IMA Journal of Applied Mathematics最新文献_第8页

Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability 多变量反应扩散系统中的平稳峰:亚临界图灵不稳定性引起的叶状蛇形

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-07-01 DOI: 10.1093/imamat/hxab029

Edgar Knobloch;Arik Yochelis

{"title":"Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability","authors":"Edgar Knobloch;Arik Yochelis","doi":"10.1093/imamat/hxab029","DOIUrl":"10.1093/imamat/hxab029","url":null,"abstract":"An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with \u0000<tex>$N$</tex>\u0000 identical equidistant peaks, \u0000<tex>$N=1,2,dots ,$</tex>\u0000, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the \u0000<tex>$N=1$</tex>\u0000 state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 5","pages":"1066-1093"},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47707535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 7

Curvature effects and radial homoclinic snaking 曲率效应与径向同宿蛇形

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-07-01 DOI: 10.1093/imamat/hxab028

Damià Gomila;Edgar Knobloch

{"title":"Curvature effects and radial homoclinic snaking","authors":"Damià Gomila;Edgar Knobloch","doi":"10.1093/imamat/hxab028","DOIUrl":"10.1093/imamat/hxab028","url":null,"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.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 5","pages":"1094-1111"},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41963536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

Origin, bifurcation structure and stability of localized states in Kerr dispersive optical cavities 克尔色散光学腔中局域态的起源、分岔结构和稳定性

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-07-01 DOI: 10.1093/imamat/hxab031

P Parra-Rivas;E Knobloch;L Gelens;D Gomila

引用次数: 13

Editorial to Homoclinic snaking at 21: in memory of Patrick Woods 21岁时对Homoclinic蛇的评论:纪念帕特里克·伍兹

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-07-01 DOI: 10.1093/imamat/hxab041

Alan Champneys

引用次数: 4

Localized states in coupled Cahn–Hilliard equations 耦合Cahn–Hilliard方程中的局部化态

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-07-01 DOI: 10.1093/imamat/hxab026

Tobias Frohoff-Hülsmann;Uwe Thiele

引用次数: 14

Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance 循环优势异斜网中螺旋波的数值延拓

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-07-01 DOI: 10.1093/imamat/hxab027

Cris R Hasan;Hinke M Osinga;Claire M Postlethwaite;Alastair M Rucklidge

{"title":"Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance","authors":"Cris R Hasan;Hinke M Osinga;Claire M Postlethwaite;Alastair M Rucklidge","doi":"10.1093/imamat/hxab027","DOIUrl":"10.1093/imamat/hxab027","url":null,"abstract":"Heteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces, and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction–diffusion model that describes the spatiotemporal evolution of three competing populations in a 2D spatial domain—much like the Rock–Paper–Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock–Paper–Scissors–Lizard–Spock game.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 5","pages":"1141-1163"},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43674659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems 局部模式和半强相互作用，反应扩散系统的统一框架

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-07-01 DOI: 10.1093/imamat/hxab036

Fahad Al Saadi;Alan Champneys;Nicolas Verschueren

{"title":"Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems","authors":"Fahad Al Saadi;Alan Champneys;Nicolas Verschueren","doi":"10.1093/imamat/hxab036","DOIUrl":"10.1093/imamat/hxab036","url":null,"abstract":"Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 5","pages":"1031-1065"},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47648841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 11

Explicit superposed and forced plane wave generalized Beltrami flows 显式叠加和强迫平面波广义Beltrami流

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-06-01 DOI: 10.1093/imamat/hxab015

Artur Prugger;Jens D M Rademacher

{"title":"Explicit superposed and forced plane wave generalized Beltrami flows","authors":"Artur Prugger;Jens D M Rademacher","doi":"10.1093/imamat/hxab015","DOIUrl":"https://doi.org/10.1093/imamat/hxab015","url":null,"abstract":"We revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier–Stokes equations on \u0000<tex>${{{mathbb{R}}}}^n$</tex>\u0000, as well as the rotating Boussinesq equations on \u0000<tex>${{{mathbb{R}}}}^3$</tex>\u0000. We cast these solutions are superpositions of certain linear plane waves of arbitrary amplitudes that also solve the nonlinear equations by constraints on wave vectors and flow directions. For \u0000<tex>$nleqslant 3$</tex>\u0000, these are explicit examples for generalized Beltrami flows. We show that forcing terms of corresponding plane wave type yield explicit solutions by linear variation of constants. We work in Eulerian coordinates and distinguish the two situations of vanishing and of gradient nonlinear terms, where the nonlinear terms modify the pressure. The methods that we introduce to find explicit solutions in nonlinear fluid models can also be used in other equations with material derivative. Our approach offers another view on known explicit solutions of different fluid models from a plane wave perspective and provides transparent nonlinear interactions between different flow components.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"761-784"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50425544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

Stationary and oscillatory localized patterns in ratio-dependent predator–prey systems 比例依赖捕食者-被捕食系统的平稳和振荡局部化模式

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-06-01 DOI: 10.1093/imamat/hxab018

Fahad Al Saadi;Alan Champneys;Annette Worthy;Ahmed Msmali

{"title":"Stationary and oscillatory localized patterns in ratio-dependent predator–prey systems","authors":"Fahad Al Saadi;Alan Champneys;Annette Worthy;Ahmed Msmali","doi":"10.1093/imamat/hxab018","DOIUrl":"10.1093/imamat/hxab018","url":null,"abstract":"Investigations are undertaken into simple predator–prey models with rational interaction terms in one and two spatial dimensions. Focusing on a case with linear interaction and saturation, an analysis for long domains in 1D is undertaken using ideas from spatial dynamics. In the limit that prey diffuses much more slowly than predator, the Turing bifurcation is found to be subcritical, which gives rise to localized patterns within a Pomeau pinning parameter region. Parameter regions for localized patterns and isolated spots are delineated. For a realistic range of parameters, a temporal Hopf bifurcation of the balanced equilibrium state occurs within the localized-pattern region. Detailed spectral computations and numerical simulations reveal how the Hopf bifurcation is inherited by the localized structures at nearby parameter values, giving rise to both temporally periodic and chaotic localized patterns. Simulation results in 2D confirm the onset of complex spatio-temporal patterns within the corresponding parameter regions. The generality of the results is confirmed by showing qualitatively the same bifurcation structure within a similar model with quadratic interaction and saturation. The implications for ecology are briefly discussed.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"808-827"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43943151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Integrable reduction and solitons of the Fokas–Lenells equation Fokas–Lenells方程的可积约化和孤立子

IF 1.2 4区数学

IMA Journal of Applied Mathematics Pub Date : 2021-06-01 DOI: 10.1093/imamat/hxab020

Theodoros P Horikis

引用次数: 1