Dynamics of spiral waves under phase feedback control in a Belousov-Zhabotinsky reaction

Feedback control of spiral waves by the phases of the spiral tip is investigated experimentally in a light-sensitive Belousov-Zhabotinsky reaction. The phases of rotation (Psi(r)) and meandering (Psi(m)) of the spiral tip are obtained in real time during experiments. It is found that, for both meandering and rigid rotating spirals, one can manipulate the spirals to move with any arbitrary paths by the feedback signals derived from Psi(r). Synchronization between meandering and rotation dynamics can be induced when both Psi(m) and Psi(r) are used simultaneously as control signals. Experimental findings are compared well with numerical simulations of the Oregonator model.     

A linear theory of resonance structures in spiral galaxies

The resonance mechanism for the formation of galactic spirals is considered. Expressions are derived for the resonance responses of disks with circular and nearly circular stellar orbits. The spiral responses produced by the central oval-shaped structures (bars) available in many galaxies are shown to have the characteristic properties of the spirals observed in these galaxies. In the most interesting case of a quasi-steady state, the spiral responses possess a similarity property: the spiral thickness and inclination are proportional to the mean size of an epicycle (an analog of the Larmor circle in plasma).     

Meandering Spiral Waves Induced by Time-Periodic Coupling Strength

Effects of time-periodic coupling strength (TPCS) on spiral waves dynamics are studied by numerical computations and mathematical analyses. We find that meandering or drifting spirals waves, which are not observed for the case of constant coupling strength, can be induced by TPCS. In particular, a transition between outward petal and inward petal meandering spirals is observed when the period of TPCS is varied. These two types of meandering spirals are separated by a drifting spiral, which can be induced by TPCS when the period of TPCS is very close to that of rigidly rotating spiral. Similar results can be obtained if the coupling strength is modulated by a rectangle wave. Furthermore, a kinetic model for spiral movement suggested by Di et al., [Phys. Rev. E 85 (2012) 046216] is applied for explaining the above findings. The theoretical results are in good qualitative agreement with numerical simulations.     

A novel numerical synthetic technique for determination of the TMon dispersion relation in a coaxial corrugated cylindrical waveguide

A novel, highly accurate numerical synthetic technique for determining the complete dispersive characteristics of electromagnetic modes in a spatially periodic structure is presented. The numerical method based on the coupling of the finite difference method in time domain with the discrete fourier transform is applied to calculate the eigenfrequencies and eigenfield distribution of a resonant cavity which is an appropriately shorted periodic slow wave circuit of N periods at both ends. The analytical synthetic technique, which is based on the intrinsic characteristic of spatially periodic structure, is used to derive the complete dispersion relation using the numerically measured resonances. The method was successfully applied for the case of TM_on modes in a coaxial corrugated waveguide and is applicable to slow wave structures of arbitrary geometry.     

Multi-armed spirals and multi-pairs antispirals in spatial rock–paper–scissors games

We study the formation of multi-armed spirals and multi-pairs antispirals in spatial rock–paper–scissors games with mobile individuals. We discover a set of seed distributions of species, which is able to produce multi-armed spirals and multi-pairs antispirals with a finite number of arms and pairs based on stochastic processes. The joint spiral waves are also predicted by a theoretical model based on partial differential equations associated with specific initial conditions. The spatial entropy of patterns is introduced to differentiate the multi-armed spirals and multi-pairs antispirals. For the given mobility, the spatial entropy of multi-armed spirals is higher than that of single armed spirals. The stability of the waves is explored with respect to individual mobility. Particularly, we find that both two armed spirals and one pair antispirals transform to the single armed spirals. Furthermore, multi-armed spirals and multi-pairs antispirals are relatively stable for intermediate mobility. The joint spirals with lower numbers of arms and pairs are relatively more stable than those with higher numbers of arms and pairs. In addition, comparing to large amount of previous work, we employ the no flux boundary conditions which enables quantitative studies of pattern formation and stability in the system of stochastic interactions in the absence of excitable media.     

Meandering Spiral Waves Induced by Time-Periodic Coupling Strength

Effects of time-periodic coupling strength （TPCS） on spiral waves dynamics are studied by numerical computations and mathematical analyses. We find that meandering or drifting spirals waves, which are not observed for the case of constant coupling strength, can be induced by TPCS. In particular, a transition between outward petal and inward petal meandering spirals is observed when the period of TPCS is varied. These two types of meandering spirals are separated by a drifting spiral, which can be induced by TPCS when the period of TPCS is very close to that of rigidly rotating spiral. Similar results can be obtained if the coupling strength is modulated by a rectangle wave. Furthermore, a kinetic model for spiral movement suggested by Diet al., [Phys. Rev. E 85 （2012） 046216] is applied for explaining the above findings. The theoretical results are in good qualitative agreement with numerical simulations.     

