Insight on a particular class of curves. (1 Viewer)

glittergal96

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Nothing all that special is going on here, you will get this kind of behaviour with a lot of functions.
I will just give a brief partial answer illustrating where some of the properties come from.

Noting that f(x)=x^x has a minimum at x=1/e, we have x^x+y^y >= 2f(1/e) = 1.3844...

So for k_1 smaller than this quantity, our first curve will vanish, and for k_1 slightly larger than this quantity we expect our curve to be close to (1/e,1/e). In fact, as we are looking at level sets near a local minimum, these curves will look like ellipses about the minimum. (You can find the aspect ratio of these ellipses in terms of the second order local behaviour of x^x+y^y. I haven't done this but it might even be circular as you are suggesting.)

A similar phenomenon is going on with g(x)=x^(-x), it is just a local maximum at 1/e instead with 2g(1/e)=2.8893...

With curves 2 and 4, we still have a stationary point at (1/e,1/e), but this time it is a saddle point. When you take the preimage of saddle points you will get things that look like hyperbolas (again you can look at second order behaviour to find exactly what these hyperbola are). This explains the two branches meeting tangentially in the limit.

Analogous things happen with your second quadruple of functions.
 
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Paradoxica

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Nothing all that special is going on here, you will get this kind of behaviour with a lot of functions.
I will just give a brief partial answer illustrating where some of the properties come from.

Noting that f(x)=x^x has a minimum at x=1/e, we have x^x+y^y >= 2f(1/e) = 1.3844...

So for k_1 smaller than this quantity, our first curve will vanish, and for k_1 slightly larger than this quantity we expect our curve to be close to (1/e,1/e). In fact, as we are looking at level sets near a local minimum, these curves will look like ellipses about the minimum. (You can find the aspect ratio of these ellipses in terms of the second order local behaviour of x^x+y^y. I haven't done this but it might even be circular as you are suggesting.)

A similar phenomenon is going on with g(x)=x^(-x), it is just a local maximum at 1/e instead with 2g(1/e)=2.8893...

With curves 2 and 4, we still have a stationary point at (1/e,1/e), but this time it is a saddle point. When you take the preimage of saddle points you will get things that look like hyperbolas (again you can look at second order behaviour to find exactly what these hyperbola are). This explains the two branches meeting tangentially in the limit.

Analogous things happen with your second quadruple of functions.
At the exact point where the hyperbola branches degenerate, what function best describes the smooth intersecting curves?
 

glittergal96

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At the exact point where the hyperbola branches degenerate, what function best describes the smooth intersecting curves?
Well if I wanted to describe/study that union of two curves I would literally just use x^y+y^x-1.3844... . The branches should be symmetric, so finding a way of representing a single one of them but not the other probably wouldn't be particularly useful.

As for what this looks like globally, I am not sure (have you tried using something like mathematica?).

I don't really see a reason that the answer will necessarily be nice (where "nice" means can be parametrised using elementary functions or is at least very simple geometrically).
 

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