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What textbook would you recommend for a bachelor in theoretical physics to study dynamical systems theory? I don't want to focus too much on chaos, just having a broad view of every interesting characteristic is enough. Physical meaning behinds equations should be explained.

Some related resources:

Qmechanic
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Ooker
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1 Answers1

8

In no specific order:

  • Alligood K.T., Sauer T.D., Yorke J.A, Chaos. An Introduction to Dynamical Systems

That's a personal favorite of mine at the undergraduate level. It's clearly written and they strike a great physics/math balance, including from (a few) mathematical proofs to "computer experiments".

  • Tél T., Gruiz M., Chaotic dynamics. An introduction based on classical mechanics

Highly recommended. Also aimed the the undergraduate level, it's very clear conceptually and strives to make the math accessible. It's a newer book (2006) that includes current topics.

  • Ott E., Chaos in Dynamical Systems

A classic that cannot be missed. It's aimed at the graduate level, but it's pretty accessible and especially useful when you need to get to the details of some specific topic.

  • Strogatz S.H., Nonlinear Dynamics And Chaos: With Applications to Physics, Biology, Chemistry, and Engineering

It's explicitly aimed at newcomers and has only calculus and introductory physics as prerequisites. The title "applications" include "love affairs" as 2-D flows and, possibly very interesting, the author lectures are available on Youtube.

  • Cvitanović P., Artuso R., Mainieri R., Tanner G., and Vattay G., Chaos: Classical and Quantum $-$ ChaosBook.org

That's a very interesting freely available on-line graduate textbook. It takes a fresh approach to the subject and "aims to bridge the gap between the physics and mathematics dynamical systems literature".

stafusa
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  • Thanks. But why are all of them about chaos? – Ooker Sep 02 '17 at 16:12
  • @Ooker, to some degree it's my own bias, since that's my field of research. But I believe that complex systems (chaos) is also the branch of physics that most regularly uses the concept of fractals, as it comes about quite often: in the boundary between phase space regions that correspond to different behaviors ("basins of attraction") and in the typical geometry of chaotic attractors, for instance. – stafusa Sep 02 '17 at 19:10
  • But as I understand from the book Complexity: A Guided Tour, complex system science does not completely deal with chaos, and it's an interdisciplinary field, not just a branch of physics. And while dynamical system theory originated from the three-body problem, it's actually a branch of mathematics, and its scope surely is broader than just chaos? Please correct me if I'm wrong. – Ooker Sep 03 '17 at 16:26
  • @Ooker, of course you're correct. But your question asks specifically for "dynamical systems" (not complex systems in general) and already mentions the mathematics and complex systems lists, so I avoided those. It also asks for "physical meaning to be explained", which is easier to find in physics texts. Besides, afaik, outside mathematics per se, physicists are the ones working most often with complex systems, even when applied to biology, economy, engineering, etc. – stafusa Sep 03 '17 at 17:16
  • oh, so you mean dynamical systems without complex systems are just chaos? My last comment was to reply to the bit that complex systems are about chaos in your first comment. – Ooker Sep 03 '17 at 19:14
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    @Ooker, oh, ok, that part of the comment is indeed misleading. The definitions are pretty vague, but usually "complex system" is the biggest category, with most of "dynamical systems" in it, together with networks, emergence, etc. And, "dynamical systems", even as done by physicists, includes more than chaos: e.g., bifurcation theory and even linear systems, but I think chaos is the most common research subject. – stafusa Sep 03 '17 at 19:43
  • Thanks. Before dwelling into those books, can you tell me the relation between this and analytical mechanics? Is one an extension of the other? – Ooker Sep 04 '17 at 06:21
  • @Ooker, Lagrangian and Hamiltonian mechanics are different formulations of mechanics (mostly) equivalent to Newtonian mechanics. Depending on the problem, one formalism or the other might be a better choice (a bit like the right choice of coordinate system might make a problem resolution easier). For conservative systems, for example, the Hamiltonian formulation is often advantageous. – stafusa Sep 04 '17 at 08:03
  • So what formalism is mostly used in dynamical systems theory? I guess that the lagrangian one cause it doesn't deal much with vectors or conservative systems, am I right? Also, is fluid dynamics a child branch of it? – Ooker Sep 04 '17 at 14:01
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    Fluids are complex systems, no doubt, and advection of particles, even in a periodic flow can be chaotic. Lagrangian is actually the formalism I've seen the least being used. While the Hamiltonian formalism dominates the description of conservative systems, usually the Newtonian mechanics is used to describe a forced pendulum or an engineering model. The field of dynamical systems only cares about the behavior of the system - which formalism one employs to obtain the equations of motion (or whether it's a mechanical system at all) is secondary. – stafusa Sep 04 '17 at 15:35
  • Since fluid is a subset of dynamical systems, do you think it would cover most topics from the latter? Would studying fluids only allow me to see analogies in other systems like biology or economics, or do I really need to learn dynamical systems theory to get some insights on them? – Ooker Sep 04 '17 at 15:58
  • Let us continue this discussion in chat. – stafusa Sep 04 '17 at 16:34