Examples of Periodic Biological Oscillators: Transition to a Six-dimensional System

Abstract

We study a genetic model (including gene regulatory networks) consisting of a system of several ordinary differential equations. This system contains a number of parameters and depends on the regulatory matrix that describes the interactions in this multicomponent network. The question of the attracting sets of this system, which depending on the parameters and elements of the regulatory matrix, isconsidered. The consideration is mainly geometric, which makes it possible to identify and classify possible network interactions. The system of differential equations contains a sigmoidal function, which allows taking into account the peculiarities of the network response to external influences. As a sigmoidal function, a logistic function is chosen, which is convenient for computer analysis. The question of constructing attractors in a system of arbitrary dimension is considered by constructing a block regulatory matrix, the blocks of which correspond to systems of lower dimension and have been studied earlier. The method is demonstrated with an example of a three-dimensional system, which is used to construct a system of dimensions twice as large. The presentation is provided with illustrations obtained as a result of computer calculations, and allowing, without going into details, to understand the formulation of the issue and ways to solve the problems that arise in this case.