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On stable and quasi-chaotic regimes in a one-dimensional unimodal mapping obtained by modeling the dynamics of a biological population
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On stable and quasi-chaotic regimes in a one-dimensional unimodal mapping obtained by modeling the dynamics of a biological population
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Journal Article
On stable and quasi-chaotic regimes in a one-dimensional unimodal mapping obtained by modeling the dynamics of a biological population
2024
Overview
The paper considers the properties of the difference equation describing the dynamics of the animal population, obtained earlier in the framework of studies of tundra communities. We have considered a special case in which the model is represented by a one-parameter difference equation that defines a one-dimensional unimodal mapping of a segment into itself, similar to the well-known triangular (tent) mapping, supplemented by a region with a constant value. A change in the mapping parameter generates a bifurcation scenario, in which stability zones arise, characterized by orbits of a constant period, interspersed with zones with more complicated, “quasi-chaotic” regimes. Based on the properties of the n -iterated triangular mapping, a necessary and sufficient condition for the localization of cyclic orbits in the considered type of unimodal mappings is formulated, which makes it possible to identify stability regions for any given period n . On the basis of this condition an algorithm for detecting stability zones is proposed. The main subject of the study is the fractal properties of the set, which is the complement of the obtained set of stability regions to the entire domain of mapping definition. The dynamics of the D H ( n ) value is obtained, the limit of which at n → ∞ is equal to the fractal dimension d H . It is shown that in the studied range of n (2 ≤ n ≤ 22), the D H ( n ) < 0.9, which suggest that d H < 1. If so, then according to the definition of a fractal set, its topological dimension is d T = 0, which means that the complement of the set of stability regions consists of isolated points.
Publisher
IOP Publishing
Subject
