The adverb doggedly means tenaciously or with strong determination. If your dog ran away, you might doggedly pursue him across the park, down the road, and through the woods.
Doggedly is the adverb form of the adjective dogged. In the 1300s, both words meant having the negative qualities of a dog, or mean and cruel. By the late 1700s, both had evolved to mean persistent and with determination. If you doggedly insisted on your innocence, despite the evidence linking you to your brother's missing baseball mitt, you might then search doggedly for the mitt, not stopping until you found it in the dog house.