Stability Properties and Spurious Period Two Solutions in a Numerical Scheme for a Reaction-Diffusion Equation with Nonlinear Diffusion

In previous work a discrete version of a generalized form ofFisher's equation was considered with both linear and a quadraticnonlinear diffusion, and the stability regions of certain periodtwo solutions were calculated. In this paper higher-order nonlinearitiesin the diffusion term are considered. It is shown that termsof order higher than cubic destabilize the period two solutions,in the sense that the stability region becomes smaller thanin the case of linear diffusion. Quadratic diffusion, on theother hand, has the largest stability region. It is also shown that for nonlinear diffusion of sufficientlyhigh order, unstable period two solutions may exist for thesame parameter values as the stable fixed point. Unlike thecase of lower orders of nonlinear diffusion, there is no longera strip of stable period two solutions adjacent to the entireboundary of the stability region of this fixed point. The results of several numerical calculations are given. Theseimply that the basin of attraction of the stable fixed pointbecomes smaller when the order of nonlinearity of the diffusionterm is increased.