Revealing Chaos Synchronization Below the Threshold in Coupled Mackey–Glass Systems

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AKHMET M., Başkan K., YEŞİL C.

Mathematics, vol.11, no.14, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 11 Issue: 14
  • Publication Date: 2023
  • Doi Number: 10.3390/math11143197
  • Journal Name: Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Keywords: delta synchronization, Mackey–Glass system, Poincaré chaos, synchronization, unpredictability
  • Middle East Technical University Affiliated: Yes

Abstract

This study presents a novel concept in chaos synchronization, delta synchronization of chaos, which reveals the presence of chaotic models evolving in unison even in the absence of generalized synchronization. Building upon an analysis of unpredictability in Poincaré chaos, we apply this approach to unilaterally coupled time-delay Mackey–Glass models. The main novelty of our investigation lies in unveiling the synchronization phenomenon for a coupling constant below the synchronization threshold, an unattainable domain for conservative methods. Furthermore, we rigorously examine the coexistence of generalized synchronization and complete synchronization of unpredictability, which is a special case of delta synchronization, above the threshold. Therefore, the threshold is no longer a requirement for the synchronization of chaos in view of the present results. Additionally, transitions to fully chaotic regimes are demonstrated via return maps, phase portraits, and a bifurcation diagram. The findings are substantiated by tables and novel numerical characteristics.