1
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Alonso A, Batiste O, Mercader I. Stationary localized solutions in binary convection in slightly inclined rectangular cells. Phys Rev E 2022; 106:055106. [PMID: 36559449 DOI: 10.1103/physreve.106.055106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2022] [Accepted: 10/25/2022] [Indexed: 06/17/2023]
Abstract
We analyze numerically the effect of a slight inclination in the lowest part of the snaking branches of convectons that are present in negative separation ratio binary mixtures in two-dimensional elongated rectangular cells. The exploration reveals the existence of novel stationary localized solutions with striking spatial features different from those of convectons. The numerical continuation of these solutions with respect to the inclination of the cell unveils the existence of even further families of localized structures that can organize in closed branches. A variety of localized solutions coexist for the same heating and inclination, depicting a highly complex scenario for solutions in the lowest part of the snaking diagrams for moderate to high heating. The different localized solutions obtained in the horizontal cell are discussed in detail.
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Affiliation(s)
- Arantxa Alonso
- Departament de Física, Universitat Politècnica de Catalunya, Mòdul B4, 08034 Barcelona, Spain
| | - Oriol Batiste
- Departament de Física, Universitat Politècnica de Catalunya, Mòdul B4, 08034 Barcelona, Spain
| | - Isabel Mercader
- Departament de Física, Universitat Politècnica de Catalunya, Mòdul B4, 08034 Barcelona, Spain
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2
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Verschueren N, Knobloch E, Uecker H. Localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk. Phys Rev E 2021; 104:014208. [PMID: 34412325 DOI: 10.1103/physreve.104.014208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2021] [Accepted: 06/16/2021] [Indexed: 11/07/2022]
Abstract
Axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation posed on a disk with Neumann boundary conditions are studied via numerical continuation and bifurcation analysis. Axisymmetric localized solutions in the form of spots and rings known from earlier studies persist and snake in the usual fashion until they begin to interact with the boundary. Depending on parameters, including the disk radius, these states may or may not connect to the branch of domain-filling target states. Secondary instabilities of localized axisymmetric states may create multiarm localized structures that grow and interact with the boundary before broadening into domain-filling states. High azimuthal wave number wall states referred to as daisy states are also found. Secondary bifurcations from these states include localized daisies, i.e., wall states localized in both radius and angle. Depending on parameters, these states may snake much as in the one-dimensional Swift-Hohenberg equation, or invade the interior of the domain, yielding states referred to as worms, or domain-filling stripes.
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Affiliation(s)
- Nicolás Verschueren
- Physics Department, University of California at Berkeley, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Physics Department, University of California at Berkeley, Berkeley, California 94720, USA
| | - Hannes Uecker
- Institute for Mathematics, Carl von Ossietzky University of Oldenburg, Oldenburg, Germany
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3
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Salewski M, Gibson JF, Schneider TM. Origin of localized snakes-and-ladders solutions of plane Couette flow. Phys Rev E 2019; 100:031102. [PMID: 31640040 DOI: 10.1103/physreve.100.031102] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2019] [Indexed: 11/07/2022]
Abstract
Spatially localized invariant solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations [Schneider, Gibson, and Burke, Phys. Rev. Lett. 104, 104501 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.104501]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of wavy-vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a close connection between pattern-formation theory and Navier-Stokes flow.
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Affiliation(s)
- Matthew Salewski
- Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany
| | - John F Gibson
- Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824, USA
| | - Tobias M Schneider
- Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
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4
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Gordillo L, Knobloch E. Fluid-supported elastic sheet under compression: Multifold solutions. Phys Rev E 2019; 99:043001. [PMID: 31108605 DOI: 10.1103/physreve.99.043001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2018] [Indexed: 11/07/2022]
Abstract
The properties of a hinged floating elastic sheet of finite length under compression are considered. Numerical continuation is used to compute spatially localized buckled states with many spatially localized folds. Both symmetric and antisymmetric states are computed and the corresponding bifurcation diagrams determined. Weakly nonlinear analysis is used to analyze the transition from periodic wrinkles to singlefold and multifold states and to compute their energy. States with the same number of folds have energies that barely differ from each other and the energy gap decreases exponentially as localization increases. The stability of the different competing states is studied and the multifold solutions are all found to be unstable. However, the decay time into solutions with fewer folds can be so slow that multifolds may appear to be stable.
