Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

Tymochko, Sarah; Munch, Elizabeth; Khasawneh, Firas A.

doi:10.3390/a13110278

Cited by 16 publications

(8 citation statements)

References 41 publications

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“…Tools for analyzing time series data and dynamical systems using topological methods have been developed by Harer, Perea, Robins, Mischaikow, Mukherjee among others (see [125,142,191,216,226] and references within). Memoli, Munch, and colleagues have been extending persistent homology theory and methods to analyse time varying systems, ranging from collective behavior in the form of dynamic point cloud, dynamic graphs, and Hopf bifurcation detection [154,156,194,264]. Growing graphs have also been analysed with nodefiltered order complexes [35].…”

Section: Applications Of Persistent Homologymentioning

confidence: 99%

What are higher-order networks?

Bick¹,

Gross²,

Harrington³

2021

Preprint

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Modeling complex systems and data using the language of graphs and networks has become an essential topic across a range of different disciplines. Arguably, this network-based perspective derives is success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.

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Section: Applications Of Persistent Homologymentioning

confidence: 99%

What are higher-order networks?

Bick¹,

Gross²,

Harrington³

2021

Preprint

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“…Zigzag persistence tracks the formation and disappearance of these homologies through a persistence diagram as a two-dimensional summary diagram consisting of persistence pairs or points (𝑏, 𝑑) where 𝑏 is the birth or formation time of a homology and 𝑑 is its death or disappearance. For example, in [45] the Hopf bifurcation is detected through zigzag persistence (i.e., a loop is detected through the one-dimensional zigzag persistence diagram).…”

Section: Introductionmentioning

confidence: 99%

Temporal Network Analysis Using Zigzag Persistence

Myers¹,

Khasawneh²,

Munch³

2022

Preprint

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This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero-and one-dimensional zigzag persistence diagrams can detect changes in the networks' shapes that are missed by traditional connectivity and centrality graph statistics.

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“…Of course, many options are available for such a question; in this work we focus on a data driven approach using topological data analysis (TDA) to measure the shape and structure of the attractor of the system as a proxy for behavior. The idea of combining methods from TDA with dynamical systems and/or time series analysis is not new [8,12,3,19,16,10,5,11]. In this work, we focus on a new way to encode information about the bifurcation, namely the CROCKER plot [17,20,21].…”

Section: Introductionmentioning

confidence: 99%

A Case Study on Identifying Bifurcation and Chaos with CROCKER Plots

Güzel¹,

Munch²,

Khasawneh³

2022

Preprint

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The CROCKER plot is a coarsened but easy to visualize representation of the data in a one-parameter varying family of persistence barcodes. In this paper, we use the CROCKER plot to view changes in the persistence under a varying bifurcation parameter. We perform experiments to support our methods using the Rössler and Lorenz system and show the relationship with common methods for bifurcation analysis such as the Lyapunov exponent.

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Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

Cited by 16 publications

References 41 publications

What are higher-order networks?

What are higher-order networks?

Temporal Network Analysis Using Zigzag Persistence

A Case Study on Identifying Bifurcation and Chaos with CROCKER Plots

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