Abstract
We study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the system parameter, with a transition from a single constant stationary state to a large range of possible stationary states. Our main result classifies the stability of (a subset of) these stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected. We further describe exact solutions for graphs with external equitable partitions and characterize the basins of attraction on tree graphs. Our technical analysis is supplemented by a study of the system on a number of prototypical networks: tree graphs, complete graphs and barbell graphs. We describe a number of qualitative properties of the dynamics on these networks, with promising modelling consequences.
Highlights
Network dynamics are widely used as a natural way to model complex processes taking place in systems of interacting components
We find that the system can be seen as (i) a set of interacting (1D) pitchfork systems, (ii) a gradient dynamical system for a potential composed of double-well potentials over the links of the network and (iii) as the dominating behaviour of a general class of nonlinear dynamics with odd coupling functions
We have introduced and studied a nonlinear dynamical system on networks inspired by the pitchfork bifurcation
Summary
Network dynamics are widely used as a natural way to model complex processes taking place in systems of interacting components. Analytical results are found for the basins of attraction (on tree graphs) of the stationary states, and an exact solution of the system is derived for certain types of graphs which include graphs with external equitable partitions The latter result adds to a long list of interesting observations of dynamics on graphs with (external) equitable partitions and related symmetries (Schaub et al 2016; Pecora et al 2014; Bonaccorsi et al 2015; Devriendt and Van Mieghem 2017; Ashwin and Swift 1992; Golubitsky and Stewart 2006). Other works on this subject, notably the results of Golubitsky and Stewart (2006), Gandhi et al (2020) and Nijholt (2018), Nijholt et al (2019), describe and characterize general classes of systems whose dynamics are constrained by a given underlying structure Their results allow to determine which dynamical features (e.g. synchronization conditions, bifurcations) are robust (generic) with respect to the network structure; in other words, it details which features can be explained purely from the network structure irrespective of the specific choice of coupling functions.
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