Abstract
This paper aims to provide bifurcation analysis for a Kuramoto model with time-delay and random coupling strength. A delay differential equation governing the system is obtained on the Ott-Antonsen’s manifold, and the bifurcation analysis is proceeded by using the characteristic equation and the normal form method. The general case where the coupling strength is chosen as a function of delay is investigated. Afterwards, the synchronization of the model with three different distributions of time delay including degenerate distribution, two-point distribution and Gamma distribution, is discussed respectively. Particularly, the coupled system of which the coupling strength and the delays are divided into two groups is studied in detail and the bifurcation results are obtained both theoretically and numerically.
Highlights
Synchronization of coupled system plays a prominent role in nonlinear sciences.1–5 Kuramoto model6 is a coupled phase model established by Kuramoto based on the research of Winfree.7 It has a wide range of applications in physical, chemical and biological systems to investigate the state of collective synchronization
This paper involves with bifurcation analysis for a Kuramoto model with time delay and random coupling strength
Detailed analysis and corresponding numerical simulations are provided for the case where time delay between two oscillators is subject to Bernoulli distribution
Summary
Synchronization of coupled system plays a prominent role in nonlinear sciences. Kuramoto model is a coupled phase model established by Kuramoto based on the research of Winfree. It has a wide range of applications in physical, chemical and biological systems to investigate the state of collective synchronization. Hong and Strogatz tried to illuminate the Daido’s14 oscillator glass transition by analyzing a much simpler model with mixed coupling The model they studied can be described as follow: θi. It is natural and necessary to discuss the properties of coupled system with time lag to improve the study for relevant domains.. It is natural and necessary to discuss the properties of coupled system with time lag to improve the study for relevant domains.19,20 Studying such a system of oscillators contributes to the theoretical development and practical application. The specific behaviors of the coupled system with time lags and positive coupling strength Kj are explored, which could be the foundation for the future study on the similar system with random time-delay and mixed couplings.
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