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Affiliation(s)
- Leonardo Gordillo
- Departamento de Física, Universidad de Santiago de Chile, Av. Ecuador 3493, Estación Central, Santiago, Chile
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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5
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Zykov VS. Spiral wave initiation in excitable media. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:rsta.2017.0385. [PMID: 30420544 DOI: 10.1098/rsta.2017.0385] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 07/19/2018] [Indexed: 05/20/2023]
Abstract
Spiral waves represent an important example of dissipative structures observed in many distributed systems in chemistry, biology and physics. By definition, excitable media occupy a stationary resting state in the absence of external perturbations. However, a perturbation exceeding a threshold results in the initiation of an excitation wave propagating through the medium. These waves, in contrast to acoustic and optical ones, disappear at the medium's boundary or after a mutual collision, and the medium returns to the resting state. Nevertheless, an initiation of a rotating spiral wave results in a self-sustained activity. Such activity unexpectedly appearing in cardiac or neuronal tissues usually destroys their dynamics which results in life-threatening diseases. In this context, an understanding of possible scenarios of spiral wave initiation is of great theoretical importance with many practical applications.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.
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Affiliation(s)
- V S Zykov
- Max Planck Institute for Dynamics and Self-Organization, Goettingen, Germany
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6
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Gandhi P, Zelnik YR, Knobloch E. Spatially localized structures in the Gray-Scott model. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:20170375. [PMID: 30420543 PMCID: PMC6232600 DOI: 10.1098/rsta.2017.0375] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 08/21/2018] [Indexed: 05/19/2023]
Abstract
Spatially localized structures in the one-dimensional Gray-Scott reaction-diffusion model are studied using a combination of numerical continuation techniques and weakly nonlinear theory, focusing on the regime in which the activator and substrate diffusivities are different but comparable. Localized states arise in three different ways: in a subcritical Turing instability present in this regime, and from folds in the branch of spatially periodic Turing states. They also arise from the fold of spatially uniform states. These three solution branches interconnect in complex ways. We use numerical continuation techniques to explore their global behaviour within a formulation of the model that has been used to describe dryland vegetation patterns on a flat terrain.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.
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Affiliation(s)
- Punit Gandhi
- Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, USA
| | - Yuval R Zelnik
- Centre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS and Paul Sabatier University, 09200 Moulis, France
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, CA 94720, USA
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7
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Wetzel D. Tristability between stripes, up-hexagons, and down-hexagons and snaking bifurcation branches of spatial connections between up- and down-hexagons. Phys Rev E 2018; 97:062221. [PMID: 30011496 DOI: 10.1103/physreve.97.062221] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2017] [Indexed: 11/07/2022]
Abstract
Third-order amplitude equations on hexagonal lattices can be used for predicting the existence and stability of stripes, up-hexagons, and down-hexagons in pattern-forming systems. These amplitude equations predict the nonexistence of bistable ranges between up- and down-hexagons and tristable ranges between stripes, up-, and down-hexagons. In the present work we use fifth-order amplitude equations for finding such bistable and tristable ranges for a generalized Swift-Hohenberg equation and discuss stationary front connections between up- and down-hexagons.
