doi:10.3850/978-981-08-7619-7_P042

ABSTRACT

Starting with the basic thought of probability density evolution, the new advances of probability density evolution method for nonlinear systems are presented. The principle of preservation of probability is a fundamental property of stochastic systems. This provides a unified basis for probability density evolution equations for different types of stochastic systems. By clarifying the physical meaning of the principle of preservation of probability from the state space description and the random event description, respectively, the logic position of the existing probability density evolution equations can be specified. What's more, a new family of generalized density evolution equation could be derived. This provides a new possibility of tackling nonlinear stochastic systems, which can be referred to as probability density evolution method (PDEM). The present paper will outline the advances of the PDEM, including the derivation of the generalized density evolution equations, the numerical algorithms to solve the generalized density evolution equation and the open problems. Some applications to engineering practice are illustrated.

Keywords: Probability density evolution method, Quasi-symmetric point method, Fluctuation analysis, Stochastic structure.

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