On chaotic and random sequences

It is demonstrated that lack of (exact) predictability in the sense defined in [H.N. Nazareno, J.A. González, I.F. Costa, Propagating nature in chaotic systems, Phys. Rev. B 57 (1998) 13583–13588; J.A. González, R. Pino, Chaotic and stochastic functions, Physica A 276 (2000) 425–440; J.A. González, L.I. Reyes, J.J. Suárez, L.E. Guerrero, G. Gutiérez, From exactly solvable chaotic maps to stochastic dynamics, Physica D 178 (2003) 26–50] does not provide a reliable characterization of chaotic and random sequences. In fact, there are many regular sequences { X n ∈ R } n ≥ 0 such that there does not exist any positive integer m and any mapping g m such that X n + 1 = g m ( X n , X n − 1 , … , X n − m + 1 ) , n ≥ m − 1 , and so such sequences are not predictable in the terminology of [H.N. Nazareno, J.A. González, I.F. Costa, Propagating nature in chaotic systems, Phys. Rev. B 57 (1998) 13583–13588; J.A. González, R. Pino, Chaotic and stochastic functions, Physica A 276 (2000) 425–440; J.A. González, L.I. Reyes, J.J. Suárez, L.E. Guerrero, G. Gutiérez, From exactly solvable chaotic maps to stochastic dynamics, Physica D 178 (2003) 26–50]. Furthermore, it is shown via elementary methods that the sequence X n = sin ⁡ 2 ( π θ z n ) , n ≥ 0 , is quite simple, regular for an infinite set of nontrivially obtained irrational numbers z > 1 , at least for certain θ values, thus providing counter-examples to claims of high irregularity and randomness made in the series of interesting papers [H.N. Nazareno, J.A. González, I.F. Costa, Propagating nature in chaotic systems, Phys. Rev. B 57 (1998) 13583–13588; J.A. González, R. Pino, Chaotic and stochastic functions, Physica A 276 (2000) 425–440; J.A. González, L.I. Reyes, J.J. Suárez, L.E. Guerrero, G. Gutiérez, From exactly solvable chaotic maps to stochastic dynamics, Physica D 178 (2003) 26–50].