The well known Fokker-Plank-Kolmogorov Equation Method has been developed to study random vibration in systems with hysteresis that often described by the stochastic integro-differential equations or differential equations with delay. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 375 - 384 Special Issue Dedicated to the Memory of Prof. Nguyen Van Dao AN APPROACH TO STUDY VIBRATION IN STOCHASTIC SYSTEMS BASED ON THE ASYMPTOTIC METHOD NGUYEN TIEN KHIEM Institute of Mechanics, VAST Abstract. The well known Fokker-Plank-Kolmogorov Equation Method has been developed to study random vibration in systems with hysteresis that often described by the stochastic integro-differential equations or differenti al equations with delay. 1. INTRODUCTION The asymptotic method is well-known as one of fundamental methods in study of weakly nonlinear systems. They have come to be effectively used for the stochastic systems via theory of diffusional processes [1]. However, many processes in practice arc not diffusional so that for those the Fokker-Plank-Kolmogorov Equation (FPKE) Method is not applicable. In the case, something like the FPKE for non-diffusional processes has been needed. Stratonovich [2] is the first who constructed approximately an equation for probability density function for arbitrary stochastic process based on its asymptotic expansion. It was in fifties of the last millennium . Later, in 1966, Khasminskii [3] had deeper studied the problem in his paper published in Journal of Theory of Probability and Its Application (in Russian). Since 1965, Professor Nguyen Van Dao [4] had published a paper in Vietnamese dealt with an application of the Stratonovich's equation to study random vibration in a weakly nonlinear system. The author of the paper in 1979, after reading Van Dao's work, has came to the idea of developments of the Stratonovich's method to study the processes given by stochastic integro-differential equations. The first result of the author were published in Ukrainian Mathematical Journal in 1983 [5]. This problem were further developed in the author's doctor of science dissertation published in 1991 [8] at the Institute ·of Mathematics, Ukrainian Academy of .
