2014/12/25 by T. Uezu, Tatsuya Uezu, T. Kimoto +5
Computer Science · Medicine · Physics and Astronomy · #Cardiac electrophysiology and arrhythmias #Class (philosophy) #Classical XY model #Kernel (algebra) #Limit (mathematics) #Neural Networks Stability and Synchronization #Nonlinear Dynamics and Pattern Formation #Order (exchange) #Randomness #Saddle #Type (biology) #cond-mat.stat-mech
paper · pdf · doi:10.7566/jpsj.84.033001
13 pages, 2 figures
arxiv created 2014/12/25 · openalex publication_date 2015/02/05 · arxiv updated 2015/06/23 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/06
We study phase oscillator networks with distributed natural frequencies and classical XY models, both of which have a class of infinite-range interactions in common. We find that the integral kernel of the self-consistent equations (SCEs) for oscillator networks corresponds to that of the saddle point equations (SPEs) for XY models, and that the quenched randomness (distributed natural frequencies) corresponds to thermal noise. We find a sufficient condition under which the probability density of natural frequency distributions is one-humped, so that the kernel in an oscillator network is strictly decreasing, as in the XY model. Furthermore, taking the uniform and Mexican-hat-type interactions, we prove the one-to-one correspondence between the solutions of the SCEs and SPEs. As an application of the correspondence, we study the associative-memory-type interaction. In the XY model with this interaction, there exists a peculiar one-parameter family of solutions. For the oscillator network, we find a nontrivial solution, i.e., a limit cycle oscillation.