Synchronization Conditions for Nonlinear Oscillator Networks

TL;DR

This work addresses the problem of identifying synchronization conditions in nonlinear oscillator networks by combining Lyapunov-Floquet theory with the Master Stability Function (MSF). It proves that for linear full-state diffusive coupling of identical oscillators, synchronization occurs if and only if the coupling gain satisfies $K>0$, and for partial-state coupling, positive $K$ guarantees asymptotic contraction of the state-space volume. The results are supported by MSF calculations on benchmark oscillators, numerical simulations for Van der Pol and repressilator networks, LTspice simulations, and experimental electronic implementations. Together, these findings provide a tight, non-conservative criterion for synchronization and offer practical guidance for designing synchronized oscillator networks. The methodology and validation across theory, simulation, and experiment underscore the robustness of the $K>0$ condition as a fundamental synchronization trigger in diffusive nonlinear networks.

Abstract

Understanding conditions for the synchronization of a network of interconnected oscillators is a challenging problem. Typically, only sufficient conditions are reported for the synchronization problem. Here, we adopted the Lyapunov-Floquet theory and the Master Stability Function approach in order to derive the synchronization conditions for a set of coupled nonlinear oscillators. We found that the positivity of the coupling constant is a necessary and sufficient condition for synchronizing linearly full-state coupled identical oscillators. Moreover, in the case of partial state coupling, the asymptotic convergence of volume in state space is ensured by a positive coupling constant. The numerical calculation of the Master Stability Function for a benchmark two-dimensional oscillator validates the synchronization corresponding to the positive coupling. The results are illustrated using numerical simulations and experimentation on benchmark oscillators.

Figures (7)

• Figure 1: Numerical computation of the Master Stability Function for a coupled Van der Pol oscillator shows that the maximum Floquet multiplier decreases as the coupling strength ($K$) increases. Calculating the Floquet multiplier numerically required calculating the period and limit cycle. Solve the matrix differential equation over one period given the Identity matrix as the initial condition
• Figure 2: Numerical simulation of three coupled Van der Pol oscillators for $\mu=1$, $\text{initial condition:}(x_{10}=[0,1],x_{20}=[2,3],x_{30}=[4,5])$ and $K=1$. (a) Synchronization with full-state coupling. (b) Synchronization with partial-state coupling
• Figure 3: Numerical simulation of three coupled repressilator for different parameters value $\alpha=1000$, $\alpha_{0}=1$, $\beta=5$, $n=2$, $\text{initial condition:}(p_{10}=[0,1,0,3,0,5],p_{20}=[0,7,0,9,0,11],p_{30}=[0,13,15,17,4,6])$ and $K=1$. (a) Synchronization with full-state coupling (b) Synchronization with partial-state coupling.
• Figure 4: Circuit diagram of a Van der Pol oscillator. (a) LT Spice design, (b) experimental implementation.
• Figure 5: Synchronization of three Van der Pol oscillators coupled with different physical parameters is given in Table 1. (a) LT Spice simulation and (b) experiment.

Theorems & Definitions (5)

• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof