Abstract: The homoclinic cycles formed by heteroclinic trajectories and stationary points exist primarily in dynamical systems with symmetries and can undergo various bifurcations. This article addresses the global aspects of a gene network model with three variables and the numerical studies of the bifurcation of a homoclinic cycle consisting of three saddles and three heteroclinic orbits. Of great interest is the fact that the bifurcation of the homoclinic cycle is characterized by high sensitivity to initial data. In this case for small perturbations of concentration we encounter a random transition from relaxation oscillations corresponding to an observed stable limit cycle close to a triangular homoclinic cycle to various stable steady states. The scenario of the destruction of the triangular homoclinic cycle has a simple biological meaning: while a gene network is generating relaxation pulses, in a neighborhood of the bifurcation values of the parameter small fluctuation spikes of concentration can knock the system off its oscillation regime and stabilize it randomly to substantially different sinks.

Cite: Chumakov G.A. , Chumakova N.A.
Homoclinic Cycles in a Gene Network Model
Математические заметки СВФУ (Mathematical Notes of NEFU). 2014. V.21. N4. P.87 - 95.

Original: Чумаков Г.А. , Чумакова Н.А.
Гомоклинические циклы в одной модели генной сети
Математические заметки СВФУ (Mathematical Notes of NEFU). 2014. Т.21. №4. С.97 - 106.

Dates:

Submitted:

Oct 30, 2014

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