Thanks. But why are all of them about chaos?

@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.

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, 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.

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, 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.

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, 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.

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?

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.

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?

No, fluids are not a subset of Dynamical Systems theory, nor the opposite: just like biology or any other discipline whose dynamics is modeled mathematically, what exists is an overlap between them. And no, only studying fluids would be very unlikely give you a good overview of dynamical systems. For example, fluids could hardly tell you much about the dynamics of billiards, or of a system of rattling gears (both traditional dynamical systems).

Dynamical Systems is a vast, vaguely defined area, mostly straddling mathematics and physics. If you have a specific application in mind, I suggest you choose a book with the appropriate focus. For example, I just googled for 'dynamical systems in economics' and one of the first results is the book "Nonlinear Dynamical Systems in Economics | Marji Lines | Springer"; for 'dynamical systems in biology' you get "Dynamical Systems for Biological Modeling: An Introduction - CRC Press".

Actually you get too much for "biology", you probably have to be more specific nowadays.

My focus is to understand any periodic system. At the moment I'm close to finish the book Vibration and Wave from French, and I'd like to find a continuing book/area of research that can answer these questions:

You're right in trying to get your answers from books, going through them will give you a truly solid knowledge, which you wouldn't easily get otherwise.