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Register a new userLong-term fluctuations in globally coupled phase oscillators with general coupling: Finite size effects
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We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D sim O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D sim O(1/N^a) with a certain constant a>0 in the coherent regime and D sim O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.
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We investigate a critical exponent related to synchronization transition in globally coupled nonidentical phase oscillators. The critical exponents of susceptibility, correlation time, and correlation size are significant quantities to characterize fluctuations in coupled oscillator systems of large but finite size and understand a universal property of synchronization. These exponents have been identified for the sinusoidal coupling but not fully studied for other coupling schemes. Herein, for a general coupling function including a negative second harmonic term in addition to the sinusoidal term, we numerically estimate the critical exponent of the correlation size, denoted by $ u_+$, in a synchronized regime of the system by employing a non-conventional statistical quantity. First, we confirm that the estimated value of $ u_+$ is approximately 5/2 for the sinusoidal coupling case, which is consistent with the well-known theoretical result. Second, we show that the value of $ u_+$ increases with an increase in the strength of the second harmonic term. Our result implies that the critical exponent characterizing synchronization transition largely depends on the coupling function.
We study the effects of delayed coupling on timing and pattern formation in spatially extended systems of dynamic oscillators. Starting from a discrete lattice of coupled oscillators, we derive a generic continuum theory for collective modes of long wavelength. We use this approach to study spatial phase profiles of cellular oscillators in the segmentation clock, a dynamic patterning system of vertebrate embryos. Collective wave patterns result from the interplay of coupling delays and moving boundary conditions. We show that the phase profiles of collective modes depend on coupling delays.
Universal scaling laws form one of the central issues in physics. A non-standard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a non-standard scaling law with respect to system size in a synchronized state of globally coupled non-identical phase oscillators [Nishikawa et al., Chaos $boldsymbol{22}$, 013133 (2012)]. However, it is still unclear how widely this non-standard scaling law is observed. In the present paper, we discuss the conditions required for the unusual scaling law in globally coupled oscillator systems, and we validate the conditions by numerical simulations of several different models.
Many studies of synchronization properties of coupled oscillators, based on the classical Kuramoto approach, focus on ensembles coupled via a mean field. Here we introduce a setup of Kuramoto-type phase oscillators coupled via two mean fields. We derive stability properties of the incoherent state and find traveling wave solutions with different locking patterns; stability properties of these waves are found numerically. Mostly nontrivial states appear when the two fields compete, i.e. one tends to synchronize oscillators while the other one desynchronizes them. Here we identify normal branches which bifurcate from the incoherent state in a usual way, and anomalous branches, appearance of which cannot be described as a bifurcation. Furthermore, hybrid branches combining properties of both are described. In the situations where no stable traveling wave exists, modulated quasiperiodic in time dynamics is observed. Our results indicate that a competition between two coupling channels can lead to a complex system behavior, providing a potential generalized framework for understanding of complex phenomena in natural oscillatory systems.
The interest in the topological properties of materials brings into question the problem of topological phase transitions. As a control parameter is varied, one may drive a system through phases with different topological properties. What is the nature of these transitions and how can we characterize them? The usual Landau approach, with the concept of an order parameter that is finite in a symmetry broken phase is not useful in this context. Topological transitions do not imply a change of symmetry and there is no obvious order parameter. A crucial observation is that they are associated with a diverging length that allows a scaling approach and to introduce critical exponents which define their universality classes. At zero temperature the critical exponents obey a quantum hyperscaling relation. We study finite size effects at topological transitions and show they exhibit universal behavior due to scaling. We discuss the possibility that they become discontinuous as a consequence of these effects and point out the relevance of our study for real systems.
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