Preface

This book is the fourth volume in a series of the Handbook of Ordinary Differential Equa-tions. This volume contains six contributions which are written by excellent mathemati-cians. We thank them for accepting our invitation to contribute to this volume and also fortheir effort and hard work on their papers. The scope of this volume is large. We hope thatit will be interesting and useful for research, learning and teaching.

A brief survey of the volume follows. First, the contributions are presented in alpha-betical authors’ names. The paper by Balanov and Krawcewicz is devoted to the Hopfbifurcation occurring in dynamical systems admitting a certain group of symmetries. Theyuse a so-called twisted equivariant degree method. Global symmetric Hopf bifurcation re-sults are presented. Applications are given to several concrete problems. The contributionof Fabbri, Johnson and Zampogni lies in linear, nonautonomous, two-dimensional differ-ential equation. For instance, they study the minimal subsets of the projective flow definedby these equations. They also discuss some recent developments in the spectral theoryand inverse spectral theory of the classical Sturm–Liouville operator. The question of thegenericity of the exponential dichotomy property is considered, as well, for cocycles gen-erated by quasi-periodic, two-dimensional linear systems. The paper by Lailne is mainlydevoted to considering growth and value distribution of meromorphic solutions of com-plex differential equations in the complex plane, as well as in the unit disc. Both linear andnonlinear equations are studied including algebraic differential equations in general andtheir relations to differential fields. A short presentation of algebroid solutions of complexdifferential equations is also given. The paper by Palmer deals with the existence of chaoticbehaviour in the neighbourhood of a transversal periodic-to-periodic homoclinic orbit forautonomous ordinary differential equations. The concept of trichotomy is essential in thisstudy. Also, a perturbation problem is considered when an unperturbed system has a non-transversal homoclinic orbit. Then it is shown that a perturbed system has a transversalorbit nearby provided that a certain Melnikov function has a simple zero. The contributionby A. Rontó and M. Miklós investigates the solvability and the approximate constructionof solutions of certain types of regular nonlinear boundary value problems for systems ofordinary differential equations on a compact interval. Several types of problems are con-sidered including periodic and multi-point problems. Parametrized and symmetric systemsare considered as well. Most of theoretical results are illustrated by examples. Some histor-ical remarks concerning the development and application of the method are presented. Fi-nally, the paper by Zoładek is devoted to the local theory of analytic differential equations.Classification of linear meromorphic systems near regular and irregular singular point isdescribed. Also, a local theory of nonlinear holomorphic equations is presented. Next, for-

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mal classification of nilpotent singularities is given and analyticity of the Takens prenormalform is proved.

We thank the Editors of Elsevier for their collaboration during the preparation of thisvolume.