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A research project on the topics of differential equations and operator equations, generalized functions, and neutron transport and diffusion systems is briefly summarized. (Author)

Topics discussed in the paper include perturbations and integral manifolds for systems of nonlinear differential equations; evolution systems and conditional stability; neutron transport and radiative transfer; and generalized functions. (Author)

The report summarizes the research efforts under this project and includes discussions of (1) periodic solutions, (2) a function equation, (3) infinitely differentiable functions, (4) equivalence problems, and (5) operations and distributions. (Author)

Research activity on mathematical analysis of resonant oscillations is summarized. Project publications and technical reports are listed.

The research conducted was concerned with resonant-type oscillations exhibited by certain dynamical systems. The main objective was to establish rigorous mathematical techniques which focus attention upon the interesting phenomena associated with resonant operating conditions. The main results of the research were reported in several papers prepared for publication in the open literature. (To illustrate the results, a discussion of the resonance phenomena for the familiar Duffing equation is appended to this report.) (Author)

Periodic solutions of nonlinear oscillators are investigated using elementary geometrical and analytical arguments. The existence of normal trajectories is established using geometrical arguments and the usual periodicity condition is rephrased so as to permit the use of simple continuity arguments in establishing the periodicity. Some preliminary concepts, as well as a basic theorem and three corollaries concerning the existence of periodic solutions, are introduced. Existence of normal trajectories is discussed. Estimates of elapsed times and the standard existence problem are considered. The general case is discussed and an examination of higher dimensional systems is presented. Certain connections with special concepts which have been employed by other investigators are established. Using these concepts the main results are included in a General Theorem which covers the higher dimensional cases. Throughout the Duffing equation serves as a concrete example. (Author)

The Duffing equation d2xdt2 + n2x - x3 = Focos t is investigated under various conditions. Asymptotic series solutions are discussed in detail. (Author)