Rotating waves in scalar equations with neumann boundary conditions

Abstract

Rotating and spiral waves occur in a variety of biological and chemical systems, such as the Belousov-Zhabotinskii reaction. Existence of such solutions is well known for a number of different coupled reaction-diffusion models, although analytical results are usually difficult to obtain. Traditionally, rotating and spiral waves have been thought to arise only in coupled systems of equations, although recent work has shown that such solutions are also possible in scalar equations if suitable boundary conditions are imposed. Such spiral or rotating boundaries are not, however, physically realistic. In this paper, we consider a class of scalar equations incorporating a discrete delay. We find that the time delay can bring about rotating wave solutions (through a Hopf bifurcation from a nontrivial spatially uniform equilibrium) in equations with homogeneous Neumann boundary conditions.