On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations

Abstract

Abstract This article consists of Chapter 2 of the author's 1954 Dissertation bearing the same title and published as Courant Institute report IMM-NYU 333, available in full on the author's personal web site. Chapter 2 consists of the first complete proof of what has come to be known as the Neimark-Sacker bifurcation theorem. It includes the reduction to normal form using weighted monomials which precludes using the Center Manifold theorem which was not known to the author and was published the same year as the dissertation and called "a Reduction Principle" by V. Pliss and later named Center Manifold theorem by A. Kelley (see previous article for citations). After reduction to normal form, the resulting functional equations for the bifurcating invariant curve is solved in detail. Stand-alone decimals provide solutions of linear functional equations, a-priori estimates and interpolation inequalities between derivatives in the sup norm. Keywords: Neimark-Sacker bifurcation theoremnormal form weighted monomialsfunctional equationsKeywords: 37C05 Acknowledgements This work was sponsored by the Office of Naval Research, Contract Nonr-285(46), the National Science Foundation and the Sloan Foundation. Reproduction in whole or in part is permitted for any purpose of the United States Government. Notes 2 Lemmas 1, 2 and 3 are given later. 1 By the smoothness assumption made after equation (Equation4), R k is well defined provided k ≤ 4.