Statistical analysis of symbolic dynamics in weakly coupled chaotic oscillators

The difference between phases of weakly coupled chaotic oscillators fluctuates around its average value similar to a random walk. For a very weak coupling strength, the phase difference has the same stochastic properties as a Brownian motion characterized by a -2 scaling exponent in its power spectrum, and for stronger coupling it behaves as a pink noise with a close to -1 power law. Ordinary methods of stochastic analysis based on the Fourier spectrum and detrended fluctuation analysis are not able to distinguish determinism from the time series of this phase drift. Nevertheless, determinism can be revealed by the method of ordinal pattern symbolization which allows finding forbidden patterns in the time series. The efficiency of this approach is proven with the Brownian motion, where non-occurring patterns are also detected. The robustness of the method to noise is demonstrated with coupled Rössler oscillators. The stochastic properties of phase fluctuations can be promising for cryptography and secure communication using chaotic systems.