Due to multiple factors, expect to potentially pay more when booking in December. For low rates, August can be a great choice for travel to Madurai. Flights to Madurai from Chicago O'Hare Intl Airport in August are about $1,043 on average during August, but can be found for as low as $686.
Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Chicago O'Hare Intl Airport to Madurai as you normally would.