Competing resonances in spatially forced pattern-forming systems

Frequency locking in auditory hair cells: Distinguishing between additive and parametric forcing

- The auditory system displays remarkable sensitivity and frequency discrimination, attributes shown to rely on an amplification process that involves a mechanical as well as a biochemical response. Models that display proximity to an oscillatory onset (also known as Hopf bifurcation) exhibit a resonant response to distinct frequencies of incoming sound, and can explain many features of the amplification phenomenology. To understand the dynamics of this resonance, frequency locking is examined in a system near the Hopf bifurcation and subject to two types of driving forces: additive and parametric. Derivation of a universal amplitude equation that contains both forcing terms enables a study of their relative impact on the hair cell response. In the parametric case, although the resonant solutions are 1 : 1 frequency locked, they show the coexistence of solutions obeying a phase shift of π, a feature typical of the 2 : 1 resonance. Different characteristics are predicted for the transition from unlocked to locked solutions, leading to smooth or abrupt dynamics in response to different types of forcing. The theoretical framework provides a more realistic model of the auditory system, which incorporates a direct modulation of the internal control parameter by an applied drive. The results presented here can be generalized to many other media, including Faraday waves, chemical reactions, and elastically driven cardiomyocytes, which are known to exhibit resonant behavior.

Non-monotonic resonance in a spatially forced Lengyel-Epstein model

We study resonant spatially periodic solutions of the Lengyel-Epstein model modified to describe the chlorine dioxide-iodine-malonic acid reaction under spatially periodic illumination. Using multiple-scale analysis and numerical simulations, we obtain the stability ranges of 2:1 resonant solutions, i.e., solutions with wavenumbers that are exactly half of the forcing wavenumber. We show that the width of resonant wavenumber response is a non-monotonic function of the forcing strength, and diminishes to zero at sufficiently strong forcing. We further show that strong forcing may result in a π/2 phase shift of the resonant solutions, and argue that the nonequilibrium Ising-Bloch front bifurcation can be reversed. We attribute these behaviors to an inherent property of forcing by periodic illumination, namely, the increase of the mean spatial illumination as the forcing amplitude is increased.

Reaction diffusion Voronoi diagrams: from sensors data to computing

In this paper, a new method to solve computational problems using reaction diffusion (RD) systems is presented. The novelty relies on the use of a model configuration that tailors its spatiotemporal dynamics to develop Voronoi diagrams (VD) as a part of the system's natural evolution. The proposed framework is deployed in a solution of related robotic problems, where the generalized VD are used to identify topological places in a grid map of the environment that is created from sensor measurements. The ability of the RD-based computation to integrate external information, like a grid map representing the environment in the model computational grid, permits a direct integration of sensor data into the model dynamics. The experimental results indicate that this method exhibits significantly less sensitivity to noisy data than the standard algorithms for determining VD in a grid. In addition, previous drawbacks of the computational algorithms based on RD models, like the generation of volatile solutions by means of excitable waves, are now overcome by final stable states.

Reversing desertification as a spatial resonance problem

An important environmental application of pattern control by periodic spatial forcing is the restoration of vegetation patterns in water-limited ecosystems that went through desertification. Vegetation restoration is often based on periodic landscape modulations that intercept overland water flow and form favorable conditions for vegetation growth. Viewing this method as a spatial resonance problem, we show that plain realizations of this method, assuming a complete vegetation response to the imposed modulation pattern, suffer from poor resilience to rainfall variability. By contrast, less intuitive realizations, based on the inherent spatial modes of vegetation growth and involving partial vegetation implantation, can be highly resilient and equally productive. We derive these results using two complementary models, a realistic vegetation model, and a simple pattern formation model that lends itself to mathematical analysis and highlights the universal aspects of the behaviors found with the vegetation model. We focus on reversing desertification as an outstanding environmental problem, but the main conclusions hold for any spatially forced system near the onset of a finite-wave-number instability that is subjected to noisy conditions.

Spatial forcing of pattern-forming systems that lack inversion symmetry

The entrainment of periodic patterns to spatially periodic parametric forcing is studied. Using a weak nonlinear analysis of a simple pattern formation model we study the resonant responses of one-dimensional systems that lack inversion symmetry. Focusing on the first three n:1 resonances, in which the system adjusts its wavenumber to one nth of the forcing wavenumber, we delineate commonalities and differences among the resonances. Surprisingly, we find that all resonances show multiplicity of stable phase states, including the 1:1 resonance. The phase states in the 2:1 and 3:1 resonances, however, differ from those in the 1:1 resonance in remaining symmetric even when the inversion symmetry is broken. This is because of the existence of a discrete translation symmetry in the forced system. As a consequence, the 2:1 and 3:1 resonances show stationary phase fronts and patterns, whereas phase fronts within the 1:1 resonance are propagating and phase patterns are transients. In addition, we find substantial differences between the 2:1 resonance and the other two resonances. While the pattern forming instability in the 2:1 resonance is supercritical, in the 1:1 and 3:1 resonances it is subcritical, and while the inversion asymmetry extends the ranges of resonant solutions in the 1:1 and 3:1 resonances, it has no effect on the 2:1 resonance range. We conclude by discussing a few open questions.

Fronts and patterns in a spatially forced CDIMA reaction

We use experiments on a chemical reaction and model analysis to study localized phase fronts in stripe patterns and their roles as building blocks of extended rectangular and oblique patterns.