PD Control of Reaction-Diffusion Systems with Neumann Boundary in Coupled Map Lattices

Abstract

Reaction-diffusion systems often exhibit complex spatiotemporal behaviors such as bifurcations and chaotic dynamics, posing challenges for effective control. This study focuses on developing a discrete control framework based on coupled map lattices (CMLs) to analyze and stabilize such behaviors. By introducing a proportional-derivative (PD) control strategy, we investigate its influence on the suppression of Flip and Neimark-Sacker bifurcations as well as Turing pattern formation. Additionally, this paper incorporates Neumann boundary conditions into CMLs system for the first time, providing a more realistic representation of boundary effects in physical and biological systems. Numerical simulations demonstrate that appropriate tuning of PD parameters can significantly delay the onset of instabilities and reduce spatial chaos. These results provide insights into the control of reaction-diffusion systems and offer references for discrete modeling and stabilization strategies in related contexts.

Appendices

A

$$\begin{aligned} F_1(W, Z, \tau ) =&\;((-a_{100} + \lambda _{2})a_{101} - a_{010}b_{101})\\&\times a_{010}(W + Z)\tau \\&+ ((-a_{100} + \lambda _{2})a_{200} - a_{010}b_{200})\\&\times a_{010}^2(W + Z)^2\\&+ ((-a_{100} + \lambda _{2})a_{201} - a_{010}b_{201})\\&\times a_{010}^2(W + Z)^2\tau \\&+ ((-a_{100} + \lambda _{2})a_{300} - a_{010}b_{300})\\&\times a_{010}^3(W + Z)^3\\&- ((-a_{100} + \lambda _{2})a_{011} - a_{010}b_{011})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})\tau \\&- ((-a_{100} + \lambda _{2})a_{110} - a_{010}b_{110})\\&\times a_{010}(W + Z)(W + (W + Z)a_{100}\\&- Z\lambda _{2}) - ((-a_{100} + \lambda _{2})a_{111} - a_{010}b_{111})\\&\times a_{010}(W + Z)(W + (W + Z)a_{100} \\&- Z\lambda _{2})\tau - ((-a_{100} + \lambda _{2})a_{210} - a_{010}b_{210})\\&\times a_{010}^2(W + Z)^2(W + (W + Z)a_{100}\\&- Z\lambda _{2}) + ((-a_{100} {+} \lambda _{2})a_{020} {-} a_{010}b_{020})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})^2\\&+ ((-a_{100} + \lambda _{2})a_{021} - a_{010}b_{021})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})^2\tau \\&+ ((-a_{100} + \lambda _{2})a_{120} - a_{010}b_{120})\\&\times a_{010}(W + Z)(W + (W + Z)a_{100}\\&- Z\lambda _{2})^2 {-} ((-a_{100} {+} \lambda _{2})a_{030} - a_{010}b_{030})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})^3\\&\times +O(|W,Z,\tau |^4), \end{aligned}$$

$$\begin{aligned} G_1(W, Z, \tau )=&\;((1+a_{100})a_{101} - a_{010}b_{101})\\&\times a_{010}(W + Z)\tau \\&+ ((1+a_{100})a_{200} - a_{010}b_{200})\\&\times a_{010}^2(W + Z)^2\\&+ ((1+a_{100})a_{201} - a_{010}b_{201})\\&\times a_{010}^2(W + Z)^2\tau \\&+ ((1+a_{100})a_{300} - a_{010}b_{300})\\&\times a_{010}^3(W + Z)^3\\&- ((1+a_{100})a_{011} - a_{010}b_{011})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})\tau \\&- ((1+a_{100})a_{110} - a_{010}b_{110})\\&\times a_{010}(W + Z)(W + (W + Z)\\&\times a_{100} - Z\lambda _{2})\\&- ((1+a_{100})a_{111} - a_{010}b_{111})\\&\times a_{010}(W + Z)(W + (W + Z)\\&\times a_{100} - Z\lambda _{2})\tau \\&- ((1+a_{100})a_{210} - a_{010}b_{210})\\&\times a_{010}^2(W + Z)^2(W + (W + Z)\\&\times a_{100} - Z\lambda _{2})\\&+ ((1+a_{100})a_{020} - a_{010}b_{020})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})^2\\&+ ((1+a_{100})a_{021} - a_{010}b_{021})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})^2\tau \\&+ ((1+a_{100})a_{120} - a_{010}b_{120})\\&\times a_{010}(W + Z)(W + (W + Z)\\&\times a_{100} - Z\lambda _{2})^2 \\&- ((1+a_{100})a_{030} - a_{010}b_{030})\\&\times (W + (W + Z)a_{100} - Z\lambda _{2})^3\\&\times +O(|W,Z,\tau |^4). \end{aligned}$$

