4,322 research outputs found
Noise-induced Turbulence in Nonlocally Coupled Oscillators
We demonstrate that nonlocally coupled limit-cycle oscillators subject to
spatiotemporally white Gaussian noise can exhibit a noise-induced transition to
turbulent states. After illustrating noise-induced turbulent states with
numerical simulations using two representative models of limit-cycle
oscillators, we develop a theory that clarifies the effective dynamical
instabilities leading to the turbulent behavior using a hierarchy of dynamical
reduction methods. We determine the parameter region where the system can
exhibit noise-induced turbulent states, which is successfully confirmed by
extensive numerical simulations at each level of the reduction.Comment: 23 pages, 17 figures, to appear in Phys. Rev.
Accurate macroscale modelling of spatial dynamics in multiple dimensions
Developments in dynamical systems theory provides new support for the
macroscale modelling of pdes and other microscale systems such as Lattice
Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically
resolving subgrid microscale dynamics the dynamical systems approach constructs
accurate closures of macroscale discretisations of the microscale system. Here
we specifically explore reaction-diffusion problems in two spatial dimensions
as a prototype of generic systems in multiple dimensions. Our approach unifies
into one the modelling of systems by a type of finite elements, and the
`equation free' macroscale modelling of microscale simulators efficiently
executing only on small patches of the spatial domain. Centre manifold theory
ensures that a closed model exist on the macroscale grid, is emergent, and is
systematically approximated. Dividing space either into overlapping finite
elements or into spatially separated small patches, the specially crafted
inter-element/patch coupling also ensures that the constructed discretisations
are consistent with the microscale system/PDE to as high an order as desired.
Computer algebra handles the considerable algebraic details as seen in the
specific application to the Ginzburg--Landau PDE. However, higher order models
in multiple dimensions require a mixed numerical and algebraic approach that is
also developed. The modelling here may be straightforwardly adapted to a wide
class of reaction-diffusion PDEs and lattice equations in multiple space
dimensions. When applied to patches of microscopic simulations our coupling
conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv
admin note: substantial text overlap with arXiv:0904.085
Noncoaxial multivortices in the complex sine-Gordon theory on the plane
We construct explicit multivortex solutions for the complex sine-Gordon
equation (the Lund-Regge model) in two Euclidean dimensions. Unlike the
previously found (coaxial) multivortices, the new solutions comprise single
vortices placed at arbitrary positions (but confined within a finite part of
the plane.) All multivortices, including the single vortex, have an infinite
number of parameters. We also show that, in contrast to the coaxial complex
sine-Gordon multivortices, the axially-symmetric solutions of the
Ginzburg-Landau model (the stationary Gross-Pitaevskii equation) {\it do not}
belong to a broader family of noncoaxial multivortex configurations.Comment: 40 pages, 7 figures in colou
Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation
We formulate and study dynamics from a complex Ginzburg-Landau system with
saturable nonlinearity, including asymmetric cross-phase modulation (XPM)
parameters. Such equations can model phenomena described by complex
Ginzburg-Landau systems under the added assumption of saturable media. When the
saturation parameter is set to zero, we recover a general complex cubic
Ginzburg-Landau system with XPM. We first derive conditions for the existence
of bounded dynamics, approximating the absorbing set for solutions. We use this
to then determine conditions for amplitude death of a single wavefunction. We
also construct exact plane wave solutions, and determine conditions for their
modulational instability. In a degenerate limit where dispersion and
nonlinearity balance, we reduce our system to a saturable nonlinear
Schr\"odinger system with XPM parameters, and we demonstrate the existence and
behavior of spatially heterogeneous stationary solutions in this limit. Using
numerical simulations we verify the aforementioned analytical results, while
also demonstrating other interesting emergent features of the dynamics, such as
spatiotemporal chaos in the presence of modulational instability. In other
regimes, coherent patterns including uniform states or banded structures arise,
corresponding to certain stable stationary states. For sufficiently large yet
equal XPM parameters, we observe a segregation of wavefunctions into different
regions of the spatial domain, while when XPM parameters are large and take
different values, one wavefunction may decay to zero in finite time over the
spatial domain (in agreement with the amplitude death predicted analytically).
While saturation will often regularize the dynamics, such transient dynamics
can still be observed - and in some cases even prolonged - as the saturability
of the media is increased, as the saturation may act to slow the timescale.Comment: 36 page
An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning
Manifold-learning techniques are routinely used in mining complex
spatiotemporal data to extract useful, parsimonious data
representations/parametrizations; these are, in turn, useful in nonlinear model
identification tasks. We focus here on the case of time series data that can
ultimately be modelled as a spatially distributed system (e.g. a partial
differential equation, PDE), but where we do not know the space in which this
PDE should be formulated. Hence, even the spatial coordinates for the
distributed system themselves need to be identified - to emerge from - the data
mining process. We will first validate this emergent space reconstruction for
time series sampled without space labels in known PDEs; this brings up the
issue of observability of physical space from temporal observation data, and
the transition from spatially resolved to lumped (order-parameter-based)
representations by tuning the scale of the data mining kernels. We will then
present actual emergent space discovery illustrations. Our illustrative
examples include chimera states (states of coexisting coherent and incoherent
dynamics), and chaotic as well as quasiperiodic spatiotemporal dynamics,
arising in partial differential equations and/or in heterogeneous networks. We
also discuss how data-driven spatial coordinates can be extracted in ways
invariant to the nature of the measuring instrument. Such gauge-invariant data
mining can go beyond the fusion of heterogeneous observations of the same
system, to the possible matching of apparently different systems
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