Unidirectional dynamics in translationally coupled bistable systems: Waves, fronts, and self-organization

Translational coupling takes place when the dynamics of a state variable at a given position depend on the state variable at a translated position, as occurs in optical experiments that present misaligned feedback. This article investigates analytically and numerically the effects of translational coupling on the dynamics of a spatially extended dissipative ϕ4 system. We predict a translational-coupling-induced Andronov–Hopf instability with non-zero wavenumber. The resulting unidirectional waves are characterized using a Ginzburg–Landau equation. The analytic solution for this wave shows excellent agreement with the numerical results. For larger values of the coupling parameter, secondary instabilities occur, resulting in spatiotemporal chaotic dynamics. Beyond the dynamics of uniform equilibria and waves, this system exhibits fronts with nonreciprocal propagation. The front speed as a function of the translational coupling parameter is obtained by analytical approximations and numerical simulations, showing good agreement between the two methods. Increasing the translational coupling values, the domains at each side of the front core exhibit a rich self-organization that goes from regular to chaotic waves. Finally, numerical bifurcation diagrams are presented.