• Links to the course website & Notes discussion database can be found at www.lasalle.edu/~didio 
• All personal e-mail to me must include HON 462:   in the subject line
• Important class news, including assignments, exam news, class notes, etc., will be posted to the class Notes database or e-mailed.
• It is your responsibility to check e-mail and the web site regularly!

This course is an exploration of the absolutely thriving fields of ‘Chaos’ & ‘Fractals’ or, in somewhat more sobering terms, ‘Non-Linear Dynamical Systems’ and ‘Surfaces with non-Integer Hausdorff Dimension’.  The approach will consist of a (non-linear) combination of empirical observations, computer simulations, and mathematical derivations, coupled with readings ranging from popularizations to original texts.

Student experimentation, hypothesis making, discussion, and writing will be emphasized by a course grade based on participation, journal keeping, computer experimentation, and a major research paper/project.  A major goal is that students gain an awareness of the new ways in which ‘chaotic/fractal’ analysis may be used to model real-world situations that in the past have seemed random, and hence intractable.  It is especially hoped that this course will prepare students for further, in-depth reading and research into the uncountably diverse facets of chaos and fractal theory and applications, particularly within their specific fields of study.

• Gleick, Chaos:  Making a New Science, Viking, New York, 1987.
• Hall (editor), Exploring Chaos, Norton, New York, 1991.

• “A Sound of Thunder” (R. Bradbury)
• “The Mathematical Universe” (J. D. Barrow)
• “Fractal Geometry:  What Is It, and What Does It Do?” (B.B. Mandelbrot)
• “Origins of Randomness In Physical Systems” (S. Wolfram)
• “Chaos and The Dynamics Of Biological Populations” (R.M. May)
• “Universal Behavior In Nonlinear Systems” (M.J. Feigenbaum)
•  “Nonlinear Dynamics, Chaos and Complex Cardiac Arrhythmias” (L. Glass, A.L. Goldberger, M. Courtemanche, and A. Shrier)
• “Sudden Death Is Not Chaos” (A.L. Goldberger and D.R. Rigney)
•  “Deterministic Nonperiodic Flow” (E.N. Lorenz)
•  “Chaos and the Making of International Security Policy” (A.M. Saperstein)
• “Controlling Cardiac Chaos” (A. Garfinkel, M.L. Spano,W.L. Ditto, J.N.Weiss)
•  “Chaos” (J.P. Crutchfield, J. D. Farmer, N.H. Packard, and R. Shaw)

Journal

Students are required to keep a journal in which outlines, handouts, comments, questions, queasy feelings, and personal responses to these readings are kept. The journal will also be the location of any notes/work connected with outside readings or personal research into your final project.  The journals will be periodically reviewed, but not graded.  Rather, the journals will be used as sources for class discussion.  Carry them with you at all times.

• Class Participation, Computer Projects, and Research Project are all weighted equally in determining the final grade. The +/- system will be used.
• Participation:  When readings are issued, 2-3 students will be assigned to lead the discussion on the readings at the following class.  In leading this discussion, these students will prepare a brief synopsis of the readings and questions for discussion to distribute to the class.
• Computer Lab Participation/Reports: Computer labs will be held weekly in which LaSalle-grown and commercial software packages are used to investigate some of the mesmerizing visual properties of chaotic and fractal systems.  There will be a several (3-4) labs in which short “lab reports” will be required.
• Research paper/project:  This project provides an opportunity for students to investigate the ways in which Chaos theory intersects their major fields of study.  The actual format and requirements of the project will be described in much more detail shortly.  Some basics can be stated, however.  The length of the paper will be in the 10-20-page range, and there will be a 15-20 minute presentation of the project at the end of the semester.  Some projects may be better suited for group work rather than individuals. 

The fact that we can describe the motions of the world using Newtonian mechanics tells us nothing about the world.  The fact that we do, does tell us something about the world. -Ludwig Wittgenstein

Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history. - Leo Tolstoy

There ought to be something very special about the boundary conditions of the universe and what can be more special than the condition that there is no boundary.  - S. W. Hawking

Weird, wacky stuff... Anon. 462 student,’91, ’93,’95, ’97

• Newton, Laplace, and the Origins of Determinism
• Problems in Determinism Land:  From Accu-Weather to Heart Attacks
• Differential Equations and Iterated Maps
• The Solar System and King Otto’s prize
• Population Modeling
• Linear vs. Non-Linear Things
• Simple Systems Yielding Complex Results:  The Binary Shift Map
• Mathematical Stretching and Folding and Sensitive Dependence on Initial Conditions (SDIC)

• Rabbit Proliferation, the Stock Market, and Structure within Structure
• Cantor Sets, Sierpinski Gaskets, and other Dusty Objects
• The Ever-Growing Coastline and New Dimensions

• Model Construction:  Populations and Exponential Growth
• Analytic vs. Numerical Solutions
• A First Visit to Phase Space:  Simple Newtonian Systems and the Linear Pendulum
• The Non-Linear pendulum
• Coupled Systems:  Romeo and Juliet Functions and the Double Pendulum
• Historical Sites in Phase Space:  Trajectories, Fixed Points, Limit Cycles, Basins of Attraction, Poincare Sections, and Strange Attractors
• Stability and SDIC
• The Lorenz System and the Butterfly Effect

• Population Growth with Limits:  The Logistic Equation
• Graphical Iteration, Asymptotic Behavior and the Bifurcation Diagram
• The Period Doubling Route to Chaos
• Universality and Feigenbaum’s Number
• Looking for the Map:  Embedding
• Finding Bounded Orbits for the Logistic Map: The Cantor Set Revisited
• two-dimensional Maps:  Fractal Structures in Henon-ville
• Circle Maps:  Walking the Devil’s Staircase

• Chicken Hearts and Arnol’d Tongues
• Dripping Faucets
• Nuclear Proliferation
• Is the Solar System  Stable, or What?:  The KAM Theorem

• Iteration in the Complex Plane: The Mandelbrot Set
• Exploding Julia Sets, and Julia’s missing nose
• Looking for Roots in all the Wrong Places:  Newton’s Method and Fractal Basin Boundaries