Nonlinear Dynamics 6: 179-191, 1994. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands. Chaotic Motion and Stochastic Excitation E BONTEMPI 1 and F. CASCIATI 2 l Ph.D. Student, Structural Engineering Course, Polytechnic of Milan, Italy; z Professor, Department of Structural Mechanics, Universi~ of Pavia, Italy (Received: 4 May 1992; accepted: 27 January 1993) Abstract. In this paper, one considers dynamic chaotic/stochastic systems and the associated Gibbs set. The behavior of these sets leads one to characterize the systems and to calculate the values of the Kolmogoroventropy. The ultimate objective is to extend an approach typical of the statistical mechanics to the analysis of systems of the mechanical engineering. Key words: Chaotic motion, Gibbs set, Kolmogoroventropy, stochastic excitation, Introduction Several nonlinear systems present a behavior which is deeply influenced by initial conditions. This sensitivity to initial conditions is exalted by the passing of time. It means that two systems, starting side by side, will follow trajectories which are as different as time is passing. If a system is sensitive to the initial conditions there is no need anymore to know the single track, but it is more important to consider many systems which start one near to the other [1 1, 16, 17]. In this way we can simulate the dispersion of the measurement of the initial status of the system and to build up a statistics of the behavior on them. Such a set is called the Gibbs set, widely employed in statistical mechanics [8]. Studying the behavior of this set, it is possible to understand the characteristics of the system. It is possible to measure the divergence between the systems of the Gibbs set both by Lyapunov exponents and Kolmogorov entropy. For the first ones it is already possible to find consolidated good numerical techniques [4], while for the second one there are still some computational, subtle problems [5, 6, 7]. In this paper, one considers dynamic chaotic/stochastic systems and the associated Gibbs sets. The behavior of these sets leads to characterize the systems and to calculate the values of the Kolmogorov entropy. The information arriving from the Kolmogorov entropy is compared with the one achievable by the Lyapunov exponents estimated. The ultimate objective is to extend an approach typical of the statistical mechanics to the analysis of systems of the mechanical engineering. 1. Governing Relations 1.1. DYNAMIC SYSTEMS Considering a dynamic system, it can evolve in time either in a continuous way, driven by the differential equation: