Objective: In this course you will learn how to build, implement, and analyze mathematical models suitable for the study of social, economic, ecological, and other large scale systems. You will discover the complexities that make these large scale systems difficult to predict and understand. You will learn to address some of these difficulties using the framework of complex systems. In particular, you will build, simulate, and analyze difference equations, differential equations, cellular automata, and network models to answer questions related to the interplay between the dynamics and structure of complex systems.

In support of the main objective of learning to model complex systems, you will learn some additional skills. You will continue to develop your communication skills by writing about models, discussing model techniques, and writing a full length journal article on a model that you create. In order to be able to directly interact with the models that you are studying, you will also learn how to write and execute Python scripts and functions.

The course involves a final project which will allow you to bring together the analytical tools, simulation methods, and writing skills that we will focus on in the class.

Course Information, Grading, and Policies

MODULES

The Course will be divided into four units (called Modules). They are listed below, with approximate time that we will spend on each Module:

Introduction -- Agent based models (2 weeks)

Goal 1: You will be able to decompose the modeling process into component tasks, compare different types of models, and generate simplistic mathematical models for real world questions.

Goal 2: You will be able to execute agent-based models using NetLogo, and in the process gain an intuition for the sources of complexity that make modeling social, economic and ecological systems challenging.

Goal 3: You will be able to describe types of behavior that are characteristic of complex systems in terms of macroscopic observables, transients, steady state behavior, equilibriums, and bifurcations.

Goal 4: You will write a journal style article describing the use of an agent based model to answer a scientific question. In support of this goal you will learn the purposes and components of abstracts, introductions, methods sections, results sections, and discussion sections.

Dynamical Systems (5 weeks)

Goal 1: You will be able to simulate discrete and continuous-time dynamical systems in one and two variables using Python and produce and interpret time series and phase space plots.

Goal 2: You will be able to find attractors and classify as either fixed point attractors, periodic attractors, or chaotic attractors.

Goal 3: You will be able to find fixed points of dynamical systems.

Goal 4: You will be able to perform linear stability analysis to classify the stability of fixed points.

Goal 5: You will be able to identify different types of bifurcations between attractors.

Goal 6: You will be able to create bifurcation diagrams.

Goal 7: You will be able to reduce complicated dynamical systems models to simpler models through techniques such as rescaling.

Goal 8: You will be able to demonstrate and describe universality by showing that different models have the same bifurcation diagrams, and understand the implication of the universality theorem for "one-hump maps" with a quadratic maximum.

Goal 9: You will be able to identify two routes to chaos: the period doubling route and the intermittency route.

Goal 10: You will understand the implications of chaos for modeling complex systems.

Goal 11: You will write a journal style article showing incremental improvement to a two dimensional dynamical systems model incorporating both analytical work and simulation to describe the stability of your modified nonlinear system.

Continuous Time and High Dimensional Systems (2 weeks)

Goal 1: You will be able to model discrete-time systems with a spatial component using Cellular Automata, and be able to describe and recognize phase transition behavior.

Goal 2: You will understand a Renormalization Group approach to finding critical points for phase transitions in cellular automata.

Goal 3: You will be able to describe the global error of numerical ODE solvers in terms of the order of the solver.

Goal 4: You will implement a Runge Kutta algorithm for the numerical solution of systems of ODE's.

Goal 5: You will be able to find fixed points, perform stability analyses, produce bifurcation diagrams, recognize different types of bifurcations, and rescale variables in systems of ODE's.

Goal 6: You will be able to define and implement both discrete and continuous time models on networks.

Goal 7: You will understand ways in which network structure impacts the dynamics of diffusion processes on networks.

What the course does NOT cover: The course does not cover the use of observational or experimental data for testing the validity of models or fitting parameters. Students interested in a data driven approach to modeling should consult QSS 15: Introduction to Data Analysis, GOV 16: Data Visualization, and MATH 50: Linear Models.