Yes, more than likely the swans you describe are the Mute Swans (orange bills).

This is very normal behavior in all species of swans. When juvenile swans are told to get lost in preparation for the next nesting season, the parents will chase the young juveniles from their habitat. This encourages the young birds to go out find a habitat of their own, choose a mate and begin their adult lives.

In the case you described, the bird chasing the other swan was probably an established adult whose habitat was being protected. Either it was an adult chasing its own young, or a wandering swan had entered into the swan's established territory and it was being chased for protection of its nesting site, mate and young. The interloper will be chased until it gets the picture that this is another swan's territory and it needs to move along. You will see this behavior more common once the mating/nesting season begins (March-April) The Regal Swan