- Open Access
Bifurcation and hysteresis in a nonlinear transport model on network motifs
Phys. Rev. Research 6, 013059 – Published 16 January, 2024
DOI: https://doi.org/10.1103/PhysRevResearch.6.013059
Abstract
Nonlinear transport has been observed in many real systems, such as international trade, business transactions, human movement, and urban traffic. To understand the fundamental properties of transport in such systems, several models have been proposed in the past few years. These models exhibit complex behavior due to the interplay between nonlinearity and the underlying network topology. We studied a transport equation on three-body network motifs both numerically and analytically, and we found bifurcation on two motifs and hysteresis of a model on one motif. These three motifs show different bifurcation points. The first two motifs have a branching node, i.e., a node with outdegree 2, and the third motif is a fully connected bidirectional triangle. In particular, the hysteresis on the bidirectional triangle is not predictable from the linear stability analysis on the uniform stationary solution. The analysis based on the self-consistency equation of the parameter [see Eq. (6) for the precise definition] measuring the ratio between sizes on two nodes reveals the following: The bifurcation on the first two motifs is due to supercritical pitchfork bifurcation. The difference between these two motifs comes from the existence of bidirectional edges. The hysteresis on the third motif is due to the combination of the saddle-node and transcritical bifurcation. Our results imply that the nonlinear transport model exhibits nontrivial behavior even in minimal settings and that the bifurcation on large-scale networks becomes far more complex and unpredictable from the underlying network structure.
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