Stability, Chaos, and Bifurcation Analysis of a Discrete Chemical System

Abstract

The local dynamics, chaos, and bifurcations of a discrete Brusselator system are investigated. It is shown that a discrete Brusselator system has an interior fixed point $P\left(1,r\right)$ if $r>0$ . Then, by linear stability theory, local dynamical characteristics are explored at interior fixed point $P\left(1,r\right)$ . Furthermore, for the discrete Brusselator system, the existence of periodic points is investigated. The existence of bifurcations around an interior fixed point is also investigated and proved that the discrete Brusselator model undergoes hopf and flip bifurcations if $\left(r,h\right)\in \mathrm{\mathscr{H}ℬ}{|}_{P\left(1,r\right)}=\left\{\left(r,h\right),h=2-r\right\}$ and $\left(r,h\right)\in \mathrm{ℱℬ}{|}_{P\left(1,r\right)}=\left\{\left(r,h\right),h=4/2-r-\sqrt{r^2-4r}\right\}$ , respectively. The next feedback control method is utilized to stabilize the chaos that exists in the discrete Brusselator system. Finally, obtained results are verified numerically.