CSCI 5446: Chaotic Dynamics

 

Instructor

Liz Bradley

Office: ECOT 747

Email: lizb@cs.colorado.edu

Lecture Time/Place: Tuesdays & Thursdays 9:30-10:45am in ECCR 139

Textbooks

  • Nonlinear Dynamics and Chaos, S. Strogatz, Addison-Wesley.
  • Nonlinear Time-Series Analysis, H. Kantz and T. Schreiber

Course Description

This course explores dynamical systems and the various ways to use a computer to investigate their behavior. It covers the standard computational and analytical tools used in nonlinear dynamics, together with their underlying theory, and concludes with a brief review of leading-edge chaos research. Examples of important topics and techniques are: state-space representation, interpretation, and surfaces of section; steady-state solutions and limit sets; numerical integration; time and frequency domain analysis; bifurcation diagrams; fractals, fractal dimension, and the link between fractals and chaos; etc. Students construct their own computational tools and use them to explore interesting chaotic systems, ranging from mechanical pendulums to biological populations to electronic circuits. 

Strogatz’s book will be the primary text. It is well-written and thorough, but gets mathematically dense now and then. The reading assignments at the beginning of each problem set will identify these places and lectures will guide you through them. The text will be supplemented by xeroxed material and current research papers, as well as a comprehensive reserve list from the Gemmill Engineering Library. A running bibliography of other interesting (but optional) material will be given in the problem sets. The Kantz/Schreiber book is a great source of both background material and algorithms for time-series analysis. Most of the algorithms described in that book are instantiated in the TISEAN package.

Note: this is an up-to-the-minute current-topics course, and a substantial percentage of the material that will be discussed is not covered in any textbook or paper. Do not take this class unless you plan to attend all the lectures. Really. You won’t be able to pass the final exam if you don’t. There are recorded versions of the lectures from previous years, and that may be available access if a student misses a class (e.g., because of illness or religious observance dates).

Assignments, workload, and expectations

The heart of this course is a sequence of thirteen problem sets. People who are registered for CSCI 5446 will do a month-long final project instead of the last three homework sets. A cumulative exam will be held during the scheduled final exam period (Wednesday 7 May from 4:30-7pm). If you have three exams scheduled on May 7th, instructor need to know before the tenth week of the semester in order to make accomodations. You can find the exam schedules here: www.colorado.edu/registrar/calendars-schedules/final-exams-schedule 

You should expect to spend at least 8-10 hours per week of prep time—outside of lecture— on this class. Roughly a quarter of that time will be spent writing code, about half playing with your code and figuring out the results, and the other quarter reading source materials and thinking about the concepts that were presented in class. 

An important point about the assignments: each one builds upon the previous ones. That means that you can’t punt a problem set; you have to get the code for every assignment at least mostly working. The final project and the final exam will also depend upon successful completion of the assignments. 

You are free to work together on homework during the initial problem-solving stages, but you should finish, polish, interpret, and write up your work on your own. You should also feel free to consult instructor or the other class members for clarifications, corroboration, suggestions, debugging help, etc., at any point in any assignment. Please do not carry this to the point where anyone avoids learning from the assignments. In particular, the final, turned-in version of all assignments should be your own work. That includes words, code, ideas, etc. Identical material will garner a zero grade for that assignment, and possibly a discussion with the Honor Code Council, for all of the people involved, regardless of who copied from whom. If you are in any doubt about where the academic honesty boundaries lie for this class, please ask instructor.

Computers

The most critical of this course’s prerequisites, besides independence and motivation, is functional computer literacy. If you are not totally comfortable hacking code — in some language of your choice that exists on a computer to which you will have access for the entire semester — you may still enroll in this course, but be prepared to spend 15+ hours per week on it instead of the 8-10 hours cited above. 

You may use any computer, language, and graphics package that you want. In the past, people have used PCs, Macs, and Unix machines; the graphics need not be whizzy, just good enough to plot a bunch of points on a scaled, labeled axis. Please meet the instructor if you do not have access to adequate computer facilities and they will make arrangements. 

Note that there is very little canned software for this course, and for a very important reason: Instructor wants you to understand the details of the algorithms, which only happens when you write them yourself from scratch. Almost all of what you turn in will be graphics output, sketches, and written analysis and interpretation of the patterns and trends you see in the pictures. Instructor will not grade your code for style; they are interested in what comes out of it, not how well-commented or well-written it is.

Grades

Grades will be based on the homework and the final exam (for everyone) and the final project (for those enrolled in 5446). Here is the formula: 

CSCI 5446: 40% homework + 40% project + 20% final exam 

CSCI 4446: 75% homework + 25% final exam 

You must get all the programs to work at a basic level, understand and explain the results, and turn in all problem sets to pass this course. You must do so elegantly, thoughtfully, and in a timely fashion to get an A. 

Note that “participation” does not just mean “attendance”— although there is a very strong correlation between attendance and final-exam performance.

