Browsing by Subject "Pattern formation"

Now showing 1 - 4 of 4
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    Defects and stability of turing patterns
    (2013-05) Wu, Qiliang
    This paper is concerned with formation mechanisms of patterns and contains two main parts. The first one is about defects of patterns. Specifically, we study grain boundaries in the Swift-Hohenberg equation. Grain boundaries arise as stationary interfaces between roll solutions of different orientations. Our analysis shows that such stationary interfaces exist near onset of instability for arbitrary angle between the roll solutions. This extends prior work in [1] where the analysis was restricted to large angles, that is, weak bending near the grain boundary. The main new difficulty stems from possible interactions of the primary modes with other resonant modes. We generalize the normal form analysis in [1] and develop a singular perturbation approach to treat resonances. In the second part, we investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in L1 ∩ L∞, with sharp rates, recovering and slightly improving on results in [2, 3].
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    Mesoscale models for soft layered materials: the role of curvatures in topological defect motion, flows and instabilities
    (2020-09) Vitral, Eduardo
    Curvature driven phenomena in soft matter involves both complex geometry at small scales and anisotropies associated with material symmetries. In particular, the class of soft modulated materials present molecules that are organized in layers, so that material properties significantly vary between the direction normal to the layer and those of the layers. This is the case of smectic liquid crystals, which behave as a solid in the direction normal to the molecular layers, while each of their layers behave as a two-dimensional fluid. Under appropriate boundary conditions, smectic layers are known to bend and form focal conic defects, whose curvatures significantly increase in magnitude as the tip of the cone is approached. Intriguingly, experiments on smectic films presenting arrays of focal conics have shown that these materials undergo unexpected morphological transitions, which are not explained by classical local equilibrium thermodynamics. For example, annealing of focal conic domains can lead to conical pyramids, changing the sign of both mean and Gaussian curvatures and exposing smectic layers at the interface. In order to understand the role played by high order curvature terms on the stability and evolution of a smectic film interface, we propose a phase-field model for a smectic-isotropic system. Through an asymptotic analysis, we generalize the classical condition of local equilibrium, the Gibbs-Thomson equation, to include contributions from surface bending and torsion and a dependence on the layer orientation at the interface. Numerical results for a diffusive evolution of the interface reproduce the focal conic to conical pyramid transition in smectic films, and we show that such morphologies can be explained in light of the derived interface equations. We then generalize this model to include flows and to allow each phase to have a different density. We derive both a quasi-incompressible and a weakly compressible smectic-isotropic model from this approach, explaining their applicability and limitations. Finally, we investigate the role of flows and defect interactions in two-dimensional active smectics, known as the spiral defect chaos state in Rayleigh-Bénard convection.
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    Pattern formation in the wake of external mechanisms
    (2016-06) Goh, Ryan
    Pattern formation in nature has intrigued humans for centuries, if not millennia. In the past few decades researchers have become interested in harnessing these processes to engineer and manufacture self-organized and self-regulated devices at various length scales. Since many natural pattern forming processes nucleate or grow from a homogeneous unstable state, they typically create defects, caused by thermal and other inherent sources of noise, which can hamper effectiveness in applications. One successful experimental method for controlling the pattern forming process is to use an external mechanism which moves through a system, transforming it from a stable state to an unstable state from which the pattern forming dynamics can take hold. In this thesis, we rigorously study partial differential equations which model how such triggering mechanisms can select and control patterns. We first use dynamical systems techniques to study the case where a spatial trigger perturbs a pattern forming freely invading front in a scalar partial differential equation. We study such perturbations for the two generic types of scalar invasion fronts, known as pulled and pushed fronts, which roughly correspond to fronts which invade either through a linear or nonlinear mechanism. Our results give the existence of perturbed fronts and provide expansions in the speed of the triggering mechanism for the wavenumber perturbation of the pattern formed. With the hope of moving towards the more complicated geometries which can arise in two spatial dimensions, where many dynamical systems methods cannot be readily applied, we also develop a functional analytic method for the study of Hopf bifurcation in the presence of continuous spectrum. Our method, while still giving computable information about the bifurcating solution, is more direct than previously proposed methods. We develop this method in the context of a triggered Cahn-Hilliard equation, in one spatial dimension, which has been used to study many triggered pattern forming systems. Furthermore, we use these abstract results to characterize an explicit example and also use our method to give a simplified proof of the bifurcation of oscillatory shock solutions in viscous conservation laws.
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    Stochastic fluctuations in signaling, gene control and pattern formation.
    (2011-09) Zheng, Likun
    Stochasticity is inherent in biochemical systems. Noise can come from internal sources such as the random motion and reactions of molecules, and external sources such as environmental fluctuations. The main purpose of this thesis is to study how fluctuations propagate in biological systems. First, we focus on how a signaling molecule called a ligand searches for and binds to its target (receptor). There exist membrane proteins that can bind to the ligand molecule and localize it near receptors, affecting the association between it and the receptor. Our analysis shows that although the membrane protein can concentrate the ligand molecule near the receptor surface, the membrane protein has to pass the localized ligand molecule to the receptor fast enough, in order to enhance signaling. Otherwise, the membrane protein inhibits signaling. Moreover, we also study the effect of localization on signal specificity. In particular, we discuss how the membrane proteins bind to ligand molecules and distribute them to different downstream signaling pathways. Upon ligand binding to receptors, bound receptors can initiate the downstream network, which may finally lead to gene expression. We then study how the noise from the initiation step of transcription propagates in the elongation step. Elongation can be interrupted by the pauses of the transcription complex on the DNA sequence. We give a condition under which the pause of the transcription complex can cause bursts of mRNA production. Finally, we use stochastic simulations to study dorsal-ventral patterning in Drosophila numerically. Our results indicate that a feedback loop can stabilize the determination of the amnioserosa boundary. We then propose a detailed single cell system for the downstream network in nuclei. Our analysis of time scales of reactions and molecular transport shows the phosphorylation of Mad and transport of mRNA across the nuclear membrane are the major limiting steps in the signal transduction pathway. Simulations results show noises are amplified at these limiting steps.