Bifurcation Analysis and Chaos Control for a Discrete Space–time Lokta–volterra Diffusion Equation With Neumann Boundary Conditions

This paper investigates a discrete Lotka–Volterra competition diffusion system subject to Neumann boundary conditions. By employing normal form theory and center manifold reduction, we show that the system exhibits a flip bifurcation at the unique positive equilibrium, whereas neither a Neimark–Sacker nor a fold bifurcation is observed, moreover, a Turing bifurcation is identified. To regulate the bifurcation behavior, both direct and indirect state feedback control strategies are implemented. Under appropriate parameter conditions, these control schemes effectively stabilize the system by suppressing chaos and bifurcation-induced instabilities. Extensive numerical simulations are conducted to validate the theoretical results. The bifurcation analysis framework developed herein is generalizable to other discrete diffusion systems with different boundary configurations.