B

$$\begin{aligned}&A_{20}=\left( a_{02} a_{10}^2 - a_{01} a_{10} a_{11} + a_{01}^2 a_{20} \right. \\&\qquad \quad \left. - 2 a_{02} a_{10} \alpha + a_{01} a_{11} \alpha + a_{02} \alpha ^2\right) \beta ,\\&A_{11}=\left( 2 a_{02} a_{10} - a_{01} a_{11} - 2 a_{02} \alpha \right) ,A_{02}= a_{02}\beta ^3,\\&A_{30}=\left( a_{01} a_{10}^2 a_{12} -a_{03} a_{10}^3 - a_{01}^2 a_{10} a_{21} \right. \\&\qquad \quad \left. + a_{01}^3 a_{30} + 3 a_{03} a_{10}^2 \alpha - 2 a_{01} a_{10} a_{12} \alpha \right. \\&\left. + a_{01}^2 a_{21} \alpha - 3 a_{03} a_{10} \alpha ^2 + a_{01} a_{12} \alpha ^2 + a_{03} \alpha ^3 \right) \beta ,\\&A_{21}=\left( 2 a_{01} a_{10} a_{12} -3 a_{03} a_{10}^2 - a_{01}^2 a_{21} \right. \\&\qquad \quad \left. + 6 a_{03} a_{10} \alpha - 2 a_{01} a_{12} \alpha - 3 a_{03} \alpha ^2\right) \beta ^2 ,\\&A_{12}=\left( a_{01} a_{12} -3 a_{03} a_{10} + 3 a_{03} \alpha \right) \beta ^3 ,\\&A_{03}=- a_{03}\beta ^4.\\&B_{20}=\frac{a_{01}}{a_{10}} \left( a_{02} a_{10}^3 + a_{01} a_{10}^2 a_{11} - a_{01}^2 a_{10} a_{20} \right. \\&\qquad \quad \left. - a_{10}^3 b_{02} + a_{01} a_{10}^2 b_{11} - a_{01}^2 a_{10} b_{20} \right. \\&\qquad \quad +3 a_{02} a_{10}^2 \alpha - 2 a_{01} a_{10} a_{11} \alpha + a_{01}^2 a_{20} \alpha \\&\qquad \quad + 2 a_{10}^2 b_{02} \alpha - a_{01} a_{10} b_{11} \alpha - 3 a_{02} a_{10} \alpha ^2 \\&\qquad \quad \left. +a_{01} a_{11} \alpha ^2 - a_{10} b_{02} \alpha ^2 + a_{02} \alpha ^3 \right) ,\\&B_{11}=\frac{a_{01} }{a_{10}} \left( 2 a_{02} a_{10}^2 + a_{01} a_{10} a_{11} \right. \\&\qquad \quad \left. - 2 a_{10}^2 b_{02} + a_{01} a_{10} b_{11} + 4 a_{02} a_{10} \alpha - a_{01} a_{11} \alpha \right. \\&\qquad \quad \left. +2 a_{10} b_{02} \alpha - 2 a_{02} \alpha ^2 \right) \beta ,\\&B_{02}= \frac{a_{01} }{a_{10}} \left( a_{02} a_{10} - a_{10} b_{02} + a_{02} \alpha \right) \beta ^2, \\&B_{30}=\frac{ a_{01} }{a_{10}} \left( a_{03} a_{10}^4 - a_{01} a_{10}^3 a_{12} + a_{01}^2 a_{10}^2 a_{21} \right. \\&\qquad \quad \left. - a_{01}^3 a_{10} a_{30} + a_{10}^4 b_{03} - a_{01} a_{10}^3 b_{12} \right. \\&\qquad \quad +a_{01}^2 a_{10}^2 b_{21} - a_{01}^3 a_{10} b_{30} - 4 a_{03} a_{10}^3 \alpha \\&\qquad \quad + 3 a_{01} a_{10}^2 a_{12} \alpha - 2 a_{01}^2 a_{10} a_{21} \alpha + a_{01}^3 a_{30} \alpha \\&\qquad \quad -3 a_{10}^3 b_{03} \alpha + 2 a_{01} a_{10}^2 b_{12} \alpha - a_{01}^2 a_{10} b_{21} \alpha \\&\qquad \quad + 6 a_{03} a_{10}^2 \alpha ^2 - 3 a_{01} a_{10} a_{12} \alpha ^2 + a_{01}^2 a_{21} \alpha ^2 \\&\qquad \quad \left. + 3 a_{10}^2 b_{03} \alpha ^2 - a_{01} a_{10} b_{12} \alpha ^2 - 4 a_{03} a_{10} \alpha ^3 \right. \\&\qquad \quad \left. + a_{01} a_{12} \alpha ^3 - a_{10} b_{03} \alpha ^3 + a_{03} \alpha ^4 \right) ,\\&B_{21}= \frac{ a_{01}}{a_{10}} \left( 3 a_{03} a_{10}^3 - 2 a_{01} a_{10}^2 a_{12} + a_{01}^2 a_{10} a_{21} \right. \\&\qquad \quad \left. + 3 a_{10}^3 b_{03} - 2 a_{01} a_{10}^2 b_{12} +a_{01}^2 a_{10} b_{21} \right. \\&\qquad \quad - 9 a_{03} a_{10}^2 \alpha + 4 a_{01} a_{10} a_{12} \alpha - a_{01}^2 a_{21} \alpha \\&\qquad \quad - 6 a_{10}^2 b_{03} \alpha + 2 a_{01} a_{10} b_{12} \alpha \\&\qquad \quad \left. +9 a_{03} a_{10} \alpha ^2 - 2 a_{01} a_{12} \alpha ^2 \right. \\&\left. + 3 a_{10} b_{03} \alpha ^2 - 3 a_{03} \alpha ^3 \right) \beta ,\\&B_{12}= \frac{a_{01} }{a_{10}} \left( 3 a_{03} a_{10}^2 - a_{01} a_{10} a_{12} + 3 a_{10}^2 b_{03} \right. \\&\qquad \quad \left. - a_{01} a_{10} b_{12}- 6 a_{03} a_{10} \alpha + a_{01} a_{12} \alpha \right. \\&\qquad \quad \left. - 3 a_{10} b_{03} \alpha + 3 a_{03} \alpha ^2 \right) \beta ^2 ,\\&B_{03}= \frac{a_{01} }{a_{10}} \left( a_{03} a_{10} + a_{10} b_{03} - a_{03} \alpha \right) \beta ^3. \end{aligned}$$