A cycle was a process that occurred in a particular sequence in a continuous loop that ended with similar results each time before starting over again. The duration of time required for a cycle to complete was known a period, while the time between the completion of one cycle and the beginning of the next was known as an interval. The number of occurrences of a cycle over a specific period of time was known as the frequency.
Before Kukulkan left Earth, he gave the Mayans a remarkably accurate calendar, from which he told them to build a city according to its cycles. However, on the date that the city was finished, when Kukulkan was supposed to return, he never appeared. (TAS: "How Sharper Than a Serpent's Tooth")
The initial reading on Vaal's power source was generating alternating cycles totaling 10020 Wortham units. (TOS: "The Apple")
In 2269, the burned out star Arachna entered its cycle of strongest emissions of radiation and volume expansion. Later, when the USS Enterprise began losing power, they were unable to transmit a mayday signal to Starbase 23, so Captain James T. Kirk had Uhura reduce mayday signals to one per ten minute cycle to conserve energy. (TAS: "The Terratin Incident")
During the USS Enterprise-D visit to Iconia in 2365, the away team, consisting of Jean-Luc Picard, Data and Worf, observed that the destination that appeared in the Iconian gateway repeated. When they learned they were stranded on the planet, they considered using the gateway to return to the Enterprise. When Picard inquired how long the interval was for the Enterprise to appear in the gateway, Worf noted that it was "About four minutes, if the cycle holds." (TNG: "Contagion")
In 2366, the star Beta Stromgren experienced the final stages of an alternating cycle of expansion and collapse, as it began to become a supernova. (TNG: "Tin Man")
Technology, such as fuel inlet servos, could be caught in a cycle. (TNG: "Hollow Pursuits") Tools such as a fractionator or tricorder could be set or locked into a continuous cycle. In the case of the tricorder, being locked in a continuous cycle meant that it would record data whether it was open or closed. (TNG: "Imaginary Friend", "Schisms")
Portions of the planet Bersallis III experienced a period of fire storm activity that happened on a seven year cycle. In 2369 the Bersallin fire storms arrived eight months early, forcing the Enterprise-D to evacuate the Federation outpost there. (TNG: "Lessons")
The relationship between the sun setting and the moon rising was likened by Picard and Worf in discussing how Masaka and Korgano switched positions, "[l]ike two powerful rulers changing places in a continuous cycle." Deanna Troi then added, "And like the sun and the moon, only one of them can be in ascendance at any given time." (TNG: "Masks")
When Operation Watson was in full swing, USS Voyager was able to receive data from Earth on a monthly basis. This was accomplished by using a cyclic pulsar to amplify signals from the MIDAS array. The cycle peaked every thirty-two days, and lasted for about seventeen hours. (VOY: "Life Line")