From chaos to order. Difference equations in one ecological problem

Каменев, Г. К.; Lyulyakin, Oleg P.; Sarancha, D. A.; Lysenko, Nikolai A.; Polyanovsky, V. O.

doi:10.1515/rnam-2016-0025

Cited by 4 publications

(8 citation statements)

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“…Unlike δ min (n) -the value of the smallest interval that does not contain cycles of period less than or equal to n, the value of g min (n) decreases strictly monotonically and has the limit lim g min (n) = 0, n   . At the same time, expressions (10) and (11) for the lower and upper boundaries of the value D(n) remain valid for g min (n).…”

Section: Coverage By G Min Segmentsmentioning

confidence: 88%

“…Note that in this paper we are only interested in the mathematical properties of this mapping, while the given population interpretation is only an example of its practical application. When studying the properties of the resulting model in [9,10], using the example of some approximation, which is a triangular mapping with a plateau (see formula (5.2), on p. 73 in [9]), the properties were described that allowed us to formulate a necessary and sufficient condition for the localization of cyclic trajectories in the considered type of unimodal mappings of a segment into itself, based on the following statements.…”

Section: The Origin Of the Problem Triangular Mapping With A Plateau ...mentioning

confidence: 99%

“…Thus, we know the set of points that make up the entire set of cyclic trajectories of period n, as well as the number of trajectories (p. 25 in [11]). However, the "partition" of this set into separate trajectories is unknown.…”

Section: Ic-msquare-2023mentioning

confidence: 99%

“…Therefore, if in a triangular mapping each period n corresponds to a finite number of unstable cycles [11], then in a triangular mapping with a plateau there is an uncountable set of stable n-cycles in each segment [ x i , x i+1 ]. If x A is the transition point of the descending side of the triangular mapping to the plateau (Fig.…”

Section: Ic-msquare-2023 Journal Of Physics: Conference Series 2701 (...mentioning

confidence: 99%

“…The values of 𝐷(𝑛) and 𝐷(𝑛) depending on the step n of constructing the set when covered with segments δ min (n) are presented in Table 1(Supplement 1) and on the graph in Figure 2. (10,11)), when covered with segments δ min (n), depending on the maximum cycle period n.…”

Section: Coverage By δ Min Segmentsmentioning

confidence: 99%

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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

Polyanovsky,

Kamenev

2024

J. Phys.: Conf. Ser.

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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 DH (n) value is obtained, the limit of which at n → ∞ is equal to the fractal dimension dH . It is shown that in the studied range of n (2 ≤ n ≤ 22), the DH (n) < 0.9, which suggest that dH < 1. If so, then according to the definition of a fractal set, its topological dimension is dT = 0, which means that the complement of the set of stability regions consists of isolated points.

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Section: Coverage By G Min Segmentsmentioning

confidence: 88%

Section: The Origin Of the Problem Triangular Mapping With A Plateau ...mentioning

confidence: 99%

Section: Ic-msquare-2023mentioning

confidence: 99%

Section: Ic-msquare-2023 Journal Of Physics: Conference Series 2701 (...mentioning

confidence: 99%

Section: Coverage By δ Min Segmentsmentioning

confidence: 99%

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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

Polyanovsky,

Kamenev

2024

J. Phys.: Conf. Ser.

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Electrical characteristics of the 3 × 6 × n cobweb cascaded LC complex impedance network

Luo

Li-jin

2023

Phys. Scr.

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For the cobweb circuit network, the previous research focuses on the single-stage cobweb. In this paper, we studied a kind of multi-stage cobweb composed of n single-stage cobwebs, namely a 3 × 6 × n cobweb cascade LC network (CCLCN). To calculate the equivalent impedance of such large-scale complex circuit networks, we used a method that combines the replacement method with the modified recursive-transformation method (referred to as the R-MRT method). The CCLCN circuit was first replaced by a purely resistive circuit with identical connections, then the purely resistive circuit was calculated by the modified recursive-transformation method, and finally, the equivalent impedance of the CCLCN circuit was obtained by parameter replacement. Utilizing the above method, we obtained the exact analytical expression of the equivalent impedance of the 3 × 6 × n CCLCN. This result shows that the equivalent impedance problem of large-scale complex circuit networks such as the multi-stage cobweb has been solved.

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Investigation of the Class of One-Dimensional Unimodal Mappings Obtained in the Modeling of the Lemming Population

2018

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From chaos to order. Difference equations in one ecological problem

Cited by 4 publications

References 0 publications

On stable and quasi-chaotic regimes in a one-dimensional unimodal mapping obtained by modeling the dynamics of a biological population

On stable and quasi-chaotic regimes in a one-dimensional unimodal mapping obtained by modeling the dynamics of a biological population

Electrical characteristics of the 3 × 6 × n cobweb cascaded LC complex impedance network

Investigation of the Class of One-Dimensional Unimodal Mappings Obtained in the Modeling of the Lemming Population

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