 with total page 515 pages. Available in PDF, EPUB and Kindle. Book excerpt: Non-Linear Transformations of Stochastic Processes focuses on the approaches, methodologies, transformations, and computations involved in the non-linear transformations of stochastic processes. The selection first underscores some problems of the theory of stochastic processes and the transmission of random functions through non-linear systems. Discussions focus on the transformation of moment functions for the general non-linear transformation; conversion formulas for correlation functions; transformation of moment functions for the simplest type of non-linear transformation; and normalization of the linear system of probability distribution laws. The text then ponders on quasi-moment functions in the theory of random processes and correlation functions in the theory of the Brownian motion generalization of the Fokker-Planck equation. The manuscript elaborates on the correlation functions of random sequences of rectangular pulses; method of determining the envelope of quasi-harmonic fluctuations; and the problem of measuring electrical fluctuations with the aid of thermoelectric devices. The book then examines the effect of signal and noise on non-linear elements and the approximate method of calculating the correlation function of stochastic signals. The selection is a dependable source of information for researchers interested in the non-linear transformations of stochastic processes.

Nonlinear and Stochastic Climate Dynamics

Author : Christian L. E. Franzke
Publisher : Cambridge University Press
ISBN 13 : 1316883213
Total Pages : 612 pages
Book Rating : 4.3/5 (168 download)

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Book Synopsis Nonlinear and Stochastic Climate Dynamics by : Christian L. E. Franzke

Download or read book Nonlinear and Stochastic Climate Dynamics written by Christian L. E. Franzke and published by Cambridge University Press. This book was released on 2017-01-19 with total page 612 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is now widely recognized that the climate system is governed by nonlinear, multi-scale processes, whereby memory effects and stochastic forcing by fast processes, such as weather and convective systems, can induce regime behavior. Motivated by present difficulties in understanding the climate system and to aid the improvement of numerical weather and climate models, this book gathers contributions from mathematics, physics and climate science to highlight the latest developments and current research questions in nonlinear and stochastic climate dynamics. Leading researchers discuss some of the most challenging and exciting areas of research in the mathematical geosciences, such as the theory of tipping points and of extreme events including spatial extremes, climate networks, data assimilation and dynamical systems. This book provides graduate students and researchers with a broad overview of the physical climate system and introduces powerful data analysis and modeling methods for climate scientists and applied mathematicians.

Nonlinear Stochastic Control and Filtering with Engineering-oriented Complexities

Author : Guoliang Wei
Publisher : CRC Press
ISBN 13 : 1315350661
Total Pages : 233 pages
Book Rating : 4.3/5 (153 download)

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Book Synopsis Nonlinear Stochastic Control and Filtering with Engineering-oriented Complexities by : Guoliang Wei

Download or read book Nonlinear Stochastic Control and Filtering with Engineering-oriented Complexities written by Guoliang Wei and published by CRC Press. This book was released on 2016-09-15 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Stochastic Control and Filtering with Engineering-oriented Complexities presents a series of control and filtering approaches for stochastic systems with traditional and emerging engineering-oriented complexities. The book begins with an overview of the relevant background, motivation, and research problems, and then: Discusses the robust stability and stabilization problems for a class of stochastic time-delay interval systems with nonlinear disturbances Investigates the robust stabilization and H∞ control problems for a class of stochastic time-delay uncertain systems with Markovian switching and nonlinear disturbances Explores the H∞ state estimator and H∞ output feedback controller design issues for stochastic time-delay systems with nonlinear disturbances, sensor nonlinearities, and Markovian jumping parameters Analyzes the H∞ performance for a general class of nonlinear stochastic systems with time delays, where the addressed systems are described by general stochastic functional differential equations Studies the filtering problem for a class of discrete-time stochastic nonlinear time-delay systems with missing measurement and stochastic disturbances Uses gain-scheduling techniques to tackle the probability-dependent control and filtering problems for time-varying nonlinear systems with incomplete information Evaluates the filtering problem for a class of discrete-time stochastic nonlinear networked control systems with multiple random communication delays and random packet losses Examines the filtering problem for a class of nonlinear genetic regulatory networks with state-dependent stochastic disturbances and state delays Considers the H∞ state estimation problem for a class of discrete-time complex networks with probabilistic missing measurements and randomly occurring coupling delays Addresses the H∞ synchronization control problem for a class of dynamical networks with randomly varying nonlinearities Nonlinear Stochastic Control and Filtering with Engineering-oriented Complexities describes novel methodologies that can be applied extensively in lab simulations, field experiments, and real-world engineering practices. Thus, this text provides a valuable reference for researchers and professionals in the signal processing and control engineering communities.

