Resonant and nonresonant patterns in forced oscillators

Molding the asymmetry of localized frequency-locking waves by a generalized forcing and implications to the inner ear

Frequency locking to an external forcing frequency is a well-known phenomenon. In the auditory system, it results in a localized traveling wave, the shape of which is essential for efficient discrimination between incoming frequencies. An amplitude equation approach is used to show that the shape of the localized traveling wave depends crucially on the relative strength of additive versus parametric forcing components; the stronger the parametric forcing, the more asymmetric is the response profile and the sharper is the traveling-wave front. The analysis qualitatively captures the empirically observed regions of linear and nonlinear responses and highlights the potential significance of parametric forcing mechanisms in shaping the resonant response in the inner ear.

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.

Weakly and strongly coupled Belousov-Zhabotinsky patterns

We investigate experimentally and numerically the synchronization of two-dimensional spiral wave patterns in the Belousov-Zhabotinsky reaction due to point-to-point coupling of two separate domains. Different synchronization modalities appear depending on the coupling strength and the initial patterns in each domain. The behavior as a function of the coupling strength falls into two qualitatively different regimes. The weakly coupled regime is characterized by inter-domain interactions that distorted but do not break wave fronts. Under weak coupling, spiral cores are pushed around by wave fronts in the other domain, resulting in an effective interaction between cores in opposite domains. In the case where each domain initially contains a single spiral, the cores form a bound pair and orbit each other at quantized distances. When the starting patterns consist of multiple randomly positioned spiral cores, the number of cores decreases with time until all that remains are a few cores that are synchronized with a partner in the other domain. The strongly coupled regime is characterized by interdomain interactions that break wave fronts. As a result, the wave patterns in both domains become identical.

Complex mixed-mode oscillatory patterns in a periodically forced excitable Belousov-Zhabotinsky reaction model

The Oregonator is the simplest chemically plausible model for the Belousov-Zhabotinsky reaction. We investigate the response of the Oregonator to sinusoidal inputs with amplitudes and frequencies within plausible ranges. We focus on a regime where the unforced Oregonator is excitable (with no sustained oscillations). We use numerical simulations and dynamical systems tools to both characterize the response patterns and explain the underlying dynamic mechanisms.

Two-cluster solutions in an ensemble of generic limit-cycle oscillators with periodic self-forcing via the mean-field

We study two-cluster solutions of an ensemble of generic limit-cycle oscillators in the vicinity of a Hopf bifurcation, i.e., Stuart-Landau oscillators, with a nonlinear global coupling. This coupling leads to conserved mean-field oscillations acting back on the individual oscillators as a forcing. A reduction to two effective equations makes a linear stability analysis of the cluster solutions possible. These equations exhibit a π-rotational symmetry, leading to a complex bifurcation structure and a wide variety of solutions. In fact, the principal bifurcation structure resembles that of a 2:1 resonance tongue, while inside the tongue we observe an 1:1 entrainment.

Competing resonances in spatially forced pattern-forming systems

Spatial periodic forcing can entrain a pattern-forming system in the same way as temporal periodic forcing can entrain an oscillator. The forcing can lock the pattern's wave number to a fraction of the forcing wave number within tonguelike domains in the forcing parameter plane, it can increase the pattern's amplitude, and it can also create patterns below their onset. We derive these results using a multiple-scale analysis of a spatially forced Swift-Hohenberg equation in one spatial dimension. In two spatial dimensions the one-dimensional forcing can induce a symmetry-breaking instability that leads to two-dimensional (2D) patterns, rectangular or oblique. These patterns resonate with the forcing by locking their wave-vector component in the forcing direction to half the forcing wave number. The range of this type of 2:1 resonance overlaps with the 1:1 resonance tongue of stripe patterns. Using a multiple-scale analysis in the overlap region we show that the 2D patterns can destabilize the 1:1 resonant stripes even at exact resonance. This result sheds new light on the use of spatial periodic forcing for controlling patterns.

Period doubling and spatiotemporal chaos in periodically forced CO oxidation on Pt(110)

Periodic forcing of chemical turbulence in the catalytic CO oxidation on Pt(110) can induce a period doubling cascade to chaos. Using a forcing frequency near the second harmonic of the system's natural frequency, and carefully increasing the forcing amplitude, the system successively exhibits spiral wave turbulence, resonant pattern formation, and chaotic oscillations. In the latter case, global coupling induces strong spatial correlation. Experimental results are presented as well as numerical simulations using a realistic model. Good agreement is found between experiment and theory. The results give further insight into the complex nature of reaction-diffusion systems and are of high importance regarding control strategies on such systems. The presented setup enhances the range of achievable dynamical states and allows for new experimental investigations on the dynamics of extended oscillatory systems.

Design and control of patterns in reaction-diffusion systems

We discuss the design of reaction-diffusion systems that display a variety of spatiotemporal patterns. We also consider how these patterns may be controlled by external perturbation, typically using photochemistry or temperature. Systems treated include the Belousov-Zhabotinsky (BZ) reaction, the chlorite-iodide-malonic acid and chlorine dioxide-malonic acid-iodine reactions, and the BZ-AOT system, i.e., the BZ reaction in a water-in-oil reverse microemulsion stabilized by the surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT).

Synchronization and firing death in the dynamics of two interacting excitable units with heterogeneous signals

We study the response of two coupled FitzHugh-Nagumo systems to heterogeneous external inputs. The latter, modeled by periodic parametric stimuli, force the uncoupled excitable systems into a regime of chaotic firing. Due to parameter dispersion involved in randomly distributed amplitudes and/or phases of the external forces the units are nonidentical and their firing events will be asynchronous. Interest is focused on mutually synchronized spikings arising through the coupling. It is demonstrated that the phase difference of the two external forces crucially affects the onset of spike synchronization as well as the resulting degree of synchrony. For large phase differences the degree of spike synchrony is constricted to a maximal possible value and cannot be enhanced upon increasing the coupling strength. We even found that overcritically strong couplings lead to suppression of firing so that the units perform synchronous subthreshold oscillations. This effect, which we call "firing death," is due to a coupling-induced modification of the excitation threshold impeding spiking of the units. In clear contrast, when only the amplitudes of the forces are distributed perfect spike synchrony is achieved for sufficiently strong coupling.