Quasi-periodic dynamics and a Neimark-Sacker bifurcation in nonlinear random walks on complex networks

TL;DR

This paper explores how directed network structures influence the emergence of quasi-periodic dynamics in nonlinear random walks, identifying a Neimark-Sacker bifurcation as a key mechanism.

Contribution

It reveals that certain directed network topologies induce quasi-periodic behavior via Neimark-Sacker bifurcations, expanding understanding of nonlinear dynamics on complex networks.

Findings

Directed structures can cause quasi-periodic dynamics.

Neimark-Sacker bifurcation explains the transition to quasi-periodicity.

Analytical case of a four-neighbor ring illustrates the phenomenon.

Abstract

We study the dynamics of nonlinear random walks on complex networks. We investigate the role and effect of directed network topologies on long-term dynamics. While a period-doubling bifurcation to alternating patterns occurs at a critical bias parameter value, we find that some directed structures give rise to a different kind of bifurcation that gives rise to quasi-periodic dynamics. This does not occur for all directed network structure, but only when the network structure is sufficiently directed. We find that the onset of quasi-periodic dynamics is the result of a Neimark-Sacker bifurcation, where a pair of complex-conjugate eigenvalues of the system Jacobian passes through the unit circle, destabilizing the stationary distribution with high-dimensional rotations. We investigate the nature of these bifurcations, study the onset of quasi-periodic dynamics as network structure is tuned to be more directed, and present an analytically tractable case of a four-neighbor ring.