Uniform ultimate boundedness and approximate synchronization of coupled reaction-diffusion systems with parameter mismatches

We present a framework for studying the uniform ultimate boundedness and approximate synchronization of coupled systems. The individual subsystems of the coupled systems are described by general reaction-diffusion (RD) systems, and we account for parameter mismatches between individual subsystems while considering a general coupling scheme. We derive several criteria for the uniform ultimate boundedness of coupled RD systems under different assumptions regarding the properties of the reaction terms and coupling schemes. Using M-matrix theory and iterative methods, we establish an approximate synchronization criterion for the coupled RD systems. This criterion allows the coupled systems to synchronize with synchronization errors arising from parameter mismatches, and perfect synchronization is achieved when there are no parameter mismatches. Moreover, the criterion provides an estimated bound for the synchronization error, which decreases as the degree of inhomogeneity caused by parameter mismatches reduces. To demonstrate the effectiveness of our approaches, we apply our theories to the FitzHugh-Nagumo (FHN) neuron model.