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This study introduces a novel method to build simplicial complexes from time series data, preserving topological features of dynamical systems. This approach aids in analyzing chaotic systems and offers advantages over complex network mapping.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Computational Topology
  • Data Analysis

Background:

  • Topological data analysis offers powerful tools for understanding complex systems.
  • Previous methods for analyzing time series data have limitations in preserving topological information.

Purpose of the Study:

  • To develop a general method for constructing simplicial complexes from time series data.
  • To analyze the topological properties of dynamical systems, particularly those exhibiting chaotic behavior.
  • To explore the relationship between persistent homology, embedding dimensions, and topological noise.

Main Methods:

  • Delay coordinate reconstruction to create a phase space representation.
  • Construction of a simplicial complex from the reconstructed phase space.
  • Computation of homology groups to characterize topological features.
  • Analysis of persistent homology to understand feature lifetimes and noise.

Main Results:

  • The constructed simplicial complex accurately preserves the topological features of the reconstructed phase space.
  • Homology computations reveal insights into the invariant sets of chaotic dynamical systems.
  • Persistent homology analysis demonstrates consistency across different dynamic regimes and embedding dimensions.
  • The method shows advantages over complex network mapping for time series analysis.

Conclusions:

  • The presented method provides a robust framework for topological analysis of time series data from dynamical systems.
  • This approach enhances the understanding of chaotic system dynamics through topological invariants.
  • The method offers a versatile tool with potential applications in various scientific fields requiring time series analysis.