How did Muhammad's death prove chaotic to the Islamic world?
The death of Muhammad in the seventh century effectively created divisions within the Islamic world that persist to this day. Muhammad's death left a power vacuum, the solution for which Muhammad did not provide. He did not name a successor explicitly. For this reason, the Islamic world, which at that point encompassed much of Central Asia and most of North Africa, had to make an important decision - a decision that would prove fateful in future centuries.
Some believed that the successor to Muhammad should have be selected from among the four caliphs; the most worthy would be named successor. Those who followed this view became known as Sunnis. Others, however, believed that the successor should come from within Muhammad's own family; only the fourth caliph and Muhammad's cousin/son-in-law, Ali, should be named his successor. Those who followed this belief became known as Shi-a. Ultimately, the chaos created by Muhammad's death which resulted in the fracturing of Islam derived from a purely political consideration.
What truly proved problematic for the Islamic world was the choosing of sides. Whether one chose to be a Sunni or a Shi-a mattered far more than whether one was Muslim or not. Rather than having the emergence of unified Islamic caliphates, Sunni and Shi-a caliphates would come and go throughout North Africa, Spain, Arabia, Central Asia, India, and elsewhere through the Near and Middle East. The division created by Muhammad's death certainly altered the course of Islamic history.