The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions

The following sections are included:

• Preface

• Contents

• General Notations

The following sections are included:

• Introduction

• General Equation of Stochastic Nonlinear Dissipative Hamiltonian Dynamical Systems

• General Considerations on the Kinetic Energy

• Modeling of the Kinetic Energy and Mass Matrix

• Generalized Potential, Potential Energy Function and Lagrangian

• Generalized Momentum and Hamiltonian

• Energy Function

• Total Energy of the Dynamical System

• Model of the Hamiltonian

• Generalized Momentum and Generalized Velocity

• Modeling of the Nonconservative Generalized Force

• Quasi-Linear Term in the Generalized Velocity

• Quasi-Linear Damping Term

• Quasi-Linear Gyroscopic Term

• External Stochastic Excitation

• Stochastic Parametric Excitation Term

The following sections are included:

• Introduction

• Second-Order Equation Associated with the Dissipative Hamiltonian Dynamical System

• Examples for Multidimensional Nonlinear Oscillators Under External Random Excitations

• Examples for Multidimensional Nonlinear Oscillators Under Parametric and External Random Excitations

• Remark Concerning Modeling of Hysteresis Loops

• Complements on the Potential Energy Function

The following sections are included:

• Introduction

• Principles of Probability and Random Variables

• Random Variable with Values in ℝⁿ

• Second-Order Random Variable with Values in ℝⁿ and -Space

• Classical Examples of Probability Laws

• Transformations of Random Variables with Values in ℝⁿ

• Convergence of a Sequence of Random Variables with Values in ℝⁿ

The following sections are included:

• Introduction

• Basic Concepts and Definition of Classical Stochastic Processes

• Continuity of Stochastic Processes

• Markov Processes

• Second-Order Stochastic Processes with Values in ℝⁿ

• Spectral Analysis of Second-Order Mean-Square Stationary Stochastic Processes Mean-Square Continuous with Values in ℝⁿ

• Mean-Square Derivative of Second-Order Stochastic Processes with Values in ℝⁿ

• Mean-Square Integral of Second-Order Stochastic Processes with Values in ℝⁿ

The following sections are included:

• Introduction

• Integral Linear Transformation with Function-Kernel of Second-Order Stochastic Processes

• Linear Filtering and Spectral Analysis of Second-Order Mean-Square Continuous and Mean-Square Stationary Stochastic Processes

• Normalized Gaussian White Noise

• Linear Filtering of Normalized Gaussian White Noise

• Physically Realizable Processes

• Mean-Square Theory of Ordinary Linear Stochastic Differential Equations

The following sections are included:

• Introduction

• Definition of a Diffusion Process

• Fokker-Planck Equation for a Diffusion Process

• Stationary Markov Processes

• Stationary Diffusion Processes

• Calculation of Second-Order Quantities of a Stationary Diffusion Processes

The following sections are included:

• Introduction

• Need for Introducing Stochastic Integrals and Their Different Types

• Itô Stochastic Integral

• Itô Stochastic Differentials

• Itô Stochastic Differential Equation

• Stratonovich Stochastic Integral and Stratonovich Stochastic Differential Equation

The following sections are included:

• Introduction

• Markov Realization of Physically Realizable Processes

• Modeling with an SDE: Case of a Gaussian White Noise

• Modeling with an SDE: Case of a Gaussian Process That Has a Finite Dimensional Markov Realization

• Modeling with an SDE: Case of a Physically Realizable Stationary Gaussian Process

• Modeling with an SDE: Case of a Second-Order Stationary Gaussian Wide-Band Physical Noise

The following sections are included:

• Introduction

• Stochastic Model of External and Parametric Excitations

• Stochastic Modeling with a Stratonovich Stochastic Differential Equation

• Transformation of the SSDE into an Itô Stochastic Differential Equation

• On the Type of Stochastic Solution Sought

• Fokker-Planck Equation for the Stochastic Dynamical System

The following sections are included:

• Introduction

• Comments on the Methodology Followed in this Chapter

• Properties of the Diffusion Matrix

• Preliminary Mathematical Results for the Uniqueness Theorems

• Uniqueness of the Invariant Measure as a Solution of the Steady State FKP Equation

• Illustration of the Use of the Uniqueness Theorem

• Existence and Uniqueness of an Invariant Measure as a Solution of the Steady State FKP Equation

• Existence and Uniqueness of a Stationary Solution

The following sections are included:

• Introduction

• Dynamical Systems Considered

• Calculation of the Constant of Normalization

• Calculation of the Characteristic Function

• Calculation of Second-Order Moments

• Total Energy Stochastic Process

The following sections are included:

• Introduction

• Results Concerning Special Functions Useful for the Developments

• Multidimensional Linear Oscillator Subject to External and Parametric Random Excitations

• Formulas for Multidimensional Linear Oscillators Subject to External Random Excitations

The following sections are included:

• Introduction

• Multidimensional Nonlinear Oscillator with Inertial Nonlinearity

• Characteristic Function of r.v.

• Second-Order Moments of r.v.

• Monte Carlo Numerical Simulation

• Application

The following sections are included:

• Introduction

• Examples for Multidimensional Nonlinear Oscillators Subject to External Random Excitations

• Examples for Multidimensional Nonlinear Oscillators Subject to Parametric and External Random Excitations

The following sections are included:

• Introduction

• Unsteady FKP Equation

• Symplectic Matrix

• Symplectic Transformations

• Poisson Bracket with a Symplectic Change of Variables

• Symplectic Change of Variables in the FKP Equation

• Application: A Nonstationary Solution of the Unsteady FKP Equation Using Infinitesimal Symplectic Transformation

The following sections are included:

• References

• Index