Self-Similar 2d Euler Solutions with Mixed-Sign Vorticity

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The boundaries between regions of positive and negative vorticity form algebraic spirals, similar to the Kaden spiral and as opposed to Prandtl’s logarithmic vortex spirals. Also unlike the Prandtl case, spirals are not initially present.     

Darwin curves and galaxy arms

In the natural world, there exists one kind of structure which is beyond the scope of human laboratorial experiment. It is the structure of galaxies which is usually composed of billions of stars. Spiral galaxies are flat disk-shaped. There are two types of spiral galaxies. The spiral galaxies with some bar-shaped pattern are called barred spirals, and the ones without the pattern are called ordinary spirals. Longer-wavelength galaxy images (infrared, for example) show that ordinary spiral galaxies are basically an axi-symmetric disk that is called exponential disk. For a planar distribution of matter, Jin He defined Darwin curves in the plane as such that the ratio of the matter densities at both sides of the curve is constant along the curve. Therefore, the arms of ordinary spiral galaxies are Darwin curves. Now an important question is that: Are the arms of barred spiral galaxies the Darwin curves too? Fortunately, Jin He designed a piece of Galaxy Anatomy graphic software. With the software, not only can people simulate the stellar density distribution of barred spiral galaxies but also can draw the Darwin curves of the simulated galaxy structure. This paper shows partial evidence that the arms of galaxy NGC 3275, 4548 and 5921 follow Darwin curves.     

Predicting the eigenmodes of a cavity containing an array of circular pipes

An array of pipes inside a cavity, as found, for example, in a shell-and-tube heat exchanger, changes the eigenfrequencies of the cavity. It can be tedious to determine the shifted eigenfrequencies with a finite-element model. Based on previous work by Meyer and Neumann, Parker proposed a simple relationship for predicting the shifted eigenfrequencies. In this paper, results obtained from this relationship are compared with eigenfrequencies obtained from very accurate finite element simulations. From the results it can be concluded that Parker's relationship gives fairly good predictions of the eigenfrequencies for the first few modes in a cavity with pipes arranged in a rectangular configuration. The predictions are not so accurate for pipes arranged in a diamond configuration, and a modified version of the relationship is suggested for this configuration. If the number of pipes in the cavity is small, the simple relationship is no longer valid.     

自适应线性神经元方法同轴相对论返波管高频特性的数值分析

 提出一种数值分析周期慢波结构色散特性的模拟法：先采用时域有限差分法和离散傅里叶变换技术计算含周期慢波结构的谐振腔的谐振频率及谐振场分布，再基于周期慢波电路色散关系的周期性质，利用几个特殊色散点由数值组合技术获得周期慢波线的完整色散关系。数值计算了同轴波纹波导中TMon模式的色散关系。亦能用于分析计算任意复杂几何结构的周期慢波线的色散关系。     

同轴相对论返波管高频特性的数值分析

提出一种数值分析周期慢波结构色散特性的模拟法，先采用时域有限差分法和离散傅里叶变换技术计算含周期慢波结构的谐振腔的谐振频率及谐振场分布，再基于周期慢波电路色散关系的周期性质，利用几个特殊色散点由数值组合技术获得周期慢波线的完整色散关系。数值计算了同轴波纹波导中ＴＭ０ｎ模式的色散关系，亦能用于分析计算任意复杂几何结构的周期慢波线的色散关系。     

自适应线性神经元方法同轴相对论返波管高频特性的数值分析

提出一种数值分析周期慢波结构色散特性的模拟法：先采用时域有限差分法和离散傅里叶变换技术计算含周期慢波结构的谐振腔的谐振频率及谐振场分布,再基于周期慢波电路色散关系的周期性质,利用几个特殊色散点由数值组合技术获得周期慢波线的完整色散关系。数值计算了同轴波纹波导中TMon模式的色散关系。亦能用于分析计算任意复杂几何结构的周期慢波线的色散关系。     

Coupled out of plane vibrations of spiral beams for micro-scale applications

An analytical method is proposed to calculate the natural frequencies and the corresponding mode shape functions of an Archimedean spiral beam. The deflection of the beam is due to both bending and torsion, which makes the problem coupled in nature. The governing partial differential equations and the boundary conditions are derived using Hamilton’s principle. Two factors make the vibrations of spirals different from oscillations of constant radius arcs. The first is the presence of terms with derivatives of the radius in the governing equations of spirals and the second is the fact that variations of radius of the beam causes the coefficients of the differential equations to be variable. It is demonstrated, using perturbation techniques that the derivative of the radius terms have negligible effect on structure’s dynamics. The spiral is then approximated with many merging constant-radius curved sections joined together to approximate the slow change of radius along the spiral. The equations of motion are formulated in non-dimensional form and the effect of all the key parameters on natural frequencies is presented. Non-dimensional curves are used to summarize the results for clarity. We also solve the governing equations using Rayleigh’s approximate method. The fundamental frequency results of the exact and Rayleigh’s method are in close agreement. This to some extent verifies the exact solutions. The results show that the vibration of spirals is mostly torsional which complicates using the spiral beam as a host for a sensor or energy harvesting device.     

