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In the work, a nonlinear reaction-diffusion model in a class of delayed differential equations on the hexagonal lattice is considered. The system includes a spatial operator of diffusion between hexagonal pixels. The main results deal with the qualitative investigation of the model. The conditions of global asymptotic stability, which are based on the Lyapunov function construction, are obtained. An estimate of the upper bound of time delay, which enables stability, is presented. The numerical study is executed with the help of the bifurcation diagram, phase trajectories, and hexagonal tile portraits. It shows the changes in qualitative behavior with respect to the growth of time delay; namely, starting from the stable focus at small delay values, then through Hopf bifurcation to limit cycles, and finally, through period doublings to deterministic chaos.where we consider a lattice Λ ⊂ R n , which can be presented as a discrete subset of R n , consisting of either a finite or infinite number of points, which are located in accordance with some regular spatial structure. The vectors u ξ , ξ ∈ Λ are the values of the state u = u ξ ξ∈Λ , calculated at the the points of the lattice, and g ξ are the right sides of the equations with the properties enabling us to find the existence of the solution.As a rule, without loss of generality, these consider Λ = Z n , which is the integer lattice in R n . The methods developed can be easily applied to a different type of lattice, namely the planar rectangular and hexagonal lattice, the crystallographic lattices in R 3 .These pay attention to the notion of delay in lattice differential equations, the so-called delayed lattice differential equations. One of the applications dealing with them is the investigation of traveling wave fronts and their stability [5]. The main results are applied to the delayed and discretely diffusive models for the population (see, e.g., [6,7]).Lattice differential equations are used as models in many applications, for example cellular neural networks, image processing, chemical kinetics, materials science, in particular metallurgy, and biology [5,8]. Lattice models are extremely attractive from the viewpoint of population dynamics, especially in the case of spatially-separated populations [5,6,[8][9][10][11].There are a few reasons for requiring the consideration of the hexagonal grid instead of rectangular one (primarily in image and vision computing); namely, the equal distances between neighboring pixels for hexagonal coordinate systems [12]; hexagonal points are packed more densely [13]; since "hexagons are 'rounder' than squares", the presentation of curves is more consistent with the help of hexagonal systems [13]; hence, mathematical operations of edge detection and shape extraction are more successful when applying hexagonal lattices [14].With the purpose of indexing hexagonal pixels, as a rule, two-(This is the so-called "skewed-axis" coordinate system) or three-element (It is also known as the "cube hex coordinate system") coordinate system...

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The developed method of steganographic hiding of text information in an audio signal based on the wavelet transform acquires a deep meaning in the conditions of the use by an attacker of deliberate unauthorized manipulations with a steganocoded audio signal to distort the text information embedded in it. Thus, increasing the robustness of the stego-system by compressing the steganocoded audio signal subject to the preservation of the integrity of text information, taking into account the features of the psychophysiological model of sound perception, is the main objective of this scientific research. The task of this scientific research is effectively solved using a multilevel discrete wavelet transform using adaptive block normalization of text information with subsequent recursive embedding in the low-frequency component of the audio signal and further scalar product of the obtained coefficients with the Daubechies wavelet filters. The results of the obtained experimental studies confirm the hypothesis, namely that it is proposed to use recursive embedding in the low-frequency component (approximating wavelet coefficients) followed by their scalar product with wavelet filters at each level of the wavelet decomposition, which will increase the average power of hidden data. It should be noted that upon analyzing the existing method, which is based on embedding text information in the high-frequency component (detailed wavelet coefficients), at the last level of the wavelet decomposition, we obtained the limit CR = 6, and in the developed, CR = 20, with full integrity of the text information in both cases. Therefore, the resistance of the stego-system is increased by 3.3 times to deliberate or passive compression of the audio signal in order to distort the embedded text information.