Appropriateness of question on delay differential equations

Question

Would my question below, posted on Mathematics Stack Exchange in May 2015 and having received no answers, just a couple of upvotes, be appropriate in level for MO? I recognize that cross posting is frowned on, but perhaps that is mitigated somewhat by waiting almost a year.

Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the delay logistic equation $\frac{dN}{dt}=rN(1-\frac{N(t-\tau)}{K})$ and one-dimensional discrete dynamical systems such as the logistic map $x_{n+1}=rx_n(1-x_n)$, whereas this does not occur for first-order autonomous ordinary differential equations like the logistic equation $\frac{dN}{dt}=rN(1-\frac{N}{K})$)? Does the time-delay $\tau$ play a similar role to the discrete time-step in allowing solutions to break free of the restrictions of monotonic flow on a line? If so, can this be expressed precisely, or it merely a loose analogy?

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Answer 1

IMO, yes, that's definitely a graduate-level topic, and so appropriate for MO in that respect. (And I say that as a grad student who's studied delay equations a little.)

AFAIK, cross-posting should not be a problem in a case like this, if disclosed properly (i.e. with links both ways). You could also consider simply deleting your question on MSE, since it hasn't received any answers there, and reposting it here. (It's still useful to briefly mention that it's a repost, just in case e.g. somebody saw it earlier on MSE and thinks "Hey, this looks familiar...")

That said, I also agree with Gerhard Paseman that your question could use some rephrasing for clarity. While I understand that it's fundamentally a "fuzzy" question (in the sense that you've noticed a vague similarity between two types of systems, and you're asking whether there's something that can be formalized behind it), you could at least try to rearrange it a bit to more clearly express what the similarity you've noticed is.

In this particular case, you might want to try structuring your question something like this:

First-order autonomous ODEs like ... have this property (that the orbits cannot cross), simplifying their analysis (especially in the 1D case), since...

However, this property does not necessarily hold for analogous discrete-time dynamical systems like ..., since...

Delay differential equations like ... also lack this property, and it seems to me that the reason is, in some sense, the same as for the discrete-time case: ... However, I'm not aware of any formal statement of this, and I'm having trouble coming up with one myself.

Is there some way to formalize this similarity between DDEs and discrete-time difference equations?

(Alas, I also kind of suspect that the answer to your question may turn out to be disappointing; it may be that the only connection between discrete-time difference equations and delay differential equations in this respect is simply that they're both not ODEs, and so lack some of the special properties of ODE systems. That said, it's also possible that some of the smart minds here might know of some deeper connection, so it may still be worth a try.)