I am afraid that the task you describe as "obtaining a clear picture" of PDE, Mathematical physics and dynamical systems is impossible. I doubt that there are people who have a "clear picture" of all these three fields. Each of them is enormous. My advise is to find an adviser who works in the broad area of analysis, and to rely on his directions for the study of some specific area. Of course you can read about the broader area, and perhaps after 40 years of work and reading you will obtain some general picture:-)

For a general introduction to dynamical systems, I recommend the book of Katok and Hasselblatt, Introduction to modern theory of dynamical systems, or another book of the same authors, A first course in dynamics, with a panorama of recent developments.

I don't think there exists a modern survey of the whole mathematical physics. But Reed and Simon 4 volume course, Methods of Mathematical Physics gives an introduction to some parts of it.