@Gro-Tsen your interesting example of the Cantor set is a motivation to assigne a real number to every positive terms series: the Hausdorff dimension of the set of all subseries values.

@Gro-Tsen Is $\sum (1/2^n) $ is some what an exception? I mean what can be said about the space of all positive sequence whith connected subseries values?

Is $\sum (1/2^n) $ is some what an exception? I mean what can be said about the space of all positive sequence whith connected subseries values?

Thank you for your answer. Under what condition the answer is affirmative? is there a terminology for positive series with connected set of all possible sub series value?

@Gro-Tsen so under what condition the answer is affirmative?

@Gro-Tsen Thank you for this example

@მამუკაჯიბლაძე I wrote a reply in your answer I think I misunderstood the OP

I am sorry for my mistake. misread some thing in the OP: I confused and misread $ax^2+bxy$ by $ax+bxy$. I belive that the two systems are not topological equivalent: The OP system is a Homogenous equation invariant under rescalling But the Lotka Volterra system has a center as wiikipedia indicate to. So a center in the first quadrant is not invariant under rescalling $(x,y)\mapsto (\lambda x, \lambda y)$

mathoverflow.net/questions/277481/…

+1 for your effort on this question. The OP concernes the explicite solution and your answer do that. However a geometric approach could be consideration of a metric compatible to the Lotka Volterra system. as you find in the wikipedia link I provided in my answer the system has a center so one can think to an explicite Riemannian ,metric compatible to the system

arxiv.org/pdf/math/0409594.pdf

@მამუკაჯიბლაძე Any way such kind of chang of coordinates is very common in ODE see for example page 2 of this paper $\pi(x,y)=(x^2,y)$

@მამუკაჯიბლაძე Any way such kind of chang of coordinates is very common in ODE see for example page 2 of this paper $\pi(x,y)=(x^2,y)$

The system in OP is itself in the Lotka Volterra form. Please see the wikipedia link