DYNAMICS OF AN INTERFACE CONNECTING A STRIPE PATTERN AND A UNIFORM STATE: AMENDED NEWELL–WHITEHEAD–SEGEL EQUATION

2009 ◽  
Vol 19 (08) ◽  
pp. 2801-2812 ◽  
Author(s):  
RENÉ G. ROJAS ◽  
RICARDO G. ELÍAS ◽  
MARCEL G. CLERC
Keyword(s):  
Amplitude Equation ◽  
Conventional Approach ◽  
Homogeneous State ◽  
Stripe Pattern ◽  
Localized Structures ◽  
Uniform State ◽  
The Rich

The dynamics of an interface connecting a stationary stripe pattern with a homogeneous state is studied. The conventional approach which describes this interface, Newell–Whitehead–Segel amplitude equation, does not account for the rich dynamics exhibited by these interfaces. By amending this amplitude equation with a nonresonate term, we can describe this interface and its dynamics in a unified manner. This model exhibits a rich and complex transversal dynamics at the interface, including front propagations, transversal patterns, locking phenomenon, and transversal localized structures.

Curvature effects and radial homoclinic snaking

2021 ◽  
Author(s):  
Damià Gomila ◽  
Edgar Knobloch
Keyword(s):  
Scaling Law ◽  
Spatial Dynamics ◽  
Two Dimensions ◽  
Localized States ◽  
Homogeneous State ◽  
Curvature Effects ◽  
Localized Structures ◽  
Homoclinic Snaking ◽  
Curvature Driven ◽  
Axisymmetric Structures

Abstract In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.

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

2009 ◽  
Vol 367 (1901) ◽  
pp. 3227-3238 ◽  
Author(s):  
Pablo Gutiérrez ◽  
Daniel Escaff ◽  
Orazio Descalzi
Keyword(s):  
Numerical Simulations ◽  
Control Parameter ◽  
Approximation Scheme ◽  
Landau Equation ◽  
Amplitude Equation ◽  
Ginzburg Landau ◽  
Analytical Expression ◽  
Localized Structures ◽  
Upper Limit ◽  
Pulse Type

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. When we study the width of pulses in that limit, the analytical expression shows that it is related to the transition between pulses and fronts. This fact is consistent with numerical simulations.

scholarly journals Localized States in Bi-Pattern Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
U. Bortolozzo ◽  
M. G. Clerc ◽  
F. Haudin ◽  
R. G. Rojas ◽  
S. Residori
Keyword(s):  
Optical System ◽  
Nonlinear Optical ◽  
Amplitude Equation ◽  
Localized States ◽  
Two Dimensional ◽  
One Dimensional ◽  
Self Organized ◽  
Localized Structures ◽  
Spatially Periodic ◽  
Different Shapes

We present a unifying description of localized states observed in systems with coexistence of two spatially periodic states, calledbi-pattern systems. Localized states are pinned over an underlying lattice that is either a self-organized pattern spontaneously generated by the system itself, or a periodic grid created by a spatial forcing. We show that localized states are generic and require only the coexistence of two spatially periodic states. Experimentally, these states have been observed in a nonlinear optical system. At the onset of the spatial bifurcation, a forced one-dimensional amplitude equation is derived for the critical modes, which accounts for the appearance of localized states. By numerical simulations, we show that localized structures persist on two-dimensional systems and exhibit different shapes depending on the symmetry of the supporting patterns.

scholarly journals Transversal interface dynamics of a front connecting a stripe pattern to a uniform state

2008 ◽  
Vol 83 (2) ◽  
pp. 28002 ◽  
Author(s):  
Marcel G. Clerc ◽  
Daniel Escaff ◽  
René Rojas
Keyword(s):  
Stripe Pattern ◽  
Interface Dynamics ◽  
Uniform State

Poor and Hidden Among the Rich and Not So Rich

PsycCRITIQUES ◽  
2006 ◽  
Vol 51 (49) ◽  
Author(s):  
Patricia M. Berliner
Keyword(s):  
The Rich

Institutional Theory of Endless Redistribution

2005 ◽  
pp. 4-18 ◽  
Author(s):  
K. Sonin
Keyword(s):  
Property Rights ◽  
Institutional Theory ◽  
Growth Rates ◽  
Rent Seeking ◽  
Economic Institutions ◽  
Wide Spread ◽  
Grass Roots ◽  
Protection Systems ◽  
Private Protection ◽  
The Rich

In unequal societies, the rich may benefit from shaping economic institutions in their favor. This paper analyzes the dynamics of institutional subversion by focusing on public protection of property rights. If this institution functions imperfectly, agents have incentives to invest in private protection of property rights. The ability to maintain private protection systems makes the rich natural opponents of public protection of property rights and precludes grass-roots demand to drive the development of the market-friendly institution. The economy becomes stuck in a bad equilibrium with low growth rates, high inequality of income, and wide-spread rent-seeking. The Russian oligarchs of the 1990s, who controlled large stakes of newly privatized property, provide motivation for this paper.

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