The classical Cahn–Hilliard (CH) equation corresponds to a gradient dynamics model that describes phase decomposition in a binary mixture. In the spinodal region, an initially homogeneous state spontaneously decomposes via a large-scale instability into drop, hole or labyrinthine concentration patterns of a typical structure length followed by a continuously ongoing coarsening process. Here, we consider the coupled CH dynamics of two concentration fields and show that non-reciprocal (or active or non-variational) coupling may induce a small-scale (Turing) instability. At the corresponding primary bifurcation, a branch of periodically patterned steady states emerges. Furthermore, there exist localized states that consist of patterned patches coexisting with a homogeneous background. The branches of steady parity-symmetric and parity-asymmetric localized states form a slanted homoclinic snaking structure typical for systems with a conservation law. In contrast to snaking structures in systems with gradient dynamics, here, Hopf instabilities occur at a sufficiently large activity, which results in oscillating and travelling localized patterns.
{"title":"Localized states in coupled Cahn–Hilliard equations","authors":"Tobias Frohoff-Hülsmann;Uwe Thiele","doi":"10.1093/imamat/hxab026","DOIUrl":"10.1093/imamat/hxab026","url":null,"abstract":"The classical Cahn–Hilliard (CH) equation corresponds to a gradient dynamics model that describes phase decomposition in a binary mixture. In the spinodal region, an initially homogeneous state spontaneously decomposes via a large-scale instability into drop, hole or labyrinthine concentration patterns of a typical structure length followed by a continuously ongoing coarsening process. Here, we consider the coupled CH dynamics of two concentration fields and show that non-reciprocal (or active or non-variational) coupling may induce a small-scale (Turing) instability. At the corresponding primary bifurcation, a branch of periodically patterned steady states emerges. Furthermore, there exist localized states that consist of patterned patches coexisting with a homogeneous background. The branches of steady parity-symmetric and parity-asymmetric localized states form a slanted homoclinic snaking structure typical for systems with a conservation law. In contrast to snaking structures in systems with gradient dynamics, here, Hopf instabilities occur at a sufficiently large activity, which results in oscillating and travelling localized patterns.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42394943","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}
Cris R Hasan;Hinke M Osinga;Claire M Postlethwaite;Alastair M Rucklidge
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
We revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier–Stokes equations on ${{{mathbb{R}}}}^n$ , as well as the rotating Boussinesq equations on ${{{mathbb{R}}}}^3$ . 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 $nleqslant 3$ , 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.
{"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":null,"pages":null},"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}
Fahad Al Saadi;Alan Champneys;Annette Worthy;Ahmed Msmali
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.
{"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":null,"pages":null},"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}
Novel soliton structures are constructed for the Fokas–Lenells equation. In so doing, and after discussing the stability of continuous waves, a multiple scales based perturbation theory is used to reduce the equation to a Korteweg–de Vries system whose single soliton solution gives rise to intricate (and rather unexpected) solutions to the original system. Both the focusing and defocusing equations are considered and it is found that dark solitons may exist in both cases while in the focusing case antidark solitons are also possible. These findings are quite surprising as the relative nonlinear Schrödinger equation does not exhibit these solutions. So far, similar abundance of solutions has only been observed in relative coupled systems.
{"title":"Integrable reduction and solitons of the Fokas–Lenells equation","authors":"Theodoros P Horikis","doi":"10.1093/imamat/hxab020","DOIUrl":"https://doi.org/10.1093/imamat/hxab020","url":null,"abstract":"Novel soliton structures are constructed for the Fokas–Lenells equation. In so doing, and after discussing the stability of continuous waves, a multiple scales based perturbation theory is used to reduce the equation to a Korteweg–de Vries system whose single soliton solution gives rise to intricate (and rather unexpected) solutions to the original system. Both the focusing and defocusing equations are considered and it is found that dark solitons may exist in both cases while in the focusing case antidark solitons are also possible. These findings are quite surprising as the relative nonlinear Schrödinger equation does not exhibit these solutions. So far, similar abundance of solutions has only been observed in relative coupled systems.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50349987","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}
Radiation of sound waves by a coaxial rigid duct with perforated screen is investigated by using the Mode Matching technique in conjunction with the Jones’ Method. The geometry of the problem consists semi-infinite outer duct and infinite inner duct. It is assumed that the duct walls are fully rigid. The solution of current study contains an infinite sets of coefficients satisfying an infinite systems of linear algebraic equations. These systems are truncated at a certain number and then solved numerically. The effects of open and perforated case, frequency and porosity on the radiation phenomenon are shown graphically. In the present study, perforated screen makes the problem more interesting when it is compared with the unperforated screen. In this context, solution of the problem is compered numerically with similar studies, which are used different method to obtain Wiener–Hopf equation, existing in the literature. As a result, it is observed that in the absence of a perforated screen, there is a perfect agreement between the two results.
