A New Methodology for the Analysis of Periodic Systems

A new numerical-analytical method for the combined global-localanalysis of nonlinear periodic systems referred to as an ExpandedPoint Mapping (EPM) is presented. This methodology combines thecell to cell mapping and point mapping methods toinvestigate the basins of attraction and stability characteristics ofequilibrium points and periodic solutions of nonlinear periodicsystems. The proposed method is applicable to multi-degrees-of-freedomsystems, multi-parameter systems, and allows analytical studies oflocal stability characteristics of steady state solutions. Inaddition, the EPM approach allows the study of stabilitycharacteristics as function of system parameters to obtain analyticalconditions for bifurcation. In the paper, the theoretical basis forthe EPM method is formulated and a procedure for the analysis ofnonlinear dynamical systems is presented. Analysis of a pendulum witha periodically excited support in the plane is used to illustrate themethod. The results demonstrate the efficiency and accuracy of theproposed approach in analyzing nonlinear periodic systems.