Turing pattern dynamics in a fractional-diffusion oregonator model under PD control

In this paper, fractional-order diffusion and proportional-derivative (PD) control are introduced in oregonator model, and the Turing pattern dynamics is investigated for the first time. We take the cross-diffusion coefficient as the bifurcation parameter and give some necessary conditions for Turing instability of the fractional-diffusion oregonator model under PD control. At the same time, we construct the amplitude equations near the bifurcation threshold and determine the pattern formation of the fractional-diffusion oregonator model under PD controller. It is observed by numerical simulations that in the absence of control, the pattern formation changes with the variation of the cross-diffusion coefficient in two-dimensional space. Meanwhile, it is verified that the PD control has a significant impact on Turing instability, and the pattern structure can be changed by manipulating the control gain parameters for the fractional-diffusion oregonator model.