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These authors have contributed equally to this work

ORCID: Rasa Smidtaite,

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

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This commentary is addressed to multidimensional discrete chaotic maps discussed in [

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 [

Let us consider an iterative map

where

The scalar variable

It is shown in [

Let us assume that the eigenvalues

where _{1}, _{2} are conjugate idempotents satisfying the following relations: det _{1} = det _{2} = 0, _{1} + _{2} = _{1}⋅ _{1} = _{1}, _{2} ⋅ _{2} = _{2},

Therefore, such a 2-dimensional discrete map splits into two scalar maps of eigenvalues [

Then, the 2-dimensional discrete map is not explosive if and only if the eigenvalues

Otherwise, if the matrix of initial conditions

where

where

Then, the 2-dimensional discrete chaotic map can become explosive even if the recurrent eigenvalue does belong to the basin of attraction of

The authors of [

The 2-dimensional discrete logistic map can be explosive even when the eigenvalues of the matrix of initial conditions do belong to the basin of attraction of the corresponding scalar logistic map. The matrix of initial conditions

It can be noted that a scalar variable in

A multidimensional discrete chaotic map may become explosive if at least two eigenvalues of the matrix of initial conditions do coincide (even though all eigenvalues of the matrix are located in the convergence domain of the corresponding scalar discrete map). The multiplicity indexes of eigenvalues are directly related to the packing codes due to the classical bin packing problem [

Packing and divergence codes at

Firstly, let us consider the case when all four eigenvalues of the matrix of initial conditions are different

Analogous derivations to those performed for the 2-dimensional discrete chaotic maps yield four uncoupled scalar discrete maps of eigenvalues (

Secondly, let us investigate the scenario when only two eigenvalues do coincide but other two are different

Next, let us consider the packing code

Let us consider the fourth packing code in Table 10 [4, p. 9]. It describes the case when three eigenvalues do coincide but the fourth is different

Finally, let us discuss the largest divergence code when all eigenvalues are equal

where

Packing and divergence codes for the

The largest divergence code for the 4 × 4 matrix

where

All four eigenvalues of

It is interesting to observe, that the divergence rate of the auxiliary parameters do depend not only on the Lyapunov coefficient, but also on their indexes. Detailed discussion of the rate of explosive divergence of multidimensional logistic maps is given in [4, p. 11].

This commentary paper demonstrates that a multidimensional discrete chaotic map can become explosive even if the eigenvalues of the matrix of initial conditions are located in the convergence domain of the corresponding scalar discrete map. The explosive divergence of the multidimensional discrete chaotic map does occur if the divergence code of the matrix of initial conditions is larger than zero (at least two eigenvalues of

This fact has important implications for the study of discrete chaotic systems when the nodal complexity of the system is increased by expanding the dimension of the scalar variable [

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication. Both authors contributed equally to this work and have approved it for publication.

This research is funded by the European Social Fund under the No 09.3.3-LMT-K-712 “Development of Competences of Scientists, other Researchers and Students through Practical Research Activities” measure (Project No. 09.3.3-LMT-K-712-23-0235).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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