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Affiliation(s)
- D Wetzel
- Institut für Mathematik, Carl-von-Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
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8
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Lazarides N, Tsironis GP. Multistable dissipative breathers and collective states in SQUID Lieb metamaterials. Phys Rev E 2018; 98:012207. [PMID: 30110756 DOI: 10.1103/physreve.98.012207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2017] [Indexed: 06/08/2023]
Abstract
A SQUID (Superconducting QUantum Interference Device) metamaterial on a Lieb lattice with nearest-neighbor coupling supports simultaneously stable dissipative breather families which are generated through a delicate balance of input power and intrinsic losses. Breather multistability is possible due to the peculiar snaking flux amplitude-frequency curve of single dissipative-driven SQUIDs, which for relatively high sinusoidal flux field amplitudes exhibits several stable and unstable solutions in a narrow frequency band around resonance. These breathers are very weakly interacting with each other, while multistability regimes with a different number of simultaneously stable breathers persist for substantial intervals of frequency, flux field amplitude, and coupling coefficients. Moreover, the emergence of chimera states as well as temporally chaotic states exhibiting spatial homogeneity within each sublattice of the Lieb lattice is demonstrated. The latter of the states emerge through an explosive hysteretic transition resembling explosive synchronization that has been reported before for various networks of oscillators.
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Affiliation(s)
- N Lazarides
- Department of Physics, University of Crete, P. O. Box 2208, 71003 Heraklion, Greece
- National University of Science and Technology "MISiS," Leninsky Prospekt 4, Moscow 119049, Russia
| | - G P Tsironis
- Department of Physics, University of Crete, P. O. Box 2208, 71003 Heraklion, Greece
- National University of Science and Technology "MISiS," Leninsky Prospekt 4, Moscow 119049, Russia
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
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9
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Parra-Rivas P, Gomila D, Gelens L, Knobloch E. Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion. Phys Rev E 2018; 97:042204. [PMID: 29758631 DOI: 10.1103/physreve.97.042204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2018] [Indexed: 06/08/2023]
Abstract
The origin, stability, and bifurcation structure of different types of bright localized structures described by the Lugiato-Lefever equation are studied. This mean field model describes the nonlinear dynamics of light circulating in fiber cavities and microresonators. In the case of anomalous group velocity dispersion and low values of the intracavity phase detuning these bright states are organized in a homoclinic snaking bifurcation structure. We describe how this bifurcation structure is destroyed when the detuning is increased across a critical value, and determine how a bifurcation structure known as foliated snaking emerges.
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Affiliation(s)
- P Parra-Rivas
- Laboratory of Dynamics in Biological Systems, KU Leuven Department of Cellular and Molecular Medicine, University of Leuven, B-3000 Leuven, Belgium
- Applied Physics Research Group, APHY, Vrije Universiteit Brussel, 1050 Brussels, Belgium
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
| | - D Gomila
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
| | - L Gelens
- Laboratory of Dynamics in Biological Systems, KU Leuven Department of Cellular and Molecular Medicine, University of Leuven, B-3000 Leuven, Belgium
- Applied Physics Research Group, APHY, Vrije Universiteit Brussel, 1050 Brussels, Belgium
| | - E Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
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10
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Kreilos T, Schneider TM. Fully localized post-buckling states of cylindrical shells under axial compression. Proc Math Phys Eng Sci 2017; 473:20170177. [PMID: 28989305 PMCID: PMC5627372 DOI: 10.1098/rspa.2017.0177] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2017] [Accepted: 08/11/2017] [Indexed: 11/12/2022] Open
Abstract
We compute nonlinear force equilibrium solutions for a clamped thin cylindrical shell under axial compression. The equilibrium solutions are dynamically unstable and located on the stability boundary of the unbuckled state. A fully localized single dimple deformation is identified as the edge state-the attractor for the dynamics restricted to the stability boundary. Under variation of the axial load, the single dimple undergoes homoclinic snaking in the azimuthal direction, creating states with multiple dimples arranged around the central circumference. Once the circumference is completely filled with a ring of dimples, snaking in the axial direction leads to further growth of the dimple pattern. These fully nonlinear solutions embedded in the stability boundary of the unbuckled state constitute critical shape deformations. The solutions may thus be a step towards explaining when the buckling and subsequent collapse of an axially loaded cylinder shell is triggered.
