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I am close to finish the book Vibration and Wave by French, and I would like to know which branches of physics can answer these groups of questions:

  • Defining questions: Can a periodic event be described as a (hidden/imaginary) oscillator, even when its causes are still unclear? When can't such system be approximated by simple harmonic oscillator?
    For example, assuming we haven't discovered the rotations of the Earth, can we theorize the periodicity of seasons (spring-summer-autumn-winter-spring...) as an oscillator or a wave?

  • Dynamical questions: Is there a phenomenon that has an underlying dynamical process that is different to its surface (like a sea)? When that process goes up, how would it be? How does turbulence spreads out in a media, and how to control it?

My feeling is that the first group is best answered by Dynamical Systems Theory, and the second group is best for Fluid Dynamics, hence I'm waving between the two. Fluid is a complex system and is dynamical, "but, so far, the methods and results of dynamical system have not been up to the job of fulling explaining turbulence." Learning both is of course preferable, but which one covers most questions?

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Your "feeling" is correct: Dynamical Systems Theory and Dynamics of Fluids will roughly cover your questions.

but which one covers most questions?

More relevant then the number of questions covered (which will anyway change as you study any subject) is what you consider the most important question? If your real interest is turbulence, than starting with fluid dynamics is the obvious path. If rather general dynamical systems interest you the most, then of course that's what you should study.

Minimum prerequisites for learning both these theories together are vector calculus and a good grasp of introductory undergraduate physics (check, e.g., Halliday, vol. 1), though you're not unlikely to need to learn at some point small bits of topology, thermodynamics, partial differential equations, statistical physics, number theory, etc. There are some general Book recommendations available, in particular: 1) Self-study book for dynamical systems theory? and 2) Book recommendations for fluid dynamics self study.

Some of the other more specific questions are a bit vague, possibly ill posed, but my take on them is:

Can a periodic event be described as a (hidden/imaginary) oscillator, even when its causes are still unclear?

Yes. You can describe periodic phenomena with the machinery of oscillators. It's perhaps ideal to obtain descriptions from known underlying physics, but that's not always possible, in which case we resort to phenomenological models (which is my interpretation of the "hidden/imaginary" bit).

When can't such system be approximated by simple harmonic oscillator?

Whenever nonlinearities aren't negligible. The simple pendulum is not approximated by the SHO for large angles. More examples and discussion: Why can some oscillations be modeled by Simple Harmonic Motion, while others cannot? and Why is the harmonic oscillator so important? .

can we theorize the periodic characteristic of seasons (spring-summer-autumn-winter-spring...) as an oscillator or a wave?

Yes. That's actually related to the fact that the projection (think "shadow") of a circular movement is a harmonic oscillation.

Is there a phenomenon that has an underlying dynamical process that is different to its surface (like a sea)? When that process goes up, how would it be?

I assume you mean whether it can happen that the underlying physics is not obvious to the observation: yes, certainly. And, by hypothesis, you can't tell in general what it'll turn out to be.

How does a turbulence spreads out in a media, and how to control it?

Lower Reynolds numbers are associated with laminar (i.e., non-turbulent) flows, so you can try to prevent turbulence from forming by reducing the Reynolds number of the flow. In general, we cannot control turbulence, but there are methods to achieve partial control. A review on the subject from an engineering perspective is Lumley and Blossey's CONTROL OF TURBULENCE.

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  • $\begingroup$ Can all dynamical systems, regardless of their equations, be described with Fourier transform? $\endgroup$ – Ooker Nov 8 '17 at 3:40
  • $\begingroup$ @Ooker, Any dynamical system produces some sort of time series, to which a Fourier transform can be applied - but that would rather be an analysis of the system, rather than a description. $\endgroup$ – stafusa Nov 8 '17 at 8:49
  • $\begingroup$ Yes, I find out the correct name for "hidden/imaginary oscillators" is Fourier series. But since dynamical systems usually involve a system of differential equations, so I wonder if Fourier analysis can be applied in them. But if it can be applied everywhere, then if I only need to know about the equilibria/fixed points of the system, isn't it a little easy too work with? $\endgroup$ – Ooker Nov 8 '17 at 14:58
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    $\begingroup$ @ooker, It really depends on what exactly you want to do. If all you need are the periodic fixed points of low dimensional system, and you're given its equations, it might be easy, but in general it isn't. $\endgroup$ – stafusa Nov 8 '17 at 16:30

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