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Multi-type synchronization for coupled van der Pol oscillator systems with multiple coupling modes.

Shuai Wang1, Xue Yang2

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This study explores synchronous solutions in coupled van der Pol oscillator systems with multiple coupling modes. Symmetry in coupling topologies dictates synchronization types, offering insights into complex oscillator networks.

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Area of Science:

  • Nonlinear Dynamics
  • Complex Systems
  • Network Science

Background:

  • Coupled van der Pol oscillators are fundamental models for studying synchronization phenomena.
  • Understanding synchronization in systems with multiple coupling modes (position, velocity, acceleration) is crucial for complex network analysis.
  • Existing theories often focus on single coupling modes, limiting applicability to multi-modal systems.

Purpose of the Study:

  • To investigate synchronous solutions in coupled van der Pol oscillator systems with multiple coupling modes.
  • To analyze how symmetry in coupling topologies influences synchronization patterns.
  • To extend the understanding of synchronization to multi-layer network contexts.

Main Methods:

  • Utilizing the theory of rotating periodic solutions to analyze synchronous states.
  • Examining systems with two or three coupling modes: position, velocity, and acceleration.
  • Applying symmetry analysis to coupling topology matrices and quotient graphs.

Main Results:

  • Synchronous solutions are determined by the intersection of symmetries in symmetric coupling topologies.
  • For asymmetric topologies, synchronization is dictated by the symmetry of quotient graphs derived from equitable partitions.
  • The findings align with synchronization behaviors observed in one-to-one connected multi-layer networks.

Conclusions:

  • The symmetry of coupling modes is a key determinant of synchronization in van der Pol oscillator networks.
  • The developed framework provides a method to predict synchronization types based on network topology.
  • This research contributes to the understanding of synchronization in complex, multi-modal, and multi-layer oscillator systems.