Efficient detection of the quasi-periodic route to chaos in discrete maps by the three-state test

The three-state test (3ST) is a method based on ordinal pattern analysis for detecting chaos and determining the period in time series. For some well-known chaotic dynamical systems we showed that the test behaves similar to Lyapunov exponents. However, the 3ST is detecting quasi-periodic motions both as regular and non-regular. In this paper, we propose to use the sensitivity of its chaos indicator λ to the time delay for clear discernment between quasi-periodic and chaotic dynamics. Simulation results obtained using the logistic map and the sine-circle map attest that the sensitivity of λ to the time delay is sufficient for the detection of the periodic and quasi-periodic route to chaos. A comparison with the permutation entropy confirms the effectiveness of the 3ST for the analysis of discrete time series data.