Stable Periodic Bifurcations of an Explicit Discretization of a Nonlinear Partial Differential Equation in Reaction Diffusion

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Abstract
The stability of an explicit discretization of Fisher's equation from reaction-diffusion is studied from the point of view of long time calculations with a fixed time step. The method is found to be stable under the same conditions as those required by the linearized scheme in the neighbourhood of the constant, stable, fixed point of the underlying partial differential equation. When these conditions are violated, it is shown that a variety of different period-doubling bifurcations can occur which extend, through the addition of a (discrete) diffusion term, known results from ordinary differential equations and maps on the line.

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