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

Abstract

Stochastic Canonical Equation of Multidimensional Nonlinear Dissipative Hamiltonian Dynamical Systems Fundamental Examples of Nonlinear Dynamical Systems and Associated Second Order Equation Brief Review of Probability and Random Variables Probabilistic Tools I. Classical Stochastic Processes Probabilistic Tools II. Mean-Square theory of linear integral transformations and of linear Differential Equations Probabilistic Tools III. Diffusion Processes and Fokker-Planck Equation Probabilistic Tools IV. Stochastic Integrals and Stochastic Differential Equations Stochastic Modelling with Stochastic Differential Equations FKP Equation for the Dissipative Hamiltonian Dynamical Systems Stationary Response of Dissipative Dynamical Systems, Existence and Uniqueness, Explicit Solution of an Invariant Measure Complements for the Normalization Condition, Characteristic Function and Moments of the Invariant Measure Application II. Multidimensional Linear Oscillators Subjected to External and Parametric Random Excitations Application III. Multidimensional Nonlinear Oscillators with Inertial Nonlinearity Subjected to External Random Excitations Application Ill. Multidimensional Nonlinear Oscillators Subjected to External and Parametric Random Excitations Symplectic Change of Variables in the Multidimensional Unsteady FKP Equation.