Range of validity and intermittent dynamics of the phase of oscillators with nonlinear self-excitation

A range of active systems, particularly of chemical nature, are known to perform self-excited oscillations coupled by diffusion. The role of the diffusion is not trivial so that the differences in the phase of the oscillations through space may persist, depending on the values of the controlling parameters of the system. First, we analyse a sixth-order nonlinear partial differential equation describing such dynamics. We evaluate the range of the parameters leading to different finite versions of the equation, specifically a version based on nonlinear excitation and a version based on linear excitation. In the second part of the work we solve the equation in two spatial dimensions by finite-difference discretization in space and subsequent numerical integration of a system of ordinary differential equations in time. A forced variant of the equation is derived and selected exact solutions are presented. They are also used to verify the numerical code. For the unforced equation, irregular dynamics intermitting with periods of slow evolution are recorded and discussed.