scholarly journals Wavenumber selection via spatial parameter jump

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170191 ◽  
Author(s):  
Arnd Scheel ◽  
Jasper Weinburd
Keyword(s):  
Spatial Dynamics ◽  
Zero Solution ◽  
Numerical Continuation ◽  
Spatial Inhomogeneity ◽  
Homogeneous State ◽  
Severe Restriction ◽  
Spatial Parameter ◽  
Parameter Case ◽  
Steady State Solutions ◽  
Type Parameter

The Swift–Hohenberg equation describes an instability which forms finite-wavenumber patterns near onset. We study this equation posed with a spatial inhomogeneity; a jump-type parameter that renders the zero solution stable for x <0 and unstable for x >0. Using normal forms and spatial dynamics, we prove the existence of a family of steady-state solutions that represent a transition in space from a homogeneous state to a striped pattern state. The wavenumbers of these stripes are contained in a narrow band whose width grows linearly with the size of the jump. This represents a severe restriction from the usual constant-parameter case, where the allowed band grows with the square root of the parameter. We corroborate our predictions using numerical continuation and illustrate implications on stability of growing patterns in direct simulations. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

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.

Design of Circularly Towed Cable-Body Systems

2011 ◽  
Author(s):  
Varma Gottimukkala ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
Bifurcation Analysis ◽  
Small Diameter ◽  
The Body ◽  
Numerical Continuation ◽  
Stable Steady State ◽  
Dimensional System ◽  
Design Algorithm ◽  
Towed Cable ◽  
Steady State Solutions

Circularly towed cable-body systems can be used to pickup and deliver payloads, provide surveillance, and tow aerial and marine vehicles. To provide a stable operating platform, the body or end mass should have a unique and stable steady state solution with small diameter so that it travels at a much slower speed than the tow vehicle. In this paper, the minimum damping is calculated that ensures the stable single valued steady state solutions as a function of non-dimensional system parameters including cable length and end mass. Steady state solutions are found using the numerical continuation and bifurcation analysis and Galerkin’s method provides the linearized vibration equations that determine stability. Bifurcation analysis is also used to find the minimum achievable end mass radius. A design algorithm is presented and demonstrated using an example.

scholarly journals Hidden Symmetry in a Kuramoto–Sivashinsky Initial-Boundary Value Problem

2017 ◽  
Vol 27 (09) ◽  
pp. 1750136
Author(s):  
Pietro-Luciano Buono ◽  
Lennaert van Veen ◽  
Eryn Frawley
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Zero Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dirichlet Boundary Conditions ◽  
Numerical Continuation ◽  
Bifurcation Structure ◽  
Hidden Symmetry ◽  
Extended Problem

We investigate the bifurcation structure of the Kuramoto–Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and [Formula: see text] symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary conditions and one that breaks a shift-reflect symmetry of the extended system. Using Lyapunov–Schmidt reduction, we show both to be supercritical. We extend the primary branches by means of numerical continuation, and show that they lose stability in pitchfork, transcritical or Hopf bifurcations. Tracking the corresponding secondary branches reveals an interval of the viscosity parameter in which up to four stable equilibria and time-periodic solutions coexist. Since the study of the extended problem is indispensable for the explanation of the bifurcation structure, the Kuramoto–Sivashinsky problem with Dirichlet boundary conditions provides an elegant manifestation of hidden symmetry.

SYNCHRONIZATION OF CHAOS AND THE TRANSITION TO WAVE TURBULENCE

2012 ◽  
Vol 22 (10) ◽  
pp. 1250234 ◽  
Author(s):  
R. L. VIANA ◽  
S. R. LOPES ◽  
J. D. SZEZECH ◽  
I. L. CALDAS
Keyword(s):  
Chaotic Dynamics ◽  
Spatial Dynamics ◽  
Wave Turbulence ◽  
Homogeneous State ◽  
Spatial Mode ◽  
Mode Excitation ◽  
Spatially Extended ◽  
Spatially Homogeneous

We investigated the transition to wave turbulence in a spatially extended three-wave interacting model, where a spatially homogeneous state undergoing chaotic dynamics undergoes spatial mode excitation. The transition to this weakly turbulent state can be regarded as the loss of synchronization of chaos of mode oscillators describing the spatial dynamics.

Sozial-räumliche Dynamiken der Agrartreibstoffe.

2016 ◽  
Vol 46 (184) ◽  
pp. 423-440 ◽  
Author(s):  
Kristina Dietz ◽  
Bettina Engels ◽  
Oliver Pye
Keyword(s):  
Political Economy ◽  
Capital Flows ◽  
Spatial Dynamics ◽  
Financial Sector ◽  
Comprehensive Analysis ◽  
Spatial Theory ◽  
Global Political Economy

This article explores the spatial dynamics of agrofuels. Building on categories from the field of critical spatial theory, it shows how these categories enable a comprehensive analysis of the spatial dynamics of agrofuels that links the macro-structures of the global political economy to concrete, place-based struggles. Four core socio-spatial dynamics of agrofuel politics are highlighted and applied to empirical findings: territorialization, the financial sector as a new scale of regulation, place-based struggles and transnational spaces of resources and capital flows.

Temporal and spatial dynamics of pulse production in India

2017 ◽  
Author(s):  
International Food Policy Research Institute (IFPRI)
Keyword(s):  
Spatial Dynamics ◽  
Temporal And Spatial ◽  
Pulse Production

Modified Y Splitting of Lateral Rectus in Treatment of Abnormal Vertical Eye Movements Combined with Globe Retraction in Duane Retraction Syndrome

2020 ◽  
pp. 1-3
Author(s):  
Ayse Gul Kocak Altintas ◽  
Ayse Gul Kocak Altintas
Keyword(s):  
Case Report ◽  
Eye Movements ◽  
Palpebral Fissure ◽  
Ocular Motility ◽  
Severe Restriction ◽  
Horizontal Movement ◽  
Retraction Syndrome ◽  
Duane Retraction Syndrome ◽  
Ocular Motility Disorders

Duane retraction syndrome is the most frequently seen restrictive ocular motility disorders. It is clinically presented with limitation of horizontal movement, variable amounts of upshoots or downshoots and globe retraction combined with narrowing of the palpebral aperture on attempted adduction. An 8-year-old patient presented with severe restriction of abduction, reciprocal upshots or downshoots, and globe retraction combined with the palpebral fissure narrowing of on adduction. After the modified Y splitting of LR and recession of both horizontal rectus operation, all cosmetically disfiguring clinical features disappeared. In this case report modified Y splitting procedure and its long-term efficacy is presented.

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