Periodicity of chaotic trajectories of single and coupled maps in realizations of finite computer precisions

A fundamental periodicity problem of chaotic trajectories in computer realization with finite computation precision is investigated systematically by taking single and coupled Logistic maps as examples. Low-dimensional chaotic trajectories have rather short periods even with double precision computation, while the period increases rapidly when the number of coupled maps increases. Empirical exponential relations of the period and transient iterations with the computation precisions and the sizes of coupled systems are obtained, which coincide with numerically measured data in wide parameter regions. This understanding is useful for possible applications of chaos, e.g., chaos cryptography in secure communication. PACS numbers: 05.45.-a 05.45.Ra 05.45.Pq