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Affiliation(s)
- Tobias Kreilos
- Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
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11
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Chong C, Li F, Yang J, Williams MO, Kevrekidis IG, Kevrekidis PG, Daraio C. Damped-driven granular chains: an ideal playground for dark breathers and multibreathers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032924. [PMID: 24730930 DOI: 10.1103/physreve.89.032924] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2013] [Indexed: 06/03/2023]
Abstract
By applying an out-of-phase actuation at the boundaries of a uniform chain of granular particles, we demonstrate experimentally that time-periodic and spatially localized structures with a nonzero background (so-called dark breathers) emerge for a wide range of parameter values and initial conditions. We demonstrate a remarkable control over the number of breathers within the multibreather pattern that can be "dialed in" by varying the frequency or amplitude of the actuation. The values of the frequency (or amplitude) where the transition between different multibreather states occurs are predicted accurately by the proposed theoretical model, which is numerically shown to support exact dark breather and multibreather solutions. Moreover, we visualize detailed temporal and spatial profiles of breathers and, especially, of multibreathers using a full-field probing technology and enable a systematic favorable comparison among theory, computation, and experiments. A detailed bifurcation analysis reveals that the dark and multibreather families are connected in a "snaking" pattern, providing a roadmap for the identification of such fundamental states and their bistability in the laboratory.
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Affiliation(s)
- C Chong
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - F Li
- Graduate Aerospace Laboratories (GALCIT), California Institute of Technology, Pasadena, California 91125, USA
| | - J Yang
- Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195-2400, USA
| | - M O Williams
- Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA
| | - I G Kevrekidis
- Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - C Daraio
- Graduate Aerospace Laboratories (GALCIT), California Institute of Technology, Pasadena, California 91125, USA and Department of Mechanical and Process Engineering (D-MAVT), Swiss Federal Institute of Technology (ETH), 8092 Zurich, Switzerland
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12
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Kao HC, Beaume C, Knobloch E. Spatial localization in heterogeneous systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012903. [PMID: 24580293 DOI: 10.1103/physreve.89.012903] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2013] [Indexed: 06/03/2023]
Abstract
We study spatial localization in the generalized Swift-Hohenberg equation with either quadratic-cubic or cubic-quintic nonlinearity subject to spatially heterogeneous forcing. Different types of forcing (sinusoidal or Gaussian) with different spatial scales are considered and the corresponding localized snaking structures are computed. The results indicate that spatial heterogeneity exerts a significant influence on the location of spatially localized structures in both parameter space and physical space, and on their stability properties. The results are expected to assist in the interpretation of experiments on localized structures where departures from spatial homogeneity are generally unavoidable.
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Affiliation(s)
| | - Cédric Beaume
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
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13
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Thiele U, Archer AJ, Robbins MJ, Gomez H, Knobloch E. Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042915. [PMID: 23679497 DOI: 10.1103/physreve.87.042915] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2013] [Revised: 03/21/2013] [Indexed: 05/11/2023]
Abstract
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.
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Affiliation(s)
- Uwe Thiele
- Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom.
