Why did Europeans, and not Asians, undertake the voyages of discovery connecting Old World with new?

Bureaucrats. China has always been extremely devoted to extremely centralized government. If you think about it, you realize that centralized government is impossible; the emperor gets the glory but government has to be turned over to bureaucrats. Bureaucrats are very sensitive about protecting their jobs. So at some point the bureaucrats felt threatened, so they scuttled the entire navy to protect their positions. China had no boats for several centuries. And that is why China never explored the oceans at all. It is only recently that China has begun to build a navy again.