326 research outputs found
Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
A chimera state is a spatio-temporal pattern in a network of identical
coupled oscillators in which synchronous and asynchronous oscillation coexist.
This state of broken symmetry, which usually coexists with a stable spatially
symmetric state, has intrigued the nonlinear dynamics community since its
discovery in the early 2000s. Recent experiments have led to increasing
interest in the origin and dynamics of these states. Here we review the history
of research on chimera states and highlight major advances in understanding
their behaviour.Comment: 26 pages, 3 figure
Phase fluctuations in the ABC model
We analyze the fluctuations of the steady state profiles in the modulated
phase of the ABC model. For a system of sites, the steady state profiles
move on a microscopic time scale of order . The variance of their
displacement is computed in terms of the macroscopic steady state profiles by
using fluctuating hydrodynamics and large deviations. Our analytical prediction
for this variance is confirmed by the results of numerical simulations
The dynamics of interacting multi-pulses in the one-dimensional quintic complex Ginzburg-Landau equation
We formulate an effective numerical scheme that can readily, and accurately,
calculate the dynamics of weakly interacting multi-pulse solutions of the
quintic complex Ginzburg-Landau equation (QCGLE) in one space dimension. The
scheme is based on a global centre-manifold reduction where one considers the
solution of the QCGLE as the composition of individual pulses plus a remainder
function, which is orthogonal to the adjoint eigenfunctions of the linearised
operator about a single pulse. This centre-manifold projection overcomes the
difficulties of other, more orthodox, numerical schemes, by yielding a
fast-slow system describing 'slow' ordinary differential equations for the
locations and phases of the individual pulses, and a 'fast' partial
differential equation for the remainder function. With small parameter
where is a constant and is the
pulse separation distance, we write the fast-slow system in terms of
first-order and second-order correction terms only, a formulation which is
solved more efficiently than the full system. This fast-slow system is
integrated numerically using adaptive time-stepping. Results are presented here
for two- and three-pulse interactions. For the two-pulse problem, cells of
periodic behaviour, separated by an infinite set of heteroclinic orbits, are
shown to 'split' under perturbation creating complex spiral behaviour. For the
case of three pulse interaction a range of dynamics, including chaotic pulse
interaction, are found. While results are presented for pulse interaction in
the QCGLE, the numerical scheme can also be applied to a wider class of
parabolic PDEs.Comment: 33 page
Stochastic population dynamics in spatially extended predator-prey systems
Spatially extended population dynamics models that incorporate intrinsic
noise serve as case studies for the role of fluctuations and correlations in
biological systems. Including spatial structure and stochastic noise in
predator-prey competition invalidates the deterministic Lotka-Volterra picture
of neutral population cycles. Stochastic models yield long-lived erratic
population oscillations stemming from a resonant amplification mechanism. In
spatially extended predator-prey systems, one observes noise-stabilized
activity and persistent correlations. Fluctuation-induced renormalizations of
the oscillation parameters can be analyzed perturbatively. The critical
dynamics and the non-equilibrium relaxation kinetics at the predator extinction
threshold are characterized by the directed percolation universality class.
Spatial or environmental variability results in more localized patches which
enhances both species densities. Affixing variable rates to individual
particles and allowing for trait inheritance subject to mutations induces fast
evolutionary dynamics for the rate distributions. Stochastic spatial variants
of cyclic competition with rock-paper-scissors interactions illustrate
connections between population dynamics and evolutionary game theory, and
demonstrate how space can help maintain diversity. In two dimensions,
three-species cyclic competition models of the May-Leonard type are
characterized by the emergence of spiral patterns whose properties are
elucidated by a mapping onto a complex Ginzburg-Landau equation. Extensions to
general food networks can be classified on the mean-field level, which provides
both a fundamental understanding of ensuing cooperativity and emergence of
alliances. Novel space-time patterns emerge as a result of the formation of
competing alliances, such as coarsening domains that each incorporate
rock-paper-scissors competition games
Dynamics of Patterns
Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction
Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities
Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the self-organization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures the viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior.
A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have an important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable. It may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time-scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in non-equilibrium physics
Fluctuations and phase transitions in Larkin-Ovchinnikov liquid crystal states of population-imbalanced resonant Fermi gas
Motivated by a realization of imbalanced Feshbach-resonant atomic Fermi
gases, we formulate a low-energy theory of the Fulde-Ferrell and the
Larkin-Ovchinnikov (LO) states and use it to analyze fluctuations, stability,
and phase transitions in these enigmatic finite momentum-paired superfluids.
Focusing on the unidirectional LO pair-density wave state, that spontaneously
breaks the continuous rotational and translational symmetries, we show that it
is characterized by two Goldstone modes, corresponding to a superfluid phase
and a smectic phonon. Because of the liquid-crystalline "softness" of the
latter, at finite temperature the 3d state is characterized by a vanishing LO
order parameter, quasi-Bragg peaks in the structure and momentum distribution
functions, and a "charge"-4, paired Cooper-pairs, off-diagonal-long-range
order, with a superfluid-stiffness anisotropy that diverges near a transition
into a nonsuperfluid state. In addition to conventional integer vortices and
dislocations the LO superfluid smectic exhibits composite half-integer
vortex-dislocation defects. A proliferation of defects leads to a rich variety
of descendant states, such as the "charge"-4 superfluid and Fermi-liquid
nematics and topologically ordered nonsuperfluid states, that generically
intervene between the LO state and the conventional superfluid and the
polarized Fermi-liquid at low and high imbalance, respectively. The fermionic
sector of the LO gapless superconductor is also quite unique, exhibiting a
Fermi surface of Bogoliubov quasiparticles associated with the Andreev band of
states, localized on the array of the LO domain-walls.Comment: 56 pages, 21 figure
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