Overview The goal of this tutorial is to outline some elementary, but beautiful aspects of random walks. Random walks are ubiquitous in nature. They naturally arise in describing the motion of microscopic particles, such as bacteria or pollen grains, whose motion is governed by being buffeted by collisions with the molecules in a surrounding fluid. …
Overview This tutorial contains excerpts from David Feldman’s Dynamical Systems course. Note that Complexity Explorer tutorials are meant to introduce students to various important techniques and to provide illustrations of their application in complex systems. A given tutorial is not meant to offer complete coverage of its topic or substitute for an entire course on that topic. …
Overview What does a JPEG have to do with economics and quantum gravity? All of them are about what happens when you simplify world-descriptions. A JPEG compresses an image by throwing out fine structure in ways a casual glance won’t detect. Economists produce theories of human behavior that gloss over the details of individual psychology. Meanwhile, even …
Overview Sometimes the best description of a system is one that acknowledges our ignorance. Maximum entropy methods provide a coherent framework for doing this in a rigorous fashion, and their success has been felt in everything from physics to ecology to machine learning. In this tutorial, Simon DeDeo introduces MaxEnt through a series of examples, …
Overview This tutorial contains excerpts from David Feldman’s Dynamical Systems course. Note that Complexity Explorer tutorials are meant to introduce students to various important techniques and to provide illustrations of their application in complex systems. A given tutorial is not meant to offer complete coverage of its topic or substitute for an entire course on that topic. …
Overview This tutorial introduces students to essential ideas related to vectors and matrices. These mathematical structures form the foundations for many key topics in complex systems, such as dynamical systems, stochastic processes, and network science. The prerequisite for this tutorial is knowledge of high-school algebra. The content of the tutorial is built, in a self-contained …
Overview This tutorial contains excerpts from Elizabeth Bradley’s Nonlinear Dynamics course. Note that Complexity Explorer tutorials are meant to introduce students to various important techniques and to provide illustrations of their application in complex systems. A given tutorial is not meant to offer complete coverage of its topic or substitute for an entire course on that topic. …
Overview This course covers several mathematical techniques that are frequently used in complex systems science. The techniques are covered in independent units, taught by different instructors. Each unit has its own prerequisites. Note that this course is meant to introduce students to various important techniques and to provide illustrations of their application in complex …
Overview In this course you’ll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time. Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from …
Overview We will begin by viewing fractals as self-similar geometric objects such as trees, ferns, clouds, mountain ranges, and river basins. Fractals are scale-free, in the sense that there is not a typical length or time scale that captures their features. A tree, for example, is made up of branches, off of which are smaller branches, …
