Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahrzeits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah. In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations.
This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years.
Adar I is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle.
The current cycle began in Jewish year 5758 (the year that began October 2, 1997).
A few years ago, I was in a synagogue, and I overheard one man ask another, "When is Chanukkah this year?