# Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems

## Abstract

We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. First, we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding. The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the Rössler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures—thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore, we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length, and network diameter are highly sensitive to the interior crisis captured in this particular data set.

- Authors:

- School of Electrical and Electronic Engineering, The University of Western Australia, Crawley WA 6009 (Australia)
- School of Mathematics and Statistics, The University of Western Australia, Crawley WA 6009 (Australia)

- Publication Date:

- OSTI Identifier:
- 22402554

- Resource Type:
- Journal Article

- Journal Name:
- Chaos (Woodbury, N. Y.)

- Additional Journal Information:
- Journal Volume: 25; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1054-1500

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CHAOS THEORY; COMPARATIVE EVALUATIONS; DYNAMICS; LYAPUNOV METHOD; MARKOV PROCESS; PARTITION FUNCTIONS; PERIODICITY; PHASE SPACE; RESONATORS; TIME DELAY

### Citation Formats

```
McCullough, Michael, Iu, Herbert Ho-Ching, Small, Michael, and Stemler, Thomas.
```*Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems*. United States: N. p., 2015.
Web. doi:10.1063/1.4919075.

```
McCullough, Michael, Iu, Herbert Ho-Ching, Small, Michael, & Stemler, Thomas.
```*Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems*. United States. doi:10.1063/1.4919075.

```
McCullough, Michael, Iu, Herbert Ho-Ching, Small, Michael, and Stemler, Thomas. Fri .
"Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems". United States. doi:10.1063/1.4919075.
```

```
@article{osti_22402554,
```

title = {Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems},

author = {McCullough, Michael and Iu, Herbert Ho-Ching and Small, Michael and Stemler, Thomas},

abstractNote = {We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. First, we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding. The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the Rössler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures—thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore, we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length, and network diameter are highly sensitive to the interior crisis captured in this particular data set.},

doi = {10.1063/1.4919075},

journal = {Chaos (Woodbury, N. Y.)},

issn = {1054-1500},

number = 5,

volume = 25,

place = {United States},

year = {2015},

month = {5}

}