ABSTRACT

The historical Li–Yorke theorem gives a criterion for the existence of nonperiodic orbits in a one-dimensional map. The topological entropy and the Lyapunov number describe the folding and stretching property of a chaotic map and are important in practice. The paper ‘Period three implies chaos’ by Li and Yorke, published in 1975, had enomous impact in the research of chaos. In reality, the Lyapunov number is related to the Kolmogorov entropy, which is different from the topological entropy. The orbit of an observable chaos, when it appears, fills a certain region densely. The positivity of the Lyapunov number is most often employed as a criterion for chaos in the analysis of an irregular wave generated by a computer, experiment or observation. In fact, this condition is an excellent tool in judging an oscillatory wave being irregular or an orbit being complex.