Modeling Complex Systems (eBook)

The purpose of this book is to illustrate how models of complex systems are built up and to provide indispensable mathematical tools for studying their dynamics. It is an introductory text directed mainly towards advanced undergraduate students in most scientific disciplines. It can also serve as an important reference book for graduate students and young researchers.

After a general introduction, followed by an overview of various modeling techniques used to explain the observed coupled oscillations of predator and prey population densities, the book is divided into two parts.

The first part describes models formulated in terms of differential equations or recurrence equations in which local interactions between the agents are replaced by uniform long-range ones, and whose solutions can only give the time evolution of spatial averages.

Despite the fact that such models offer rudimentary representations of multi-agent systems, they are often able to give a useful qualitative picture of the system's behavior.

The second part is devoted to models formulated in terms of automata networks in which the local character of the interactions between the individual agents is explicitly taken into account. Chapters of both parts include a few exercises that are meant to challenge the reader and complement the material in the text.

This second edition text includes more recent research results and bibliographic references, extra footnotes on the biographies of important, cited scientists who have made significant contributions to the field, and many new and improved worked out examples and exercises to aid a student’s comprehension of the content studied in courses on mathematical and simulation modeling.

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The essential points of this ?rst chapter are * The de?nition of a complex system * The notion of emergence * The de?nition of a model * The notion of dynamical system This book is about the dynamics of complex systems. Roughly speaking, a system is a collection of interacting elements making up a whole such as, for instance, a mechanical clock. While many systems may be quite complicated, they are not necessarily considered to be complex. Today, most authors agree on the essential properties a system has to possess to be called complex. The ?rst section is devoted to the description of these properties. To interpret the time evolution of a system, scientists build up models, which are simpli?ed mathematical representations of the system. The exact purpose of a model and what its essential features should be is explained in the second section. The mathematical models that will be discussed in this book are dynami- 1 calsystems. A dynamical system is essentially a set of equations whose- lutiondescribesthe evolution,asafunction oftime, ofthe state ofthe system. There exist di?erent types of dynamical systems. Some of them are de?ned in the third section. 1 There is an extensive literature on mathematical modeling. The reader may, for example, consult [11,88,163,233].