Averaging theory offers constructive tools for the analysis of oscillatory ordinary differential equations with time-discontinuous right-hand sides, if requirements of solution differentiability are relaxed. This theory forms a rigorous framework for evaluating, refining, and extending heuristic averaged models used in power electronics, and provides direct techniques for recovering oscillatory effects such as ripple from the averaged model. An overview of this theory is presented, and DC-DC converters are analyzed as examples. >

On the use of averaging for the analysis of power electronic systems

(1974). PERIODIC OPERATION OF CHEMICAL REACTORS: A REVIEW. Chemical Engineering Communications: Vol. 1, No. 3, pp. 111-124.

Periodic operation of chemical reactors: a review

Conditions are given under which an integral manifold ("slow manifold") exists for a nonlinear system representing a broad class of adaptive algorithms. The parameter update equation restricted to the manifold is an exact description of the slow adaptation process which can be approximately analyzed by averaging. Stability properties of the slow motion in the manifold are extended off the manifold by a set of conditions under which the manifold is shown to be attractive. This two-time-scale analysis provides a geometrical visualization of instability phenomena observed in adaptive systems and generalizes earlier local stability results.

Integral manifolds of slow adaptation

In the first part of this work [1], the criteria for the existence of stabilizing parametric oscillations have been derived. In the present paper, the problem of choosing the stabilizing vibrations is addressed and the properties of transient behavior of vibrationally controlled nonlinear systems are analyzed.

Vibrational control of nonlinear systems: Vibrational controllability and transient behavior

Pat-hological dynamic behavior of forced periodic chemical processes

where x is an n vector, e the same small parameter as in (1.1) and / an n vector function periodic or almost periodic in t. The results in the literature, valid for all time, for either systems (1.1) or (1.2), are concerned with the existence and stability of periodic or almost periodic solutions and usually a technique for obtaining approximations to these solutions is also given. See, for example, the books by Hale [1], Coddington and Levinson [2], and Bogoliuboff and Mitropolskiy [3]. Bogoliuboff and Mitropolskiy [4] also discuss solutions other than periodic or almost periodic solutions, for systems of the type (1.2), by comparing the solution of (1.2) with the solution of the associated "averaged equation", but their results are valid only for a finite interval of time. Our study is concerned with the solutions of (1.2), valid for all time, but which are not restricted to only periodic or almost periodic solutions. Our main result gives sufficient conditions for the solutions of (1.2) to approach stable almost periodic solutions of (1.2) as time goes to infinity. As in [4], we also compare the solutions of (1.2) with the solutions of the associated "averaged equations" but in our case asymptotically valid estimates for solutions of (1.2) are given for all time. On the basis of this main result, it is possible to give, with the aid of a theorem of LaSalle [5], results regarding the domains of attraction of almost periodic solutions of (1.2). These results regarding the domains of attraction are generalizations of the results for periodic systems given by Loud and Sethna in [6]. In [6] explicit estimates are given for the domains of attraction for second order systems. In the present study we give explicit estimates for the domains of attraction of almost periodic solutions of dynamical systems of order greater then two but in the special case of systems

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P. R. Sethna

Papers

Some nonlocal results for weakly nonlinear dynamical systems