MODELING COMPLEX SYSTEMS

By DANIEL MARTINEZ

CONTENTS INTRODUCTION WHAT IS A COMPLEX SYSTEM? WHAT IS A MODEL? WHAT IS A DYNAMICAL SYSTEM? HOW TO BUILD UP A MODEL LOTKA-VOLTERRA MODEL MORE REALISTIC PREDATOR-PREY MODELS A MODEL WITH A STABLE LIMIT CYCLE FLUCTUATING ENVIRONMENTS HUTCHINSON’S TIME-DELAY MODEL DISCRETE-TIME MODELS LATTICE MODELS PART I MEAN-FIELD TYPE MODELS DI?ERENTIAL EQUATIONS FLOWS LINEARIZATION AND STABILITY LINEAR SYSTEMS NONLINEAR SYSTEMS GRAPHICAL STUDY OF TWO-DIMENSIONAL SYSTEMS STRUCTURAL STABILITY LOCAL BIFURCATIONS OF VECTOR ?ELDS ONE-DIMENSIONAL VECTOR ?ELDS EQUIVALENT FAMILIES OF VECTOR ?ELDS HOPF BIFURCATION CATASTROPHES IN?UENCE OF DI?USION RANDOM WALK AND DI?USION ONE-POPULATION DYNAMICS WITH DISPERSAL CRITICAL PATCH SIZE DI?USION-INDUCED INSTABILITY EXERCISES SOLUTIONS RECURRENCE EQUATIONS ITERATION OF MAPS STABILITY POINCAR´E MAPS LOCAL BIFURCATIONS OF MAPS MAPS ON R THE HOPF BIFURCATION SEQUENCES OF PERIOD-DOUBLING BIFURCATIONS LOGISTIC MODEL UNIVERSALITY CHAOS DE?NING CHAOS DYNAMICS OF THE LOGISTIC MAP F4 DE?NITION OF CHAOS ROUTES TO CHAOS CHARACTERIZING CHAOS STOCHASTIC PROPERTIES LYAPUNOV EXPONENT “PERIOD THREE IMPLIES CHAOS” STRANGE ATTRACTORS CHAOTIC DISCRETE-TIME MODELS ONE-POPULATION MODELS THE H´ENON MAP CHAOTIC CONTINUOUS-TIME MODELS THE LORENZ MODEL PART II AGENT-BASED MODELS CELLULAR AUTOMATA CELLULAR AUTOMATON RULES NUMBER-CONSERVING CELLULAR AUTOMATA APPROXIMATE METHODS GENERALIZED CELLULAR AUTOMATA KINETIC GROWTH PHENOMENA SITE-EXCHANGE CELLULAR AUTOMATA ARTI?CIAL SOCIETIES NETWORKS THE SMALL-WORLD PHENOMENON GRAPHS RANDOM NETWORKS SMALL-WORLD NETWORKS WATTS-STROGATZ MODEL NEWMAN-WATTS MODEL HIGHLY CONNECTED EXTRA VERTEX MODEL SCALE-FREE NETWORKS EMPIRICAL RESULTS A FEW MODELS POWER-LAW DISTRIBUTIONS CLASSICAL EXAMPLES A FEW NOTIONS OF PROBABILITY THEORY BASIC DE?NITIONS CENTRAL LIMIT THEOREM LOGNORMAL DISTRIBUTION L´EVY DISTRIBUTIONS TRUNCATED L´EVY DISTRIBUTIONS STUDENT’S T-DISTRIBUTION A WORD ABOUT STATISTICS EMPIRICAL RESULTS AND TENTATIVE MODELS FINANCIAL MARKETS DEMOGRAPHIC AND AREA DISTRIBUTION FAMILY NAMES DISTRIBUTION OF VOTES SELF-ORGANIZED CRITICALITY THE SANDPILE MODEL DROSSEL-SCHWABL FOREST ?RE MODEL PUNCTUATED EQUILIBRIA AND DARWINIAN EVOLUTION REAL LIFE PHENOMENA NOTATIONS REFERENCES