chase (someone or something) around
To chase someone or something in what becomes a circuitous pursuit. I can't chase the dog around all morning, I have to get to work! After chasing the criminal around for an hour, the police were finally able to catch him.
chase someone or something around
to follow someone or something around in pursuit. (There is an implication that the person or thing pursued is attempting to elude whatever is in pursuit.) The dog chased us around in play. It chased around all the children.