{"title":"Radiation of sound waves from a coaxial duct with perforated screen","authors":"Burhan Tiryakioglu;Ayse Tiryakioglu","doi":"10.1093/imamat/hxab016","DOIUrl":"10.1093/imamat/hxab016","url":null,"abstract":"Radiation of sound waves by a coaxial rigid duct with perforated screen is investigated by using the Mode Matching technique in conjunction with the Jones’ Method. The geometry of the problem consists semi-infinite outer duct and infinite inner duct. It is assumed that the duct walls are fully rigid. The solution of current study contains an infinite sets of coefficients satisfying an infinite systems of linear algebraic equations. These systems are truncated at a certain number and then solved numerically. The effects of open and perforated case, frequency and porosity on the radiation phenomenon are shown graphically. In the present study, perforated screen makes the problem more interesting when it is compared with the unperforated screen. In this context, solution of the problem is compered numerically with similar studies, which are used different method to obtain Wiener–Hopf equation, existing in the literature. As a result, it is observed that in the absence of a perforated screen, there is a perfect agreement between the two results.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45606456","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}
We use the classical factorization method proposed firstly by Kirsch to reconstruct the support of the mixed inhomogeneous medium associated with complex valued refractive indexes and different transmission boundary conditions. We will show that for well-chosen inhomogeneous backgrounds, one obtains a necessary and sufficient condition characterizing the support of the medium via the eigensystem of a self-adjoint operator, which is related to the far field operator. Moreover, for completeness of our problem, the variational method is applied to solve the direct scattering problem. And, we present a variant of numerical examples in 2D to verify the effectiveness and robustness of the proposed inverse algorithms.
{"title":"The factorization method for the scattering by a mixed inhomogeneous medium","authors":"Jianli Xiang;Guozheng Yan","doi":"10.1093/imamat/hxab017","DOIUrl":"10.1093/imamat/hxab017","url":null,"abstract":"We use the classical factorization method proposed firstly by Kirsch to reconstruct the support of the mixed inhomogeneous medium associated with complex valued refractive indexes and different transmission boundary conditions. We will show that for well-chosen inhomogeneous backgrounds, one obtains a necessary and sufficient condition characterizing the support of the medium via the eigensystem of a self-adjoint operator, which is related to the far field operator. Moreover, for completeness of our problem, the variational method is applied to solve the direct scattering problem. And, we present a variant of numerical examples in 2D to verify the effectiveness and robustness of the proposed inverse algorithms.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43048568","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}
We optimize the spring constants $k^{i,j}$ (stiffness) of circular spring-mass systems with nearest-neighbour (NN) and next-nearest-neighbour (NNN) springs only. In this optimization problem, such systems are also subjected to damping and periodic external forces. The function to be minimized is the average ratio of the square norm of the on-site internal forces (response) to the square norm of the external on-site forces (input). Under the average of this response/input ratio is meant the average over time and over all configurations of external forces. As main result, it is established that the optimum stiffness matrix converges to the discrete and periodic bi-Laplacian operator as the size $n$ of the system increases. Such a result is obtained under the following assumptions: (a) the system has the natural mode shape (eigenvector) of alternating $1$ s and $-1$ s; and (b) the (external) forcing frequency is at least $1.095$ times higher than the highest natural frequency. It is remarkable that this optimum stiffness matrix exhibits negative stiffness for the springs linking NNN point masses. More specifically, as $n$ increases, $0> k^{i,i+2} , , = , , - tfrac{1}{4} , k^{i,i+1}$ is the relation between the optimum NNN spring constant and the optimum NN spring constant. Such systems illustrate that the introduction of negative stiffness springs in some specific positions does in fact reduce the average response/input ratio. Numerical tables illustrating the main result are given.
