Discretization, bifurcation analysis and chaos control for Schnakenberg model. (English) Zbl 1447.39008
Summary: Schnakenberg model is a system showing sustained oscillations for a simple model of glycolysis in which a metabolic process that converts glucose to provide energy for metabolism. Euler approximation is implemented to obtain discrete version of Schnakenberg model. It is proved that discrete-time system via Euler approximation undergoes Neimark-Sacker bifurcation as well as period-doubling bifurcation is also examined at its unique positive steady-state. Keeping in view the dynamical consistency for continuous models, a nonstandard finite difference scheme is proposed for Schnakenberg model. It is proved that continuous system undergoes Hopf bifurcation at its interior equilibrium, whereas discrete-time system via nonstandard finite difference scheme undergoes Neimark-Sacker bifurcation at its interior fixed point. Some chaos and bifurcation control methods are implemented to both discrete-time models. Numerical simulation is provided to strengthen our theoretical discussion.
MSC:
| 39A60 | Applications of difference equations |
| 39A28 | Bifurcation theory for difference equations |
| 92B05 | General biology and biomathematics |
| 92C50 | Medical applications (general) |
Keywords:
Schnakenberg model; nonstandard finite difference scheme; stability; Neimark-Sacker bifurcation; period-doubling bifurcation; chaos controlReferences:
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