Modelling the emergence of polarity patterns for the intercellular transport of auxin in plants

The hormone auxin is actively transported throughout plants via protein machineries including the dedicated transporter known as PIN. The associated transport is ordered with nearby cells driving auxin flux in similar directions. Here, we provide a model of both the auxin transport and of the dynamics of cellular polarization based on flux sensing. Our main findings are: (i) spontaneous intracellular PIN polarization arises if PIN recycling dynamics are sufficiently nonlinear, (ii) there is no need for an auxin concentration gradient and (iii) ordered multi-cellular patterns of PIN polarization are favoured by molecular noise.     

Snakes and isolas in non-reversible conservative systems

Reversible variational partial differential equations such as the Swift–Hohenberg equation can admit localized stationary roll structures whose solution branches are bounded in parameter space but unbounded in function space, with the width of the roll plateaus increasing without bound along the branch: this scenario is commonly referred to as snaking. In this work, the structure of the bifurcation diagrams of localized rolls is investigated for variational but non-reversible systems, and conditions are derived that guarantee snaking or result in diagrams that either consist entirely of isolas.     

弹性支承下弧齿锥齿轮系统参数平面中的域界结构研究

弹性支承下弧齿锥齿轮系统的振动分析属于高维非线性动力学问题,其参数平面中的域界结构研究对实施振动抑制与控制具有重要意义。以某新型航空发动机的中心轴弧齿锥齿轮传动系统为研究对象,建立了7自由度非线性动力学模型。采用Adomian分解算法求解振动方程的分级响应。基于胞映射和庞加莱截面的思想,在支承刚度与啮合频率、支承阻尼与啮合频率以及支承刚度与支承阻尼所构成的参数平面中,分析了弧齿锥齿轮非线性系统的不同周期振动之间以及周期振动和混沌振动之间的域界。借助平面域界图,研究了参数平面内系统振动分岔类型与分岔结构。研究结果为弧齿锥齿轮系统的支承参数设计与振动主动控制提供了依据,并可根据系统参数所处的范围对弹性支承下弧齿锥齿轮系统的振动形态进行预测。     

Hidden symmetries and pattern formation in Lapwood convection

We study Lapwood convection (convection of a fluid in a porous medium) on a two-dimensional rectangular domain. The linearized eigenmodes are symmetric pxq cellular patterns, which we call (p, q) modes. Numerical calculations of the branching structure near mode interaction points have derived bifurcation diagrams for the (3, 1)/(1, 1) and (3, l)/(2, 2) mode interactions which are non-generic, even when the rectangular symmetry of the domain is taken into account. This has raised questions about the accuracy of the numerical method used, a finite-element Galerkin approximation implemented using Harwell's ENTWIFE code. We show that this apparent lack of genericity is partly a consequence of 'hidden' translational symmetries, which arise when the problem is extended to one with periodic boundary conditions. This extension procedure has become standard for partial differential equations (PDEs) with Neumann or Dirichlet boundary conditions, and it reveals restrictions on the Liapunov-Schmidt reduced bifurcation equations and the resulting singularity-theoretic normal forms. Its application to Lapwood convection is unusual in that the PDE involves a mixture of both Neumann and Dirichlet boundary conditions. Specifically, on the vertical sidewalls the stream function satisfies Dirichlet boundary conditions (is zero), but the temperature satisfies Neumann (no-flux) boundary onditions. Nevertheless, we show that for abstract group-theoretical reasons the same symmetry constraints that occur for purely Neumann boundary conditions are imposed on the Liapunov-Schmidt reduced bifurcation equations, and therefore the same list of normal forms is valid. The hidden symmetries force certain terms in the reduced bifurcation equations to be zero and change the generic branching geometry. With the aid of MACSYMA, we determine a small number of low-order coefficients of the reduced bifurcation equations which are needed to find the correct normal form. We show that in some cases the normal form is more degenerate than might be anticipated, but that when these degeneracies are taken into account the resulting branching geometry reproduces that found in the earlier numerical approach. In particular, we obtain an analytic vindication of the numerical method     

