Convergence of analytic gradient-type systems with periodicity and its applications in Kuramoto models. (April 2019)

• Record Type: Journal Article
• Authors: Li, Zhuchun; Xue, Xiaoping
• Is Part Of: Applied mathematics letters. Volume 90(2019)
• Page Start: 194
• Page End: 201
• Publication Date: 2019-04
• Subjects: Gradient systems -- Periodicity -- Analyticity -- Convergence -- Łojasiewicz inequality -- Kuramoto model; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2018.10.015
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: We consider the convergence of gradient-type systems with periodic and analytic potentials. The main tool is the celebrated Łojasiewicz inequality which is valid for any analytic function. Our results show that the convergence of such systems with periodic and analytic potentials is unconditional to the initial data; in other words, any trajectory converges to some equilibrium. As direct applications, we can show that any trajectory converges to phase-locked state for the first- and second-order Kuramoto models on a symmetric network with attractive–repulsive forces and identical natural frequencies. In particular, the inertial Kuramoto model with identical oscillators converges to phase-locked state for any initial configuration.