I find the logistic map absolutely fascinating. Both in itself (because I love fractal) and because it is observed in nature (see: https://www.youtube.com/watch?v=ovJcsL7vyrk).
I'm wondering if anyone tried it as an activation function in some way or another with any kind of success.
I like it because it has some kind of "I'm not sure what to do" above ~3.0 and the less confidence the more chaotic the response is. It gives the possibility to explore some other solution to escape a local optimum (not sure I use this word correctly). And below 3 it's still a nice and smooth activation function like, eg, a tanh.
Eg : the reward i got isn't the reward i expect, and the higher the difference the more i'll explore other solution. But it's still gradual, from 1 choice, to 2 choice, 4, 8, 16, ... until it become chaotic. (giving the possibility to experiment some pseudo-random choice). And below this threshold it still act as a usable "good old" activation function.
Another good side is that it's gpu-friendly and don't need many iteration for this application since a little bit of uncertainty (even below the threshold) isn't undesirable. see : https://upload.wikimedia.org/wikipedia/commons/6/63/Logistic_Map_Animation.gif
Edit : so, ok, i tested it on my extremely naive racetrack. (feedforward, no feedback, no error, no fitness, only genetic selection for the car that didn't crash). It does work, for sure. I don't see any advantage in practive but with such a naive NN, there isn't much i can tell.
My implementation :
def logi(r): x = .6 # the initial population doesn't matter so i took .6 for _ in range(random.randrange(10,40)): x = r * x * (1 - x) return x
The activation take 8% of my laptop cpu (while is was invisible on my radar with leaky leru)