Patterns on growing square domains via mode interactions

Numerical simulations of reaction-diffusion systems with Neumann boundary conditions (NBC) on growing square domains by Maini et al. exhibit square and stripe (or roll) patterns that are usually associated with bifurcations from a trivial equilibrium on a square lattice. However, these patterns change as the domain grows. In this article we discuss several of these transitions; namely, transitions between different types of squares and between squares and stripes (or rolls). We show that these transitions can be understood by tracing paths through the unfoldings of certain co-dimension two mode interactions. To understand these transitions, we need to discuss two issues: the speed at which the domain size changes and the relations between NBC and periodic boundary conditions (PBC) on a square. First, in the simulations, the domain growth takes place on a time scale that is longer than the one needed for pattern formation. Therefore, we can assume that the domain growth is identified with quasistatic variation of time; that is, the domain grows slowly enough that the PDE solution of the time-dependent system tracks equilibria of the reaction-diffusion systems posed on a fixed size domain. Second, as is well-known, NBC problems on a square of side length l can be embedded in PBC problems on a square with side length 2l. The PBC problem has translation symmetries that are not present in the NBC problem. These additional symmetries are called hidden symmetries in the literature. Moreover, solutions to PBC that restrict to the smaller square and satisfy NBC are just those solutions that satisfy certain symmetry constraints. We show further that the transitions between different patterns can be understood by considering relevant mode interaction bifurcation problems on the larger square and then restricting to the smaller square. We have found that a generic continuous transition can occur between two types of squares. Also, the transition between squares and stripes can generically occur either via steady states and time-periodic states (standing waves) or via a jump. Interestingly, in mode interactions, the symmetry constraints induced by NBC are important in understanding which solutions exist and which solutions are stable. For example, diagonal stripes cannot occur as a primary branch in the NBC problem but do in the PBC problem. Also, patterns can be stable in the NBC problem that are not stable in the PBC problem. As a consequence, in the NBC problem we see standing wave time-periodic solutions as stable patterns leading to stable stripes, whereas in the PBC problem we see wavy rolls steady states as stable patterns leading to stable stripes. In principle, a classification of all transitions in NBC mode interactions is possible. However, we concentrate only on those transitions that are relevant to the numerically observed transitions.     

Catastrophe models for cognitive workload and fatigue

We reconceptualised several problems concerning the measurement of cognitive workload – fixed versus variable limits on channel capacity, work volume versus time pressure, adaptive strategies, resources demanded by tasks when performed simultaneously, and unclear distinctions between workload and fatigue effects – as two cusp catastrophe models: buckling stress resulting from acute workload, and fatigue resulting from extended engagement. Experimental participants completed a task that was intensive on non-verbal episodic memory and had an automatically speeded component. For buckling stress, the epoch of maximum (speeded) performance was the asymmetry parameter; however, anxiety did not contribute to bifurcation as expected. For fatigue, the bifurcation factor was the total work accomplished, and arithmetic, a compensatory ability, was the asymmetry parameter; R ² for the cusp models outperformed the linear comparison models in both cases. A research programme is outlined that revolves around the two models with different types of task and resource configurations.     

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Banded vegetation is a characteristic feature of semi-arid environments. It occurs on gentle slopes, with alternating stripes of vegetation and bare ground running parallel to the contours. A number of mathematical models have been proposed to investigate the mechanisms underlying these patterns, and how they might be affected by changes in environmental conditions. One of the most widely used models is due to Rietkerk and co-workers, and is based on a water redistribution hypothesis, with the key feedback being that the rate of rainwater infiltration into the soil is an increasing function of plant biomass. Here, for the first time, we present a detailed study of the existence and stability of pattern solutions of the Rietkerk model on slopes, using the software package wavetrain (www.ma.hw.ac.uk/wavetrain). Specifically, we calculate the region of the rainfall–migration speed parameter plane in which patterns exist, and the sub-region in which these patterns are stable as solutions of the model partial differential equations. We then perform a detailed simulation-based study of the way in which patterns evolve when the rainfall parameter is slowly varied. This reveals complex behaviour, with sudden jumps in pattern wavelength, and hysteresis; we show that these jumps occur when the contours of constant pattern wavelength leave the parameter region giving stable patterns. Finally, we extend our results to the case in which a diffusion term for surface water is added to the model equations. The parameter regions for pattern existence and stability are relatively insensitive to small or moderate levels of surface water diffusion, but larger diffusion coefficients significantly change the subdivision into stable and unstable patterns.     

Vortex-like current states in Josephson ballistic point contacts

The stationary Josephson effect is studied theoretically in the situation when there is externally injected transport supercurrent which flows in the banks parallel to the contact interface. Coexistence of this supercurrent with the order parameter phase difference between the banks of the contact results in the appearance of quasiparticles in the weak link, which create the current, localized in the vicinity of the contact and counter-flowing to the transport supercurrent. We review the results of our previous study of weak links between current-carrying conventional and unconventional (d-wave) superconductors. And further we study the weak link in the form of double point contact between current-carrying superconductors. Current distribution patterns containing vortex-like states are obtained.     

