Contraction Maps and Equivalent Linearization

The method of equivalent linearization is a most valuable technique to investigate nonlinear phenomena, particularly nonlinear oscillations. It has its roots in the method of Krvlov and Bogoliubov and is related to for equivalent to, depending on the specific definitions) the method of harmonic balance, Galerkin's method, and the describing function method used by control engineers. The purpose of the present study is to develop a new technique for investigating the method of equivalent linearization. We shall be primarily concerned with the following question: If the method of equivalent linearization indicates the existence of a periodic solution .r,,, is there actually a periodic solution near .r,,? To answer this question we first introduce a convenient modification of the contraction mapping fixed point theorem which is actually more general than just applicable to the question posed above.f We apply * Taken from a dissertation submitted to the Faculty of I lie Polytechnic Institute of Brooklyn in partial fulfillment of the requirements for the degree of Doctor of Philosophy (System Science), 1967. t Appendix A contains some reading suggestions for engineers interested in this work but who are not familiar with the mathematics used. 2405