{"title":"Optimizing the spring constants of forced, damped and circular spring-mass systems—characterization of the discrete and periodic bi-Laplacian operator","authors":"L L A de Oliveira;M V Travaglia","doi":"10.1093/imamat/hxab021","DOIUrl":"10.1093/imamat/hxab021","url":null,"abstract":"We optimize the spring constants \u0000<tex>$k^{i,j}$</tex>\u0000 (stiffness) of circular spring-mass systems with nearest-neighbour (NN) and next-nearest-neighbour (NNN) springs only. In this optimization problem, such systems are also subjected to damping and periodic external forces. The function to be minimized is the average ratio of the square norm of the on-site internal forces (response) to the square norm of the external on-site forces (input). Under the average of this response/input ratio is meant the average over time and over all configurations of external forces. As main result, it is established that the optimum stiffness matrix converges to the discrete and periodic bi-Laplacian operator as the size \u0000<tex>$n$</tex>\u0000 of the system increases. Such a result is obtained under the following assumptions: (a) the system has the natural mode shape (eigenvector) of alternating \u0000<tex>$1$</tex>\u0000s and \u0000<tex>$-1$</tex>\u0000s; and (b) the (external) forcing frequency is at least \u0000<tex>$1.095$</tex>\u0000 times higher than the highest natural frequency. It is remarkable that this optimum stiffness matrix exhibits negative stiffness for the springs linking NNN point masses. More specifically, as \u0000<tex>$n$</tex>\u0000 increases, \u0000<tex>$0> k^{i,i+2} , , = , , - tfrac{1}{4} , k^{i,i+1}$</tex>\u0000 is the relation between the optimum NNN spring constant and the optimum NN spring constant. Such systems illustrate that the introduction of negative stiffness springs in some specific positions does in fact reduce the average response/input ratio. Numerical tables illustrating the main result are given.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45366354","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}
We study a hybrid impulsive reaction-diffusion equation composed with a discrete-time map in bounded domain $varOmega $ in space dimension $nin mathbb N$ . We assume that the exterior of domain is not lethal (not completely hostile) but hostile. We consider Robin boundary conditions which are used for mixed or reactive or semipermeable boundaries. Given geometry of the domain $varOmega $ , we establish critical domain sizes for the persistence and extinction of a species. Specifically, for habitats with the shape of $n$ -hypercube and ball of fixed radius, we formulate the critical domain sizes depending on parameters of the model, including $h$ , i.e. a measure of the hostility of the external (to $varOmega $ ) environment. For a general habitat, called Lipschitz domains, we apply isoperimetric inequalities and variational methods to find the associated critical domain sizes. We also provide applications of the main results in marine reserve, terrestrial reserve and insect pest outbreaks.
研究了在空间维数为$n mathbb n $的有界域$varOmega $上由一个离散时间映射组成的混合脉冲反应扩散方程。我们假设领域的外部不是致命的(不是完全敌对的)，而是敌对的。我们考虑Robin边界条件，它用于混合边界或反应边界或半渗透边界。给定域$varOmega $的几何形状，我们建立了物种持续和灭绝的临界域大小。具体来说，对于形状为$n$-超立方体和固定半径球的栖息地，我们根据模型参数(包括$h$)制定了关键域尺寸，即外部(对$varOmega $)环境敌意的度量。对于称为Lipschitz域的一般生境，我们应用等周不等式和变分方法来找到相关的临界域大小。我们还提供了主要结果在海洋保护区、陆地保护区和虫害暴发中的应用。
{"title":"Critical domain sizes of a discrete-map hybrid and reaction-diffusion model on hostile exterior domains","authors":"Mostafa Fazly","doi":"10.1093/imamat/hxab019","DOIUrl":"10.1093/imamat/hxab019","url":null,"abstract":"We study a hybrid impulsive reaction-diffusion equation composed with a discrete-time map in bounded domain \u0000<tex>$varOmega $</tex>\u0000 in space dimension \u0000<tex>$nin mathbb N$</tex>\u0000. We assume that the exterior of domain is not lethal (not completely hostile) but hostile. We consider Robin boundary conditions which are used for mixed or reactive or semipermeable boundaries. Given geometry of the domain \u0000<tex>$varOmega $</tex>\u0000, we establish critical domain sizes for the persistence and extinction of a species. Specifically, for habitats with the shape of \u0000<tex>$n$</tex>\u0000-hypercube and ball of fixed radius, we formulate the critical domain sizes depending on parameters of the model, including \u0000<tex>$h$</tex>\u0000, i.e. a measure of the hostility of the external (to \u0000<tex>$varOmega $</tex>\u0000) environment. For a general habitat, called Lipschitz domains, we apply isoperimetric inequalities and variational methods to find the associated critical domain sizes. We also provide applications of the main results in marine reserve, terrestrial reserve and insect pest outbreaks.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43174103","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}
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