具有时滞反馈控制的非线性主动悬架系统的稳定性、分岔和混沌

研究了具有时滞反馈控制的非线性主动悬架系统模型,该模型考虑了悬架弹簧和阻尼的非线性特性。运用广义Sturm准则推导了时滞无关稳定区域的临界增益和稳定性开关的临界时滞。在不同稳定性区间内选取参数组合进行数值模拟,验证理论分析的有效性。在动力学方程的基础上,利用分岔图、庞加莱映射图和时域图,研究了在路面激励下的悬架系统的非线性动力学行为。结果表明,在增益系数和阻尼系数g~ζ1平面内存在一个小的参数区间来实现时滞无关稳定性,并且区间范围随着悬架阻尼系数的增加而增大。当受控系统不具有时滞无关稳定性时,系统会随着时滞的变化而发生稳定性切换,这些稳定性开关对应时滞跨越临界值时发生的Hopf分岔。数值仿真验证了理论分析的正确性。时滞作为分岔参数,观察到系统由准周期运动通往混沌运动的途径:准周期环面破裂。     

A method for identifying instrument faults in nuclear power plants possibly leading to wrong situation assessment

In this paper, we propose a method for identifying instrument faults that could potentially affect an operators’ situation assessment capability in nuclear power plants (NPPs), an issue which has received a lot of attention recently. In the proposed method, patterns of selected plant parameter trends in selected plant states and NPP operators’ patterns of selected plant parameter trends in selected plant states are analyzed, and a comparison between the two kinds of patterns is performed to identify instrument faults which could potentially affect a NPP operators’ capability to correctly assess a plant's conditions. An example application is presented to demonstrate how the proposed method can be used to identify the possibilities of operators’ developing a wrong situation assessment because of instrument faults and to identify the corresponding safety concerns. We conclude that in order to get more accurate results, an analysis with a full-scope NPP simulator and interviews with NPP operators will be necessary.     

Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation

We study a scalar reaction-diffusion equation which contains a nonlocal term in the form of an integral convolution in the spatial variable and demonstrate, using asymptotic, analytical and numerical techniques, that this scalar equation is capable of producing spatio-temporal patterns. Fisher's equation is a particular case of this equation. An asymptotic expansion is obtained for a travelling wavefront connecting the two uniform steady states and qualitative differences to the corresponding solution of Fisher's equation are noted. A stability analysis combined with numerical integration of the equation show that under certain circumstances nonuniform solutions are formed in the wake of this front. Using global bifurcation theory, we prove the existence of such non-uniform steady state solutions for a wide range of parameter values. Numerical bifurcation studies of the behaviour of steady state solutions as a certain parameter is varied, are also presented.     

Autonomous Boolean modelling of developmental gene regulatory networks

During early embryonic development, a network of regulatory interactions among genes dynamically determines a pattern of differentiated tissues. We show that important timing information associated with the interactions can be faithfully represented in autonomous Boolean models in which binary variables representing expression levels are updated in continuous time, and that such models can provide a direct insight into features that are difficult to extract from ordinary differential equation (ODE) models. As an application, we model the experimentally well-studied network controlling fly body segmentation. The Boolean model successfully generates the patterns formed in normal and genetically perturbed fly embryos, permits the derivation of constraints on the time delay parameters, clarifies the logic associated with different ODE parameter sets and provides a platform for studying connectivity and robustness in parameter space. By elucidating the role of regulatory time delays in pattern formation, the results suggest new types of experimental measurements in early embryonic development.     

具有非线性运动约束输液曲管振动的分岔

研究了具何非线性运动约束输液曲管在系统参数区域内振动的分岔现象。基于牛顿法推导出了输液曲管模型面内振动的非线性控制方程，利用微分求积法将系统的偏微分方程转化为关于时间域的上阶常微分方程组；在此基础之上，采用数值迭代技术求解了输液曲管的非线性动力学方程。数值模拟表明，输液曲管在系统多种参数区域内俘有复杂的分岔现象；在这些参数区域内，系统将以静态变形、周期运动和混沌等形式作复杂的振动。     

