Abstract
The dynamics of networks of interacting dynamical systems depend on the nature of the coupling between individual units. We explore networks of oscillatory units with coupling functions that have ‘dead zones’, that is the coupling functions are zero on sets with interior. For such networks, it is convenient to look at the effective interactions between units rather than the (fixed) structural connectivity to understand the network dynamics. For example, oscillators may effectively decouple in particular phase configurations. Along trajectories, the effective interactions are not necessarily static, but the effective coupling may evolve in time. Here, we formalize the concepts of dead zones and effective interactions. We elucidate how the coupling function shapes the possible effective interaction schemes and how they evolve in time.This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.
Highlights
Many systems in applied sciences can be seen as systems of coupled units that mutually influence each other, such as interacting neurons of an animal’s nervous system
The dynamical systems we investigate have state-dependent interactions that arise through dead zones in the coupling function; this concept has been previously identified
Given a coupling function g, which properties of g imply certain effective coupling graphs realized by g? On the other hand, given θ ∈ C, a structural coupling graph A and H ∈ H(A), how can one construct a coupling function g such that H = Gg,A(θ )? Among the different parameters characterizing the coupling function g, the number of dead zones plays a major role in these questions, since it determines the shapes of the resulting effective coupling graphs
Summary
Many systems in applied sciences can be seen as systems of coupled units that mutually influence each other, such as interacting neurons of an animal’s nervous system. Note that it is not necessarily sufficient to consider the structural coupling graph A to determine dynamical properties: this is the case if the coupling function g has dead zones, i.e. if it is zero over some interval of phase differences. In the presence of dead zones, we will define an effective coupling graph of (1.1) as a subgraph of A, which encodes the effective interactions between oscillators at a particular point in phase space. It seems that the interaction of dynamics and dead zones may be quite complex and so we explore some examples. There is a residual action of ZN := Z/NZ on the CIR and Θsplay is the fixed point of this action [50]
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