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Design of parametrically forced patterns and quasipatterns
 SIAM J. Appl. Dyn. Syst
, 2009
"... Abstract. The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear threewave interactions between driven and weakly damped modes play a key role in determ ..."
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Abstract. The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear threewave interactions between driven and weakly damped modes play a key role in determining which patterns are favoured. We use this idea to design single and multifrequency forcing functions that produce examples of superlattice patterns and quasipatterns in a new model PDE with parametric forcing. We make quantitative comparisons between the predicted patterns and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very close to onset. The reason that the range of validity is limited is that the theory requires strong damping of all modes apart from the driven patternforming modes. This is in conflict with the requirement for weak damping if threewave coupling is to influence pattern selection effectively. We distinguish the two different ways that threewave interactions can be used to stabilise quasipatterns, and present examples of 12, 14 and 20fold approximate quasipatterns. We identify which computational domains provide the most accurate approximations to 12fold quasipatterns, and systematically investigate the Fourier spectra of the most accurate approximations. Key words. Pattern formation, quasipatterns, superlattice patterns, mode interactions, Faraday waves. AMS subject classifications. 35B32, 37G40, 52C23, 70K28, 76B15 1. Introduction. The
PhaseFields and the renormalization group: A continuum approach to multiscale modelling of materials” Phd thesis Pg 2
, 2006
"... Important phenomena in materials processing, such as dendritic growth during solidification, involve a wide range of length scales from the atomic level up to product dimensions. The phasefield approach, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with a part of this ..."
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Important phenomena in materials processing, such as dendritic growth during solidification, involve a wide range of length scales from the atomic level up to product dimensions. The phasefield approach, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with a part of this range of scales, from few tens of microns to millimeters, and provides an effective continuum modeling technique for moving boundary problems. A serious limitation of the usual representation of the phasefield model however, is that it fails to keep track of the underlying crystallographic anisotropy, and thus is unable to capture lattice defects and model polycrystalline microstructure without nontrivial modifications. The phasefield crystal (PFC) model on the other hand, is a phase field equation with periodic solutions that represent the atomic density. It natively incorporates elasticity, and can model formation of polycrystalline films, dislocation motion and plasticity, and nonequilibrium dynamics of phase transitions in real materials. Because it describes matter at the atomic length scale however, it is unsuitable for coping with the range of length scales in problems of serious interest. This
Destabilization and Localization of Traveling Waves by an Advected Field
, 2008
"... We study a model of smallamplitude traveling waves arising in a supercritical Hopfbifurcation, that are coupled to a slowly varying, real field. The field is advected by the waves and, in turn, affects their stability via a coupling to the growth rate. In the absence of dispersion we identify two d ..."
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We study a model of smallamplitude traveling waves arising in a supercritical Hopfbifurcation, that are coupled to a slowly varying, real field. The field is advected by the waves and, in turn, affects their stability via a coupling to the growth rate. In the absence of dispersion we identify two distinct shortwave instabilities. One instability induces a phase slip of the waves and a corresponding reduction of the winding number, while the other leads to a modulated wave structure. The bifurcation to modulated waves can be either forward or backward, in the latter case permitting the existence of localized, traveling pulses which are bistable with the basic, conductive state. 1
Robust heteroclinic cycles in the onedimensional complex ginzburglandau equation
 Physica D
, 2005
"... Numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the onedimensional complex GinzburgLandau equation (CGL) on the unit, spatially periodic domain. These cycles connect di®erent spatially and temporally inhomogeneous timeperiodic solutio ..."
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Numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the onedimensional complex GinzburgLandau equation (CGL) on the unit, spatially periodic domain. These cycles connect di®erent spatially and temporally inhomogeneous timeperiodic solutions as t! §1. A careful analysis of the connections is made using a projection onto 5 complex Fourier modes. It is shown ¯rst that the timeperiodic solutions can be treated as (relative) equilibria after consideration of the symmetries of the CGL. Second, the cycles are shown to be robust since the individual heteroclinic connections exist in invariant subspaces. Thirdly, after constructing appropriate Poincar¶e maps around the cycle, a criteria for temporal stability is established, which is shown numerically to hold in speci¯c parameter regions where the cycles are found to be of Shil'nikov type. This criterion is also applied to a much highermode Fourier truncation where similar results are found. In regions where instability of the cycles occurs, either Shilni'kovHopf or blowout bifurcations are observed, with numerical evidence of competing attractors. Implications for observed spatiotemporal intermittency in situations modelled by the CGL are discussed.
Localized Pattern Formation with a LargeScale Mode: Slanted Snaking *
, 2008
"... Abstract. Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical "patternforming" instability and in ..."
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Abstract. Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical "patternforming" instability and intertwine in a manner often referred to as "homoclinic snaking." In this paper we consider an extension of this paradigm where, in addition to the patternforming instability (with nonzero wavenumber), a largescale neutral mode exists, having zero growth rate at zero wavenumber. Such a situation naturally arises in the presence of a conservation law; we give examples of physical systems in which this arises, in particular, thermal convection in a horizontal fluid layer with a vertical magnetic field. We introduce a novel scaling that allows the derivation of a nonlocal GinzburgLandau equation to describe the formation of localized states. Our results show that the existence of the largescale mode substantially enlarges the region of parameter space where localized states exist and are stable.
A mathematical model of liver cell aggregation in vitro
, 2008
"... The behaviour of mammalian cells within threedimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multice ..."
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The behaviour of mammalian cells within threedimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that, provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cellECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work.
A Mathematical Model of Liver Cell Aggregation In Vitro
"... Abstract The behavior of mammalian cells within threedimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal ..."
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Abstract The behavior of mammalian cells within threedimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cellECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work.
unknown title
, 1999
"... Compressible magnetoconvection in three dimensions: pattern formation in a strongly strati ed layer ..."
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Compressible magnetoconvection in three dimensions: pattern formation in a strongly strati ed layer
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"... Stability of patterns with arbitrary period for a GinzburgLandau equation with a mean field ..."
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Stability of patterns with arbitrary period for a GinzburgLandau equation with a mean field
Annual Research Briefs 2003
"... The e®ect of surface topography on the nonlinear dynamics of Rossby waves ..."
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The e®ect of surface topography on the nonlinear dynamics of Rossby waves