TY - JOUR

T1 - Periodic Trajectories of Nonlinear Circular Gene Network Models

AU - Minushkina, L. S.

N1 - The study was supported by the Russian Science Foundation (Grant no. 23-21-00019).

PY - 2024/5

Y1 - 2024/5

N2 - The article addresses the qualitative analysis of the two dynamical systems simulating circular gene network functioning.The equations of a three-dimensionaldynamical system contain some monotonically decreasing smooth functions that describe negative feedback.A six-dimensional dynamical system consists of three equations with monotonically decreasing smooth functionsand three equations with monotonically increasing smooth functions that characterize negative and positive feedbacks.In both models the process of degradation is described by smooth nonlinear functions.We construct invariants domains in order to localize cycles for both systems,show that each of the two systems has a unique stationary pointin the invariant domain, and find the conditions for this point to be hyperbolic.The main result is the proof of existence of a cycle in the invariant subdomainfrom which the trajectories cannot pass to other subdomains obtained by discretization of the phase portrait.The cycles of three- and six-dimensional systems bound thetwo-dimensional invariant surfaces including the trajectories of the systems.

AB - The article addresses the qualitative analysis of the two dynamical systems simulating circular gene network functioning.The equations of a three-dimensionaldynamical system contain some monotonically decreasing smooth functions that describe negative feedback.A six-dimensional dynamical system consists of three equations with monotonically decreasing smooth functionsand three equations with monotonically increasing smooth functions that characterize negative and positive feedbacks.In both models the process of degradation is described by smooth nonlinear functions.We construct invariants domains in order to localize cycles for both systems,show that each of the two systems has a unique stationary pointin the invariant domain, and find the conditions for this point to be hyperbolic.The main result is the proof of existence of a cycle in the invariant subdomainfrom which the trajectories cannot pass to other subdomains obtained by discretization of the phase portrait.The cycles of three- and six-dimensional systems bound thetwo-dimensional invariant surfaces including the trajectories of the systems.

KW - 517.938

KW - circular gene network

KW - cycles

KW - invariant domains and surfaces

KW - mathematical models

KW - nonlinear degradation

KW - positive and negative feedback

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85195146760&origin=inward&txGid=7afe46ad1accd45e96da9f8bb0e6b7bf

UR - https://www.mendeley.com/catalogue/307123b8-2b0d-3f26-8bac-eb6f1d30460a/

U2 - 10.1134/S0037446624030212

DO - 10.1134/S0037446624030212

M3 - Article

VL - 65

SP - 718

EP - 724

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 3

ER -