Eckhaus instability in systems with large delay

The dynamical behavior of various physical and biological systems under the influence of delayed feedback or coupling can be modeled by including terms with delayed arguments in the equations of motion. In particular, the case of long delay may lead to complicated and high-dimensional dynamics. We investigate the effects of delay in systems that display an oscillatory instability (Hopf bifurcation) in the absence of delay. We show by analytical and numerical methods that the dynamical scenario includes the coexistence of multiple stable periodic solutions and can be described in terms of the Eckhaus instability, which is well known in the context of spatially extended systems.     

On the Eckhaus instability for spatially periodic patterns

The range of stable wavevectors is near the threshold for appearance of periodic patterns in quasi-one-dimensional systems limited by the long-wavelength Eckhaus instability. At this instability saddle-point solutions characterizing the wavelength-changing processes inside the stable range merge with the periodic solutions. We first analyse this bifurcation near threshold using the amplitude expansion in lowest order. Then a nonlinear equation for the evolution of slow modulations of the periodic pattern far from threshold but near the Eckhaus instability is derived and used to analyse the universal properties of the Eckhaus bifurcation. More detailed information concerning the spatial symmetry of saddle-point solutions is obtained by numerical integration of simple model systems.     

双组分混合流体中Eckhaus不稳定调谐的行波对流

在双组分混合流体的Rayleigh-Bénard对流系统中，数值模拟获得了周期性时空位错缺陷调谐的扩展行进波对流状态.研究了该状态的时空演化特性.结果表明，Eckhaus不稳定触发的对流涡卷对的产生是调谐的起源，对流涡卷在边界处生成与湮没频率的不同，又使系统返回不稳定区域，系统在稳定与不稳定的波数之间振荡.还探讨了一个周期内传热与混合特性的改变及其控制参数的变化规律. 关键词： 双组分混合流体 行波对流 时空位错缺陷 Eckhaus不稳定调谐     

Convective and absolute Eckhaus instability leading to modulated waves in a finite box

We report an experimental study of the secondary modulational instability of a one-dimensional nonlinear traveling wave in a long bounded channel. Two qualitatively different instability regimes involving fronts of spatiotemporal defects are linked to the convective and absolute nature of the instability. Both transitions appear to be subcritical. The spatiotemporal defects control the global mode structure.     

Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures

The Eckhaus stability boundaries of travelling periodic roll patterns arising in binary fluid convection is analysed using high-resolution numerical methods. We present results corresponding to three different values of the separation ratio used in experiments. Our results show that the subcritical branches of travelling waves bifurcating at the onset of convection suffer sideband instabilities that are restabilised further away in the branch. If this restabilisation is produced after the turning point of the travelling-wave branch, these waves do not become stable in a saddle node bifurcation as would have been the case in a smaller domain. In the regions of instability of the uniform travelling waves we expect to find either transitions between states of different wave number or modulated travelling waves arising in these bifurcations.     

Metals near a magnetic instability

Zero-temperature magnetic phase transitions exhibit an abundance of nearly critical magnetic fluctuations that allow to probe the traditional concepts of the metallic state. For the prototypical heavy-fermion compound, CeCu_6−x Au _x , a breakdown of the Fermi-liquid properties may be tuned by Au concentration, hydrostatic pressure, or magnetic field. The d-electron weak itinerant ferromagnet ZrZn₂, on the other hand, was recently found to display superconductivity in coexistence with ferromagnetism.     

Quantum critical behavior near a density-wave instability in an isotropic fermi liquid

We study the quantum critical behavior in an isotropic Fermi liquid in the vicinity of a zero-temperature density-wave transition at a finite wave vector qc. We show that, near the transition, the Landau damping of the soft bosonic mode yields a crossover in the fermionic self-energy from Sigma(k,omega) approximately Sigma(k) to Sigma(k,omega) approximately Sigma(omega), where k and omega are momentum and frequency. Because of this self-generated locality, the fermionic effective mass diverges right at the quantum critical point, not before; i.e., the Fermi liquid survives up to the critical point.     

Fluctuations provide strong selection in Ostwald ripening

A selection problem that appears in the Lifshitz-Slyozov (LS) theory of Ostwald ripening is reexamined. The problem concerns selection of a self-similar distribution function (DF) of the minority domains with respect to their sizes from a whole one-parameter family of solutions. A strong selection rule is found via an account of fluctuations. Fluctuations produce an infinite tail in the DF and drive the DF towards the "limiting solution" of LS or its analogs for other growth mechanisms.     

Pair correlation function near a convective instability

The pair correlation function immediately below and above the first hydrodynamic instability in a Bénard cell is computed. In both cases a modified Ornstein-Zernike-type behavior is found that couldbe detected by microwave experiments.     

Critical dynamics near a hard mode instability

Scaling hypothesis and a renormalization group procedure are formulated in the vicinity of the bifurcation point, where the behaviour is governed by inhomogeneous fluctuations. The working of the general ideas is illustrated in a model system in which the number of components of the complex order parameter field goes to infinity.An account of this work was reported at the Eighth International Seminar on Phase Transitions and Critical Phenomena (MECO), Saarbrücken, FRG, March 23–25, 1981