Perturbation Methods in Non Linear Systems This volume is intended to provide a comprehensive treatment of recent developments in methods of perturbation for nonlinear systems of ordinary differ­ ential equations. In this respect it appears to be a unique work. The main goal is to describe perturbation techniques discuss their ad­ vantages and limitations and give some examples. The approach is founded on analytical and numerical methods of nonlinear mechanics. Attention has been given to the extension of methods to high orders of approximation required now by the increased accuracy of measurements in all fields of science and technology. The main theorems relevant to each perturbation technique are outlined but they only provide a foundation and are not the objective of these notes. Each chapter concludes with a detailed survey of the pertinent literature supplemental information and more examples to complement the text when necessary for better comprehension. The references are intended to provide a guide for background information and for the reader who wishes to analyze any particular point in more detail. The main sources referenced are in the fields of differential equations nonlinear oscillations and celestial mechanics. Thanks are due to Katherine MacDougall and Sandra Spinacci for their patience and competence in typing these notes. Partial support from the Mathematics Program of the Office of Naval Research is gratefully acknowledged. I. Canonical Transformation Theory and Generalizations 1. Introduction 2. Canonical Transformations 3. Hamilton Jacobi Equation. Generalizations 4. Lie Series and Lie Transforms 5. Lie Transform Depending on a Parameter 6. Equivalence Relations 7. General Transformations Induced by Lie Series Notes References II. Perturbation Methods for Hamiltonian Systems. Generalizations 1. Introduction 2. Convergence of a Classical Method of Iteration 3. Secular Terms. Lindstedt’s Device 4. Poincaré’s Method (Lindstedt’s Method) 5. Fast and Slow Variables 6. Generalization of the Averaging Procedure Birkoff’s Normalization and Adelphic Integrals 7. The Solution of Poincaré’s Problem in Poisson’s Parentheses. Elimination of Secular Terms from Adelphic Integrals 8. Perturbation Techniques Based on Lie Transforms 9. Perturbation Methods of Non Hamiltonian Systems Based on Lie Transforms Notes References III. Perturbations of Integrable Systemsl 1. Motion of an Integrable System 2. Perturbations of an Integral System 3. Degenerate Systems 4. Perturbed Linear Oscillations 5. Linear Periodic Perturbations Notes References IV. Perturbations of Area Preserving Mappings 1. Preliminary Considerations 2. Regions of Motion. Perturbation of a Truncated Birkoff’s Normal Form 3.Moser’s Theorem 4. System with n Degree of Freedom 5. Degenerate Systems Notes References V. Resonance 1. Introduction 2. Motion in the Neighborhood of an Equilibrium Point 3. Solution by Formal Series28l 4. Equivalence with the Problem of Perturbation of a Linear System 5. Nonlinear Resonance 6. Asymptotic Expansion to Any Order 7. Extended Theory and the Ideal Resonance Problem 8. Several Degrees of Freedom 9. Coupling of Two Harmonic Oscillators Notes References Appendix. Remarks Some Open Questions and Research Topics References.