Analysis of XY Model with Mexican-Hat Interaction on a Circle –Derivation of Saddle Point Equations and Study of Bifurcation Structure–

Abstract

In our previous study, we investigated a classical XY model on a circle by adopting the Mexican-hat type interaction, which is composed of uniform and location-dependent interactions. We solved the saddle point equations numerically and found three nontrivial solutions. In this study, we determined the phases of complex order parameters and derived the saddle point equations for stable and unstable nontrivial solutions and the formula of boundaries of bistable regions analytically. We performed Markov Chain Monte Carlo simulations and confirmed that the numerical and theoretical results agree well.KEYWORDS: XY model, Mexican-hat interaction, saddle point equations, bistability §1. IntroductionOver these past years, we have been studying the synchronization -desynchronization phase transition of oscillator networks. 1 In particular, we have studied the phase oscillator network [2][3][4] with the Mexican-hat type interaction on a circle. This type of interaction was introduced to model the feature extraction cells in neurosciences 5,6 and to express effects of excitation of nearby neurons and inhibition of distant neurons.In the course of the analysis of the phase oscillator network, it turned out that information on the phases of complex order parameters is necessary. Therefore, we studied the XY model on a circle with the same interaction as the phase oscillator network, because both models coincide with each other under some conditions. In the XY model, we found three nontrivial solutions of the saddle point equations (SPEs), the uniform (U), spinning (S), and pendulum (Pn) solutions. 7 We confirmed the agreement between the theoretical and numerical results, and drew phase diagrams by performing numerical simulations.In this study, we theoretically determined the phases of complex order parameters that enabled us to derive the self-consistent equations (SCEs) of the amplitudes of complex order parameters in the phase oscillator network. We derived the SPEs of the amplitudes of complex order param- *

Analysis of a solvable model of a phase oscillator network on a circle with infinite-range Mexican-hat-type interaction

“…To obtain self-consistent equations, information about the differences between the phases of the complex order parameters is necessary. Previously, for this purpose, we studied the classical XY model to which the phase oscillator network reduces when all oscillators have the same natural frequency [29,30]. The relevant order parameters are the same in the oscillator network and the XY model.…”

“…The order parameters that characterize the solutions are R and R 1 . The saddle point equations (SPEs) for the XY model were obtained and the differences between the phases of the complex order parameters were determined analytically [30]. So far, we have used information on the phases of the complex order parameters in the XY model to solve the SCEs for the phase oscillator network.…”