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14
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Tzou JC, Ma YP, Bayliss A, Matkowsky BJ, Volpert VA. Homoclinic snaking near a codimension-two Turing-Hopf bifurcation point in the Brusselator model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022908. [PMID: 23496592 DOI: 10.1103/physreve.87.022908] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2012] [Indexed: 06/01/2023]
Abstract
Spatiotemporal Turing-Hopf pinning solutions near the codimension-two Turing-Hopf point of the one-dimensional Brusselator model are studied. Both the Turing and Hopf bifurcations are supercritical and stable. The pinning solutions exhibit coexistence of stationary stripes of near critical wavelength and time-periodic oscillations near the characteristic Hopf frequency. Such solutions of this nonvariational problem are in contrast to the stationary pinning solutions found in the subcritical Turing regime for the variational Swift-Hohenberg equations, characterized by a spatially periodic pattern embedded in a spatially homogeneous background state. Numerical continuation was used to solve periodic boundary value problems in time for the Fourier amplitudes of the spatiotemporal Turing-Hopf pinning solutions. The solution branches are organized in a series of saddle-node bifurcations similar to the known snaking structures of stationary pinning solutions. We find two intertwined pairs of such branches, one with a defect in the middle of the striped region, and one without. Solutions on one branch of one pair differ from those on the other branch by a π phase shift in the spatially periodic region, i.e., locations of local minima of solutions on one branch correspond to locations of maxima of solutions on the other branch. These branches are connected to branches exhibiting collapsed snaking behavior, where the snaking region collapses to almost a single value in the bifurcation parameter. Solutions along various parts of the branches are described in detail. Time dependent depinning dynamics outside the saddle nodes are illustrated, and a time scale for the depinning transitions is numerically established. Wavelength variation within the snaking region is discussed, and reasons for the variation are given in the context of amplitude equations. Finally, we compare the pinning region to the Maxwell line found numerically by time evolving the amplitude equations.
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Affiliation(s)
- J C Tzou
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.
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15
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Kao HC, Knobloch E. Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:026211. [PMID: 22463303 DOI: 10.1103/physreve.85.026211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2011] [Indexed: 05/31/2023]
Abstract
The transition from subcritical to supercritical stationary periodic patterns is described by the one-dimensional cubic-quintic Ginzburg-Landau equation A(t) = μA + A(xx) + i(a(1)|A|(2)A(x) + a(2)A(2)A(x)*) + b|A|(2)|A - |A|(4)A, where A(x,t) represents the pattern amplitude and the coefficients μ, a(1), a(2), and b are real. The conditions for Eckhaus instability of periodic solutions are determined, and the resulting spatially modulated states are computed. Some of these evolve into spatially localized structures in the vicinity of a Maxwell point, while others resemble defect states. The results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during the transition from subcriticality to supercriticality.
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Affiliation(s)
- Hsien-Ching Kao
- Department of Physics, University of California, Berkeley, California 94720, USA.
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17
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Dawes JHP. The emergence of a coherent structure for coherent structures: localized states in nonlinear systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2010; 368:3519-3534. [PMID: 20603365 DOI: 10.1098/rsta.2010.0057] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Coherent structures emerge from the dynamics of many kinds of dissipative, externally driven, nonlinear systems, and continue to provoke new questions that challenge our physical and mathematical understanding. In one specific subclass of such problems, in which a pattern-forming, or 'Turing', instability occurs, rapid progress has been made recently in our understanding of the formation of localized states: patches of regular pattern surrounded by the unpatterned homogeneous background state. This short review article surveys the progress that has been made for localized states and proposes three areas of application for these ideas that would take the theory in new directions and ultimately be of substantial benefit to areas of applied science. Finally, I offer speculations for future work, based on localized states, that may help researchers to understand coherent structures more generally.
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Affiliation(s)
- J H P Dawes
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
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18
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Schneider TM, Gibson JF, Burke J. Snakes and ladders: localized solutions of plane Couette flow. PHYSICAL REVIEW LETTERS 2010; 104:104501. [PMID: 20366430 DOI: 10.1103/physrevlett.104.104501] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2009] [Indexed: 05/29/2023]
Abstract
We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in one spatial dimension. Solutions of different size are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations. These new solutions are a step towards extending the dynamical systems view of transitional turbulence to spatially extended flows.
