A global study of nonlinear dynamical systems by a combined numerical-analytical approach

A new numerical-analytical method for the combined global-local analysis of nonlinear periodic systems referred to as an Expanded Point Mapping (EPM) is presented. This methodology combines the cell to cell mapping and point mapping methods to investigate the basins of attraction and stability characteristics of equilibrium points and periodic solutions of nonlinear periodic systems. The proposed method is applicable to multi-degree of freedom systems, multi-parameter systems, and allows analytical studies of local stability characteristics of steady state solutions. In addition, the EPM approach allows the study of stability characteristics as a function of system parameters to obtain analytical conditions for bifurcation. In this dissertation, the theoretical basis for the EPM method is formulated and a procedure for the analysis of nonlinear dynamical systems is presented. Two analytical studies are performed. The first analysis consists of a study of a pendulum with a vertical moving support. Next a more general investigation of a pendulum with a periodically excited support in the plane is used to illustrate the method. The results demonstrate the efficiency and accuracy of the proposed approach in analyzing nonlinear periodic systems.