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<p>The combination–combination synchronisation (CCS) of fractional-order ecological systems is studied in this paper using two distinct approaches. By designing an appropriate controller, we attain CCS among chaotic systems. In addition, calculating the bounds of a chaotic system is a significant task in chaos theory. It reveals fundamental features of the system. We also estimated the ultimate bound, Mittag–Leffler positive invariant set and globally attracting set of the ecological system using fractional-order and complex system. The chaotic system’s bounds are demonstrated using simulation. Finally, the numerical results confirm the efficacy of the suggested scheme.</p>

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Combination–combination synchronisation and ultimate bound of the fractional-order ecological system

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Vijay K Shukla,
  • Mahesh C Joshi,
  • Juan E Nápoles Valdes,
  • Prashant K Mishra,
  • Changjin Xu

摘要

The combination–combination synchronisation (CCS) of fractional-order ecological systems is studied in this paper using two distinct approaches. By designing an appropriate controller, we attain CCS among chaotic systems. In addition, calculating the bounds of a chaotic system is a significant task in chaos theory. It reveals fundamental features of the system. We also estimated the ultimate bound, Mittag–Leffler positive invariant set and globally attracting set of the ecological system using fractional-order and complex system. The chaotic system’s bounds are demonstrated using simulation. Finally, the numerical results confirm the efficacy of the suggested scheme.