True and false forbidden patterns in deterministic and random dynamicsJ. M. Amigó1, S. Zambrano2 and M. A. F. Sanjuán2
1 Centro de Investigación Operativa, Universidad Miguel Hernández - Avda. de la Universidad s/n. 03202 Elche, Spain
2 Departamento de Fásica, Universidad Rey Juan Carlos - Tulipáns/n 28933 Madrid, Spain
received 27 April 2007; accepted in final form 5 July 2007; published September 2007
published online 27 July 2007
In this letter we discuss some properties of order patterns both in deterministic and random orbit generation. As it turns out, the orbits of one-dimensional maps have always forbidden patterns, i.e., order patterns that cannot occur, in contrast with random time series, in which any order pattern appears with probability one. However, finite random sequences may exhibit "false" forbidden patterns with non-vanishing probability. In this case, forbidden patterns decay with the sequence length, thus unveiling the random nature of the sequence. Last but not least, true forbidden patterns are robust against noise and disintegrate with a rate that depends on the noise level. These properties can be embodied in a simple method to distinguish deterministic, finite time series with very high levels of observational noise, from random ones. We present numerical evidence for white noise.
05.45.Tp - Time series analysis.
05.40.Ca - Noise.
02.50.-r - Probability theory, stochastic processes, and statistics.
© Europhysics Letters Association 2007