Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications
Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications

Author: Johan Grasman

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 224

ISBN-13: 3662038579

DOWNLOAD EBOOK

Asymptotic methods are of great importance for practical applications, especially in dealing with boundary value problems for small stochastic perturbations. This book deals with nonlinear dynamical systems perturbed by noise. It addresses problems in which noise leads to qualitative changes, escape from the attraction domain, or extinction in population dynamics. The most likely exit point and expected escape time are determined with singular perturbation methods for the corresponding Fokker-Planck equation. The authors indicate how their techniques relate to the Itô calculus applied to the Langevin equation. The book will be useful to researchers and graduate students.

Nonlinear Fokker-Planck Equations
Nonlinear Fokker-Planck Equations

Author: T.D. Frank

Publisher: Springer Science & Business Media

Published: 2005-01-07

Total Pages: 414

ISBN-13: 3540212647

DOWNLOAD EBOOK

Centered around the natural phenomena of relaxations and fluctuations, this monograph provides readers with a solid foundation in the linear and nonlinear Fokker-Planck equations that describe the evolution of distribution functions. It emphasizes principles and notions of the theory (e.g. self-organization, stochastic feedback, free energy, and Markov processes), while also illustrating the wide applicability (e.g. collective behavior, multistability, front dynamics, and quantum particle distribution). The focus is on relaxation processes in homogeneous many-body systems describable by nonlinear Fokker-Planck equations. Also treated are Langevin equations and correlation functions. Since these phenomena are exhibited by a diverse spectrum of systems, examples and applications span the fields of physics, biology and neurophysics, mathematics, psychology, and biomechanics.

Fokker-Planck-Kolmogorov Equations
Fokker-Planck-Kolmogorov Equations

Author: Vladimir I. Bogachev

Publisher: American Mathematical Soc.

Published: 2015-12-17

Total Pages: 495

ISBN-13: 1470425580

DOWNLOAD EBOOK

This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields
An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields

Author: Marco Bramanti

Publisher: Springer Science & Business Media

Published: 2013-11-20

Total Pages: 157

ISBN-13: 3319020870

DOWNLOAD EBOOK

​Hörmander's operators are an important class of linear elliptic-parabolic degenerate partial differential operators with smooth coefficients, which have been intensively studied since the late 1960s and are still an active field of research. This text provides the reader with a general overview of the field, with its motivations and problems, some of its fundamental results, and some recent lines of development.

Encyclopedia of Nonlinear Science
Encyclopedia of Nonlinear Science

Author: Alwyn Scott

Publisher: Routledge

Published: 2006-05-17

Total Pages: 2881

ISBN-13: 1135455570

DOWNLOAD EBOOK

In 438 alphabetically-arranged essays, this work provides a useful overview of the core mathematical background for nonlinear science, as well as its applications to key problems in ecology and biological systems, chemical reaction-diffusion problems, geophysics, economics, electrical and mechanical oscillations in engineering systems, lasers and nonlinear optics, fluid mechanics and turbulence, and condensed matter physics, among others.

An Introduction to Stochastic Dynamics
An Introduction to Stochastic Dynamics

Author: Jinqiao Duan

Publisher: Cambridge University Press

Published: 2015-04-13

Total Pages: 313

ISBN-13: 1107075394

DOWNLOAD EBOOK

An accessible introduction for applied mathematicians to concepts and techniques for describing, quantifying, and understanding dynamics under uncertainty.

Historical Developments in Singular Perturbations
Historical Developments in Singular Perturbations

Author: Robert E. O'Malley

Publisher: Springer

Published: 2014-11-19

Total Pages: 263

ISBN-13: 3319119249

DOWNLOAD EBOOK

This engaging text describes the development of singular perturbations, including its history, accumulating literature, and its current status. While the approach of the text is sophisticated, the literature is accessible to a broad audience. A particularly valuable bonus are the historical remarks. These remarks are found throughout the manuscript. They demonstrate the growth of mathematical thinking on this topic by engineers and mathematicians. The book focuses on detailing how the various methods are to be applied. These are illustrated by a number and variety of examples. Readers are expected to have a working knowledge of elementary ordinary differential equations, including some familiarity with power series techniques, and of some advanced calculus. Dr. O'Malley has written a number of books on singular perturbations. This book has developed from many of his works in the field of perturbation theory.

Modeling Aggregate Behavior and Fluctuations in Economics
Modeling Aggregate Behavior and Fluctuations in Economics

Author: Masanao Aoki

Publisher: Cambridge University Press

Published: 2001-12-20

Total Pages: 283

ISBN-13: 1139431870

DOWNLOAD EBOOK

This book has two components: stochastic dynamics and stochastic random combinatorial analysis. The first discusses evolving patterns of interactions of a large but finite number of agents of several types. Changes of agent types or their choices or decisions over time are formulated as jump Markov processes with suitably specified transition rates: optimisations by agents make these rates generally endogenous. Probabilistic equilibrium selection rules are also discussed, together with the distributions of relative sizes of the bases of attraction. As the number of agents approaches infinity, we recover deterministic macroeconomic relations of more conventional economic models. The second component analyses how agents form clusters of various sizes. This has applications for discussing sizes or shares of markets by various agents which involve some combinatorial analysis patterned after the population genetics literature. These are shown to be relevant to distributions of returns to assets, volatility of returns, and power laws.

Resonance
Resonance

Author: Jan Awrejcewicz

Publisher: BoD – Books on Demand

Published: 2017-11-29

Total Pages: 256

ISBN-13: 953513633X

DOWNLOAD EBOOK

Resonance is a common phenomenon, which is observed both in nature and in numerous devices and structures. It occurs in literally all types of vibrations. To mention just a few examples, acoustic, mechanical, or electromagnetic resonance can be distinguished. In the present book, 12 chapters dealing with different aspects of resonance phenomena have been presented.