## 8 Miscellaneous
### The Use of the Lozi Map as a Benchmark  
In certain cases, the Lozi map has been used as a benchmark to test numerical methods, for example, in [158], in which the goal of Skrynnyke et al. was to demonstrate the effectiveness of the cycle search averaged predictive control method on certain dynamical systems widely referred to in technical reference sources. Their article demonstrated the efficiency of the large-length cycle searching method based on the stabilization of unstable and a priori unknown periodic orbits of these systems using several well-known examples of discrete-time nonlinear systems. Research in this area has mainly focused on Lozi systems. As an example, they computed two different cycles of period 1001 and one cycle of period 1111 for \(a=1.7\) and \(b=0.5\). They suggested that the proposed algorithm with the phenomenological rule of the choice of control parameter makes it possible to hope for the possibility of finding a large number of giga- and even tera-cycles. In [159], Shang et al. proposed a Multi-Dimensional Scaling (MDS) method based on the Complexity-Invariant Distance (CID) and Generalized Complexity-Invariant Distance (GCID) to analyze and classify complex time series such as traffic signals and financial stock indexes. Shang et al. adopted a new method for measuring dissimilarity called the CID along with an advanced version called the GCID. Both methods were applied in MDS for the classification of time series, including artificial and real-world time series. First, in order to test the efficiency of the MDS method based on CID and GCID, they used three types of artificial time series: the \(\alpha\) map model, the 2D Henon map model, and the Lozi map model (with \(a=0.95\) and \(b=0.12\) for the first pair of parameters and \(a=1.0\), \(b=0.14\) for the second), which they used to observe the spatial distributions created by the MDS with the CID and GCID and examine their plausibility by calculating the stress. Additionally, three more sets of artificial time series were created with longer data length to demonstrate that their method is not likely to be affected by data length. Jacob et al. [160] proposed a novel measure of degree heterogeneity for unweighted and undirected complex networks which requires only the degree distribution of the network for its computation. They showed that the proposed measure can be applied to all types of network topology with ease and that it increases with the diversity of the node degrees in the network. As a specific application, they showed that the proposed measure can be used to compare the heterogeneity of recurrence networks constructed from time series of several low-dimensional chaotic attractors, thereby providing a single index to compare the structural complexity of chaotic attractors. They compared the Lorenz, Rossler, and Duffing continuous attractors and the Henon, Lozi, and Cat maps. Their results indicated that the Lorenz attractor is the most structurally complex among the continuous systems, while in the case of two-dimensional discrete systems the Lozi attractor is the most diverse in terms of probability density variations. In [161], Amigo and Kennel showed that the topological permutation entropy rate \((h^{*}_{top}(f))\) for expansive maps equals the conventional topological entropy rate \((h_{top}(f))\) from symbolic dynamics. After providing rigorous theoretical proofs of this equality, they sought to check it numerically. This task was found to be quite difficult; they stated that \"Estimation of topological entropies from naive numerical simulation of long orbits is notoriously difficult. Metric entropy by itself can be quite tricky and difficult, requiring very long data sets for increasing L (the number of the joints distributions of a finite-state stochastic process), but topological entropy is worse yet, because it weights each pattern equally. This means that patterns which are exceptionally infrequent on the natural measure of the attractor can still have a significant influence on the result.