Donnie Darko doesn't do any time traveling... he has a premonition of what will happen if he doesn't stay in bed.
The only thing that does any time traveling is the bit of the plane that gets sucked through a worm hole. In the future his mum and sister would have been on that plane, his girlfriend would have been crushed by the car and he would have shot the guy dressed as a rabbit. All because of events he put into motion.
He sacrifices himself, he basically has a dream that will become reality if he doesn't. He somehow dreams a little piece of Chaos theory (probably because there is a giant wormhole above his bedroom!!)
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