The asymptotic deterministic randomness

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 x_n = 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.