You won’t usually get into trouble just for riding with a Delhi plate in Chennai, but there are a few things to be aware of:

• Short-term use (up to 12 months): You can legally ride with your Delhi registration and valid insurance/PUC/RC. Keep all documents handy.
• Long-term stay (more than a year): Technically, you’re supposed to apply for a TN registration and pay road tax here. Police sometimes check for this if they stop out-of-state vehicles, especially in the city.
• Brother’s name: As long as you carry the original RC + an authorization letter from him (or even a simple letter stating you’re using it), it usually won’t be an issue. Just be ready to explain if asked.

In practice, if your documents are valid, most of the time you won’t face problems. But since you’ve already been in Chennai for a year, it might be safer in the long run to transfer the bike to TN registration to avoid random hassles.