Zykluszahl

Zykluszahl, in German, literally means "cycle number" and is used in mathematics and computer science to denote the length of a repeating cycle in a discrete process. In a sequence x0, x1, x2, ... defined by x_{n+1} = f(x_n) on a finite or bounded state space, the trajectory eventually becomes periodic: after a transient part, the values repeat with some period p. The smallest positive p with x_{n+p} = x_n for all n large enough is called the Zykluszahl of the orbit (or the cycle length). If the initial state lies on a cycle, the Zykluszahl is simply the length of that cycle.

In finite-state systems, every orbit eventually enters a cycle; the pair (mu, p) describes the structure: mu is the length of the transient phase, p the cycle length; the total time until the orbit repeats from the start is mu + p. The Zykluszahl thus captures how long it takes for repetition to occur and how long the repeating pattern lasts.

Applications include analysis of pseudorandom number generators, cellular automata, and time-series models where cycle behavior is relevant. The concept helps assess predictability and structure of the dynamics: a longer Zykluszahl often implies longer mixing or less predictability, though other factors matter.

Cycle-detection methods, such as Floyd's Tortoise and Hare and Brent's algorithm, are used to determine the Zykluszahl efficiently without enumerating all states. Terminology-wise, Zykluszahl is a direct translation of "cycle number"; in English-language literature, terms like cycle length or period are more common.