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Complexity Explorer

Nonlinear Dynamics: Mathematical and Computational Approaches

Santa Fe Institute via Complexity Explorer

Overview

This course provides a broad introduction to the field of nonlinear dynamics, focusing both on the mathematics and the computational tools that are so important in the study of chaotic systems. The course is aimed at students who have had at least one semester of college-level calculus and physics, and who can program in at least one high-level language. After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend several weeks thinking about how to solve them numerically and what challenges arise in that process. We finish by learning about techniques and tools for applying all of this theory to real-world data.

Taught by

Liz Bradley

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Reviews

4.8 rating, based on 62 reviews

Start your review of Nonlinear Dynamics: Mathematical and Computational Approaches

  • Anonymous
    I come from a System Dynamics background where we utilize the concepts of stocks, flows and feedback loops to understand the dynamics of a system. I was trying to understand concepts like eigenvalues and eigenvectors when I chanced upon information that...
  • Anonymous

    Anonymous is taking this course right now.

    Before reviewing the course, a bit about myself. I am a final semester under-grad student from India, pursuing Maths and Computer Science as my Major Subject. Since I am currently doing my project on the characteristics of Lorenz Attractors, the topics,...
  • Leo Römers completed this course, spending 2 hours a week on it and found the course difficulty to be medium.

    I found the course very interesting as it used a rather down-to-earth approach with quite clear explanations of essential concepts and also caveats in practice. Much of the required mathematics is explained in this course focussing on concepts rather...
  • Anonymous
    This is a well-balanced course providing you with intuition, computational methods, and pointers to rigorous proofs and resources. It showcases great applications and interviews with invited professors. Perks like artistic references, jokes, and philosophical sidenotes are also provided.
  • Anonymous
    Very good course. I recommend it for those who want to start or review the topics of nonlinear dynamics with a practical approach. I also recommend the course for the excellent explanations of the instructor, since she presents difficult topics in a pleasant way that facilitates learning.
  • Profile image for Hugo Campos
    Hugo Campos
    Very attractive for a large audience since they focus on understanding key ideas rather and tediously cover each topic. Plenty of quizes and tests to keep track of your understanding as well as homeworks for you to ponder with.
  • Juan Cabrera García-Olmedo completed this course.

    Stunning course with a high pedagogic motivation and a practical orientation which is very much appreciated. I come from a non-physics background and the maths and programming were challenging at the beginning but ended up wanting MORE videos about how to calculate manifolds in practice and more homework on Lyapunov exponents etc. For me it was just GREAT!
  • Profile image for Dasha Pinakhina
    Dasha Pinakhina
    This is one the most fascinating subjects I've ever encountered in my life. The instructors are the best.
  • Anonymous

    Anonymous completed this course.

    The course was excellent and I feel fortunate to have had the opportunity. I am working to develop a better understanding of complexity theory while also building foundation knowledge in physics and math. In part, I took the course to understand what...
  • Anonymous

    Anonymous is taking this course right now.

    This course is a great gateway to a difficult subject. Professor Bradley explained the essence of technical concepts without dumbing them down, without ignoring or glossing over their mathematical cores, but also without distractingly excessive detail...
  • Anonymous

    Anonymous completed this course.

    The course is very interesting because besides the basic theoretical aspects of dynamical systems it allows you to undestand the mathematics behind the behavior of such dynamical systems. Also allows you to implement your own programs in order to calculate...
  • Anonymous

    Anonymous completed this course.

    A fantastic and engaging introduction to the ideas, implementation, and applications of nonlinear dynamics. The course perfectly mixes the underlying theory and math with practical problems and challenges to implement various standard algorithms. You can either take it as an overview course, or take the opportunities given at every stage to dive into the material, reading the literature and improving on the solutions and algorithms presented in lecture. Prof. Bradley is a clear and entertaining lecturer, and she and her TAs were friendly, responsive, and happy to explain and discuss anything related to the course on the course forum.

    Highly recommended!
  • Anonymous

    Anonymous completed this course.

    I completed this course last year. The course provides broad ranges of topic and requires little maturity of mathematical backgrounds. The homework problems are of high-quality and makes you acquainted with the principal concepts in nonlinear dynamics. Especially, the fascinating part was that there are videos which experts from various fields give a talk about their work regarding the concepts that were taught in the chapter. This gives a motivation about how to apply the learned things to what I am concerned with.
  • Profile image for Felipe Morelli
    Felipe Morelli
    excellent teacher. Very didactic.
    I managed to make all the codes.
    The proposed exercises help a lot in learning.
  • Anonymous

    Anonymous completed this course.

    I wanted to learn something new in the field of advanced mathematics, something that I never studied in grad school or in real life. I am a retired professor of Mechanical Engineering. I wanted to keep learning something new as just as I did when I was working. This course was excellent. She is an excellent professor and through some correspondence with her a very nice woman. She provided some great examples. I hope she will have more advanced courses in the coming year.
    Sincerely
    Professor Jake Trexel
  • Anonymous

    Anonymous completed this course.

    1. Perfect course for first timers who want to learn about the field of Nonlinear Dynamics.

    2. Course offered by a Master Instructor making the course interesting and non-mundane.

    3. Tests, home-works etc., complement the learning beautifully.

    4. Course is not easy-peasy. The videos need to watched, re-watched and contemplated upon for the ideas to sink in.

    5. A good command in mathematical computation (using Mathlab/FreeMat, Python, Mathematica, etc., ) is essential for learning the ideas presented in this course.
  • Tristan Hardy completed this course, spending 2 hours a week on it and found the course difficulty to be easy.

    Loved the course. Great introduction to the math behind several concepts that were, in turn, hinted at in the Introduction to Complexity course on Complexity Explorer. I agree with other reviewers that are interested in more in-depth courses or tutorials that delve into more examples on how to apply these concepts, rather than simply understand them.
  • Anonymous

    Anonymous completed this course.

    Great introduction and a broad sweep of the field. For its objectives, time commitment, and scope, it works well. Worth taking to get a good perspective on nonlinear dynamics. Would tempt you to dig deeper in at least one area.

    Credit to Liz Bradley for the character of the coverage. The support/Q&A forum was superb.

    Maybe not 5star yet, but better than a 4star. The next iteration should be better.
  • Anonymous

    Anonymous is taking this course right now.

    As an individual pursuing graduate work in engineering, I often mused about taking a fun math course. This fit the bill in so many ways. I learned so much and my mathematical knowledge has been enriched in a significant way. Thank you for the introduction to nonlinear dynamics!
  • Anonymous

    Anonymous completed this course.

    I gained a lot from this course. Was so excited by the material that I bought a whole bunch of complexity explorer gear afterwards and got the Strogatz materials to audit his class and learn more!

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