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Dynamics Reported is a series of books dedicated to the exposition of the mathematics of dynamcial systems. Its aim is to make the recent research accessible to advanced students and younger researchers. The series is also a medium for mathematicians to use to keep up-to-date with the work being done in neighboring fields. The style is best described as expository, but complete. Thus, there is an emphasis on examples and explanations, but also theorems normally occur with their proofs. The focus is on the analytic approach to dynamical systems, emphasizing the origins of the subject in the theory of differential equations. Dynamics Reported provides an excellent foundation for seminars on dynamical systems.



Bifurcational Aspects of Parametric Resonance

Generic nonlinear oscillators with parametric forcing are considered near resonance. This can be seen as a case-study in the bifurcation theory of Hamiltonian systems with or without certain discrete symmetries. In the analysis, among other things, structure preserving normal form or averaging techniques are used, as well as equivariant singularity theory and theory of flat perturbations.
H. W. Broer, G. Vegter

A Survey of Normalization Techniques Applied to Perturbed Keplerian Systems

This article is a survey of the use of normal form techniques in the study of Hamiltonian perturbations of the two body problem in three space. We treat the following examples in some detail: the quadratic Zeeman effect, orbit­ing dust, a three dimensional lunar problem, and the main problem of artificial satellite theory.
Richard H. Cushman

On Littlewood’s Counterexample of Unbounded Motions in Superquadratic Potentials

Littlewood [11] constructed an example of an equation of the form + V’(t) with \( \frac{{V'(x)}}{x} \to \infty \) and yet possessing unbounded solutions. This note contains a considerable simplification of Littlewood's construction together with rather precise information on the behavior of \( \frac{{V'(x)}}{x} \to \infty \) under which the resonances of the type constructed by Littlewood are possible. The original construction [11] contained an error, as was pointed out by Yiming Long, who also corrected the original proof [12].
Mark Levi

Center Manifold Theory in Infinite Dimensions

Center manifold theory forms one of the cornerstones of the theory of dynamical systems. This is already true for finite-dimensional systems, but it holds a fortiori in the infinite-dimensional case. In its simplest form center manifold theory reduces the study of a system near a (non-hyperbolic) equilibrium point to that of an ordinary differential equation on a low-dimensional invariant center manifold. For finite-dimensional systems this means a (sometimes considerable) reduction of the dimension, leading to simpler calculations and a better geometric insight. When the starting point is an infinite-dimensional problem, such as a partial, a functional or an integro differential equation, then the reduction forms also a qualitative simplification. Indeed, most infinite-dimensional systems lack some of the nice properties which we use almost automatically in the case of finite-dimensional flows. For example, the initial value problem may not be well posed, or backward solutions may not exist; and one has to worry about the domains of operators or the regularity of solutions. Therefore the reduction to a finite-dimensional center manifold, when it is possible, forms a most welcome tool, since it allows us to recover the familiar and easy setting of an ordinary differential equation.
A. Vanderbauwhede, G. Iooss

Oscillations in Singularly Perturbed Delay Equations

This paper presents some recent results on the scalar singularly perturbed differentila delay equation
$$[v\dot x(t) + x(t) = f(x(t - 1)).$$
A. F. Ivanov, A. N. Sharkovsky

Topological Approach to Differential Inclusions on Closed Subset of ℝn

The present paper is a survey of the current results concerning the existence problem, topological characterization of the set of solutions and periodic solutions of differential inclusions on subsets of euclidean spaces. First, based on [5, 15, 17, 19, 21], we give a topological approach to these questions on whole ℝn or on balls in ℝn. It is presented in Sects. 5 and 6. Later, in Sects. 7, 8 and 9 we give some new results. Namely, we are trying to tackle the mentioned problems on compact subsets of euclidean spaces called by us sets with property p. Let us remark that in particular convex sets and smooth manifolds with boundary or without boundary have property p. Note that our results obtained in Sects. 7, 8 and 9 provide an application and generalization of respective results given in [3, 7, 13, 20, 21 and 25]. We would like to add that in our considerations, we use the topological degree methods only. Finally let us remark that instead of the topological degree the analytical methods in the theory of differential inclusions are well developed (cf. [3] and [30]).
R. Bielawski, L. Górniewicz, S. Plaskacz
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