Abstract
We analyze a network of non-identical Rayleigh–van der Pol (RvdP) oscillators interconnected through either diffusive or nonlinear coupling functions. The work presented here extends existing results on the case of two nonlinearly coupled RvdP oscillators to the problem of considering a network of three or more of them. Specifically, we study synchronization and entrainment in networks of heterogeneous RvdP oscillators and contrast the effects of diffusive linear coupling strategies with the nonlinear Haken–Kelso–Bunz coupling, originally introduced to study human bimanual experiments. We show how convergence of the error among the nodes’ trajectories toward a bounded region is possible with both linear and nonlinear coupling functions. Under the assumption that the network is connected, simple, and undirected, analytical results are obtained to prove boundedness of the error when the oscillators are coupled diffusively. All results are illustrated by way of numerical examples and compared with the experimental findings available in the literature on synchronization of people rocking chairs, confirming the effectiveness of the model we propose to capture some of the features of human group synchronization observed experimentally in the previous literature.
Highlights
Interpersonal coordination and synchronization between the motion of two individuals have been extensively studied over the past few decades (Schmidt and Turvey 1994; Varlet et al 2011)
We will explore whether and how the model of coupled Rayleigh– van der Pol (RvdP) oscillators we propose in this paper can reproduce the key features of the observed experimental results
Mi ≤ (1 + αM p2M + βM v2M )vM + ω2M pM := M. This means that the fourth hypothesis of Theorem 2 is always satisfied in the case of RvdP oscillators, and the bound Mis defined in Eq 46
Summary
Interpersonal coordination and synchronization between the motion of two individuals have been extensively studied over the past few decades (Schmidt and Turvey 1994; Varlet et al 2011). Some of the existing works on coordination of multiple human players include studies on choir singers during a concert (Himberg and Thompson 2009), rhythmic activities as, for example, “the cup game” and marching tasks (Iqbal and Riek 2015), rocking chairs (Frank and Richardson 2010; Richardson et al 2012), and coordination of rowers’ movements during a race (Wing and Woodburn 1995) In these papers, the authors provide several experimental results in order to analyze the behavior of a group of people performing some coordinated activities, but a rigorous mathematical model capable of capturing the observed results and explaining the features of the movement coordination among them is still missing.
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