Introduction and Overview

Abstract

Nonlinear systems appear in many scientific disciplines, including engineering, physics, chemistry, biology, economics, and demography. Methods of analysis of nonlinear systems, which can provide a good understanding of their behavior, have, therefore, wide applications. In the classical mathematical analysis of nonlinear systems (Lefschetz [1957], Bogoliubov and Mitropolsky [1961], Minorsky [1962], Cesari [1963], Hayashi [1964], Andronov et al. [1973], and Nayfeh and Mook [1979]), once the equation of motion for a system has been formulated, one usually tries first to locate all the possible equilibrium states and periodic motions of the system. Second, one studies the stability characteristics of these solutions. As the third task, one may also study how these equilibrium states and periodic solutions evolve as the system parameters are changed. This leads to various theories of bifurcation (Marsden and McCracken [1976], boss and Joseph [1980], and Chow and Hale [1982]). Each of these tasks can be a very substantial and difficult one. In recent years the existence and importance of strange attractors have been recognized, and one now attempts to include these more exotic system responses, if they exist, in the first three types of investigations (Lichtenberg and Lieberman [1982] and Guckenheimer and Holmes [1983]).