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I am currently an undergrad math major taking a gap year (because my university is entirely online this semester). For this year, I have signed up for a "Directed Reading Program" with a graduate student whose specialty involves dynamical systems. For this program, I am supposed to read through a textbook that we can discuss. Two of her suggestions were Nonlinear Dynamics and Chaos by Steven H. Strogatz and Differential Equations, Dynamical Systems, and an Introduction to Chaos by Hirsch, Smale, and Devaney.

As I took a look at those books, I realized an additional reason why reading a book such as those could be useful: although I took a differential equations course at my local community college when I was in high school, I don't remember them all that well. My university's math department is very theory-oriented, so I may never have the opportunity to take a DiffEQ course as an undergrad, though, as a math major who may want to go into something more applied, I feel as though a high level of comfort with differential equations would be nice to have. Looking through the two textbooks online, neither appears to cover Laplace transforms, which I remember to have been an entire unit in my community-college course. Because of this, I am having doubts about the efficacy of the two books with respect to giving me said comfort. However, the books seem to be fantastic with respect to gaining a deeper understanding of the material, so I am not trying to criticize.

Two questions:

  1. Between Strogatz and Hirsch/Smale/Devaney, which would you recommend?
  2. In light of the above (the lack of coverage of topics such as Laplace transforms), do you think I ought to, in addition to one of those two books, spend time with Ordinary Differential Equations by Tenenbaum and Pollard (which I got for Christmas or something awhile back but haven't spent much time with)?
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  • $\begingroup$ I bought Jordan and Smith (third edition) as this site sees lots of questions on periodic solutions, in any case several variable $\endgroup$
    – Will Jagy
    Oct 21, 2020 at 21:02
  • $\begingroup$ google.com/books/edition/… has preview $\endgroup$
    – Will Jagy
    Oct 21, 2020 at 21:13
  • $\begingroup$ Those books are both great. Read them both, or at least read parts of both. (In general you don't have to commit to a single math book. You can read multiple books simultaneously.) Also, I suspect you don't need the Laplace transform at all. It is likely to be irrelevant to what this grad student wants to work on. I say don't get sidetracked by the Laplace transform. Focus on the two books the grad student recommended. $\endgroup$
    – littleO
    Nov 6, 2020 at 6:57

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Let me preface by saying that analysis of ODEs runs along two different directions. The first is to solve ODEs, in the sense that you end up with exact or approximate solutions of the ODE. This is taught in a first course, in terms of manipulation of simpler ODEs such as first and second order with constant coefficients or numerical approximations of solutions. Application of Laplace transforms is also limited to linear ODEs afaik.

The second direction takes an abstract setting and fits into a larger study called dynamical systems. One no longer looks for solutions of the ODEs in the sense of a closed form description of a function. Instead the focus is on the qualitative behavior of solutions. The first two books you listed mostly take this approach. The reason for the abstract setting is two fold: It gives you a simple way of dealing with a large class of equations. But more importantly, once you start looking at nonlinear ODEs, the "mode" of analysis changes drastically. For example, instead of solutions, one looks to qualify solutions based on their long term behavior. This leads to notions of stable and unstable solutions.

Although I cannot answer your first question, as I haven't read these books, I can say that I've heard good things about both. In any case, spending time with basic books like Tenenbaum and Pollard will be beneficial, especially if you do not remember much from your first course.

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