Stochastic analysis of time-variant nonlinear dynamic systems. Part 1: The Fokker-Planck-Kolmogorov equation approach in stochastic mechanics

The probabilistic description and analysis of the response of time-invariant nonlinear dynamic systems driven by stochastic processes is usually treated by means of evaluation of statistical moments and cumulants of the response. The background of these methods is the Fokker-Planck-Kolmogorov (FPK) equation for a probability density function or the Pugachev equation for a characteristic function, respectively. The exact solutions of these equations are obtained only for isolated cases. For engineering probabilistic analysis of a complex nonlinear systems, different mixed (hybrid) methods in these cases are used. In this study a ‘benchmark’ solution is obtained on the basis of the FPK equation in conjunction with the method of statistical moments for nonlinear mechanical system with colored parametric excitations. In Part 1 (this part), an exact solution of FPK equation on the basis of asymptotic analysis of nonlinear dynamic behavior of parametric excitation system is discussed. In Parts 2 and 3, applications of this method to stochasticity and stability analysis of nonlinear time-variant systems are considered. A comparison with the accuracy of different statistical methods is discussed. In Parts 4 and 5, a method of stochastic analysis of relativistic and quantum dynamic systems is described on the basis of a generalized stochastic Hamilton-Jacobi equations on a differential manifold as Riemanian geometry. This involves the task of relativistic navigation and dissipative quantum models of a nonlinear parametric oscillator in the presence of stochastic excitations on a differential manifold with different metric tensors of the space-time continuum.