Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions

Abstract

The Belousov–Zhabotinskii reaction is considered in one and two-dimensional reaction–diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.

Appendix: Expressions for the semi-analytical ODEs

This appendix presents relevant expression for for the semi-analytical models. The $M_i$ for the 1-D model (6) are

$$\begin{aligned}&M_1=\frac{2H}{\epsilon }u_dv_d+\frac{2H}{\epsilon }u_d^2-\frac{20H}{3\epsilon \pi }u_d v_d -\frac{1}{\epsilon }u_d+\frac{16u_1}{3\epsilon \pi }u_d+\frac{16u_2}{15\epsilon \pi }u_d\\&\qquad \quad +\,\frac{8u_1}{3\epsilon \pi }v_d +\frac{k \pi ^2}{4}u_d-\frac{q}{\epsilon }v_d \\&\qquad \quad +\,\frac{8v_2}{15\epsilon \pi }u_d-\frac{v_1}{\epsilon }u_d -\frac{20H}{3\epsilon \pi }u_d^2-\frac{2u_1}{\epsilon }u_d-\frac{u_1}{\epsilon }v_d, \\&M_2=\frac{2H}{\delta }u_d v_d+\frac{8u_1}{3\delta \pi }v_d-\frac{20H}{3\delta \pi }u_d v_d +\frac{8v_2}{15\delta \pi }u_d-\frac{u_1}{\delta }v_d\\&\quad \quad \quad -\,\frac{4q}{\delta \pi }v_d+\frac{8v_1}{3\delta \pi }u_d +\frac{k \pi ^2}{4}v_d-\frac{v_1}{\delta }u_d\\&\quad -\,\frac{f}{\delta }w_d+\frac{8u_2}{15\delta \pi }v_d +\frac{4f}{\delta \pi }w_d+\frac{q}{\delta }v_d, \\&M_3=\frac{4}{\pi }u_d-u_d+w_d+\frac{k \pi ^2}{4}w_d-\frac{4}{\pi }w_d,\\&M_4=\frac{8v_1}{15\epsilon \pi }u_d+\frac{8u_1}{15\epsilon \pi }v_d+\frac{144}{35\epsilon \pi }u_d-\frac{4}{3\epsilon \pi }u_d +\frac{4H}{5\epsilon \pi }u_d v_d\\&\qquad \quad +\,\frac{4H}{5\epsilon \pi }u_d^2+\frac{72 u_2}{35\epsilon \pi }v_d -\frac{2u_2}{\epsilon }u_d+\frac{16u_1}{15\epsilon \pi }u_d\\&\quad \qquad +\,\frac{72v_2}{35\epsilon \pi }u_d -\frac{v_2}{\epsilon }u_d-\frac{u_2}{\epsilon }v_d+\frac{4q}{3\epsilon \pi }v_d,\\&M_5=\frac{4q}{3\delta \pi }v_d-\frac{4f}{3\delta \pi }w_d+\frac{72v_2}{35\delta \pi }u_d+\frac{72u_2}{35\delta \pi }v_d +\frac{4}{5\delta \pi }u_d v_d+\frac{8u_1}{15\delta \pi }v_d\\&\quad \qquad +\,\frac{8v_1}{15\delta \pi }u_d -\frac{u_2}{\delta }v_d-\frac{v_2}{\delta } u_d, \\&M_6=\frac{4}{3\pi }w_d-\frac{4}{3\pi }u_d. \end{aligned}$$

