Commentary: Multidimensional discrete chaotic maps

This commentary is addressed to multidimensional discrete chaotic maps discussed in [1]. The authors of the commented paper [1] note that the iterative logistic map of matrices is introduced in [2, 3]. Then, the authors of [1] introduce a multidimensional discrete chaotic map, define when this map is explosive, and prove two theorems describing the conditions when the map is not explosive, and when a chaotic behavior is observed for each scalar variable. There are two issues commented in this commentary paper. The first one is related to the fact that multidimensional discrete chaotic maps have been already introduced in [4]. Secondly, both theorems describing the conditions when multidimensional discrete chaotic maps are not explosive, and when a chaotic behavior is observed for each scalar variable, do not hold true. Correct conditions when multidimensional discrete chaotic map is explosive are given in [2, 3] (for 2-dimensional discrete chaotic maps) and in [4] (for n-dimensional discrete chaotic maps). This commentary demonstrates that a multidimensional discrete chaotic map can be explosive even if the eigenvalues of the matrix of initial conditions are located in the convergence domain of the corresponding scalar discrete map. Necessary and sufficient conditions for a multidimensional discrete chaotic map to become explosive are discussed in [2, Theorem 3.2, p. 935], [3, Definition 3.3, p. 4433], [3, Comment 3, p. 4433], [3, Corollary 4, p. 4434], [4, Eq. 30, p. 7]. Moreover, a numerical example of a 2dimensional discrete chaotic map is used to illustrate the fact that theorems 1 and 2 in [1] are incorrect. OPEN ACCESS