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Keywords = conley theory

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25 pages, 609 KB  
Article
Planar Bistable Structures Detection via the Conley Index and Applications to Biological Systems
by Junbo Jia, Pan Yang, Huaiping Zhu, Zhen Jin, Jinqiao Duan and Xinchu Fu
Mathematics 2023, 11(19), 4139; https://doi.org/10.3390/math11194139 - 30 Sep 2023
Viewed by 2172
Abstract
Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in two-dimensional dynamical systems are studied: one is two one-point attractors, another is a one-point attractor accompanied by a cycle attractor. By the Conley index theory, we [...] Read more.
Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in two-dimensional dynamical systems are studied: one is two one-point attractors, another is a one-point attractor accompanied by a cycle attractor. By the Conley index theory, we prove that there exist other isolated invariant sets besides the two attractors, and also obtain the possible components and their configuration. Moreover, we find that there is always a separatrix or cycle separatrix, which separates the two attractors. Finally, the biological meanings and implications of these structures are given and discussed. Full article
(This article belongs to the Special Issue Statistical and Mathematical Modelling of Infectious Diseases)
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19 pages, 363 KB  
Article
Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator
by Nikolai Sidorov, Denis Sidorov and Aliona Dreglea
Symmetry 2020, 12(6), 912; https://doi.org/10.3390/sym12060912 - 1 Jun 2020
Cited by 4 | Viewed by 3859
Abstract
The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, [...] Read more.
The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations. Full article
22 pages, 601 KB  
Article
Coarse Dynamics for Coarse Modeling: An Example From Population Biology
by Justin Bush and Konstantin Mischaikow
Entropy 2014, 16(6), 3379-3400; https://doi.org/10.3390/e16063379 - 19 Jun 2014
Cited by 10 | Viewed by 5825
Abstract
Networks have become a popular way to concisely represent complex nonlinear systems where the interactions and parameters are imprecisely known. One challenge is how best to describe the associated dynamics, which can exhibit complicated behavior sensitive to small changes in parameters. A recently [...] Read more.
Networks have become a popular way to concisely represent complex nonlinear systems where the interactions and parameters are imprecisely known. One challenge is how best to describe the associated dynamics, which can exhibit complicated behavior sensitive to small changes in parameters. A recently developed computational approach that we refer to as a database for dynamics provides a robust and mathematically rigorous description of global dynamics over large ranges of parameter space. To demonstrate the potential of this approach we consider two classical age-structured population models that share the same network diagram and have a similar nonlinear overcompensatory term, but nevertheless yield different patterns of qualitative behavior as a function of parameters. Using a generalization of these models we relate the different structure of the dynamics that are observed in the context of biologically relevant questions such as stable oscillations in populations, bistability, and permanence. Full article
(This article belongs to the Special Issue Information in Dynamical Systems and Complex Systems)
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