Information Theory and Stochastics for Multiscale Nonlinear Systems

Author : Andrew Majda
Publisher : American Mathematical Soc.
ISBN 13 : 9780821869864
Total Pages : 152 pages
Book Rating : 4.8/5 (698 download)

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Book Synopsis Information Theory and Stochastics for Multiscale Nonlinear Systems by : Andrew Majda

Download or read book Information Theory and Stochastics for Multiscale Nonlinear Systems written by Andrew Majda and published by American Mathematical Soc.. This book was released on with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces mathematicians to the fascinating mathematical interplay between ideas from stochastics and information theory and practical issues in studying complex multiscale nonlinear systems. It emphasizes the serendipity between modern applied mathematics and applications where rigorous analysis, the development of qualitative and/or asymptotic models, and numerical modeling all interact to explain complex phenomena. After a brief introduction to the emerging issues in multiscale modeling, the book has three main chapters. The first chapter is an introduction to information theory with novel applications to statistical mechanics, predictability, and Jupiter's Red Spot for geophysical flows. The second chapter discusses new mathematical issues regarding fluctuation-dissipation theorems for complex nonlinear systems including information flow, various approximations, and illustrates applications to various mathematical models. The third chapter discusses stochastic modeling of complex nonlinear systems. After a general discussion, a new elementary model, motivated by issues in climate dynamics, is utilized to develop a self-contained example of stochastic mode reduction. Based on A. Majda's Aisenstadt lectures at the University of Montreal, the book is appropriate for both pure and applied mathematics graduate students, postdocs and faculty, as well as interested researchers in other scientific disciplines. No background in geophysical flows is required. About the authors: Andrew Majda is a member of the National Academy of Sciences and has received numerous honors and awards, including the National Academy of Science Prize in Applied Mathematics, the John von Neumann Prize of the Society of Industrial and Applied Mathematics, the Gibbs Prize of the American Mathematical Society, and the Medal of the College de France. In the past several years at the Courant Institute, Majda and a multi-disciplinary faculty have created the Center for Atmosphere Ocean Science to promote cross-disciplinary research with modern applied mathematics in climate modeling and prediction. R.V. Abramov is a young researcher; he received his PhD in 2002. M. J. Grote received his Ph.D. under Joseph B. Keller at Stanford University in 1995.

Stochastic Nonlinear Systems in Physics, Chemistry, and Biology

Author : L. Arnold
Publisher :
ISBN 13 :
Total Pages : 237 pages
Book Rating : 4.:/5 (897 download)

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Book Synopsis Stochastic Nonlinear Systems in Physics, Chemistry, and Biology by : L. Arnold

Download or read book Stochastic Nonlinear Systems in Physics, Chemistry, and Biology written by L. Arnold and published by . This book was released on 1981 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Nonlinear Stochastic Systems Theory and Applications to Physics

Author : G. Adomian
Publisher : Springer Science & Business Media
ISBN 13 : 902772525X
Total Pages : 248 pages
Book Rating : 4.0/5 (277 download)

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Book Synopsis Nonlinear Stochastic Systems Theory and Applications to Physics by : G. Adomian

Download or read book Nonlinear Stochastic Systems Theory and Applications to Physics written by G. Adomian and published by Springer Science & Business Media. This book was released on 1988-12-31 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approach your problems from the right end and begin with the answers. Then one day, perhaps you will find the final answer. "The Hermit Clad In Crane Feathers" In R. van Gullk's The Chinese Haze Hurders. It Isn't that they can't see the solution. It IS that they can't see the problem. G. K. Chesterton. The Scandal of Father Brown. "The POint of a Pin." Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of k now ledge of m athemat i cs and re I ated fie I ds does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, COding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And In addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely Integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the eXisting classificatIOn schemes.

