We know that present Vyasa is Krishna Dwaipayana. There is a detail about him in Mahabharata. But who is Badarayana Vyasa and what is his story? In which mahayuga was he the Vyasa?
Badarayana and Veda Vyasa are one and the same.The reason why Vyasa came to be known by the Badarayana name is explained in the following passage:
In the Brahmasutra, on which there are commentaries according to the various philosophical schools, Vyasa presents in an extremely terse form the substance of the ten (principal) Upanisads. Since he dwelt under the badari tree (jujube) he came to be called "Badarayana" and his work became well-known as "Badarayana-sutra".