Front interaction induces excitable behavior.

Spatially extended systems can support local transient excitations in which just a part of the system is excited. The mechanisms reported so far are local excitability and excitation of a localized structure. Here we introduce an alternative mechanism based on the coexistence of two homogeneous stable states and spatial coupling. We show the existence of a threshold for perturbations of the homogeneous state. Subthreshold perturbations decay exponentially. Superthreshold perturbations induce the emergence of a long-lived structure formed by two back to back fronts that join the two homogeneous states. While in typical excitability the trajectory follows the remnants of a limit cycle, here reinjection is provided by front interaction, such that fronts slowly approach each other until eventually annihilating. This front-mediated mechanism shows that extended systems with no oscillatory regimes can display excitability.

Transition from oscillatory to excitable regime in a system forced at three times its natural frequency.

The effect of a temporal modulation at three times the critical frequency on a Hopf bifurcation is studied in the framework of amplitude equations. We consider a complex Ginzburg-Landau equation with an extra quadratic term, resulting from the strong coupling between the external field and the unstable modes. We show that, by increasing the intensity of the forcing, one passes from an oscillatory regime to an excitable one with three equivalent frequency-locked states. In the oscillatory regime, topological defects are one-armed phase spirals, while in the excitable regime they correspond to three-armed excitable amplitude spirals. Analytical results show that the transition between these two regimes occurs at a critical value of the forcing intensity. The transition between phase and amplitude spirals is confirmed by numerical analysis and it might be observed in periodically forced reaction-diffusion systems.

Walk-off and pattern selection in optical parametric oscillators.

The effect of walk-off in pattern selection in optical parametric oscillators is theoretically examined. We show that a dynamic mechanism also allows us to observe the formation of structures for positive signal detunings. In this regime the pattern that is generated is a periodic array of kinks that separate regions in which one of two stable steady states is alternately selected. This structure can be regarded as a train of dark soliton stripes because the two solutions have opposite signs. The wavelength of the selected pattern is theoretically predicted, and the prediction agrees with the results of the numerical solutions of the equations governing the device.

Growth dynamics of noise-sustained structures in nonlinear optical resonators.

The existence of macroscopic noise-sustained structures in nonlinear optics is theoretically predicted and numerically observed, in the regime of convective instability. The advection-like term, necessary to turn the instability to convective for the parameter region where advection overwhelms the growth, can stem from pump beam tilting or birefringence induced walk-off. The growth dynamics of both noise-sustained and deterministic patterns is exemplified by means of movies. This allows to observe the process of formation of these structures and to confirm the analytical predictions. The amplification of quantum noise by several orders of magnitude is predicted. The qualitative analysis of the near- and far-field is given. It suffices to distinguish noise-sustained from deterministic structures; quantitative informations can be obtained in terms of the statistical properties of the spectra.

From oscillations to excitability: A case study in spatially extended systems.

This volume is devoted to the presentation of the main contributions to the workshop "From oscillations to excitability: A case study in spatially extended systems," organized by the authors in Nice in June 1993. It gives an overview of the current research on spatiotemporal patterns in a wide range of systems that display self-oscillatory or excitable behavior. It tries to give a better understanding of the transition from the oscillatory to the excitable regime and of its effect on the properties of spiral waves, and to fill the gap between the theories and concepts used to describe both regimes in the so-called "active media."

The Ginzburg-Landau approach to oscillatory media.

Close to a supercritical Hopf bifurcation, oscillatory media may be described, by the complex Ginzburg-Landau equation. The most important spatiotemporal behaviors associated with this dynamics are reviewed here. It is shown, on a few concrete examples, how real chemical oscillators may be described by this equation, and how its coefficients may be obtained from the experimental data. Furthermore, the effect of natural forcings, induced by the experimental realization of chemical oscillators in batch reactors, may also be studied in the framework of complex Ginzburg-Landau equations and its associated phase dynamics. We show, in particular, how such forcings may locally transform oscillatory media into excitable ones and trigger the formation of complex spatiotemporal patterns.

Spatial patterns and double diffusion in chemical reactions.

Spatial ordering has been observed recently during various photochemical reactions. Convoluted concentration bands first appear near the surface of shallow irradiated solutions. They thereafter extend into the bulk, and finger-like structures spontaneously develop. We discuss here the possible role of double-diffusion effects in the onset of this phenomenon. Indeed, chemical reactions occurring near the surface or evaporation of the solvent, or both, induce in the bulk adverse gradients of a pair of properties (concentrations of solute or concentration and temperature) having different diffusivities. This difference can then destabilize the homogeneous solution and trigger the observed patterns.