Stochastic Systems with Time Delay

Author : Sarah A.M. Loos
Publisher : Springer Nature
ISBN 13 : 3030807711
Total Pages : 296 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Stochastic Systems with Time Delay by : Sarah A.M. Loos

Download or read book Stochastic Systems with Time Delay written by Sarah A.M. Loos and published by Springer Nature. This book was released on 2021-09-18 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: The nonequilibrium behavior of nanoscopic and biological systems, which are typically strongly fluctuating, is a major focus of current research. Lately, much progress has been made in understanding such systems from a thermodynamic perspective. However, new theoretical challenges emerge when the fluctuating system is additionally subject to time delay, e.g. due to the presence of feedback loops. This thesis advances this young and vibrant research field in several directions. The first main contribution concerns the probabilistic description of time-delayed systems; e.g. by introducing a versatile approximation scheme for nonlinear delay systems. Second, it reveals that delay can induce intriguing thermodynamic properties such as anomalous (reversed) heat flow. More generally, the thesis shows how to treat the thermodynamics of non-Markovian systems by introducing auxiliary variables. It turns out that delayed feedback is inextricably linked to nonreciprocal coupling, information flow, and to net energy input on the fluctuating level.

Instabilities and Nonequilibrium Structures VI

Author : E. Tirapegui
Publisher : Springer Science & Business Media
ISBN 13 : 9780792361299
Total Pages : 432 pages
Book Rating : 4.3/5 (612 download)

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Book Synopsis Instabilities and Nonequilibrium Structures VI by : E. Tirapegui

Download or read book Instabilities and Nonequilibrium Structures VI written by E. Tirapegui and published by Springer Science & Business Media. This book was released on 2000-01-31 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains two introductory papers on important topics of nonlinear physics. The first one, by M. San Miguel et al., refers to the effect of noise in nonequilibrium systems. The second, by M.E. Brachet, is a modern introduction to turbulence in fluids. The material can be very useful for short courses and is presented accordingly. The authors have made their texts self-contained. The volume also contains a selection of the invited seminars given at the Sixth International Workshop on Instabilities and Nonequilibrium Structures. Audience: This book should be of interest to graduate students and scientists interested in the fascinating problems of nonlinear physics.

A stochastic particle system

Author : Peter Dittrich
Publisher :
ISBN 13 :
Total Pages : 23 pages
Book Rating : 4.:/5 (46 download)

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Book Synopsis A stochastic particle system by : Peter Dittrich

Download or read book A stochastic particle system written by Peter Dittrich and published by . This book was released on 1987 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt:

High-dimensional Nonlinear Diffusion Stochastic Processes

Author : Yevgeny Mamontov
Publisher : World Scientific
ISBN 13 : 9814492590
Total Pages : 322 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis High-dimensional Nonlinear Diffusion Stochastic Processes by : Yevgeny Mamontov

Download or read book High-dimensional Nonlinear Diffusion Stochastic Processes written by Yevgeny Mamontov and published by World Scientific. This book was released on 2001-01-19 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the first one devoted to high-dimensional (or large-scale) diffusion stochastic processes (DSPs) with nonlinear coefficients. These processes are closely associated with nonlinear Ito's stochastic ordinary differential equations (ISODEs) and with the space-discretized versions of nonlinear Ito's stochastic partial integro-differential equations. The latter models include Ito's stochastic partial differential equations (ISPDEs).The book presents the new analytical treatment which can serve as the basis of a combined, analytical-numerical approach to greater computational efficiency in engineering problems. A few examples discussed in the book include: the high-dimensional DSPs described with the ISODE systems for semiconductor circuits; the nonrandom model for stochastic resonance (and other noise-induced phenomena) in high-dimensional DSPs; the modification of the well-known stochastic-adaptive-interpolation method by means of bases of function spaces; ISPDEs as the tool to consistently model non-Markov phenomena; the ISPDE system for semiconductor devices; the corresponding classification of charge transport in macroscale, mesoscale and microscale semiconductor regions based on the wave-diffusion equation; the fully time-domain nonlinear-friction aware analytical model for the velocity covariance of particle of uniform fluid, simple or dispersed; the specific time-domain analytics for the long, non-exponential “tails” of the velocity in case of the hard-sphere fluid.These examples demonstrate not only the capabilities of the developed techniques but also emphasize the usefulness of the complex-system-related approaches to solve some problems which have not been solved with the traditional, statistical-physics methods yet. From this veiwpoint, the book can be regarded as a kind of complement to such books as “Introduction to the Physics of Complex Systems. The Mesoscopic Approach to Fluctuations, Nonlinearity and Self-Organization” by Serra, Andretta, Compiani and Zanarini, “Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis” and “Statistical Physics: An Advanced Approach with Applications” by Honerkamp which deal with physics of complex systems, some of the corresponding analysis methods and an innovative, stochastics-based vision of theoretical physics.To facilitate the reading by nonmathematicians, the introductory chapter outlines the basic notions and results of theory of Markov and diffusion stochastic processes without involving the measure-theoretical approach. This presentation is based on probability densities commonly used in engineering and applied sciences.

