For many years, I have been interested in global analysis of nonlinear systems. The original interest stemmed from the study of snap-through stability and jump phenomena in structures. For systems of this kind, where there exist multiple stable equilibrium states or periodic motions, it is important to examine the domains of attraction of these responses in the state space. It was through work in this direction that the cell-to-cell mapping methods were introduced. These methods have received considerable development in the last few years, and have also been applied to some concrete problems. The results look very encouraging and promising. However, up to now, the effort of developing these methods has been by a very small number of people. There was, therefore, a suggestion that the published material, scattered now in various journal articles, could perhaps be pulled together into book form, thus making it more readily available to the general audience in the field of nonlinear oscillations and nonlinear dynamical systems. Conceivably, this might facilitate getting more people interested in working on this topic. On the other hand, there is always a question as to whether a topic (a) holds enough promise for the future, and (b) has gained enough maturity to be put into book form. With regard to (a), only the future will tell. With regard to (b), I believe that, from the point of view of both foundation and methodology, the methods are far from mature.

Klappentext

The intended audience of the book is the group of scientists and engineers who need to deal with nonlinear systems and who are particularly interested in studying the global behavior of these systems. This book introduces such a reader to the methods of cell-to-cell mapping. These methods are believed to provide a new framework of global analysis for nonlinear systems. They are based upon the idea of discretizing a continuum state space into cells, and casting the evolution of a system in the form of a cell-to-cell mapping. Up to now, two kinds of cell-mapping, simple and generalized, have been introduced and studied. These methods allow us to perform the task of locating all the attractors and domains of attraction in an effective manner. Generalized cell-mapping is particularly attractive because it can deal not only with fractally dimensioned entities of deterministic systems, but also with stochastic systems. The main purpose of the book is to make the scattered published results on cell-mapping readily available in one source. The reader, after seeing the power and potential of this new approach, will hopefully want to explore various possibilities of cell-mapping to develop new methodologies for use in his own field of research.

Inhalt
1 Introduction and Overview.- 2 Point Mapping.- 3 Analysis of Impulsive Parametric Excitation Problems by Point Mapping.- 4 Cell State Space and Simple Cell Mapping.- 5 Singularities of Cell Functions.- 6 A Theory of Index for Cell Functions.- 7 Characteristics of Singular Entities of Simple Cell Mappings.- 8 Algorithms for Simple Cell Mappings.- 9 Examples of Global Analysis by Simple Cell Mapping.- 10 Theory of Generalized Cell Mapping.- 11 Algorithms for Analyzing Generalized Cell Mappings.- 12 An Iterative Method, from Large to Small.- 13 Study of Strange Attractors by Generalized Cell Mapping.- 14 Other Topics of Study Using the Cell State Space Concept.- References.- List of Symbols.