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Synchronization and locking in oscillators with flexible periods.

Mariya Savinov1, David Swigon2, Bard Ermentrout2

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This study explores how an oscillator synchronizes with external forces, revealing complex multi-stability and intricate attraction basins in entrainment patterns. The rate of change in external forcing significantly influences the final synchronization state.

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Area of Science:

  • Nonlinear Dynamics
  • Complex Systems
  • Mathematical Modeling

Background:

  • Entrainment of nonlinear oscillators by periodic forces is a fundamental problem in nonlinear dynamics.
  • The circle map model illustrates N:M entrainment, where an oscillator completes N cycles for every M stimulus cycles.
  • Experimental evidence suggests entrainment involves phase shifts and intrinsic period adjustments.

Purpose of the Study:

  • To investigate a two-dimensional map model where both phase and period update based on stimulus phase.
  • To analyze the number and stability of fixed points in various N:M locking regions (e.g., 1:1, 1:2, 2:3).
  • To understand how stimulus sensitivity and preferred oscillator period affect entrainment dynamics.

Main Methods:

  • Developed a two-dimensional map model for oscillator phase and period updates.
  • Characterized fixed points and their stability across different N:M locking regions.
  • Investigated the impact of varying stimulus sensitivities and preferred periods.
  • Analyzed the effect of changing forcing periods on the final locking pattern.

Main Results:

  • Observed significant multi-stability of locking modes even within limited explored regimes.
  • Found that basins of attraction for different entrainment patterns are complex and riddled.
  • Demonstrated that the rate of change in forcing period critically influences the final synchronization pattern.

Conclusions:

  • The explored two-dimensional map exhibits rich dynamics, including multi-stability and complex basins of attraction.
  • Oscillator entrainment is sensitive to stimulus properties and the rate of change in external forcing.
  • This model provides insights into the complex synchronization behaviors observed in physical and biological systems.