Discretization, bifurcation analysis and chaos control for Schnakenberg model

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.