A "cycle menu" simply refers to a set of meals that rotates through a period of time. The senior living complex where my parents lived had a four week menu cycle in the dining room. Week one had different meal offerings for breakfasts, lunches and dinners each day than did the menus for weeks two, three or four. Obviously, the length of time covered by the cycle can be established by the facility or restaurant or wherever the food is being provided.
A "fixed menu" presents a selection of meals for a set price. There is generally no substitution allowed regarding what is included in the meal, but the price of the meal is usually considered extremely reasonable. The concept is common in Europe, usually known by the French name of "prix fixe" (price fixed) menu.
Contrasting the two is difficult because they are address different concerns. A restaurant could develop a series of fixed menus that are presented on a cyclical basis.