Self-organization of anti-aligning active particles: Waving pattern formation and chaos.

Recently, it has been shown that purely anti-aligning interaction between active particles may induce a finite wavelength instability. The formed patterns display intricate spatiotemporal dynamics, suggesting the presence of chaos. Here, we propose a quasi-one-dimensional simplification of the particle interaction model.

Anti-aligning interaction between active particles induces a finite wavelength instability: The dancing hexagons.

By considering a simple model for self-propelled particle interaction, we show that anti-aligning forces induce a finite wavelength instability. Consequently, the system exhibits pattern formation. The formed pattern involves, let us say, a choreographic movement of the active entities.

Solitonic-like interactions of counter-propagating clusters of active particles.

This report considers a set of interacting self-propelled particles immersed in a viscous and noisy environment. The explored particle interaction does not distinguish between alignments and anti-alignments of the self-propulsion forces. More specifically, we considered a set of self-propelled apolar aligning attractive particles.

Self-organized spiral patterns at the edge of an order-disorder nonequilibrium phase transition.

We present a spatially extended version of the Wood-Van den Broeck-Kawai-Lindenberg stochastic phase-coupled oscillator model. Our model is embedded in two-dimensional (2d) array with a range-dependent interaction. The Wood-Van den Broeck-Kawai-Lindenberg model is known to present a phase transition from a disordered state to a globally oscillatory phase in which the majority of the units are in the same discrete phase.

Flocking transition within the framework of Kuramoto paradigm for synchronization: Clustering and the role of the range of interaction.

A Kuramoto-type approach to address flocking phenomena is presented. First, we analyze a simple generalization of the Kuramoto model for interacting active particles, which is able to show the flocking transition (the emergence of coordinated movements in a group of interacting self-propelled agents). In the case of all-to-all interaction, the proposed model reduces to the Kuramoto model for phase synchronization of identical motionless noisy oscillators.

Synchronization and fluctuations: Coupling a finite number of stochastic units.

It is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit (infinite number of elements). In fact, since the early work of Kuramoto, mean-field theory has been used to analyze this transition. In contrast, work that directly deals with finite arrays is relatively scarce in the context of synchronization.

A continuous-time persistent random walk model for flocking.

A classical random walker is characterized by a random position and velocity. This sort of random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a walker represents an inanimate particle driven by environmental fluctuations.

Arrays of two-state stochastic oscillators: Roles of tail and range of interactions.

We study the role of the tail and the range of interaction in a spatially structured population of two-state on-off units governed by Markovian transition rates. The coupling among the oscillators is evidenced by the dependence of the transition rates of each unit on the states of the units to which it is coupled. Tuning the tail or range of the interactions, we observe a transition from an ordered global state (long-range interactions) to a disordered one (short-range interactions).

Synchronization of coupled noisy oscillators: Coarse graining from continuous to discrete phases.

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse graining continuous phase units.

Arrays of stochastic oscillators: Nonlocal coupling, clustering, and wave formation.

We consider an array of units each of which can be in one of three states. Unidirectional transitions between these states are governed by Markovian rate processes. The interactions between units occur through a dependence of the transition rates of a unit on the states of the units with which it interacts.

Globally coupled stochastic two-state oscillators: fluctuations due to finite numbers.

Infinite arrays of coupled two-state stochastic oscillators exhibit well-defined steady states. We study the fluctuations that occur when the number N of oscillators in the array is finite. We choose a particular form of global coupling that in the infinite array leads to a pitchfork bifurcation from a monostable to a bistable steady state, the latter with two equally probable stationary states.

Synchronization of globally coupled two-state stochastic oscillators with a state-dependent refractory period.

We present a model of identical coupled two-state stochastic units, each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state.

Pattern formation via intermittence from microscopic deterministic dynamics.

We propose a one-dimensional lattice model, inspired by population dynamics interaction. The model combines a variable coupling range with the Allee effect. The system is capable of exhibiting pattern formation that is similar to what occurs in similar continuous models for population dynamics.

Moving breathing pulses in the one-dimensional complex cubic-quintic Ginzburg-Landau equation.

We show and characterize numerically moving breathing pulses in the one-dimensional complex cubic-quintic Ginzburg-Landau equation. This class of stable moving breathing pulses has not been described before for this prototype envelope equation as it arises near the weakly hysteretic onset of traveling waves.

Transition from pulses to fronts in the cubic-quintic complex Ginzburg-Landau equation.

The cubic-quintic complex Ginzburg-Landau is the amplitude equation for systems in the vicinity of an oscillatory sub-critical bifurcation (Andronov-Hopf), and it shows different localized structures. For pulse-type localized structures, we review an approximation scheme that enables us to compute some properties of the structures, like their existence range. From that scheme, we obtain conditions for the existence of pulses in the upper limit of a control parameter.

Noise induces partial annihilation of colliding dissipative solitons.

Partial annihilation of two counterpropagating dissipative solitons, with only one pulse surviving the collision, has been widely observed in different experimental contexts, over a large range of parameters, from hydrodynamics to chemical reactions. However, a generic picture accounting for partial annihilation is missing. Based on our results for coupled complex cubic-quintic Ginzburg-Landau equations as well as for the FitzHugh-Nagumo equation we conjecture that noise induces partial annihilation of colliding dissipative solitons in many systems.