Author: C Soize

Publisher: World Scientific

Published: 1994-05-16

Total Pages: 340

ISBN-13: 9814502022

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This is an analysis of multidimensional nonlinear dissipative Hamiltonian dynamical systems subjected to parametric and external stochastic excitations by the Fokker-Planck equation method. The author answers three types of questions concerning this area. First, what probabilistic tools are necessary for constructing a stochastic model and deriving the FKP equation for nonlinear stochastic dynamical systems? Secondly, what are the main results concerning the existence and uniqueness of an invariant measure and its associated stationary response? Finally, what is the class of multidimensional dynamical systems that have an explicit invariant measure and what are the fundamental examples for applications? Contents:Stochastic Canonical Equation of Multidimensional Nonlinear Dissipative Hamiltonian Dynamical SystemsFundamental Examples of Nonlinear Dynamical Systems and Associated Second-Order EquationBrief Review of Probability and Random VariablesProbabilistic Tools I. Classical Stochastic ProcessesProbabilistic Tools II. Mean-Square Theory of Linear Integral Transformations and of Linear Differential EquationsProbabilistic Tools III. Diffusion Processes and Fokker-Planck EquationProbabilistic Tools IV. Stochastic Integrals and Stochastic Differential EquationsStochastic Modeling with Stochastic Differential EquationsFKP Equation for the Dissipative Hamiltonian Dynamical SystemsStationary Response of Dissipative Dynamical Systems, Existence and Uniqueness, Explicit Solution of an Invariant MeasureComplements for the Normalization Condition, Characteristic Function and Moments of the Invariant MeasureApplication I. Multidimensional Linear Oscillators Subject to External and Parametric Random ExcitationsApplication II. Multidimensional Nonlinear Oscillators with Inertial Nonlinearity Subject to External Random ExcitationsApplication III. Multidimensional Nonlinear Oscillators Subject to External and Parametric Random ExcitationsSymplectic Change of Variables in the Multidimensional Unsteady FKP Equation ReferencesIndex Readership: Applied mathematicians. keywords:Fokker–Planck Equation;Stochastic Dynamics;Diffusion Process;Stochastic Methods;Random Vibration;Random Process;Stochastic Differential Equation;Hamiltonian Dynamical System;Stochastic Process;Probabilistic Methods “This is a timely volume summarizing and unifying 30 years of search for explicit solutions of (stationary) FPE's. New articles in this area, which continue to appear, have to explain in which way they extend Soize's presentation. As such, this book is a useful reference for the random vibrations community.” Mathematics Abstracts

Download or read book The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions written by C Soize and published by World Scientific. This book was released on 1994-05-16 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an analysis of multidimensional nonlinear dissipative Hamiltonian dynamical systems subjected to parametric and external stochastic excitations by the Fokker-Planck equation method. The author answers three types of questions concerning this area. First, what probabilistic tools are necessary for constructing a stochastic model and deriving the FKP equation for nonlinear stochastic dynamical systems? Secondly, what are the main results concerning the existence and uniqueness of an invariant measure and its associated stationary response? Finally, what is the class of multidimensional dynamical systems that have an explicit invariant measure and what are the fundamental examples for applications? Contents:Stochastic Canonical Equation of Multidimensional Nonlinear Dissipative Hamiltonian Dynamical SystemsFundamental Examples of Nonlinear Dynamical Systems and Associated Second-Order EquationBrief Review of Probability and Random VariablesProbabilistic Tools I. Classical Stochastic ProcessesProbabilistic Tools II. Mean-Square Theory of Linear Integral Transformations and of Linear Differential EquationsProbabilistic Tools III. Diffusion Processes and Fokker-Planck EquationProbabilistic Tools IV. Stochastic Integrals and Stochastic Differential EquationsStochastic Modeling with Stochastic Differential EquationsFKP Equation for the Dissipative Hamiltonian Dynamical SystemsStationary Response of Dissipative Dynamical Systems, Existence and Uniqueness, Explicit Solution of an Invariant MeasureComplements for the Normalization Condition, Characteristic Function and Moments of the Invariant MeasureApplication I. Multidimensional Linear Oscillators Subject to External and Parametric Random ExcitationsApplication II. Multidimensional Nonlinear Oscillators with Inertial Nonlinearity Subject to External Random ExcitationsApplication III. Multidimensional Nonlinear Oscillators Subject to External and Parametric Random ExcitationsSymplectic Change of Variables in the Multidimensional Unsteady FKP Equation ReferencesIndex Readership: Applied mathematicians. keywords:Fokker–Planck Equation;Stochastic Dynamics;Diffusion Process;Stochastic Methods;Random Vibration;Random Process;Stochastic Differential Equation;Hamiltonian Dynamical System;Stochastic Process;Probabilistic Methods “This is a timely volume summarizing and unifying 30 years of search for explicit solutions of (stationary) FPE's. New articles in this area, which continue to appear, have to explain in which way they extend Soize's presentation. As such, this book is a useful reference for the random vibrations community.” Mathematics Abstracts