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Persistent homology approach for uncovering transitions to Chaos

Author

Listed:
  • Shah, W. Hussain
  • Jaimes-Reátegui, R.
  • Huerta-Cuellar, G.
  • García-López, J.H.
  • Pisarchik, A.N.

Abstract

Traditional methods for distinguishing between periodic and chaotic time series are cumbersome and unclear. In this study, we examined the time series of the Rössler system for various values of the natural frequency, which served as a control parameter. First, we analyzed the topological structure by constructing Betti vectors for each persistence diagram and visualized them using a CROCKER plot. This innovative topological technique effectively captures changes in the time series as the control parameter is varied. Next, we investigated the transition to chaos by identifying global features using Betti curves. Specifically, we derived a physical law that related the control parameter value at which transitions to periodicity occur to the mean L1-norm of the Betti curves. Additionally, we calculated the Lyapunov exponent and compared it with the L1-norm of Betti vectors to explore their relationship. We also computed the persistence landscape to characterize loopy structures within the phase space. Our findings provide a comprehensive framework for understanding the transition to chaos in dynamical systems through topological data analysis.

Suggested Citation

  • Shah, W. Hussain & Jaimes-Reátegui, R. & Huerta-Cuellar, G. & García-López, J.H. & Pisarchik, A.N., 2025. "Persistent homology approach for uncovering transitions to Chaos," Chaos, Solitons & Fractals, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077925000670
    DOI: 10.1016/j.chaos.2025.116054
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