Spatially Periodic Patterns of Synchrony in Lattice Networks

2009 ◽  
Vol 8 (2) ◽  
pp. 641-675 ◽  
Author(s):  
Ana Paula S. Dias ◽  
Eliana Manuel Pinho
Keyword(s):  
Periodic Patterns ◽  
Spatially Periodic

Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations

1995 ◽  
Vol 352 (1698) ◽  
pp. 125-168 ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Pattern ◽  
Invariant Equilibrium ◽  
Periodic Patterns ◽  
Planar Lattice ◽  
Diffusive Convection ◽  
Spatially Periodic ◽  
Spatio Temporal ◽  
Boussinesq Fluid ◽  
Time Periodic

In Rayleigh-Bénard convection, the spatially uniform motionless state of a fluid loses stability as the Rayleigh number is increased beyond a critical value. In the simplest case of convection in a pure Boussinesq fluid, the instability is a symmetry-breaking steady-state bifurcation that leads to the formation of spatially periodic patterns. However, in many double-diffusive convection systems the heat-conduction solution actually loses stability via Hopf bifurcation. These hydrodynamic systems provide motivation for the present study of spatiotemporally periodic pattern formation in Euclidean equivariant systems. We call such patterns planforms . We classify, according to spatio-temporal symmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a group-invariant equilibrium. Instead of focusing on plan- forms periodic with respect to a specified planar lattice, as has been done in previous investigations, we consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular dif­ferential equation.

scholarly journals Peculiarities in the Director Reorientation and the NMR Spectra Evolution in a Nematic Liquid Crystals under the Effect of Crossed Electric and Magnetic Fields

Crystals ◽  
2019 ◽  
Vol 9 (5) ◽  
pp. 262 ◽  
Author(s):  
Alex V. Zakharov ◽  
Izabela Sliwa
Keyword(s):  
Nematic Phase ◽  
Viscosity Coefficient ◽  
Threshold Value ◽  
Periodic Patterns ◽  
Director Field ◽  
Electric And Magnetic Fields ◽  
Spatially Periodic ◽  
Rotational Viscosity ◽  
Director Reorientation ◽  
Reorientational Dynamics

The illustrative description of the field-induced peculiarities of the director reorientation in the microsized nematic volumes under the effect of crossed magnetic B and electric E fields have been proposed. The most interesting feature of such configuration is that the nematic phase becomes unstable after applying the strong E . The theoretical analysis of the reorientational dynamics of the director field provides an evidence for the appearance of the spatially periodic patterns in response to applied large E directed at an angle α to B . The feature of this approach is that the periodic distortions arise spontaneously from a homogeneously aligned nematic sample that ultimately induces a faster response than in the uniform mode. The nonuniform rotational modes involve additional internal elastic distortions of the conservative nematic system and, as a result, these deformations decrease of the viscous contribution U vis to the total energy U of the nematic phase. In turn, that decreasing of U vis leads to decrease of the effective rotational viscosity coefficient γ eff ( α ) . That is, a lower value of γ eff ( α ) , which is less than one in the bulk nematic phase, gives the less relaxation time τ on ( α ) ∼ γ eff ( α ) , when α is bigger than the threshold value α th . The results obtained by Deuterium NMR spectroscopy confirm theoretically obtained dependencies of τ on ( α ) on α .

