Bifurcations and Dynamics of Spiral Waves

(N) . In this article, it is shown that the dynamics near meandering spiral waves or other patterns is determined by a finite-dimensional vector field that has a certain skew-product structure over the group SESE(N) . This generalizes our earlier work on center-manifold theory near rigidly rotating spiral waves to meandering spirals. In particular, for meandering spirals, it is much more sophisticated to extract the aforementioned skew-product structure since spatio-temporal rather than only spatial symmetries have to be accounted for. Another difficulty is that the action of the Euclidean symmetry group on the underlying function space is not differentiable, and in fact may be discontinuous. Using this center-manifold reduction, Hopf bifurcations and periodic forcing of spiral waves are then investigated. The results explain the transitions to patterns with two or more temporal frequencies that have been observed in various experiments and numerical simulations. Received December 8, 1997; accepted May 19, 1996     

A Forcing Mechanism for Spiral Density Waves in Galaxies

In this paper, we consider gas dynamical models of spiral galaxies in which there is a rigidly rotating, weakly barlike structure in the central regions. It is found that, in the neighborhood of the corotation circle, this barlike structure forces a trailing spiral wave. Such a driven wave could then propagate inwards to the bar and complete a feedback loop to maintain the spiral structure. For basic distributions typical of our Galaxy, if a ten-per-cent oval distortion is assumed, the strength of the induced spiral field is found to be of the order of a few per cent of the axisymmetric field, in agreement with observational data.     

Optimal Spatial Harvesting Strategy and Symmetry-Breaking

A reaction-diffusion model with logistic growth and constant effort harvesting is considered. By minimizing an intrinsic biological energy function, we obtain an optimal spatial harvesting strategy which will benefit the population the most. The symmetry properties of the optimal strategy are also discussed, and related symmetry preserving and symmetry breaking phenomena are shown with several typical examples of habitats.     

计算对称破缺Takens-Bogdanov点的有效方法

我们将提出一种直接方法来计算对称破缺Takens-Bogdanov分歧点,这种方法构造了不引进零向量作为变量的小扩张系统,从而减少了计算量并节约了内存,数值例子的计算成功地说明了方法的有效性。     

Symmetry-Breaking Convective Dynamos in Spherical Shells

Summary. The convective dynamo is the generation of a magnetic field by the convective motion of an electrically conducting fluid. We assume a spherical domain and spherically invariant basic equations and boundary conditions. The initial state of rest is then spherically symmetric. A first instability leads to purely convective flows, the pattern of which is selected according to the known classification of O(3) -symmetry-breaking bifurcation theory. A second instability can then lead to the dynamo effect. Computing this instability is now a purely numerical problem, because the convective flow is known only by its numerical approximation. However, since the convective flow can still possess a nontrivial symmetry group G ₀ , this is again a symmetry-breaking bifurcation problem. After having determined numerically the critical linear magnetic modes, we determine the action of G ₀ in the space of these critical modes. Applying methods of equivariant bifurcation theory, we can classify the pattern selection rules in the dynamo bifurcation. We consider various aspect ratios of the spherical fluid domain, corresponding to different convective patterns, and we are able to describe the symmetry and generic properties of the bifurcated magnetic fields. Received December 3, 1996; second revision received June 5, 1997; final version received January 23, 1998     

Symmetry-Breaking Bifurcations of Wreath Product Systems

Summary. Patterns formed through steady-state and Hopf bifurcations in wreath product systems depend on both the internal and global symmetries. In this paper we explore some features of this dependence related to general constraints on commuting matrices. We describe the stability of steady states and periodic solutions of wreath product systems obtained from the Equivariant Branching Lemma and the Equivariant Hopf Theorem. Received April 14, 1998; revised October 22, 1998; accepted November 16, 1998     

Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.     

Spatio-Temporal Organization in a Morphochemical Electrodeposition Model: Analysis and Numerical Simulation of Spiral Waves

In this paper we derive Hopf instability conditions for the morphochemical mathematical model for alloy electrodeposition introduced and experimentally validated in Bozzini et al. (J. Solid State Electrochem., 17:467–479, 2013). Using normal form theory we show that in the neighborhood of the Hopf bifurcation, essential features of the system dynamics are captured by a specific Complex Ginzburg-Landau Equation (CGLE). The derived CGLE yields analytical results on the existence and stability of spiral waves. Moreover, the arising of spiral instability is discussed in terms of the relevant system parameters and the related phenomenology is investigated numerically. To face with the numerical approximation of the spiral structures and of their longtime oscillating behavior we apply an Alternating Direction Implicit (ADI) method based on high order finite differences in space.     

