I only know that the Timer Ball increases it's catch rate every so often, but how much originally, how much does it increase every turn, and is there a max?
Gens 3 and 4:
> ((number of turns passed in battle + 10) / 10)×, maximum 4×
Gen 5 onward:
> (1 + number of turns passed in battle * 1229/4096)×, maximum 4×