Question

Is zero-length line with an arrow in TikZ a bug?

documentclass[tikz,border=12pt,12pt]{standalone}

begin{document}

begin{tikzpicture}

    draw[->](1,1)--(1,1);

end{tikzpicture}

end{document}

Note: PSTricks behaves the same.

I am trying to find whether it is mathematically useful (at least for me) but in fact it is not.

The direction of the arrow does not converge. :-)

documentclass[tikz,border=12pt,12pt]{standalone}

begin{document}

foreach i in {6,5,...,1}{%

begin{tikzpicture}

     draw (0,0) grid (6,6);

     draw[->,ultra thick,red](45:1)--(45:i);

end{tikzpicture}}

end{document}

In agreement with the comments by Joseph Wright and Harald Hanche-Olsen, I would say that the answer to your question is

no, it is not a bug.

Usually, a bug is some behavior that is different than what one expects from what is written in the manual. As pointed out by Paul Gaborit (Thanks!!!), on p. 183 of the pgfmanual there is a distinction between proper and non-proper tips. All you need to do is to add tips=proper to get

documentclass[tikz,border=12pt]{standalone}

begin{document}

begin{tikzpicture}

draw[->](1,1)--(1,1);

begin{scope}[tips=proper,xshift=1cm]

    draw[->](1,1)--(1,1);

end{scope} 

end{tikzpicture}

end{document}

As you can see, the second arrow gets suppressed completely, which is, judging from your question and comments, the behavior you feel is appropriate. So all you need to do in your TikZ pictures is to add this style. (If you want it for all your pictures, just do tikzset{every picture/.append style={tips=proper}}.)

Naively, you may think that TikZ could produce an error if the information is not sufficient, as it is if one does not specify the direction of the arrow. However, there is no error message. Rather, if there is no well-defined direction, TikZ will find one for you. I'd call that a feature.

Also, I cannot see the point of the animation. Clearly, TikZ (nor any other tool I know of) has knowledge on what happened in the previous frame (without further ado) . The last frame of your animation shows an arrow of length 0, regardless of whether the previous frames had arrows or ducks in. How is there a limit?