Simple Rules, Complex Dynamics – Part I: Foundations & Intuition
Table of Contents Introduction Some simple ideas and baselines - growth, local linearity and equilibrium Exponential and logistic growth Oscillators and local linearity Invariants and conservation Oscillations from feedback and delay Lotka-Volterra predator-prey model SEIR model with vital dynamics Thresholds, alignment, and phase transitions Granovetter’s threshold model for collective behavior The Ising model: alignment, noise and criticality Schelling’s model of segregation Reinforcement, herding, and heavy tails Polya’s urn: reinforcement and path dependence Kirman-Folmer herding: bistability Sandpiles: self-organized criticality and avalanches Multiplicative growth and Kesten tails Swarms and distributed coordination Vicsek alignment: headings, noise, and a mean-field self-consistency Cucker-Smale: continuous-time velocity alignment and convergence Hydrodynamics and long-range order Selection dynamics Replicator dynamics Replicator-mutator dynamics Ricardian trade as selection of specializations Spatial structure: patterns from local rules Reaction-diffusion and diffusion-driven instability Traveling fronts: Fisher-KPP Other patterns A toolbox Main objects of study Linearization and local analysis 2D phase-plane portraits Lyapunov functions and LaSalle’s invariance Bifurcations Discrete-time systems and chaos Delays: compartments vs true delay differential equations Invariants, positivity, comparison, monotone structure Networks and coupling Non-dimensionalization and scaling Stochastic dynamics and metastability Traveling fronts Contraction analysis Singular perturbations and slow-fast decompositions Koopman operator and data-driven surrogates Ergodic theory and mixing Non-smooth and hybrid dynamics (thresholds and edges) Control and feedback Numerical methods that respect structure References and Further Reading Acknowledgements Introduction Simple local interactions can lead to a large variety of global behavior. Many examples come from physics, biology, economics, and ML, but the machinery used to look at them remains essentially the same. Looking at things through a certain lens - asking questions such as “What’s the feedback?”, “How are things coupled?”, “Is there noise?”, “What are the timescales?” - can help intuit the macroscopic outcomes of these systems: growth, cycles, alignment and discontinuities, swarms, heavy tails, and different spatio-temporal patterns. ...