Stability and Bifurcation in a Fractional Order Brusselator Model And Its Discretization

Abstract

The Periodic temporal oscillation and formation of a spatial pattern in chemical reactions are major part of nonlinear chemical dynamics. Although the history of oscillatory reactions in chemical kinetics is long enough, it was less than quarter century ago that chemical oscillator was characterized. The key features of such oscillating reactions are the auto catalysis. In this present work, a fractional order Brusselator model is proposed. First we prove the existence and uniqueness of the solutions of fractional order dynamical system and with discretization process its discrete version is obtained. Local stability of the fixed point of Brusselator model has been studied in both commensurate and incommensurate fractional order system. Also we discuss the bifurcation parameters; it is proved that the discrete fractional order system undergoes a flip (period doubling) and Neimark – Sacker bifurcation at the
interior (coexistence) fixed point. Finally numerical examples are provided to validate the analytical results and the rich dynamical nature of the discretized system is exhibited.