一类具有食饵恐惧和避难所的反应扩散捕食模型的稳定性与 Hopf 分支

在捕食生态系统中,恐惧因子和食饵避难所都有重要的作用。为此,对一类带恐惧因子和食饵避难所的捕食-食饵反应扩散模型进行了研究。通过分析平衡点特征方程,得到了平衡点的局部渐近稳定性;将不受保护食饵比例作为分支参数,给出了正平衡点 Hopf 分支存在的条件。结果表明：避难所的存在会导致 Hopf 分支,产生空间齐次周期解。扩散的加入会产生新的Hopf分支点,产生空间非齐次周期解。这说明通过设立适当的食饵避难所或者减小捕食者的扩散,有助于物种共存。最后,利用 Matlab 进行数值模拟验证了所得的结论。     

Tracing the equilibrium path beyond simple critical points

A unified presentation of some popular continuation procedures used in the non-linear finite element analysis of structural mechanics is introduced. An extension of the elliptical constraint equation proposed by Crisfield is given. It is shown that in the proposed procedure real roots can always be obtained in solving the iterative change of the load parameter. Updated weighting factors are introduced in the constraint equation in order to get better convergence characteristics in the case when localized deformations occur. For bifurcation points a modification of Rheinboldt's branching procedure is presented. Post critical response after limit and bifurcation points is determined in some numerical examples.     

Isolas, cusps and global bifurcations in an electronic oscillator

The aim of the present work is to describe the bifurcation behaviour of a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch: that is, their bifurcation diagram showed an eight-shaped isola, with a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its destruction in an isola centre with a cusp of periodic orbits. Finally, the introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifurcation) and to homoclinic connections of the non-trivial equilibria. The isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil?nikov homoclinic and heteroclinic connections, related to the symmetric and asymmetric periodic orbits, emerge from T-points and end at Shil?nikov-Hopf singularities     

Localized Fourier transform filter for noise removal in electronic speckle pattern interferometry wrapped phase patterns

This article is concerned with frequency filtering for electronic speckle pattern interferometry wrapped phase patterns. We propose a robust localized Fourier transform filter which is an extension of the root filtering method (RFM). We improve the RFM from a simple technical process and a filter function in the frequency domain. In our method, the proposed filter function is taken as the power spectrum of the convolution of an image and a Gaussian function to the power α. We test the proposed method on two computer-simulated wrapped phase fringe patterns and one experimentally obtained wrapped phase pattern, and compare our models with the widely used, well-known RFM and windowed Fourier filtering (WFF). The experimental results have demonstrated that our localized Fourier transform filter outperforms the RFM and is comparable to WFF. Our method depends on fewer parameters, as compared with WFF, and can achieve a better balance between the computational complexity and the filtered results.     

Double singularity-induced bifurcation points and singular Hopf bifurcations

The singularity-induced bifurcation and singular Hopf bifurcation theorems and the degeneracies that arise when Newton's laws are coupled to Kirchhoff's laws are explored. Such models are used in the electrical engineering literature to describe electrical power systems and they can take the form of either an index-1 differential-algebraic equation (DAE) or a singularly perturbed ordinary differential equation (ODE). As a consequence of the debate in the engineering literature as to which class of system is the 'true' representation of power systems, a discussion is included of the consequences of the power engineer's 'load-flow singularity' for both ODE and DAE.     

碰撞振动系统四阶共振下的Hopf分岔和次谐分岔

建立了一类三自由度含间隙碰撞振动系统的力学模型,求解了系统六维n?1周期运动的周期解及其Poincaré映射。通过理论分析和数值模拟相结合的方法,分析了该系统在强共振点附近,系统两参数控制的局部动力学行为。即在两参数平面上共振点的附近变化两控制参数,进行数值模拟并划分两参数平面的拓扑区域;分析了以“四方形”和“四叶形”异宿轨道为特征的存在于强共振点附近的Hopf分岔不变圈和次谐分岔4?4周期运动,并进一步分析了四阶次谐分岔向混沌的演化过程。     

Vortex-like Current States in Josephson Ballistic Point Contacts

No Heading The stationary Josephson effect is studied theoretically in the situation when there is externally injected transport supercurrent which flows in the banks parallel to the contact interface. Coexistence of this supercurrent with the order parameter phase difference between the banks of the contact results in the appearance of quasiparticles in the weak link, which create the current, localized in the vicinity of the contact and counter-flowing to the transport supercurrent. We review the results of our previous study of weak links between current-carrying conventional and unconventional (d-wave) superconductors. And further we study the weak link in the form of double point contact between current-carrying superconductors. Current distribution patterns containing vortex-like states are obtained.PACS numbers: 74.50.+r, 74.76.Bz, 74.80.Fp.