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Master stability function for networks of coupled piecewise-smooth oscillators.
Volodymyr Denysenko1, Marek Balcerzak2, Artur Dabrowski2
1Division of Dynamics, Lodz University of Technology, 90-924, Lodz, Poland. volodymyr.denysenko@dokt.p.lodz.pl.
Scientific Reports
|July 15, 2026
Summary
This study extends the Master Stability Function (MSF) framework for analyzing synchronization stability in networks of identical piecewise-smooth oscillators. The novel method enables stability analysis for systems with non-differentiable dynamics, overcoming previous limitations.
Area of Science:
- Nonlinear dynamics and control systems engineering.
- Complex systems analysis.
- Theoretical physics and applied mathematics.
Background:
- The Master Stability Function (MSF) is a standard method for analyzing synchronization stability in oscillator networks.
- Existing MSF methods struggle with systems exhibiting piecewise-smooth dynamics due to non-differentiable points.
- This limitation hinders the analysis of many real-world oscillatory systems.
Purpose of the Study:
- To develop a novel extension of the Master Stability Function (MSF) framework.
- To enable the analysis of synchronization stability in networks of identical piecewise-smooth oscillators.
- To address the limitations of current MSF methods for systems with non-differentiable dynamics.
Main Methods:
- Extension of the MSF framework using a Jacobian matrix estimation technique.
- Application of the method to piecewise-smooth systems, including linear and nonlinear impact oscillators.
- Detailed formulation and discussion of the method's advantages and disadvantages.
Main Results:
- Successfully adapted the MSF framework for piecewise-smooth oscillator networks.
- Computed MSF for linear and nonlinear impact oscillator systems.
- Demonstrated the method's effectiveness through validation against an alternative estimation approach.
Conclusions:
- The proposed MSF extension effectively analyzes synchronization stability in piecewise-smooth oscillator networks.
- This advancement expands the applicability of MSF to a broader range of complex systems.
- The method provides a valuable tool for understanding synchronization phenomena in systems with non-smooth dynamics.
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