localized states analogues The excellent collaboration with Francesco Marino and Gianni Giacomelli has brought its first result: our paper about front pinning and localized states has been published as a Physical Review Letter. Spatially localized structures (in optics but also in many other contexts) are often analyzed as the result of the pinning of fronts connecting coexisting states. One favourable situation for the emergence of spatially localized states is therefore the coexistence of two stationary states, one of which is modulated in space. While this vision works well in the transverse dimension in optical systems, its application to localized structures which form along the propagation direction is tempting but not obvious. In this paper, we analyze phenomena analogue to front pinning and localized states in a system where only temporal effects may take place: a delayed dynamical system.

It has long been known that delay dynamical systems can have strong similarities (to the point of formal equivalence in some specific cases) with spatially extended systems. In order to visualize this equivalence in the dynamics of both kinds of systems, one can display time series of a delayed dynamical system in a space-time like diagram. This diagram can be obtained by splitting the time series in segments whose duration is equal to the delay present in the system, which are then stacked on top of each other. The evolution of the segment (which constitute the space-like dimension of the system) then slowly takes place along the vertical axis. This is the way the picture above was constructed. Our experiment (and numerical analysis) is guided by previous work showing a coarsening-like dynamics in a bistable system with delay. In surprisingly good analogy with the behavior of spatially extended systems in which two uniform states coexist, what was found in that paper is that the most stable of the states invades the whole "spatial" extension, with the fronts moving away from each other at constant velocity (provided they are not too close to each other). From that point, the idea of exploring analogues of localized states is immediate: in order to obtain a "pinning" of the fronts in the space like dimension, it may be sufficient to apply a periodic forcing in the pseudo space (ie in time, in the end...). We did just that in an experimental setup based on polarization bistability in a vertical cavity surface emitting laser with optoelectronic feedback.

We observe front pinning and localized states in the pseudo space, and even the bifurcations leading to the destabilization of localized states are strongly reminiscent of the dynamics of stationary spatially extended systems. In spite of this almost perfect analogy, the absence of parity symmetry in the space like dimension (which is time...) has a very strong impact and broad implications that we begin to explore in this work.

The paper is on our publications page: it is about Front pinning and localized states analogues in long-delayed bistable systems.