No abstract

No abstract

Some analytical results are obtained for a large population of limit·cycle oscillators modelled by a set of deterministic equations 1>i = Wi-N-1 K'2;/j~lsin(

No abstract

No abstract

No abstract

Rotating spiral waves with a central core composed of phase-randomized oscillators can arise in reaction-diffusion systems if some of the chemical components involved are diffusion-free. This peculiar phenomenon is demonstrated for a paradigmatic three-component reaction-diffusion model. The origin of this anomalous spiral dynamics is the effective non-locality in coupling, whose effect is stronger for weaker coupling. There exists a critical coupling strength which is estimated from a simple argument. Detailed mathematical and numerical analyses are carried out in the extreme case of weak coupling for which the phase reduction method is applicable. Under the assumption that the mean field pattern keeps to rotate steadily as a result of a statistical cancellation of the incoherence, we derive a functional self-consistency equation to be satisfied by this space-time dependent quantity. Its solution and the resulting effective frequencies of the individual oscillators are found to agree excellently with the numerical simulation.

In this paper we review experimental and theoretical results on higher electronic multipoles in solids with strong correlations. Recent experiments and their theoretical interpretation have confirmed the ordering of octupoles and even higher multipoles in rare-earth and actinide compounds with f electrons. The concept of multipoles is critically examined in point groups where spherical tensors of different ranks mix. Using a phenomenological approach, we demonstrate how linear and nonlinear couplings of different multipoles lead to rich phase diagrams and anomalies in physical observables. As actual representative systems, we first consider Ce x La 1Àx B 6 , for which resonant X-ray scattering probed the octupole order for the first time, and NpO 2 , where quadrupoles induced by the octupole order have been observed. We then consider a class of compounds called skutterudites as the most convenient system for systematic study. Particular emphasis is placed on the ordering of scalar components from fourth-rank tensors (hexadecapoles) and sixth-rank tensors (hexacontatetrapoles). A comparison of a skutterudite PrFe 4 P 12 and URu 2 Si 2 is made, where much fewer carriers remain in the ordered states than in the disordered phase. The even number (two) of f electrons per site in Pr 3þ or U 4þ makes the system free from the Kramers degeneracy, in contrast to standard models for Mott transitions. Hence, it is pointed out that multipole orders, particularly the scalar order, should provide a new route for studying the dichotomy between the itinerant and localized behaviors of electrons.