James Murdock

Normal forms and unfoldings for local dynamical systems

Preface.- 1. Two Examples.- 2. The splitting problem for linear operators.- 3. Linear Normal Forms.- 4. Nonlinear Normal Forms.- 5. Geometrical Structures in Normal Forms.- 6. Selected Topics in Local Bifurcation Theory.- Appendix A: Rings.- Appendix B: Modules.- Appendix C: Format 2b: Generated Recursive (Hori).- Appendix D: Format 2c: Generated Recursive (Deprit).- Appendix E: On Some Algorithms in Linear Algebra.- Bibliography.- Index.

Qualitative Theory of Nonlinear Resonance by Averaging and Dynamical Systems Methods

This paper is a self-contained exposition of the theory of averaging for periodic and quasiperiodic systems, with the emphasis being on the author’s research (part of it joint work with Clark Robinson) on qualitative aspects of nonlinear resonance. Many topics in averaging theory are not covered, among them: averaging for systems more general than quasiperiodic; relations between averaging and multiple time-scale methods; Eckhaus’s approach to averaging; combinations of averaging with matching of asymptotic expansions. The principal question which is addressed is: when does averaging (to first or higher order) lead to an accurate qualitative description of the solutions of the original (unaveraged) equation? By qualitative description we mean both locally (existence and stability of certain invariant sets such as periodic orbits, or almost invariant ‘lingering’ and globally (connecting orbits between invariant sets, or claims that in certain large regions all orbits drift in a certain direction).

Qualitative dynamics from asymptotic expansions: Local theory

Perturbation problems in nonlinear oscillations have been studied from two points of view. One may seek asymptotic expansions of the solution in a small parameter, and give error estimates valid on some interval of time, generally an expanding interval of the form 0 ,( t .< L/C for some positive constants L, a. Or one may attempt to prove existence of periodic solutions for all time, and investigate their stability. These approaches overlap, in that a well-known theorem (stated below) enables one to deduce existence and stability properties of periodic solutions from the first term of an asymptotic expansion when that first term satisfies a certain condition. The purpose of the present paper is to extend the theory to provide stability criteria for periodic solutions which are known to exist but for which the first term of the asymptotic expansion does not determine stability. Some of our results have been announced in [5], which also poses a larger problem about the topological conjugacy of a map with its asymptotic approximations; one aspect of this problem is addressed in Section 4 below. Both problems arose from [4], in which the qualitative features of a certain spin/orbit resonance problem are shown to depend upon second-order terms in the asymptotic expansion. Section 5 of the present paper contains an additional application, a system of two coupled Duffing equations. More specifically, we shall consider a system of familiar form

Validity of the multiple scale method for very long intervals

AbstractFor a class of nonlinear oscillation problems containing a small parameter, it is known that a two-scale method using timest and ɛt gives results valid to any desired order for time

A Shadowing Approach to Passage Through Resonance

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A Note on the Asymptotic Expansion of Eigenvalues

(1/ɛ). We ask when results can be obtained which are valid for

Block Stanley Decompositions I. Elementary and gnomon decompositions

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