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We studied how partial order parameter adaptation affects coupled Kuramoto oscillators. Different adaptation functions, including power law and polynomial, lead to diverse synchronization transitions like double-jump and intermediate states.

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Area of Science:

  • Complex systems
  • Nonlinear dynamics
  • Network science

Background:

  • The Kuramoto model is a standard framework for studying synchronization in coupled oscillators.
  • Understanding synchronization transitions in complex networks is crucial for various scientific fields.
  • Hypergraph models offer a more general representation of interactions beyond pairwise connections.

Purpose of the Study:

  • To investigate the impact of partial order parameter adaptation using general functions on oscillator synchronization.
  • To explore synchronization behaviors in coupled Kuramoto oscillators on random hypergraphs with pairwise and triangular interactions.
  • To analyze the emergence of diverse synchronization patterns under different adaptation strategies.

Main Methods:

  • Utilizing the Ott-Antonsen ansatz to derive self-consistent equations for the order parameter.
  • Modeling interactions between oscillators as pairwise and triangular within random hypergraph structures.
  • Analyzing the effects of various generalized adaptation functions (e.g., power law, polynomial, Gaussian) on synchronization.

Main Results:

  • Observed a wide range of synchronization transitions due to the interplay of adaptation, function type, and coupling strength.
  • Identified a double-jump transition specifically under a power law adaptation function.
  • Discovered the emergence of intermediate synchronization states with polynomial adaptation and specific coefficient combinations.
  • Demonstrated that synchronization can be continuous or explosive based on pairwise coupling strength variations, also observed with Gaussian adaptation.

Conclusions:

  • Partial order parameter adaptation in generalized forms significantly diversifies synchronization behaviors in coupled Kuramoto oscillators on hypergraphs.
  • The choice of adaptation function critically influences the type and characteristics of observed synchronization transitions.
  • The study highlights the rich dynamics and potential for complex synchronization phenomena in systems with higher-order interactions.