The $N_i$ for the 2-D model (8) are

$$\begin{aligned}&N_1=\frac{64u_1}{9\epsilon \pi ^2}v_d+\frac{128v_2}{45\epsilon \pi ^2}u_d+\frac{16 q}{\epsilon \pi ^2}v_d^2+\frac{128u_2}{45\epsilon \pi ^2}u_d +\,\frac{2H}{\epsilon }u_d^2+\frac{64v_1}{9\epsilon \pi ^2}u_d-\frac{q}{\epsilon }v_d\\&\qquad \quad +\frac{k\pi ^2}{2}u_d +\,\frac{256 u_2}{45 \epsilon \pi ^2}u_d \\&\quad \qquad +\,\frac{16}{\epsilon \pi ^2}u_d -\frac{208H}{9\epsilon \pi ^2}u_dv_d +\frac{2H}{\epsilon }u_dv_d-\frac{208H}{9\epsilon \pi ^2}u_d^2-\frac{2u_1}{\epsilon }u_d +\frac{128u_1}{9\epsilon \pi ^2}u_d\\&\quad \qquad -\,\frac{v_1}{\epsilon }u_d-\frac{u_1}{\epsilon }v_d-\frac{1}{\epsilon }u_d, \\&N_2=-\frac{208H}{9\delta \pi ^2}v_du_d+\frac{2H}{\delta }v_du_d+\frac{64v_1}{9\delta \pi ^2 }u_d +\frac{k\pi ^2}{2}v_d+\frac{128v_2}{45\delta \pi ^2 }u_d+\frac{64u_1}{9\delta \pi ^2 }v_d\\&\qquad \quad -\,\frac{v_1}{\delta }u_d -\frac{16q}{\delta \pi ^2 }v_d+\frac{q}{\delta \pi ^2 }v_d\\&\quad \qquad +\,\frac{16 f}{\delta \pi ^2 }w_d-\frac{f}{\delta }w_d+\frac{128u_2}{45\delta \pi ^2 }v_d-\frac{u_1}{\delta }v_d,\\&N_3=\frac{k\pi ^2}{2}w_d-\frac{16}{\pi ^2}w_d+\frac{16}{\pi ^2}u_d+w_d-u_d,\\&N_4=-\frac{16q}{3\epsilon \pi ^2}v_d-\frac{2u_2}{\epsilon }u_d+\frac{9088u_2}{1575 \epsilon \pi ^2}v_d-\frac{v_2}{\epsilon }u_d +\frac{176H}{45\epsilon \pi ^2}u_d^2+\frac{176H}{45\epsilon \pi ^2}u_dv_d\\&\quad \qquad +\,\frac{64v_1}{45\epsilon \pi ^2}u_d +\frac{9088 v_2}{1575\epsilon \pi ^2}u_d \\&\quad \qquad +\,\frac{64u_1}{45\epsilon \pi ^2}v_d+\frac{18176u_2}{1575\epsilon \pi ^2}u_d +\frac{128u_1}{45\epsilon \pi ^2}u_d -\frac{u_2}{\epsilon }v_d-\frac{16}{3\epsilon \pi ^2}u_d, \\&N_5=\frac{16 q}{3\delta \pi ^2 }v_d-\frac{v_2}{\delta }u_d+\frac{64v_1}{45\delta \pi ^2 }u_d+\frac{9088u_2}{1575\delta \pi ^2}v_d +\frac{176H}{45\delta \pi ^2}u_dv_d+\frac{64u_1}{45\delta \pi ^2}v_d\\&\quad \qquad -\,\frac{16 f}{3\delta \pi ^2}w_d +\frac{9088 v_2}{1575\delta \pi ^2}u_d -\frac{u_2}{\delta } v_d, \\&N_6=\frac{16}{3\pi ^2}w_d-\frac{16}{3\pi ^2}u_d. \end{aligned}$$

The delay terms are defined as

$$\begin{aligned}&u_d=|u_{1s}+u_{2s}\!-\!u_1(t-\tau )\!-\!u_2(t-\tau )|, \ \ v_d\!=\!|v_{1s}\!+\!v_{2s}-v_1(t-\tau )-v_2(t-\tau )|, \\&w_d=|w_{1s}+w_{2s}-w_1(t-\tau )-w_2(t-\tau )|. \end{aligned}$$