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Complexity Explorer

Introduction to Dynamical Systems and Chaos

Santa Fe Institute via Complexity Explorer


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 the study of dynamical systems that are of particular relevance to complex systems:

1.  Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's

2.  Deterministic dynamical systems can behave randomly.  This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.

3.  Disordered behavior can be stable.  Non-periodic systems with the butterfly effect can have stable average properties.  So the average or statistical properties of a system can be predictable, even if its details are not.

4.  Complex behavior can arise from simple rules.  Simple dynamical systems do not necessarily lead to simple results.  In particular, we will see that simple rules can produce patterns and structures of surprising complexity.

Taught by

David Feldman


4.9 rating, based on 122 Class Central reviews

Start your review of Introduction to Dynamical Systems and Chaos

  • Aleta Duvall

    Aleta Duvall completed this course.

    I appreciated the structured approach to the course material and the painstaking development of foundational concepts. Dr. Feldman presents the course in an informal, across-the-desk manner. Each lecture feels like you are experiencing an individual tutoring...
  • Davide Madrisan completed this course.

    This course is a awesome mine of information and knowledge on dynamical systems and chaos. It starts at the very beginning by explaining the mathematical bricks necessary to understand the following units, but it's not a heavy mathematical course. Quite...
  • Maxime Zabiégo completed this course, spending 2 hours a week on it and found the course difficulty to be very easy.

    This course is perfect for beginners who want to get a grasp of the field of chaos with minimal mathematical/numerical material, but yet enough to have a flavor of the underlying world, beyond the bases. Dave Feldman is a great teacher, very dedicated...
  • Profile image for Navyanth Kusam
    Navyanth Kusam

    Navyanth Kusam completed this course, spending 27 hours a week on it and found the course difficulty to be medium.

    Thanks for making this course available for all. It really changed the way I see the computational world. I had a masters degree in Computational engineering where I learned about the fundamentals of Continuum mechanics and its applications in Structural...
  • Anonymous
    Incredibly organized. My undergrad students can follow this course. What a great job breaking down a very complex topic into digestible bits with exercises and explanations.
  • Class Central user

    Class Central user completed this course, spending 2 hours a week on it and found the course difficulty to be medium.

    This is a fun class which I would highly recommend. Incredibly accessible course which introduced an interesting branch of mathematics.

    If you've done a bit of computer science or programming, this class shows a very interesting result from the simple idea of iteration.

    The professor is interesting and makes the material intuitive. Regardless of your level of mathematical comfortability, this class is well paced and provides all the tools you need to proceed through it.

    Don't miss this class!!
  • Anonymous
    An excellent course, and Dr. Feldman provides clear instructions that are easy to follow along. Being able to work at my own pace was excellent as the father of a newborn/toddler (it took me almost a year to work through the course given the baby's schedule). But I feel confident that I know what I'm doing, and honestly feel that this course will benefit me in my line of work.
  • Anonymous

    Anonymous is taking this course right now.

    This was the missing link for me. I had read many popular science books about dynamical systems and chaos, and was eager to gain a more nuanced and technical understanding of the field. The only thing holding me back, of course, was a lack of background...
  • Profile image for Facu Rod
    Facu Rod
    You learn this theme from the very beginning, starting with dynamical systems, which are nothing other than differential equations and iterative maps you get to know the most basic systems that show chaotic behaviour. Then you go on to higher dimensions, up to 3d dynamica system like the Lorenz Equations and the Rössler Attractor. Universality and renormalization are very good introduced and give you a grasp, why Dynamical Systems and Chaos are omnipresent. At the end you learn why making dough is related to complex chaotic behaviour , because the stretch and fold.
  • Anonymous
    Amazing course....really made me think about so many mundane things in a new perspective. Will come back for more courses...
  • Sabine Hiller

    Sabine Hiller completed this course.

    A fascinating course, I absolutely loved it! There was a bit of maths involved, but Dave Feldman walked us through it step by step - so slowly that even somebody who has no idea whatsoever of maths should grasp it. Sitting in front of my laptop on the other side of the globe, I felt he deeply cared that you understood everything - as somebody said above, it almost felt like a personal tutorial! As I had already done Melanie Mitchell's course (equally excellent!), I was familiar with much of the territory, but Dave's approach helped me to understand some of the more difficult concepts. I think the two courses complement each other very nicely. Highly recommended!
  • Anonymous
    I really enjoyed doing this course. It gave me a great introduction to the dynamical systems and chaos field. The presentations were very clear, well presented with lots of examples, both mathematical and graphical. The lectures were delivered at the...
  • David

    David completed this course.

    This class was just amazing and great. Mr. Feldman knows how to really get you to see the 'big idea' and that's the most import part of any intro class; to get the big picture. Math was light and not terrifying. Deadline is the class day of class. Quizzes were light and the homework was challenging enough if you wanted it to be. I absolutely loved this class.
  • Abhijit Borah

    Abhijit Borah completed this course.

    A great point for a young person, with knowledge about calculus to start if he or she wishes to pursue this field. This course starts step by step, and is excellent in making the learning about nonlinear-dynamics an almost effortless process. The homework at the end of each unit are a must do to enjoy the course. Knowing computer programing is really a necessity I would say so.
  • Anonymous

    Anonymous completed this course.

    I completed this course a few years ago just to learn about chaos theory. I loved it! I am a math teacher and this year have an honors precalculus course. I am letting my students complete it for extra credit and the ones doing it are enjoying it so much next year I will require it - perhaps work on it as a class each Friday. Thank you so much for a great course!
  • Gaetano Pagani completed this course, spending 5 hours a week on it and found the course difficulty to be medium.

    I had a great time having this course and Prof. Feldman was so dedicated in diving us into this amazing subject. I think that the course gave us a solid platform in order to go deeper in the study of Dynamical Systems and Complexity.
  • Narasimhan Mg completed this course.

    Complexity and Dynamical Systems is a fascinating field of mathematics. There are linear, predictable and well-ordered systems, like simple physical motion. There are non-linear systems, like the weather. What are the latter like? Can they be studied...
  • Anonymous

    Anonymous completed this course.

    I enjoyed the course very much. The teacher is great, he can explain both familiar and new (complicated) matter in a fresh and understandable way. I helped me to understand dynamical systems deeper, as I do dynamical modeling in economics.
  • Profile image for Moses Paul
    Moses Paul

    Moses Paul completed this course, spending 5 hours a week on it and found the course difficulty to be easy.

    The course is well structured. The instructor is well equipped with handy demonstrations and visual tools to help further understanding. The content taught is taught with the learner being considered an absolute beginner, this may seem tedious and repetitive for certain learners, but it comes in really handy because every nuance is reasoned and spoken for. I would definitely recommend it to the newcomers and to people needing a refresher course.
  • Anonymous

    Anonymous is taking this course right now.

    The effort put into the course becomes clear as soon as you start. A very accessible design of the material without loosing the required depth for a proper understanding of the essence.

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