Nonlocal boundary dynamics of traveling spots in a reaction-diffusion system

Stationary broken parity states in active matter models

We demonstrate that several nonvariational continuum models commonly used to describe active matter as well as other active systems exhibit nongeneric behavior: each model supports asymmetric but stationary localized states even in the absence of pinning at heterogeneities. Moreover, such states only begin to drift following a drift-transcritical bifurcation as the activity increases. Asymmetric stationary states should only exist in variational systems, i.e., in models with gradient structure. In other words, such states are expected in passive systems, but not in active systems where the gradient structure of the model is broken by activity. We identify a "spurious" gradient dynamics structure of these models that is responsible for this nongeneric behavior, and determine the types of additional terms that render the models generic, i.e., with asymmetric states that appear via drift-pitchfork bifurcations and are generically moving. We provide detailed illustrations of our results using numerical continuation of resting and steadily drifting states in both generic and nongeneric cases.

Two-dimensional localized states in an active phase-field-crystal model

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

Resting and traveling localized states in an active phase-field-crystal model

The conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles, Menzel and Löwen [Phys. Rev. Lett. 110, 055702 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.055702] obtained a model for crystallization (swarm formation) in active systems. Here, we study the occurrence of resting and traveling localized states, i.e., crystalline clusters, within the resulting active PFC model. Based on linear stability analyses and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of periodic and localized, resting and traveling states in a one-dimensional active PFC model. This allows us, for instance, to explore how the slanted homoclinic snaking of steady localized states found for the passive PFC model is amended by activity. A particular focus lies on the onset of motion, where we show that it occurs either through a drift-pitchfork or a drift-transcritical bifurcation. A corresponding general analytical criterion is derived.

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Breathing instability versus drift instability in a two-component reaction-diffusion system

We investigate the stability of an excited domain in a two-component reaction-diffusion system in two dimensions and correct the previous results obtained by one of the authors [L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics (Springer, Berlin, 2006); L. M. Pismen, Phys. Rev. Lett. 86, 548 (2001)].

Deformable self-propelled particles with a global coupling

We have proposed a model of deformable self-propelled particles in which the time-evolution equations are given in terms of the center-of-mass velocity and a nematic order parameter representing the motion-induced deformation [T. Ohta and T. Ohkuma, Phys. Rev. Lett. 102, 154101 (2009)]. We investigate its many-body problem applying a global orientational coupling. Depending on the strength of the interaction, the self-propelled particles exhibit various kinds of collective dynamics and chaotic behavior: a ballistic procession state, a scattered state, a coherently phase synchronized state, two types of in-phase synchronized state, and an anomalously diffusive state. The phase reduction method for the weak coupling regime reveals the bifurcations between the secular collective motions. The phase boundary among the chaos regime and the synchronized regimes is determined by the linear stability analysis of the synchronized states.

Deformation of a self-propelled domain in an excitable reaction-diffusion system

We formulate a theory for a self-propelled domain in an excitable reaction-diffusion system in two dimensions where the domain deforms from a circular shape when the propagation velocity is increased. In the singular limit where the width of the domain boundary is infinitesimally thin, we derive a set of equations of motion for the center of gravity and two fundamental deformation modes. The deformed shapes of a steadily propagating domain are obtained. The set of time-evolution equations exhibits a bifurcation from a straight motion to a circular motion by changing the system parameters.

Synergistic effect of two inhibitors on one activator in a reaction-diffusion system

We investigate a three-component reaction-diffusion system that describes the interaction of one activator and two inhibitors where one inhibitor acts as a traveling pulse generator of the activator and the other acts as a lateral inhibition localizer. It is numerically shown that the synergistic effect of these two inhibitors on one activator induces several spatiotemporal patterns such as destabilization and nonannihilation of traveling pulses and the occurrence and splitting of traveling spots. By using singular perturbation procedures, the stability of radially symmetric equilibrium solutions is discussed. Furthermore, we discuss how such dynamics are caused under the synergistic effect of two inhibitors.

Perturbation theory for traveling droplets

The motion of chemically driven droplets is analyzed by applying a solvability condition of perturbed hydrodynamic equations affected by the adsorbate concentration. Conditions for traveling bifurcation analogous to a similar transition in activator-inhibitorsystems are obtained. It is shown that interaction of droplets leads to either scattering of mobile droplets or the formation of regular patterns, respectively, at low or high adsorbate diffusivity. The same method is applied to droplets running on growing terrace edges during surface freezing.

Pattern formation capacity of spatially extended systems

We analyze a class of spatially extended systems which are capable of generating many complicated patterns. These systems are given by the Ginzburg-Landau equation coupled with a system of two linear equations and describe nonlinear media with localized defects. We find a connection between these systems and spin-glass systems. We show that the system is capable to produce many patterns and describe patterning algorithms.

Noise-covered drift bifurcation of dissipative solitons in a planar gas-discharge system

The trajectories of propagating self-organized, well-localized solitary patterns (dissipative solitons) in the form of electrical current filaments are experimentally investigated in a planar quasi-two-dimensional dc gas-discharge system with high Ohmic semiconductor barrier. Earlier phenomenological models qualitatively describing the experimental observations in terms of a particle model predict a transition from stationary filaments to filaments traveling with constant finite speed due to an appropriate change of the system parameters. This prediction motivates a search for a drift bifurcation in the experimental system, but a direct comparison of experimentally recorded trajectories with theoretical predictions is impossible due to the strong influence of noise. To solve this problem, the filament dynamics is modeled using an appropriate Langevin equation, allowing for the application of a stochastic data analysis technique to separate deterministic and stochastic parts of the dynamics. Simulations carried out with the particle model demonstrate the efficiency of the method. Applying the technique to the experimentally recorded trajectories yields good agreement with the predictions of the model equations. Finally, the predicted drift bifurcation is found using the semiconductor resistivity as control parameter. In the resulting bifurcation diagram, the square of the equilibrium velocity scales linearly with the control parameter.

Reconstruction and roughening of a catalytic Pt(110) surface coupled to kinetic oscillations

Three-dimensional reconstruction and roughening of a Pt(110) surface is studied with the help of a qualitative Monte Carlo model. A distinct CO adsorption uptake on different surface phases is taken into account. The computations show that a "missing row" structure with defects relaxes to a more stable (111)-faceted structure. The CO+O2 reaction kinetics is modeled by a phenomenological equation with a cubic nonlinearity reproducing a correct qualitative picture of oscillations. The surface roughening developing under the reaction conditions causes slow changes in catalytic activity of the surface. A nanoscale front between the 1x1 and 1x2 phases disintegrates due to repeated phase transitions caused by CO coverage oscillations. Defect formation and roughening dominate the dynamics of surface phase transitions. A one-dimensional extension of the model reproduces microscopic traveling waves on the CO diffusion scale.