| Chaotic systems appear in many different scientific disciplines, such as engineering, physics, chemistry, economics, biology and even political science as well as other social problems. When these systems are casually examined, some of them may repeat themselves periodically and others may present random-like chaotic behavior. The study of chaos will thus provide us with new tools to have a better understanding of such complex behavior. Since the chaotic behavior is rooted in nonlinear dynamic systems, the study of a system's nonlinearity becomes essential. Indeed, much effort has been devoted to this and the field is experiencing a rapid expansion in the past few decades.;The purpose of this dissertation is to study the behavior of nonlinear and chaotic systems. As a start, the influence of mass variation on mechanical systems will be investigated first. It will be shown that for this linear problem, a simple application of the WKB method will obtain a fairly good approximate solution. Next, a weakly nonlinear Coulomb damped system will be studied. A time varying change of variable, or the averaging method, will be employed to reduce the nonautonomous system to an autonomous one so that the response of a system with Coulomb damping can be studied in the plane described by the averaged equation.;For a highly nonlinear system, the oscillations of a pendulum with circular rotating support will be investigated. It will be shown that the system exhibits chaotic behavior. The lower bound of the chaotic regions can be determined by the Melnikov heteroclinic bifurcation criteria. The stable and unstable manifold as well as bifurcation diagram will be numerically simulated to verify the Melnikov criteria. The Liapunov spectrum will also be calculated in the forcing-frequency plane to determine the boundary of chaotic motion. It will be shown that when the system's damping is small, the Melnikov function will provide a good estimate of the heteroclinic bifurcation. For a fixed damping coefficient, the chaotic behavior will only appear for moderate values of forcing amplitude and frequency. An analysis of the chaotic region in the forcing-frequency plane will show that the effect of damping will compress the chaotic region, thus stabilize the motion. Finally, the well known Kolmogrov-Arnold-Moser (KAM) theorem will then be applied to explore the chaotic behavior of the nonintegrable Hamiltonian systems. |
