TY - JOUR

T1 - The asymptotic deterministic randomness

AU - Wang, Kai

AU - Pei, Wenjiang

AU - Zou, Liuhua

AU - CHEUNG, Yiu Ming

AU - He, Zhenya

N1 - Funding Information:
This work was supported by the Natural Science Foundation of China under Grant 60672095, the National Information Security Program of China Grant 2005A14, the National High Technology Project of China under Grant 2002AA143010 and 2003AA143040, the Excellent Young Teachers Program of Southeast University.

PY - 2007/8/13

Y1 - 2007/8/13

N2 - In this Letter, we focus on the deterministic randomness theory. Based on Lissajous map, which is constructed by the skewed parabola map and the non-invertible nonlinearity transform, we present conditions for generating asymptotic deterministic randomness. We rectify several popular statements, such as the function xn = sin2 (θ T zn) cannot generate deterministic randomness, and the corresponding Lyapunov exponent is ln z, etc. In other words, we prove that such function can generate deterministic randomness only when the value of parameter z belongs to some relative prime fraction number which is larger than one. We further prove that the well-known autonomous system that has been stated to generate deterministic randomness can only act as an approximative asymptotic realizable model and any realizable models for deterministic randomness will degenerate to some special high dimensional chaotic system. Furthermore, we analyze the underlying dynamics such as the fixed point, bifurcation process, Lyapunov exponent spectrum, and symbolic dynamics, etc. in detail.

AB - In this Letter, we focus on the deterministic randomness theory. Based on Lissajous map, which is constructed by the skewed parabola map and the non-invertible nonlinearity transform, we present conditions for generating asymptotic deterministic randomness. We rectify several popular statements, such as the function xn = sin2 (θ T zn) cannot generate deterministic randomness, and the corresponding Lyapunov exponent is ln z, etc. In other words, we prove that such function can generate deterministic randomness only when the value of parameter z belongs to some relative prime fraction number which is larger than one. We further prove that the well-known autonomous system that has been stated to generate deterministic randomness can only act as an approximative asymptotic realizable model and any realizable models for deterministic randomness will degenerate to some special high dimensional chaotic system. Furthermore, we analyze the underlying dynamics such as the fixed point, bifurcation process, Lyapunov exponent spectrum, and symbolic dynamics, etc. in detail.

KW - Chaos

KW - Deterministic randomness

KW - Lissajous map

UR - http://www.scopus.com/inward/record.url?scp=34547414975&partnerID=8YFLogxK

U2 - 10.1016/j.physleta.2007.03.050

DO - 10.1016/j.physleta.2007.03.050

M3 - Journal article

AN - SCOPUS:34547414975

SN - 0375-9601

VL - 368

SP - 38

EP - 47

JO - Physics Letters A

JF - Physics Letters A

IS - 1-2

ER -