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

Abstract

We consider two four-dimensional piecewise linear dynamical systems of chemical kinetics. For one of them, we give an explicit construction of a hypersurface that separates the attraction basins of two stable equilibrium points and contains an unstable cycle of this system. For the other system, we prove the existence of a trajectory not contained in the attraction basin of the stable cycle of this system described earlier by Glass and Pasternack. The homotopy types of the phase portraits of these two systems are compared.

Limit cycles in models of circular gene networks regulated by negative feedback loops

“…Piecewise linear systems of the type ( 10 ) in different dimensions were studied in gene networks modeling earlier, see for example [ 47 – 49 ]. As it was shown in [ 50 ], the system ( 10 ) has a unique limit cycle C with the periodic coordinate functions x ( t ), y ( t ), z ( t ); let T be its period, and [0, T ] be a segment on the t -axis subdivided by the points 0< t 1 < t 5 < t 2 < t 3 < t 4 < T so that the graph of the function y ( t ) is depicted on the Fig.…”

“…In the case n = 3, the solution is given in Proposition 1. Higher-dimensional versions of this inverse problem require more combinatorial efforts even in the case n = 4, see [49].…”

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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