The problem of periodic patterns in embryos

1981 ◽  
Vol 295 (1078) ◽  
pp. 509-524 ◽  
Keyword(s):  
Pattern Formation ◽  
Linear Dimension ◽  
Cell Layer ◽  
Body Plan ◽  
Body Structure ◽  
Periodic Patterns ◽  
Mesodermal Cell ◽  
Somite Formation ◽  
Spatially Periodic ◽  
Derivatives Of

A segmented body-plan has developed at least twice during metazoan evolution: in the lineage including annelids and arthropods, where the segment is the unit of body structure, and in the ancestors of vertebrates, where a primary segmentation of the middle, mesodermal cell layer of the embryo imposes a spatially periodic character upon derivatives of other layers. The mechanism controlling the development of these periodic patterns has the property that the number of the serially homologous structures formed within each species is largely independent of the linear dimension, or scale, at which pattern formation occurs in individual cases. In this they contrast with other patterns of dispersed, homologous structures occurring in animal epidermis and dermis. The performance of various classes of model for the control of number in vertebrate somite formation are compared, in the light of experimentally and naturally observable properties of this aspect of pattern.

scholarly journals Spatially periodic calcium profiles around single channels

2016 ◽  
Author(s):  
S. L. Mironov
Keyword(s):  
Calcium Binding ◽  
Linear Models ◽  
Single Channel ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Cell Physiology ◽  
Single Channels ◽  
Periodic Patterns ◽  
Calcium Diffusion ◽  
Spatially Periodic

AbstractThe concept of calcium nanodomains established around the sites of calcium entry into the cell is fundamental for mechanistic consideration of key physiological responses. It stems from linear models of calcium diffusion from single channel into the cytoplasm, but is only valid for calcium increases smaller than the concentration of calcium-binding species. Recent experiments indicate much higher calcium levels in the vicinity of channel exit that should cause buffer saturation. I here derive explicit solutions of respective non-linear reaction-diffusion problem and found dichotomous solution - for small fluxes the steady state calcium profiles have quasi-exponential form, whereas in the case of buffer saturation calcium distributions show spatial periodicity. These non-trivial and novel spatial calcium profiles are supported by Monte-Carlo simulations. Imaging of 1D- and radial distributions around single α-synuclein channels measured in cell-free conditions supports the theory. I suggest that periodic patterns may arise under different physiological conditions and play specific role in cell physiology.

Spatially periodic patterns in a dc gas-discharge system

1992 ◽  
Vol 45 (4) ◽  
pp. 2546-2557 ◽  
Author(s):  
C. Radehaus ◽  
H. Willebrand ◽  
R. Dohmen ◽  
F.-J. Niedernostheide ◽  
G. Bengel ◽  
...  
Keyword(s):  
Gas Discharge ◽  
Periodic Patterns ◽  
Discharge System ◽  
Spatially Periodic

Short-Wave Instability in Extended Systems with Additional Symmetry

1997 ◽  
Vol 07 (05) ◽  
pp. 997-1006 ◽  
Author(s):  
Michael I. Tribelsky
Keyword(s):  
Seismic Waves ◽  
Short Wave ◽  
Continuous Group ◽  
Periodic Patterns ◽  
Wave Instability ◽  
Coupled Modes ◽  
Additional Symmetry ◽  
Pattern Stability ◽  
Spatially Periodic ◽  
The Stability

Stability of steady spatially periodic patterns in systems with an additional continuous group of symmetry is discussed. It is shown that different systems with the same dimensionality of the continuous group of symmetry display remarkable similarity in all qualitative features of the pattern stability problem. Attention is called to the fact that, beside an extra band of slowly varying modes, the additional symmetry may yield a mixture of different scales in the final dispersion equation for pattern's perturbations, so that the stability conditions become unusually sensitive to very fine details of the problem. A one-dimensional partial differential equation governing seismic waves in viscoelastic media is considered as a particular example. The equation exhibits short-wave instability and additional invariance under the transformation u → u + const. , where the order parameter u(x, t) is associated with the displacement velocity. The analytical study of the equation is supplemented by computer simulations. The simulations show that the system undergoes a bifurcation from the trivial state with u ≡ 0 to well-developed chaos directly and the transition to the chaos is smooth, without any discontinuity. The chaos is characterized by excitation of a big number (in a boundless system — continuum) of coupled modes localized generally in two narrow subbands, centered around the critical wavenumber for the short-wave instability and the wavenumber equals zero, respectively.

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