I'm assuming that by rivalry, you mean civil wars, and not some sort of recorded rivalry in a different field (athletics, games, etc.). Throughout the Late Republican Period and the Empire, civil wars were pretty common, such as during the Fall of the Republic when Julius Caesar defeated Pompeii, and Octavian defeated Mark Antony, or the Crisis of the Third Century, two parts of Roman history that really stick out and are just plagued by Civil War. the precedent for civil war was set by Lucius Sulla, who was the first Roman general to march on Rome, against Gaius Marius. The rivalries in the Empire's history pre-476, definitely culminated in 271AD, when the Empire was actually split into three groups; the Gallic Empire in Gaul and Britannia, the Palmyrene Empire in the East, and the Roman Empire retaining Italy, the Balkans, Iberia and North Africa, except Egypt. The two other Empire's did not last long; after Gallienus, ruler in Rome, died in 268, was replaced by Cladius II who ruled until 270, then Quintillus who ruled for a couple months (about 170 days) in 270, then finally an elder, respected Roman general came to the throne by the name Aurelian, and by 273, he would re-unite Rome while defeating several German tribes.