# Entropy and stability of phase synchronisation of oscillators on networks

## Abstract

I examine the role of entropy in the transition from incoherence to phase synchronisation in the Kuramoto model of N coupled phase oscillators on a general undirected network. In a Hamiltonian ‘action-angle’ formulation, auxiliary variables J{sub i} combine with the phases θ{sub i} to determine a conserved system with a 2N dimensional phase space. In the vicinity of the fixed point for phase synchronisation, θ{sub i}≈θ{sub j}, which is known to be stable, the auxiliary variables J{sub i} exhibit instability. This manifests Liouville’s Theorem in the phase synchronised regime in that contraction in the θ{sub i} parts of phase space are compensated for by expansion in the auxiliary dimensions. I formulate an entropy rate based on the projection of the J{sub i} onto eigenvectors of the graph Laplacian that satisfies Pesin’s Theorem. This leads to the insight that the evolution to phase synchronisation of the Kuramoto model is equivalent to the approach to a state of monotonically increasing entropy. Indeed, for unequal intrinsic frequencies on the nodes, the networks that achieve the closest to exact phase synchronisation are those which enjoy the highest entropy production. I compare numerical results for a range of networks.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22403382

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 348; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENVECTORS; ENTROPY; HAMILTONIANS; LAPLACIAN; OSCILLATORS; PHASE OSCILLATIONS; PHASE SPACE; STABILITY; SYNCHRONIZATION

### Citation Formats

```
Kalloniatis, Alexander C., E-mail: alexander.kalloniatis@dsto.defence.gov.au.
```*Entropy and stability of phase synchronisation of oscillators on networks*. United States: N. p., 2014.
Web. doi:10.1016/J.AOP.2014.05.012.

```
Kalloniatis, Alexander C., E-mail: alexander.kalloniatis@dsto.defence.gov.au.
```*Entropy and stability of phase synchronisation of oscillators on networks*. United States. doi:10.1016/J.AOP.2014.05.012.

```
Kalloniatis, Alexander C., E-mail: alexander.kalloniatis@dsto.defence.gov.au. Mon .
"Entropy and stability of phase synchronisation of oscillators on networks". United States. doi:10.1016/J.AOP.2014.05.012.
```

```
@article{osti_22403382,
```

title = {Entropy and stability of phase synchronisation of oscillators on networks},

author = {Kalloniatis, Alexander C., E-mail: alexander.kalloniatis@dsto.defence.gov.au},

abstractNote = {I examine the role of entropy in the transition from incoherence to phase synchronisation in the Kuramoto model of N coupled phase oscillators on a general undirected network. In a Hamiltonian ‘action-angle’ formulation, auxiliary variables J{sub i} combine with the phases θ{sub i} to determine a conserved system with a 2N dimensional phase space. In the vicinity of the fixed point for phase synchronisation, θ{sub i}≈θ{sub j}, which is known to be stable, the auxiliary variables J{sub i} exhibit instability. This manifests Liouville’s Theorem in the phase synchronised regime in that contraction in the θ{sub i} parts of phase space are compensated for by expansion in the auxiliary dimensions. I formulate an entropy rate based on the projection of the J{sub i} onto eigenvectors of the graph Laplacian that satisfies Pesin’s Theorem. This leads to the insight that the evolution to phase synchronisation of the Kuramoto model is equivalent to the approach to a state of monotonically increasing entropy. Indeed, for unequal intrinsic frequencies on the nodes, the networks that achieve the closest to exact phase synchronisation are those which enjoy the highest entropy production. I compare numerical results for a range of networks.},

doi = {10.1016/J.AOP.2014.05.012},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = ,

volume = 348,

place = {United States},

year = {2014},

month = {9}

}