A 2D-Model of Cell Sorting Induced by Propagation of Chemical Signals Along Spiral Waves

We study a model for cell sorting based in the presence of differential chemotactic sensitivities. The chemical waves which are responsible for the cell motion propagate along some spiral waves. We prove rigorously that cells with larger chemotactic sensitivity are trapped in a region close to the center of the spiral waves if these propagate along some archimedian spirals.     

Symmetry-Breaking Bifurcation for Free Elastic Shell of Biological Cluster,Part 2.

We will be concerned with a two-dimensional mathematical model for a free elastic shell of biological cluster. The cluster boundary is connected with its kernel by elastic links. The inside part is filled with compressed gas or fluid. Equilibrium forms of the shell of biological cluster may be found as solutions of a certain nonlinear functional-differential equation with several physical parameters. For each multiparameter this equation has a radially symmetric solution. Our goal is to study the bifurcation which breaks symmetry. In order to establish critical values of bifurcation parameter and buckling modes we will investigate an appropriate linear problem. Our main result on the existence of symmetrybreaking bifurcation will be proved by the use of a variational version of the Crandall-Rabinowitz theorem.     

On Translational Clouds for a Convex Body

For a d-dimensional convex body K let C(K) denote the minimum size of translational clouds for K. That is, C(K) is the minimum number of mutually non-overlapping translates of K which do not overlap K and block all the light rays emanating from any point of K. In this paper we prove the general upper bound . Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show . Finally, for the d-dimensional ball B^d we obtain the bounds .     

Hamiltonian Equations for Scale-Invariant Waves

We use a Hamiltonian formalism to derive equations for weakly nonlinear scale-invariant waves. We apply the results to nonlinear surface waves in elasticity and magnetohydrodynamics that satisfy nonlocal generalizations of the inviscid Burgers' equation.     

三维可压Euler方程简单波及双重波结构

本文研究了三维可压欧拉方程简单波解和双重波解的结构.给出了简单波的流动区域是被一族相互独立的平面所覆盖,沿着每个平面u,v,w从而p,ρ,c均为常数.双重波的流动区域是被一族相互独立的直线所覆盖,沿着每个特征平面u,v,w从而p,ρ,c均为常数.     

血液动力学中血管流激波与驻波的相互作用

血液动力学问题是生物力学心血管系统中的重要研究课题.血管内斑块处,血管截面和血管壁的材质发生变化,对血液流动产生重要影响.血液流动中基本波及其相互作用对探究血液流动的规律、生理学意义及与疾病的关系有着重要的意义.本文研究血液动力学血液流动简化数学模型的基本波的相互作用.血管流模型是3×3非严格双曲型方程组.构造性地得到了初值为三段常状态时,血管流问题的解,即解决了激波与驻波的相互作用问题.特别地,给出四种后前激波与驻波的相互作用的结果.     

Periodic Plane Waves for the Oregonator

The Oregonator is a set of differential equations proposed by R. J. Field and R. M. Noyes as a model for the oscillating chemical reaction first studied by B. P. Belousov and A. M. Zhabotinskii. In this paper it is shown that the associated diffusion equations have periodic plane waves for parameter values not covered in earlier work. This amounts to studying a singularly perturbed system when nothing is known about the stability of periodic solutions for the reduced system.     

Travelling Waves for Reaction Diffusion Equations

In this paper the travelling waves for the reaction diffusion equation in most general case is considered. The existence of travelling wave solutions is proved under very weak conditions, which are also necessary for the nonlinear term. A difference method is suggested and Leray-Scbauder fixed point theorem is used to prove the existence of discrete travelling waves. Then the convergence is shown and so the solution for the differential equation is obtained.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

Solitary Waves for Nonconvex FPU Lattices

Solitary waves in a one-dimensional chain of atoms are investigated. The potential energy is required to be monotone and grow super-quadratically. The existence of solitary waves with a prescribed asymptotic strain is shown under certain assumptions on the asymptotic strain and the wave speed. It is demonstrated that the invariance of the equations allows one to transform a system with nonconvex potential energy density to the situation under consideration.     

A Model Equation for Water Waves

Water waves are usually modeled as solutions of Laplace's equation in the fluid domain with appropriate nonlinear boundary conditions. Here we present a simple differential equation on the mean free surface, whose solutions behave like water waves.