As you accelerate its increases power somewhat gradually then at a certain point the car goes into a type of boost almost. What is happening at this point?
Is it a VTEC engine? Turbo?
I agree that this sounds like an engine with variable valve timing, or…perhaps…an engine with a turbo charger.
Is there any chance that the OP would like to reveal the make, model, and model year of his/her mystery vehicle, so that a more conclusive answer is possible?
Maybe the automatic transmission is downshifting to provide more acceleration. When the driver pushes hard on the pedal, that’s what it is supposed to do. Do you notice the engine rpm increase rapidly when this happens?
Until the OP provides real information suggestions are basically worthless.