Clustering coefficient and periodic orbits in flow networks

Rodriguez-Mendez, V.; Ser-Giacomi, E.; Hernandez-Garcia, E.

Chaos** 27**, 035803 (1-9) (2017)

Chaos

We show that the clustering coefficient, a standard measure in

network theory, when applied to flow networks, i.e. graph

representations of fluid flows in which links between nodes

represent fluid transport between spatial regions, identifies

approximate locations of periodic trajectories in the flow

system. This is true for steady flows and for periodic ones in

which the time interval tau used to construct the network is

the period of the flow or a multiple of it. In other situations

the clustering coefficient still identifies cyclic motion

between regions of the fluid. Besides the fluid context, these

ideas apply equally well to general dynamical systems. By

varying the value of tau used to construct the network, a

kind of spectroscopy can be performed so that the observation

of high values of mean clustering at a value of tau reveals

the presence of periodic orbits of period 3tau which impact

phase space significantly. These results are illustrated with

examples of increasing complexity, namely a steady and a

periodically perturbed model two-dimensional fluid flow, the

three-dimensional Lorenz system, and the turbulent surface flow

obtained from a numerical model of circulation in the

Mediterranean sea.

network theory, when applied to flow networks, i.e. graph

representations of fluid flows in which links between nodes

represent fluid transport between spatial regions, identifies

approximate locations of periodic trajectories in the flow

system. This is true for steady flows and for periodic ones in

which the time interval tau used to construct the network is

the period of the flow or a multiple of it. In other situations

the clustering coefficient still identifies cyclic motion

between regions of the fluid. Besides the fluid context, these

ideas apply equally well to general dynamical systems. By

varying the value of tau used to construct the network, a

kind of spectroscopy can be performed so that the observation

of high values of mean clustering at a value of tau reveals

the presence of periodic orbits of period 3tau which impact

phase space significantly. These results are illustrated with

examples of increasing complexity, namely a steady and a

periodically perturbed model two-dimensional fluid flow, the

three-dimensional Lorenz system, and the turbulent surface flow

obtained from a numerical model of circulation in the

Mediterranean sea.