The Periodic Orbits of a Dynamical System Associated with a Family of QRT-Maps

Bastien, Guy; Rogalski, Marc

doi:10.1007/s12346-020-00393-2

Cited by 1 publication

(2 citation statements)

References 16 publications

(37 reference statements)

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“…Assuming M 2 in diagonal form (space is oriented by the eigenvectors of M 2 ), two parameters of M 2 and three parameters of M 3 were varied. According to (2) and ( 6), only antisymmetric and closed curves were inspected [34,35].…”

Section: Methodsmentioning

confidence: 99%

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Topological Approach for Material Structure Analyses in Terms of R2 Orientation Distribution Function

Smirnova,

Semenova,

Prunov

et al. 2023

Mathematics

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The application of solid mechanics theory for material behavior faces the discrete nature of modern or biological material. Despite the developed methods of homogenization, there are deviations between simulated and experiments results. The reason is homogenization, which mathematically involves a type of interpolation. The situation gets worse for complex structured materials. On the other hand, a topological approach can help in such analysis, but such an approach has computational costs. At the same time, increasing modern computational capabilities remove this barrier. This study is focused on building a method to analyze material structure in a topological sense. The orientation distribution function was used to describe the structure of the material. The plane case was investigated. Quadratic and biquadratic forms of interpolant were investigated. The persistent homology approach was used for topology analysis. For this purpose, a persistence diagram for quadratic and biquadratic forms was found and analyzed. In this study, it is shown how scaling the origin point cloud influences H1 points in the persistence diagram. It was assumed that the topology of the biquadratic form can be understood as a superposition of quadratic forms. Quantitative estimates are given for ellipticity and H1 points. A dataset of micro photos was processed using the proposed method. Furthermore, the supply criteria for the interpolation choice in quadratic or biquadratic forms was formulated.

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Section: Methodsmentioning

confidence: 99%

“…The birth of graph cycles depends only on unique distances. So, all birth points will be multiplied by scale s. Consider the overlapping Equation (34) in scaled axes:…”

Section: Methodsmentioning

confidence: 99%