Siberian Electronic Mathematical Reports http://semr.math.nsc.ru Òîì IVD ID ñòðF RTR!RUQ @PHPIA ÓÄÊ SIRFURSFVP hys IHFQQHRVGsemiFPHPIFIVFHQP wg QUxPS ON UNIQUENESS AND STABILITY OF A CYCLE IN ONE GENE NETWORK FF qyvfexsuyD vFFwsxrusxeAbstract. e desrie neessry nd su'ient onditions for uniqueE ness nd stility of yle in n invrint domin of phse portrit of one qlssEsternk type lokEliner dynmil system tht simuE ltes funtioning of one nturl gene networkF ixistene of suh yleD geometry nd omintoris of phse portrits of similr systems were studied in our previous pulitionsF Keywords: irulr gene networkD (xed pointsD ylesD pieewise liner dynmil systemsD phse portritsD invrint dominsD oinr¡ e mpF

No abstract

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.

No abstract

Статья посвящена исследованию динамической системы с разрывными функциями в правых частях, моделирующей функционирование кольцевой генной сети с положительными и отрицательными обратными связями. В первой части описано построение инвариантной области и дальнейшее разбиение ее на подобласти, содержащие блоки различных валентных уровней. В подобласти с блоками минимальной валентности рассмотрены вопросы существования, единственности и устойчивости цикла, а также свойства отображения Пуанкаре этого цикла. Во второй части изучены поведение и геометрические особенности траекторий, содержащихся в неинвариантной подобласти с максимальным валентным уровнем блоков, описано построение кусочно-линейной инвариантной поверхности траекторий системы в симметричном относительно циклической перестановки переменных случае. Показано, что данная область не содержит периодических траекторий этой системы.