最後，由於在參數首次進入混沌前的倍週期區間，水平視線演算法是無法分辨不可逆性的，促使我們將差值的因素放入演算法中（ modified-Horizontal Visibility graph ），使得前述的倍週期不可逆性可以顯現，但差值的效果卻也破壞了原本演算法所顯現的碎形特徵。;In recent years, the so-called visibility algorithm has been used in many areas in nonlinear sciences by capturing the correlations of the time series through constructing the corresponding network. Using the directed horizontal visibility algorithm, the time-series can be mapped to a directed network system, and the in-degree and out-degree distributions can be calculated, and it is proposed that the irreversibility of the dynamics can be measured quantitatively by the Kullback-Leibler divergences of these degree distributions.
Here, by using the directed horizontal visibility graph, we focus on the irreversible dynamics of several nonlinear dynamical and chaotic systems, including the Tent map and the Logistic map.
With the increasing of the relevant parameter in the chaotic system, the dynamics globally become more and more irreversible. For the Tent map, we observe that the divergence value doubling in each band′s merging point shows a fractal structure. In the Logistic map, the fractal features demonstrated by the intermittency can reveal the symmetric structure between interval $[RL^{\alpha}R^{\infty},RL^{\alpha+1}R^{\infty}]$ inside the fully chaotic band.
Furthermore, we found that the horizontal visibility algorithm fails to identify the irreversibility of the period-doubling region before the relevant parameter is varied into the chaotic regime for the Logistic map, we thus propose a modified-Horizontal Visibility Graph method, which then can reveal the irreversibility for these periodic dynamics, but the associated fractal structure from the former algorithm will be destroyed.
