Pulses and global bifurcations in a nonlocal reaction-diffusion system.

Abstract

Experiments and simulations of simple reaction-diffusion systems in bounded domains with spatially nonlocal coupling display interesting pulse motions that are absent without the nonlocal effect. These include pulses that stick to the boundaries, ``bounce'' off them, or disappear at one boundary and reappear at the other, as if the domain was periodic. We numerically show that, for a two-variable model system, the transitions between these motions occur through global bifurcations. The transition from a wall-bound stationary front to a ``back-and-forth'' moving (bouncing) pulse occurs through a symmetric crisis. This motion evolves into a ``unidirectional'' motion, in which a pulse disappears at one boundary as a new one is born at the other, through a gluing bifurcation. The relationship between the spatiotemporal behavior and its phase-space representation is shown, as well as the importance of the nonlocal effect in creating the required global phase-space structure.