Welcome to Ma 4/104: Introduction to Mathematical Chaos! [Introductory remarks]

Dynamical systems are mathematical models for the evolution of quantities in time. In applications, they appear as systems of differential equations (these are models with continuous time) or as iterations of a single map from a set S to itself (discrete time).

An important discovery, that can be traced back to Poincaré (ca. 1890), is that a deterministic dynamical system—i.e. one that does not include any “random” noise—can show extreme sensitivity to its initial conditions, which renders impossible the exact prediction of its long-term behavior.

Poincaré’s work remained poorly known, and chaos would only become a widespread concept in the 1960s, when the meteorologist Lorenz introduced a simple system of three ordinary differential equations (for variables x,y,z) with chaotic (unpredictable) evolution, and observed that, when t grows, all solutions (x(t), y(t), z(t)) are “attracted” to a curious set that looks like a butterfly (see the Wiki). This butterfly, a “strange attractor”, is an example of a “fractal”: fractals are peculiar geometrical objects that are self-similar and have noninteger dimensions (as opposed to lines or surfaces, for instance); they naturally appear as attractors of dynamical system and also as a result of certain recursive constructions (examples are the Sierpinski triangle and the Cantor ternary set). These are just some of the ideas that we’ll explore in this course.

Concerning methodology, I want to make explicit some guiding principles behind the design of this course:

A. Active learning generates better learning outcomes than passive learning (please see: https://www.pnas.org/doi/10.1073/pnas.1821936116Links to an external site.). Hence this course won’t be based solely on lectures, but also on discussions and collaborative problem solving.

B. Grades have detrimental effect on several aspects of learning: creative thinking, long-term retention, interest in learning, and preference for challenging tasks (see https://www.alfiekohn.org/article/risks-rewards/Links to an external site., https://www.alfiekohn.org/article/case-grades/Links to an external site. and references there). I want to minimize the role of grading in this course. Weekly, you’ll keep track of your work, submit solutions to exercises, and give feedback on other students’ solutions. None of this will be graded, but I’ll give feedback. At the end of the course, you’ll gather all your work in a portfolio, reflect on what you did and the ways feedback affected your learning, and grade yourself accordingly. During the first weeks, we shall define together tcriteria and expectations for this self-evaluation.

C. Autonomy and freedom are key for intrinsic motivation. In this course, you’ll have to think about:

  • Your own learning goals
  • The sources are the most helpful to you
  • The problems are more interesting, appealing or challenging, according to your own level.

The structure within which these decisions are taken is provided by the me, teacher. I have some expert knowledge in the subject (although I don’t have all the answers!) and more experience with pedagogy, hence I am well placed to:

  • Propose broad learning goals (a starting point for your own)
  • Propose a feasible path towards the learning goals
  • Summarize and present the main results of the theory
  • Guide collective discussions or attempts to solve problems
  • Give personalized feedback

However, you remain responsible of your own learning.

These three elements will be discussed and elaborated during the first week of class. You can find some logistic details in the Syllabus section.

Syllabus

Ma 4 / Ma 104: Introduction to Mathematical Chaos Spring, 2024 Math Department, PMA, California Institute of Technology

Course Description

From the catalogue: Ma 4/104. Introduction to Mathematical Chaos. 9 units (3-0-6); third term. An introduction to the mathematics of “chaos.” Period doubling universality, and related topics; interval maps, symbolic itineraries, stable/unstable manifold theorem, strange attractors, iteration of complex analytic maps, applications to multidimensional dynamics systems and real-world problems. Possibly some additional topics, such as Sarkovski’s theorem, absolutely continuous invariant measures, sensitivity to initial conditions, and the horseshoe map. 

Learning Outcomes

Upon successful completion of this course, I hope you’ll be able to:

  • Determine the qualitative aspects of orbits that arise under iterations of real-valued maps by means of algebraic, analytical and graphical techniques, assisted by a computer when necessary. 
  • Identify and describe bifurcations (tangent and period-doubling bifurcations) for parametric families of real-valued maps.
  • Prove the presence of chaos in some dynamical systems by means of symbolic dynamics, using the notions of conjugation and semi-conjugation. 
  • Use measure-theoretic tools in the description of strange attractors of dynamical-systems. 
  • Develop and execute an independent program of study based on personal interests. 

