Геометрия и топология фазовых портретов интегрируемых блочно-линейных динамических систем

Abstract

We consider 3-dimensional block-linear dynamical systems as models of two classes of circular gene networks. The gene networks of the first class are regulated by negative feedbacks only. Networks of the second class are regulated by positive feedbacks. Investigation of principles of functioning of these gene networks allows to control biochemical processes in living organisms. Combinatorial structure of the considered gene networks is a circular one, and we study the case when the rate of change of concentration of any substance in the gene network depends on the concentration of the «previous» substance monotonically. Sufficient conditions of existence and uniqueness of a cycle are elaborated for the first class of these dynamical systems. These systems are integrable, and we construct for each of them an integral manifold containing a cycle. In the case of the systems symmetric with respect to cyclic permutations of the variables, we find a solution of an inverse problem of parameter identification when some information about a period of this cycle is known. For another class of similar blocklinear dynamical systems, we show the non-existence of a cycle which is symmetric to that permutation of the variables.