On the uniqueness of a cycle in an asymmetric three-dimensional model of a molecular repressilator

On existence of a piecewise smooth cycle in one asymmetric gene network model with piecewise linear equations

“…Conclusion: In contrast with [2], where the particular case m 1 = m 2 = m 1 = 1 was studied, the shifts along trajectories of the system (1) are not described by projective transformations of the faces of adjacent blocks B k which contain C. Thus, the uniqueness of this cycle does not follow from the geometric arguments used in [2,3].…”

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Motivation and Aim:We construct a simple piecewise linear dynamical system which simulates one gene network regulated by negative feedbacks in order to find conditions of existence of periodic regimes (cycles) of its functioning and to describe location of these cycles in the phase portrait of the system. Methods and Algorithms: Some approaches to modelling of similar gene networks and description of combinatorial structures of discretizations (State Transition Diagram) of the phase portraits of corresponding nonlinear dynamical systems are presented in [1][2][3]. Results: For positive parameters m j , A j and α j , where A j > α j , j = 1, 2, 3, we consider 3Ddynamical system dx dt = L 1 (z) -m 1 x; dy dt = L 2 (z) -m 2 y; dy dt = L 2 (y) -m 3 z.(1)

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On two classes of nonlinear dynamical systems: The four-dimensional case

“…and that w * satisfies both conditions (13). Thus, the process of constructing three-dimensional planes in the domains of diagram (7) Proof.…”

“…Three-and five-dimensional versions of (4) were considered in [1,9,12,13]. Our constructions can also be carried over to dynamical systems of higher dimension (see [7,10]).…”

An inverse problem in modelling of a symmetric gene network regulated by negative feedbacks

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Motivation and Aim:We study one piece-wise linear dynamical system which describes functioning of a gene network regulated by negative feedbacks in order to find conditions of existence and uniqueness of periodic regimes of its functioning and show existence and uniqueness of solution of an inverse problem of identification of parameters of this system. Methods and Algorithms: The approaches to modelling of similar gene networks, description of phase portraits of corresponding dynamical systems and detection of their periodic trajectories (cycles) are presented in [1,2]. For some other non-linear dynamical systems, similar constructions were described in [3].

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