Bibliography

The main references will be my lecture notes (see the “Modules” section) and the book: 

  • Kathleen T. Alligood, Tim D. Sauer, James A. Yorke. Chaos: An introduction to dynamical systems. Springer, 1996. [A virtual copy of this book can be easily obtained through Caltech’s library.]

My lecture notes are heavily based on 

  • Robert L. Devaney. A first course in chaotic dynamical systems: theory an experiment. Adisson-Wesley Publishing company, 1992.

For those that want a more theoretical treatment, that goes deeper into some mathematical aspects of chaos, a good introductory reference is: 

  • Yakov Pesin, Vaughn Climenhaga. Lectures on Fractal Geometry and Dynamical Systems. AMS, 2009

Coursework

The first week, you complete Assignment 1.

On weeks 2 to 6 and 8 to 10, you upload:

  • A weekly report including:
    • Time spent on the course.
    • A brief description of the way that time was spent.
    • Answers to a short questionnaire whose purpose is to motivate a reflection and evaluation of your own learning.
    • The solution of the two most challenging problems you solved that week (I’ll propose several and you can choose your own).

There will be a dedicated assignment for each submission. The deadline for both documents is Monday, 23:59. There’s no expectation of working until then! You upload your documents whenever you’re done with that week’s work. Please respect the deadline.

Over Tuesday-Thursday, you give feedback on your peers’ work and the instructor gives feedback too. We shall discuss in class the aspects that are important in feedback and the best way of giving feedback.

On May 7, instead of the usual weekly report, you’ll submit a midterm portfolio, composed mainly of a self-evaluation questionnaire. (Besides that, you’ll submit solutions to two problems as usual.)

On June 11, you’ll submit a final portfolio, including a self-evaluation questionnaire. (No need to upload solution to two problems then.) 

Grading

At the end of the term, you examine all your written submissions, summarize your participation in class and on Ed Discussion, reflect on your improvements in relation with your learning goals, and give yourself a grade (according to standards we’ll discuss together). There will be a self-evaluation questionnaire that’ll guide you through this process.

As a teacher, I reserve the right to decide a “breach of contract” (failure to meet minimum criteria to pass the course—to be discussed), or to dissent with your judgement (in which case we’ll talk and reach an agreement). But I don’t expect this will happen.

Attendance to lectures is optional and not graded. However, the attendance to the collaborative problem-solving sessions should factor into the final grade. 

That said, I encourage you to attend the lectures and participate actively in class: I’m trying to put students needs at the center of the process and for this interactions and feedback need to happen.

Collaboration Policy

Full discussion of problems and their solutions is allowed. This includes talking about the concepts relevant to the problem, as well as the details of the solution. And you can consult any reference you want.

That said, you should write down your own version of the solution, and explicitly cite your collaborators and sources (if this applies). As a guideline for the collaboration policy, you should be able to reproduce any solution you hand in without help from anyone else and without consulting external sources.

Academic Integrity

Caltech’s Honor Code: “No member of the Caltech community shall take unfair advantage of any other member of the Caltech community.”

Understanding and Avoiding Plagiarism: Plagiarism is the appropriation of another person’s ideas, processes, results, or words without giving appropriate credit, and it violates the honor code in a fundamental way. You can find more information at: http://writing.caltech.edu/resources/plagiarism.

All instances of plagiarism or other academic misconduct will be referred to the Board of Control for undergraduates. For graduate students, contact the Graduate Office

Contents 

Over the first seven weeks, I’d like to discuss some basic aspects of the theory, restricting our attention to iterations of real-valued maps:

  1. Orbit analysis: fixed points, periodic points, stability.
  2. Bifurcations: tangent and period-doubling. 
  3. Cantor set and other fractals.  
  4. Symbolic dynamics: continuity of shift map, conjugation/symbolic coding, including elements of the theory of metric spaces.
  5. Definition of chaos: density, transitivity, sensitive dependence. 
  6. Two dimensional systems, stable and unstable manifolds, the horseshoe map. 
  7. Strange attractors

Beyond this, there are several directions that we could explore (possibly in parallel), depending on your interests. Examples:

  • Continuous-time systems (differential equations). 
  • Dimension theory and entropy theory.
  • Iterations of complex-valued maps.
  • Applications.