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We investigated the effect of time delays on phase configurations in a set of two-dimensional coupled phase oscillators. Each oscillator is allowed to interact with its neighbors located within a finite radius, which serves as a control parameter in this study. It is found that distance-dependent time-delays induce various patterns including traveling rolls, square-like and rhombus-like patterns, spirals, and targets. We analyzed the stability boundaries of the emerging patterns and briefly pointed out the possible empirical implications of such time-delayed patterns.

We investigate the dynamics of a two-dimensional array of oscillators with phase-shifted coupling. Each oscillator is allowed to interact with its neighbors within a finite radius. The system exhibits various patterns including squarelike pinwheels, (anti)spirals with phase-randomized cores, and antiferro patterns embedded in (anti)spirals. We consider the symmetry properties of the system to explain the observed behaviors, and estimate the wavelengths of the patterns by linear analysis. Finally, we point out the implications of our work for biological neural networks.PACS numbers: 05.45. Xt, 89.75.Kd, 82.40.Ck, 87.19.La Large systems of interacting oscillators have been used to explain the cooperative behaviors of numerous physical, chemical, and biological systems [1,2,3,4]. When the coupling between oscillators is sufficiently weak, we can describe the dynamics of the system by phase variables defined on the limit cycles [2,3]. In the phaseoscillator models, the coupling proportional to the sine of the phase difference between oscillators has been exploited due to the mathematical tractability [2,4]. In spite of several successes in explaining the synchronization phenomena, the simple sinusoidal coupling fails to account for the collective frequency higher than natural frequency [5], dephasing phenomena [6], and effects of time-delayed interactions [7]. To resolve these problems, phase shifts in the sinusoidal coupling have been considered. Most importantly, a nonzero phase shift is naturally contributed by the broken odd symmetry of a coupling function [8]. The previous works, however, studied only the cases of phase shifts in the limited range. Moreover, they considered only limited interactions via nearest-neighbor coupling [5,7,8] or all-to-all global coupling [9] which seem to be too restrictive. In the neurobiological systems, for example, it is believed that actual coupling takes the forms between these two extremes [10].In this Rapid Communication, we investigate the effect of phase-shifted coupling on the dynamics of a twodimensional array of coupled oscillators. Here we introduce a finite interaction radius as realistic coupling and study over the whole range of phase shifts. We find that various spatial patterns come to emerge, and unravel that the symmetry properties of the system play an important role on the formation of patterns.We start with the equations of two coupled phase oscillators [5]where θ 1 and θ 2 represent the phases of oscillators, respectively, ω the natural frequency, K the coupling strength, and α the phase shift. Phase shift |α| < π/2 leads to inphasing of the two oscillators, whereas |α| > π/2 leads to their antiphasing. We find that this separation by |α| = π/2 still holds for an array of oscillators, but in a rather sophisticated manner as shown below.To investigate the phase-shift effect on the spatially extended systems, we study the following model equations:where θ ij denotes the phase of the oscillator at position (i, j) on a two-dimensional lattice, and mn ′ ≡ mn, ...

We introduce a 2-layer network model for the study of the immunization dynamics in epidemics. Spreading of an epidemic is modeled as an excitatory process in a small-world network (body layer) while immunization by prevention for the disease as a dynamic process in a scale-free network (head layer). It is shown that prevention indeed turns periodic rages of an epidemic into small fluctuation. The study also reveals that, in a certain situation, prevention actually plays an adverse role and helps the disease survive. We argue that the presence of two different characteristic time scales contributes to the immunization dynamics observed.

We investigate scaling and memory effects in return intervals between price volatilities above a certain threshold q for the Japanese stock market using daily and intraday data sets. We find that the distribution of return intervals can be approximated by a scaling function that depends only on the ratio between the return interval τ and its mean τ . We also find memory effects such that a large (or small) return interval follows a large (or small) interval by investigating the conditional distribution and mean return interval. The results are similar to previous studies of other markets and indicate that similar statistical features appear in different financial markets.We also compare our results between the period before and after the big crash at the end of 1989.We find that scaling and memory effects of the return intervals show similar features although the statistical properties of the returns are different.