Dynamics of hierarchical weighted networks of van der Pol oscillators

Abstract

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.

Timing Configurations Affect the Macro-Properties of Multi-Scale Feedback Systems

“…As highlighted in [35], natural oscillatory processes tend to follow a multi-scale organisation, with macro-scale frequencies affecting micro-scale behaviour. In most oscillators, communication and adaptation are not instantaneous [39] [16].…”

Synchronization or cluster synchronization in coupled Van der Pol oscillators networks with different topological types

“…[15] , J C. Neu [16]. The dynamic behavior of van der Pol oscillator coupled oscillator systems with more complex network structure have also been studied, such as hierarchical networks in [17] and [18], conjugate coupled networks in [19], starlike networks in [20,21]. The rotating periodic solution theory can be used to explain the synchronization phenomenon in regular networks.…”