Your tenant and you are bound by the signed lease. The pet addendum you sent to you is merely an offer to modify contract. As long as you send him a written withdrawal of offer, the addendum would be invalid.
What you need to do is to write a letter of demand and hand delivery to him, as follows.
Dear Tenant,
[basic format, cite lease as the reason for your demand, then tell him what he did wrong]
According to lease, you are not allowed to have dogs without my permission (find section in the lease and cite section number). Since you are keeping a dog and didn't sign the pet addendum, I am asking you to stop keeping the dog on premise. The pet addendum I sent on AA/BB date is hereby withdrawn.
According to lease, the curing period for lease violation is XX days (find curing section in the lease and cite section number)). Therefore, if you continue to keep dog on premise after such and such date (today + XX), I will start eviction process according to lease ( find eviction section in the lease and cite section number)
The communication has to be in writing to be effective.