Stochastic Game Strategies and their Applications

Author : Bor-Sen Chen
Publisher : CRC Press
ISBN 13 : 0429780508
Total Pages : 333 pages
Book Rating : 4.4/5 (297 download)

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Book Synopsis Stochastic Game Strategies and their Applications by : Bor-Sen Chen

Download or read book Stochastic Game Strategies and their Applications written by Bor-Sen Chen and published by CRC Press. This book was released on 2019-07-31 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: Game theory involves multi-person decision making and differential dynamic game theory has been widely applied to n-person decision making problems, which are stimulated by a vast number of applications. This book addresses the gap to discuss general stochastic n-person noncooperative and cooperative game theory with wide applications to control systems, signal processing systems, communication systems, managements, financial systems, and biological systems. H∞ game strategy, n-person cooperative and noncooperative game strategy are discussed for linear and nonlinear stochastic systems along with some computational algorithms developed to efficiently solve these game strategies.

Lectures on Dynamics of Stochastic Systems

Author : Valery I. Klyatskin
Publisher : Elsevier
ISBN 13 : 0123849675
Total Pages : 411 pages
Book Rating : 4.1/5 (238 download)

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Book Synopsis Lectures on Dynamics of Stochastic Systems by : Valery I. Klyatskin

Download or read book Lectures on Dynamics of Stochastic Systems written by Valery I. Klyatskin and published by Elsevier. This book was released on 2010-09-09 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. Models naturally render to statistical description, where random processes and fields express the input parameters and solutions. The fundamental problem of stochastic dynamics is to identify the essential characteristics of the system (its state and evolution), and relate those to the input parameters of the system and initial data. This book is a revised and more comprehensive version of Dynamics of Stochastic Systems. Part I provides an introduction to the topic. Part II is devoted to the general theory of statistical analysis of dynamic systems with fluctuating parameters described by differential and integral equations. Part III deals with the analysis of specific physical problems associated with coherent phenomena. A comprehensive update of Dynamics of Stochastic Systems Develops mathematical tools of stochastic analysis and applies them to a wide range of physical models of particles, fluids and waves Includes problems for the reader to solve

Noise in Nonlinear Dynamical Systems

Author : Frank Moss
Publisher : Cambridge University Press
ISBN 13 : 9780521118507
Total Pages : 372 pages
Book Rating : 4.1/5 (185 download)

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Book Synopsis Noise in Nonlinear Dynamical Systems by : Frank Moss

Download or read book Noise in Nonlinear Dynamical Systems written by Frank Moss and published by Cambridge University Press. This book was released on 2007-08-20 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nature is inherently noisy and nonlinear. It is noisy in the sense that all macroscopic systems are subject to the fluctuations of their environments and also to internal fluctuations. It is nonlinear in the sense that the restoring force on a system displaced from equilibrium does not usually vary linearly with the size of the displacement. To calculate the properties of stochastic (noisy) nonlinear systems is in general extremely difficult, although considerable progress has been made in the past. The three volumes that make up Noise in Nonlinear Dynamical Systems comprise a collection of specially written authoritative reviews on all aspects of the subject, representative of all the major practitioners in the field. The first volume deals with the basic theory of stochastic nonlinear systems. It includes an historical overview of the origins of the field, chapters covering some developed theoretical techniques for the study of coloured noise, and the first English-language translation of the landmark 1933 paper by Pontriagin, Andronov and Vitt.