Syllabus

  • General introduction to chaos (1 1/2 lectures)
  • Dynamics of iterated maps (2 1/2 lectures)
  • Fractals and fractal dimension (1-2 lectures)
  • Continuous-time dynamics (3 lectures)
  • Applications (1 lecture)
  • Poincar´e sections (2 lectures)
  • Attractor characterization (6 lectures)
  • Hamiltonian chaos (3 lectures)
  • Prediction, modeling, and noise (1-2 lectures)
  • Stability (2-3 lectures)
  • Current topics (1 lecture)
  • Project presentations (1-2 lectures)

Accomodations and other policies

If you qualify for accommodations because of a disability, please submit a letter to instructor from Disability Services by the end of the second class period so that your needs can be addressed. Disability Services determines accommodations based on documented disabilities. You can contact them at 303-492-8671, in Willard 322, or by email at dsinfo@colorado.edu. If you have a temporary medical condition or injury, send an email to instructor as soon as you can so they can figure out how to address your needs. (The “Temporary Injuries” guidelines under the Quick Links at the Disability Services website may also be useful in these situations.)

If you have religious obligations that will cause logistical issues (e.g., with turning in an assignment or attending a class), please let the instructor know in advance—at least two weeks ahead of time, please—so they can work out alternatives.

Students and faculty each have responsibility for maintaining an appropriate learning environment. Those who fail to adhere to such behavioral standards may be subject to discipline. Professional courtesy and sensitivity are especially important with respect to individuals and topics dealing with differences of race, color, culture, religion, creed, politics, veteran’s status, sexual orientation, gender, gender identity and gender expression, age, disability, and nationalities. Class rosters are provided to the instructor with the student’s legal name. They will gladly honor your request to address you by an alternate name or gender pronoun. Please advise them of this preference early in the semester so that they may make appropriate changes to their records. For more information, see the policies on classroom behavior and the student code.

CU Boulder is committed to maintaining a positive learning, working, and living environment. CU Boulder will not tolerate acts of sexual misconduct, discrimination, harassment or related retaliation against or by any employee or student. CU’s Sexual Misconduct Policy prohibits sexual assault, sexual exploitation, sexual harassment, intimate partner abuse (dating or do- mestic violence), stalking or related retaliation. CU Boulder’s Discrimination and Harassment Policy prohibits discrimination, harassment or related retaliation based on race, color, national origin, sex, pregnancy, age, disability, creed, religion, sexual orientation, gender identity, gender expression, veteran status, political affiliation or political philosophy. Individuals who believe they have been subject to misconduct under either policy should contact the Office of Institutional Equity and Compliance (OIEC) at 303-492-2127. Information about the OIEC, the above referenced policies, and the campus resources available to assist individuals regarding sexual misconduct, discrimination, harassment or related retaliation can be found at the OIEC website.

Materials on Reserve in the Gemmill Engineering Library

  • H. Abarbanel, Analysis of Observed Chaotic Data, Springer, 1995.
  • R. Abraham and C. Shaw, Dynamics: The Geometry of Behavior, 2nd edition, Addison Wesley, 1992.
  • G. Baker and J. Gollub, Chaotic Dynamics: An Introduction, Cambridge, 1996.
  • M. Barnsley, Fractals Everywhere, Academic Press, 1993.
  • T. Bass, The Eudaemonic Pie, Houghton-Mifflin, 1985.
  • D. Campbell et al., eds., Nonlinear Science: The Next Decade, M.I.T. Press, Cambridge MA, 1992.
  • D. Campbell, ed., Chaos/XAOC : Soviet-American perspectives on nonlinear science, American Institute of Physics, New York, 1990.
  • M. Casdagli and S. Eubank, eds., Nonlinear Modeling and Forecasting, Addison Wesley, 1992.
  • P. Cvitanovic, Universality in Chaos, Adam Hilger, 1989.
  • J. Danby, Fundamentals of Celestial Mechanics, Willmann-Bell, second edition, 1992.
  • J. Gleick, Chaos: Making a New Science, Viking, 1987.
  • H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, 1997
  • G.Mayer-Kress ed., Dimensions and Entropies in Chaotic Systems, Springer-Verlag, Berlin, 1985.
  • E. Ott, T. Sauer, and J. Yorke, eds., Coping With Chaos, Wiley, 1994.
  • E. Ott, Chaos in Dynamical Systems, Cambridge, 1993.
  • T. Parker and L. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer- Verlag, 1989.
  • I. Peterson, Newton’s Clock: Chaos in the Solar System, Freeman, 1993.
  • I. Stewart, Does God Play Dice? The Mathematics of Chaos, Blackwell, 1989.
  • S. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, 1994.
  • L. Taff, Celestial Mechanics: A Computational Guide for the Practitioner, Wiley, 1985.
  • A. Weigend and N. Gershenfeld, eds., Time Series Prediction: Forecasting the Future and Understanding the Past, Santa Fe Institute Studies in the Sciences of Complexity, Santa Fe, NM, 1993.