A safety switch in the furnace typically is the cause of the air handler blowing and stopping like yours does. If something malfunctions during the ignition of the furnace, there is a safety that turns off the gas valve and turns the fan on to cool the heat exchanger down. It differs between manufacturers, but sometimes after the fan runs for a short while it cycles everything off, then tries the ignition sequence again. If it malfunctions again, it will repeat the cycle.