- K. Alligood, T. Sauer, and J. Yorke, Chaos An Introduction to Dynamical Systems, (Springer, 1996). A well-written undergraduate text that the members of the class who like careful mathematical statements and proofs will enjoy.
- Robert Devaney, An Introduction to Chaotic Dynamical Systems, 3rd edition. An undergraduate-level book intended for people who like or want mathematically precise definitions of key nonlinear dynamical concepts and rigorous and often elegant arguments. Note: the book only discusses maps and does not discuss how mathematical results compare with experimental data.
- Robert Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (2nd edn) (Oxford, 2000). A nice broad survey, more descriptive than mathematical (good for getting the gist of many topics).
- Hiroki Sayama, Introduction to the Modeling and Analysis of Complex Systems (2015). A free PDF version is available here .
- Steven
Strogatz,
Nonlinear Dynamics and Chaos, second edition (2014). The most
widely used introductory textbook, mainly at a sophomore-junior
undergraduate level. You should own a copy for this course and as a
reference.
-
Note: there are free online video lectures by
Strogatz
here that closely follow his textbook, a good resource for
this course.
- Fred Brauer and Christopher Kribs, Dynamical Systems for Biological Modeling An Introduction (2016). A more gentle undergraduate introduction intended for life science students who know just calculus. Available online via the Duke Library .
- John Guckenheimer and Philip Holmes, Nonlinear oscilations, Dynamical Systems, and Bifurcations of Vector Fields (1990). A graduate-level text that is more oriented to mathematicians.
- Jens Feder, Fractals (1988). Good accessible discussion of fractals at the upper-undergraduate level.
- Edward Ott, Chaos in Dynamical Systems, second edition. A sophisticated and mainly theoretical overview of nonlinear dynamics at mainly the graduate level. By a physicist who is a leader and innovator of modern nonlinear dynamics.
- Arkady Pikovsky, Michael Rosenblum, and Jurgen Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge Nonlinear Science Series Book (2003). 513 will only discuss just a little bit about the fascinating science and mathematics of synchronization, but this is a standard introductory (although not easy) text on the subject.
- Philip Ball, The Self-Made Tapestry: Pattern Formation (Oxford University Press, 1999). A non-technical overview of pattern formation and self-organization, what happens when spatial structure is included as well as temporal structure.
- James Gleick (pronounced "glick"), Chaos: Making a New Science (Penguin Books, 2008). A rare scientific best-seller. Book introduces and discusses the subject by discussing key pioneers in the field. Everyone taking 513 should read this interesting book.
- Manfred Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Dover, 2009).
- Steven Strogatz, Sync: How Order Emerges from Chaos In the Universe, Nature, and Daily Life (Hachette Books, 2012).
- Mathematica
- Stephen Wolfram, An Elementary Introduction to the Wolfram Language, 2nd edition. Available free on the Web.
- Mathematica documentation center
- Mathematics:
- Calculus: Openstax calculus books, volumes 1, 2, and 3. Volume 3 discusses multivariate calculus so is a convenient free reference.
- Not online but a popular undergraduate textbook that you might like to buy as a reference for 513 is the book Mathematical Methods in the Physical Sciences, 3rd Edition by Mary Boas.
- Mathematical Tools for Physics, free online source that can help you review some of the math used in 513.
- Physics:
- Openstax University (calculus-based) Physics, volumes 1, 2, and 3.
- The Feynman Lectures on Physics, an advanced (and unfortunately increasingly dated) introduction to physics with many keen insights. Chapters 40 and 41 in Volume II on fluid dynamics are good introductions for those of you who have not studied any fluid dynamics.
- Physics for Scientists and Engineers: A Strategic Approach with Modern Physics, 4th Edition by Randall Knight. If you need to learn or review some basic physics mentioned in class, this is a well written introductory physics textbook.
Some other courses about nonlinear dynamics and complex systems
- Liz Bradley video course on "Nonlinear Dynamics: Mathematical and Computational Approaches". See also the webpage of her non-video course on nonlinear dynamics.
- Michael Cross, Physics 161: Introduction to Chaos .
- David Feldman, Introduction to Dynamical Systems and Chaos, a free online course offered through the Santa Fe Institute.
- Rodolfo Fosales, Nonlinear Dynamics and Chaos , MIT OpenCourseWare (2004)
- Alan Garfinkel, Modeling Life videos, course (with textbook) about nonlinear dynamics applied to biology, especially cardiology.
- Melanie Mitchell, Introduction to Complexity a free online course offered through the Santa Fe Institute, an organization devoted to exploring different aspects of complex systems.
- Clint Sprott, Chaos and Time Series Analysis
- Steven Strogatz 2014 video course on nonlinear dynamics and chaos
Web demos related to nonlinear dynamics
- Double pendulum simulation. The website myphysicslab.com has several other nice dynamical and chaotic simulations.
Preprints and Journals related to nonlinear dynamics:
- arXiv nonlinear dynamics preprint database.
- Chaos.
- Journal of Fluid Dynamics.
- Journal of Theoretical Biology.
- Nature
- Physical Review E Online. Main journal where physicists and many others publish basic papers related to nonlinear dynamics.
- Physical Review Letters Online. The premier physics journal for new important results.
- Science The premier American journal for important results in all areas of science.
- SIAM Journal on Applied Dynamical Systems
Fiction and movies related to nonlinear dynamics
- Ray Bradbury's 1953 science fiction short
story
A Sound of Thunder, is one of the earliest and most often
mentioned stories about
the
butterfly effect, that an infinitesimal perturbation to a
chaotic dynamical system can cause major changes later in time. I
don't know but think it is unlikely that Bradbury knew about the
technical concept of chaos when he wrote his story in the 1950s
since few people knew about chaos outside of a small community of
theorists.
- The Isaac Asimov science fiction novels called
the
Foundation series, recently made into a 2021
TV series, do not mention chaos theory but do discuss the
possibility of long-term accurate forecasting related to the rise
and fall of interstellar civilizations over hundreds of thousands of
years.
- There are many websites that mention popular TV shows and
movies (e.g., the 1981 Polish
movie Blind
Chance, the 1993
movie
Jurassic Park and the 1998
movie Sliding
Doors) that are based on the butterfly effect. One of my
favorite examples
is The
Plateau, Episode 3 of Season 3 of the TV
show
Fringe, in which a mysterious fictional drug can so greatly
enhances a person's cognition that that person can mentally
calculate in real-time the consequences of causing a simple tiny
perturbation and so carry out what seem to be unsolvable crimes,
liking causing someone to get hurt by just putting a pen in a
certain location at a certain time (see the opening
minutes here). These
shows do not mention the dynamical mechanism that underlies the
source of chaos that causes amplification of small perturbations,
and it is unlikely that a person's life is technically chaotic (can
you see why?).
Other discussions of the butterfly effect in popular culture are mentioned in this Wikipedia site.
