Stratifications and foliations in phase portraits of gene network models
Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive...
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| Format: | Article |
| Language: | English |
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Siberian Branch of the Russian Academy of Sciences, Federal Research Center Institute of Cytology and Genetics, The Vavilov Society of Geneticists and Breeders
2023-01-01
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| Series: | Вавиловский журнал генетики и селекции |
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| Online Access: | https://vavilov.elpub.ru/jour/article/view/3576 |
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| author | V. P. Golubyatnikov A. A. Akinshin N. B. Ayupova L. S. Minushkina |
| author_facet | V. P. Golubyatnikov A. A. Akinshin N. B. Ayupova L. S. Minushkina |
| author_sort | V. P. Golubyatnikov |
| collection | DOAJ |
| description | Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section “Investigation of functions and plot of their graphs” of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations. |
| format | Article |
| id | doaj-art-9ee8a47956974832adab29116b8bbbfb |
| institution | Kabale University |
| issn | 2500-3259 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Siberian Branch of the Russian Academy of Sciences, Federal Research Center Institute of Cytology and Genetics, The Vavilov Society of Geneticists and Breeders |
| record_format | Article |
| series | Вавиловский журнал генетики и селекции |
| spelling | doaj-art-9ee8a47956974832adab29116b8bbbfb2025-02-01T09:58:11ZengSiberian Branch of the Russian Academy of Sciences, Federal Research Center Institute of Cytology and Genetics, The Vavilov Society of Geneticists and BreedersВавиловский журнал генетики и селекции2500-32592023-01-0126875876410.18699/VJGB-22-911313Stratifications and foliations in phase portraits of gene network modelsV. P. Golubyatnikov0A. A. Akinshin1N. B. Ayupova2L. S. Minushkina3Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences; Novosibirsk State UniversityHuawei Russian Research InstituteSobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences; Novosibirsk State UniversityNovosibirsk State UniversityPeriodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section “Investigation of functions and plot of their graphs” of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.https://vavilov.elpub.ru/jour/article/view/3576oscillationspositive and negative feedbacksgene network modelsphase portraitsinvariant domains and surfacesinvariant foliationspoincaré mapgrobman–hartman theoremfrobenius–perron theorem |
| spellingShingle | V. P. Golubyatnikov A. A. Akinshin N. B. Ayupova L. S. Minushkina Stratifications and foliations in phase portraits of gene network models Вавиловский журнал генетики и селекции oscillations positive and negative feedbacks gene network models phase portraits invariant domains and surfaces invariant foliations poincaré map grobman–hartman theorem frobenius–perron theorem |
| title | Stratifications and foliations in phase portraits of gene network models |
| title_full | Stratifications and foliations in phase portraits of gene network models |
| title_fullStr | Stratifications and foliations in phase portraits of gene network models |
| title_full_unstemmed | Stratifications and foliations in phase portraits of gene network models |
| title_short | Stratifications and foliations in phase portraits of gene network models |
| title_sort | stratifications and foliations in phase portraits of gene network models |
| topic | oscillations positive and negative feedbacks gene network models phase portraits invariant domains and surfaces invariant foliations poincaré map grobman–hartman theorem frobenius–perron theorem |
| url | https://vavilov.elpub.ru/jour/article/view/3576 |
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