## Modeling Complex SystemsThe preface is that part of a book which is written last, placed ?rst, and read least. Alfred J. Lotka Elements of Physical Biology Baltimore: Williams & Wilkins Company 1925 The purpose of this book is to show how models of complex systems are built up and to provide the mathematical tools indispensable for studying their dynamics. This is not, however, a book on the theory of dynamical systems illustrated with some applications; the focus is on modeling, so, in prese- ing the essential results of dynamical system theory, technical proofs of th- rems are omitted, but references for the interested reader are indicated. While mathematical results on dynamical systems such as di?erential equations or recurrence equations abound, this is far from being the case for spatially - tendedsystemssuchasautomatanetworks,whosetheoryisstillinitsinfancy. Many illustrative examples taken from a variety of disciplines, ranging from ecology and epidemiology to sociology and seismology, are given. This is an introductory text directed mainly to advanced undergraduate students in most scienti?c disciplines, but it could also serve as a reference book for graduate students and young researchers. The material has been ́ taught to junior students at the Ecole de Physique et de Chimie in Paris and the University of Illinois at Chicago. It assumes that the reader has certain fundamental mathematical skills, such as calculus. |

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### Contents

1 | |

How to Build Up a Model | 17 |

3 | 39 |

Recurrence Equations | 107 |

Chaos | 145 |

Cellular Automata 191 | 190 |

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### Common terms and phrases

asymptotically stable attractor automata average bifurcation diagram bifurcation point Boccara cells cellular automaton cellular automaton rule characteristic path length clustering coefficient configuration defined degree probability distribution denotes differential equation edges eigenvalues empty epidemic equal equilibrium point Example exhibits exists exponent exponential finite fixed point hyperbolic equilibrium point infective individuals integer interactions interval iterations Lévy limit cycle linear logistic map Lyapunov map f Mathematical matrix mean-field move neighborhood neighbors nonhyperbolic nonzero one-dimensional parameter pedestrian percolation period-doubling bifurcations phase space phase transition Physical Review population positive power-law behavior predator prey probability density function probability distribution random variable random walkers recurrence equation resp result rule f satisfies self-organized criticality sequence shows small-world small-world network solution species square lattice strategy susceptible theorem transcritical bifurcation two-dimensional unstable vector field vertex degree vertex degree probability vertices zero