Identifying explicit expression of response probability density of nonlinear stochastic system: Information-theoretic method

摘要

For a general nonlinear stochastic system, the response probability density possesses crucial significance for response evaluation and reliability design. The existing literatures derive the response probability density by accurately or approximately solving the Fokker-Planck- Kolmogorov equation, by searching for the equivalent system according to some criterion and approximating the probability density of the original system by that of the equivalent one, or by numerically solving the original equation of motion and giving discrete information statistically. Herein, a novel procedure is established to identify the explicit expression of the stationary response probability density of a general nonlinear system directly from the discrete data of samples. The stationary probability density is expressed as an exponent form with the exponential power being linear combination of base functions elaborately selected. The Shannon information entropy is then arranged as a nonlinear function of the statistical moments of base functions which are evaluated by the discrete data of samples. The undetermined coefficients of base functions are finally determined by minimizing the Shannon information entropy. Three typical examples, i.e., Duffing oscillator, van der Pol system and a frictional system (as representatives of smooth systems, electric systems and non-smooth systems, respectively), are investigated to illustrate the accuracy of this procedure, the robustness to data noise, the insensitivity to sampling interval and strength of nonlinearity, and the low requirement on the amount of data.