气体放电系统中多臂螺旋波的数值分析

采用Purwins的三变量模型, 在二维空间对气体放电系统中多臂螺旋波的形成和转化进行了数值研究. 通过分析方程参数对系统空间的影响, 确定了系统获得稳定螺旋波的参数空间; 得到了斑图由简单静态六边形到螺旋波的演化过程, 分析了螺旋波的形成机制和时空特性; 进一步获得六种不同臂数的多臂螺旋波斑图(例如: 双臂、三臂、四臂、五臂、六臂和七臂螺旋波). 结果表明: 螺旋波斑图出现在图灵-霍普夫(Turing-Hopf)空间, 是Turing模和Hopf模相互竞争、相互作用的结果; 不同臂数的螺旋波波头均在持续地旋转运动, 其运动速度随螺旋波臂数的增加而增大; 随着螺旋波臂数的增加, 其波头的运动形式愈加复杂; 由于受微扰及边界条件的影响, 多臂螺旋波可以向臂数少一的螺旋波发生转变, 数值模拟结果与实验结果符合较好. 关键词： 螺旋波 数值模拟 气体放电     

Gradient induced spiral drift in heterogeneous excitable media

Nonlinear excitable systems far from equilibrium can exhibit pattern formation such as spirals, target patterns, etc. One such system is the heterogeneous catalytic reaction of CO with oxygen on platinum single crystals. It has been established that the resonant periodic forcing of spirals in such excitable systems can cause a spiral drift. Here, we investigate the effects of a linear thermal gradient on the spiral dynamics during CO oxidation on platinum (110) for the first time, both in simulations and with experiments. Our results suggest that a spatial thermal gradient established across the surface can act as an internal forcing drive and cause the spiral patterns to drift. This drift has components both parallel and perpendicular to the external gradient.     

Geometric diagnostics of complex patterns: spiral defect chaos

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems, we present an automated approach that aims at characterizing quantitatively spiral-like elements in complex stripe-like patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arc length and their winding number. In addition, it yields the number of pattern components (Betti number of order 1), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh-Benard convection and find that the arc length of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is nonmonotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number shows approximately an exponential decay. It depends only weakly on the heating, but strongly on the Prandtl number. Large spirals arise only for larger Prandtl numbers (Pr approximately > 1). In this regime the joint distribution for the spiral length and the winding number exhibits a three-peak structure, indicating the dominance of Archimedean spirals of opposite sign and relatively straight sections. For small Prandtl numbers the distribution function reveals a large number of small compact pattern components.     

复Ginzburg-Landau方程中模螺旋波的稳定性研究

研究了复Ginzburg-Landau方程系统中模螺旋波与其他斑图在同一平面内的竞争行为,发现演化结果在系统参数平面内可分为四个主要区域:在I区和III区中,模螺旋波与相螺旋波相比稳定性较差,模螺旋波的空间被相螺旋波所入侵.在II区中,模螺旋波具有较强的稳定性,相螺旋波的空间被模螺旋波所入侵.在IV区内,由于时空混沌所导致的频率不稳定性,演化的结果较为复杂.我们通过对模螺旋波、相螺旋波以及时空混沌的频率分析,发现当模螺旋波的系统参数为α1=-1.34,β1=0.35时,较高频率的模螺旋波具有较好的稳定性,高频模螺旋波可以入侵低频斑图空间.竞争结果主要受系统变量实部的频率影响,频率分析所得到的理论结果与数值实验结果符合得非常好.     

Eigenschwingungen von Doppelspiralen aus diskreten Massenpunkten

We discuss the dynamics of infinitely long double spirals of discret mass points. We use harmonic lattice theory. The secular equation of degree 3 is derived and the eigenfrequencies and eigenvibrations are calculated. The appendix contains a useful method to separate the optical from the acoustical modes to get the elastical limit.     

Dynamics of superparamagnetic filaments with finite magnetic relaxation time

In this paper we formulate a model of superparamagnetic filaments with internal dissipative torques due to the action of a rotating magnetic field. It is shown that spirals are formed at both ends of the filament due to the action of the internal torques. These spirals propagate to the center of the filament and collide, forming a compact cluster that rotates in accordance with the rotating magnetic field. These results are in agreement with recent experiments with chains of superparamagnetic beads in a rotating magnetic field.