Physics - Geology 30:  Fractals, Chaos and Complexity

Instructor:                  John Rundle, Professor of Physics and Geology

David Peak and Michael Frame, Chaos Under Control, WH Freeman, NY, 1994

Currently out of print, but can be obtained from the following vendors:

Amazon

http://www.amazon.com/Chaos-Under-Control-Science-Complexity/dp/0716724294/ref=sr_1_1?ie=UTF8&s=books&qid=1195085929&sr=1-1 

Barnes and Noble

https://www.barnesandnoble.com/w/chaos-under-control-david-peak/1119266031?ean=9780716724292

AbeBooks

https://www.abebooks.com/9780716724292/Chaos-Under-Control-Art-Science-0716724294/plp

Highly Recommended Texts:

General Comments:

               This course will introduce students to the ideas of Fractals, Chaos, Complexity and Computation.  We will begin with the examples of objects, such as trees, river networks, coastlines, weather, earthquakes, the human body, the stock market, evolution, and others that display examples of fractal geometry.  We will then explore many of the fascinating ideas popularized by B. Mandelbrot and others about self-similarity across different geometric scales.  Chaos, how it arises in familiar everyday systems, and the link with fractal geometry, will be discussed.  We will talk about how processes of "self-organization" arise in systems with feedback, and the ways in which those processes lead to the emergence of coherent space-time structures for systems with no natural length or time scales.  We will discuss the idea of Cellular Automata and its relationship to computation.  We will examine how chaos and order are inextricably linked with a kind of strange duality.  Many of these ideas are having a profound effect in fields far from their point of origin.  As a result, we will explore the profound philosophical implications of these ideas, including their effects on modern art and architecture, and especially on the definition of life itself.

Course Content

Topics to be Covered Include:

                                                          1.     Geometry, self similarity, and patterns

                                                          2.     Making fractals through recursive iteration

                                                          3.     Measuring fractals - fractal dimension

                                                          4.     Chaos, randomness, and noise - similarities and differences

                                                          5.     Iterated maps - the logistic and tent maps - fixed points

                                                          6.     Complex numbers and the Mandelbrot set

                                                          7.     Edge of chaos, fractal boundaries, and fractal domains

                                                          8.     Cellular automata and information processing

                                                          9.     Cryptocurrencies and Decentralized Finance (DeFi)

                                                          10.     Complexity and the Climate System

                                                          11.     A topic in Econophysics

                                                          12.     Applications to other real systems

Homework and Grading:

                                                                                           1.     First Paper/Project --   50%            

                                                                                           2.     Second Paper/Project --   50%            

Late Papers will be accepted (within reason)

Class Project

1-paragraph description of the project -
Should be a paper of 3-5 pages of text or longer researching some topic in chaos/complexity/fractals, possibly involving some computer calculation/graphics, demonstrating and understanding of the basic scientific ideas. See the paper guidelines. It can also be an application to a real system.

Examples for the first paper/project might include 

1.  A discussion of the fractal nature of river networks, trees, bronchial tubes, or the like.

2.  A small project on chaotic maps, such as the logistic map, and how they can be applied to real systems

3.  A project on fractal art, generating images like trees, mountains, rivers, or other fractals

4.  An investigation of neural network learning models, and how these can be used in real applications

5.  A research project on the theory of computation, and how dynamical systems can carry out computation

Examples for the second paper/project might include 

1.  A discussion of a self-organizing complex system such as the weather, earthquakes, social systems, the world wide web, the economy, the financial markets, biological systems, ecologies, evolution, etc., including emergent coherent structures, space-time patterns, phases of the system and phase transitions

2.  A computer code that addresses some other aspect of complex systems not covered in the course