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Affiliation(s)
- Tobias M Schneider
- School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, Massachusetts 02138, USA
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19
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Kozyreff G, Assemat P, Chapman SJ. Influence of boundaries on localized patterns. PHYSICAL REVIEW LETTERS 2009; 103:164501. [PMID: 19905698 DOI: 10.1103/physrevlett.103.164501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2009] [Indexed: 05/28/2023]
Abstract
We analytically study the influence of boundaries on distant localized patterns generated by a Turing instability. To this end, we use the Swift-Hohenberg model with arbitrary boundary conditions. We find that the bifurcation diagram of these localized structures generally involves four homoclinic snaking branches, rather than two for infinite or periodic domains. Second, steady localized patterns only exist at discrete locations, and only at the center of the domain if their size exceeds a critical value. Third, reducing the domain size increases the pinning range.
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Affiliation(s)
- G Kozyreff
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles (U.L.B.), CP 231, Belgium
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20
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Gelens L, Knobloch E. Faceting and coarsening dynamics in the complex Swift-Hohenberg equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:046221. [PMID: 19905429 DOI: 10.1103/physreve.80.046221] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2009] [Indexed: 05/28/2023]
Abstract
The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and finds applications in lasers, optical parametric oscillators, and photorefractive oscillators. We show that with real coefficients this equation exhibits two classes of localized states: localized in amplitude only or localized in both amplitude and phase. The latter are associated with phase-winding states in which the real and imaginary parts of the order parameter oscillate periodically but with a constant phase difference between them. The localized states take the form of defects connecting phase-winding states with equal and opposite phase lag, and can be stable over a wide range of parameters. The formation of these defects leads to faceting of states with initially spatially uniform phase. Depending on parameters these facets may either coarsen indefinitely, as described by a Cahn-Hilliard equation, or the coarsening ceases leading to a frozen faceted structure.
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Affiliation(s)
- Lendert Gelens
- Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Brussel, Belgium.
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Yochelis A, Sheintuch M. Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach. Phys Chem Chem Phys 2009; 11:9210-23. [PMID: 19812842 DOI: 10.1039/b903266e] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We present a theoretical study of nonlinear pattern selection mechanisms in a model of bounded reaction-diffusion-advection system. The model describes the activator-inhibitor type dynamics of a membrane reactor characterized by a differential advection and a single diffusion; the latter excludes any finite wave number instability in the absence of advection. The focus is on three types of different behaviors, and the respective sensitivity to boundary and initial conditions: traveling waves, stationary periodic states, and excitable pulses. The theoretical methodology centers on the spatial dynamics approach, i.e. bifurcation theory of nonuniform solutions. These solutions coexist in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The results provide an efficient understanding of the pattern selection mechanisms that operate under realistic boundary conditions, such as Danckwerts type. The applicability of the results to broader reaction-diffusion-advection contexts is also discussed.
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Affiliation(s)
- Arik Yochelis
- Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
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Houghton SM, Knobloch E. Homoclinic snaking in bounded domains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:026210. [PMID: 19792234 DOI: 10.1103/physreve.80.026210] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2009] [Revised: 05/27/2009] [Indexed: 05/28/2023]
Abstract
Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of "snaking without bistability," recently observed in simulations of binary fluid convection by Mercader et al. Phys. Rev. E 80, 025201 (2009).
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Affiliation(s)
- S M Houghton
- School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom.
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Mercader I, Batiste O, Alonso A, Knobloch E. Localized pinning states in closed containers: Homoclinic snaking without bistability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:025201. [PMID: 19792185 DOI: 10.1103/physreve.80.025201] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2008] [Revised: 06/25/2009] [Indexed: 05/28/2023]
Abstract
Binary mixtures with a negative separation ratio are known to exhibit time-independent spatially localized convection when heated from below. Numerical continuation of such states in a closed two-dimensional container with experimental boundary conditions and parameter values reveals the presence of a pinning region in Rayleigh number with multiple stable localized states but no bistability between the conduction state and an independent container-filling state. An explanation for this unusual behavior is offered.
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Affiliation(s)
- Isabel Mercader
- Departament de Física Aplicada, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
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