In this study, the jump phenomena and the chaotic motion are investigated. Firstly, it is found that the jump phenomenon occurs under the harmonic excitation by means of computer simulation, and that it occurs under the narrow band random excitation by the averaging method, too. And the conditions for the occurrence of jump phenomena, the probability density function of dynamic response and the resonance curves are shown, respectively. Furthermore, the phenomena are considered from the viewpoint of mean square value, where the statistical linearization method and Liapunov matrix equation are used. Consequently, it seems that the method of equivalent linearization for smooth hysteretic systems under random excitation is versatile and has a great potential in the analysis of hysteretic system, but that it is impertinent to the research of jump phenomena. Secondly, the occurrence of chaotic motion is disscussed by means of the method of mapping and the averaging method. As the results, it is found that the chaotic motion occurs under the various parameters.