The explosive divergence in iterative maps of matrices

Navickas, Zenonas; Ragulskis, Minvydas; Vainoras, Alfonsas; Smidtaite, Rasa

doi:10.1016/j.cnsns.2012.03.018

Cited by 10 publications

(8 citation statements)

References 17 publications

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“…of scalar variables x (k) 11 , x (k) 12 , x (k) 21 , x (k) 22 [2,3]. The 2-dimensional discrete map then reads: X…”

Section: -Dimensional Discrete Chaotic Mapsmentioning

confidence: 99%

“…It is shown in [2,3] that the dynamics of the 2-dimensional discrete chaotic map depends not only on the fact that the eigenvalues of the matrix of initial conditions do (or do not) belong to the basin of attraction of the corresponding scalar map. A 2-dimensional discrete chaotic map can be explosive even if both eigenvalues of the matrix of initial conditions do belong to the basin of attraction of the corresponding scalar map.…”

Section: -Dimensional Discrete Chaotic Mapsmentioning

confidence: 99%

“…1. The discrete chaotic map becomes explosive if the Lyapunov exponent of the original scalar map is positive and the matrix of initial conditions is a nilpotent matrix [3].…”

Section: -Dimensional Discrete Chaotic Mapsmentioning

confidence: 99%

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Commentary: Multidimensional discrete chaotic maps

Smidtaite

Ragulskis

2022

Front. Phys.

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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 [

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“…of scalar variables x (k) 11 , x (k) 12 , x (k) 21 , x (k) 22 [2,3]. The 2-dimensional discrete map then reads: X…”

Section: -Dimensional Discrete Chaotic Mapsmentioning

confidence: 99%

Section: -Dimensional Discrete Chaotic Mapsmentioning

confidence: 99%

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Commentary: Multidimensional discrete chaotic maps

Smidtaite

Ragulskis

2022

Front. Phys.

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“…􏼢 􏼣, where each element of the matrix is a separate scalar variable: 61,62]. Ten, the iterative map of matrices reads…”

mentioning

confidence: 99%

“…It appears that the dynamics of such an iterative map of matrices depends on the matrix of initial conditions X (0) [61,62]. If the eigenvalues of X (0) are diferent (λ (0)…”

mentioning

confidence: 99%

Hyper Coupled Map Lattices for Hiding Multiple Images

et al. 2023

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The concept of a hyper coupled map lattice (CML) is presented in this paper. The complexity of the lattice is increased not by adding another spatial dimension of the lattice but by replacing scalar nodal variables by multidimensional square matrices of iterative variables. The proposed scheme exploits the nonlinear effects of the spatiotemporal divergence induced by nilpotent nodal matrices to generate separate secret images at different discrete moments of time during the evolution of the CML. The time variable plays the primary role in the decoding stage of the scheme. The carrying capacity of the proposed scheme is n − 1 different dichotomous digital images, where n is the dimension of the nilpotent nodal matrices. Computational experiments are used to demonstrate the efficacy of the proposed scheme.

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The Effect of Explosive Divergence in a Coupled Map Lattice of Matrices

Smidtaite

Navickas

et al. 2018

Chaos, Solitons & Fractals

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The explosive divergence in iterative maps of matrices

Cited by 10 publications

References 17 publications

Commentary: Multidimensional discrete chaotic maps

Commentary: Multidimensional discrete chaotic maps

Hyper Coupled Map Lattices for Hiding Multiple Images

The Effect of Explosive Divergence in a Coupled Map Lattice of Matrices

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