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Symmetry-protected delay spectroscopy in oscillator networks.
Ehsan Bolhasani1, Seyed Hamed Aboutalebi2,3, Matjaž Perc4,5,6,7
1Department of Physics, University of Isfahan, Isfahan 81746-73441, Iran.
Chaos (Woodbury, N.Y.)
|July 6, 2026
Summary
We developed a method using discrete symmetry to identify specific path delays in oscillator networks from frequency response measurements. This technique allows for precise delay determination, crucial for understanding complex network dynamics.
Area of Science:
- Nonlinear dynamics
- Network science
- Systems engineering
Background:
- Time delays significantly influence collective dynamics in coupled oscillator networks.
- Identifying specific path delays from overall frequency response is challenging due to combined signals from multiple routes.
Purpose of the Study:
- To develop a method for identifying path-specific delays in delay-coupled oscillator networks.
- To resolve the inverse problem of determining individual delay paths from global network measurements.
Main Methods:
- Utilizing discrete symmetry in delay-coupled Kuramoto populations.
- Employing a symmetry-preserving operating point to create a detector-source response zero.
- Inducing symmetry breaking to transform the zero into a ladder of nodal crossings.
Main Results:
- A general theorem for finite retarded delay networks was proven.
- The minimal four-population motif was explicitly solved, showing nodal ladder spacing directly relates to detector-selected delay.
- High accuracy was achieved, with the baseline motif matching predicted delay within 0.08%.
Conclusions:
- Discrete symmetry provides a powerful tool for delay spectroscopy in complex networks.
- The proposed swept-frequency protocol enables precise measurement of path-specific delays.
- This method requires only selected linear detector-source response, simplifying experimental requirements.
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