Reentrance of Synchronization in Oscillator Networks

• Reentrance of synchronization is a nonmonotonic phenomenon where collective coherence is unexpectedly restored in oscillator networks at high frequency heterogeneity, despite initial loss due to detuning.
• The analytical model uses globally coupled Stuart–Landau oscillators divided into active and damped populations to derive precise phase boundaries via linear stability analysis and Hopf bifurcation conditions.
• Numerical simulations reveal that as detuning increases, synchrony is first lost, then reemerges, and finally collapses, highlighting the complex interplay between active oscillators and inactive (damped) units.

Reentrance of synchronization denotes a nonmonotonic phenomenon in networks of coupled oscillators where synchrony, initially lost due to increased heterogeneity (e.g., in natural frequencies), is restored at larger heterogeneity before ultimately being destroyed at even greater parameter separation. This reentrance defies the conventional expectation that increased frequency detuning monotonically suppresses collective synchrony, and arises under specific conditions—in particular, when a significant fraction of oscillators in the network are rendered inactive (damped), modeling "damaged" elements that cannot sustain autonomous oscillations. The key mechanism was systematically analyzed for globally coupled Stuart–Landau oscillators partitioned into two distinct active populations (limit-cycle oscillators with different natural frequencies) and one population of damped (inactive) oscillators (Inagawa et al., 16 Nov 2025).

1. Mathematical Model of Mixed Active–Damped Oscillator Networks

The canonical model employs a population of $N$ globally coupled Stuart–Landau oscillators described by

$$\frac{d z_j}{dt} = (\alpha_j + i\omega_j)z_j - (\beta_j + i \gamma_j)|z_j|^2 z_j + \frac{K}{N}\sum_{k=1}^N (z_k - z_j)$$

where $z_j \in \mathbb{C}$ is the complex amplitude of oscillator $j$; $\omega_j$ denotes its intrinsic frequency; $\alpha_j$ determines growth ($\alpha_j > 0$ for limit-cycle dynamics, $\alpha_j < 0$ for damped/inactive units); and $K$ is the diffusive coupling strength.

Oscillators are grouped into three distinct subpopulations:

• Type-A$_1$: active limit-cycle oscillators with natural frequency $\omega_{A1}$ (fraction $(1-p)/2$).
• Type-A$_2$: active with frequency $\omega_{A2}$ (fraction $(1-p)/2$).
• Type-I: inactive/damped oscillators with $\alpha_I < 0$, and frequency $\omega_I$ (fraction $p$).

Under assumptions of within-block synchrony, time and amplitude rescaling, and $N\rightarrow\infty$, the dynamical system is reduced to three coupled ODEs for the collective complex amplitudes $A_1,\, A_2,\, I$: $$\begin{aligned} \dot A_1 &= (1 + i\omega_{A1})\,A_1 - (1 + i\gamma_A)|A_1|^2 A_1 \ &\quad + K\left[ p (I-A_1) + \tfrac{1-p}{2}(A_2 - A_1) \right] \ \dot A_2 &= (1 + i\omega_{A2})\,A_2 - (1 + i\gamma_A)|A_2|^2 A_2 \ &\quad + K\left[ p (I-A_2) + \tfrac{1-p}{2}(A_1 - A_2) \right] \ \dot I &= (-\alpha + i\omega_I) I - (\beta + i\gamma_I)|I|^2 I \ &\quad + K\left[ \tfrac{1-p}{2}(A_1 - I) + \tfrac{1-p}{2}(A_2 - I)\right] \end{aligned}$$ with $\alpha = |\alpha_I/\alpha_A|$ and $\beta = \beta_I/\beta_A$; typically $\beta=1$ and $\gamma_A = \gamma_I = 0$.

2. Linear Stability Analysis and Emergence of Hopf Modes

Stability of global synchrony is probed by linearizing at the amplitude death state $(A_1,A_2,I)=(0,0,0)$. The Jacobian assumes the form

$$J = \begin{pmatrix} 1 + i\omega_{A1} - \frac{K(p+1)}{2} & K \frac{1-p}{2} & K p \ K \frac{1-p}{2} & 1 + i\omega_{A2} - \frac{K(p+1)}{2} & K p \ K \frac{1-p}{2} & K \frac{1-p}{2} & -\alpha + i\omega_I - K(1-p) \end{pmatrix}$$

with a characteristic polynomial $\lambda^3 + a\lambda^2 + b\lambda + c=0$.

Two principal Hopf conditions arise:

• $c=0$ signals a real eigenvalue crossing zero—the first loss of stability (onset of asynchrony).
• $c=ab$ signals a complex conjugate pair crossing the imaginary axis—the reentrance of synchrony.

In the symmetric frequency case $\omega_{A1} = \omega_1 + \Delta\Omega$, $\omega_{A2} = \omega_1 - \Delta\Omega$, $\omega_I = \omega_1$ (after setting $\omega_1=0$), analytical expressions for the critical boundaries in $(p,\Delta\Omega)$ parameter space are derived: $$p_{\rm i}(\Delta\Omega) = \frac{(\Delta\Omega^2 + 1 - K)(K + \alpha)}{K[\,\Delta\Omega^2 + (1-K)(1+\alpha)]}$$

$$p_{\rm ii}(\Delta\Omega) = \frac{\{\,\Delta\Omega^2 + 2K^2 + (1-\alpha)(1-3K+\alpha)\}(2-K)}{K[\Delta\Omega^2 - (1+\alpha)(1+K-\alpha)]}$$

These boundaries demarcate, for fixed damping $p$, where synchrony (first) is lost and then—counterintuitively—recovers as $\Delta\Omega$ increases further before eventual amplitude death.

3. Numerical Investigation and Order Parameter Diagnostics

Numerical simulations use large system sizes ($N=1000$) and typical parameter regimes ($p\in[0.6,0.9]$, $K=1.2$, $\alpha=1.0$, $\beta=1$, $\gamma_A=\gamma_I=0$). Synchronization is quantified via a Kuramoto-like order parameter

$R e^{i\Phi} = \frac1N\sum_{j=1}^N z_j$

with $R\approx 1$ denoting synchrony, $R\ll 1$ indicating desynchronization, and $R\to 0$ corresponding to amplitude death.

Empirically, as $|\Delta\omega| = |\omega_{A2} - \omega_{A1}|$ increases:

• For small $|\Delta\omega|$, synchrony persists.
• For moderate $|\Delta\omega|$, synchrony is lost as the system enters an asynchronous regime.
• For larger $|\Delta\omega|$, synchrony reemerges ("reenters") due to the effective decoupling of active and damped populations.
• For yet larger $|\Delta\omega|$ or high $p$, the system collapses to amplitude death.

Figure 1(a) in (Inagawa et al., 16 Nov 2025) visualizes these regions in the $(p, \Delta\omega)$ plane.

Parameter Region	Dynamical Regime	Order Parameter $R$
Small $|\Delta\omega|$	Synchrony	$R \approx 1$
Intermediate $|\Delta\omega|$	Desynchrony	$R \ll 1$
Large $|\Delta\omega|$, moderate $p$	Reentrance of Synchrony	$R \approx 1$
Large $|\Delta\omega|$, large $p$	Amplitude Death	$R \to 0$

Boundaries obtained numerically agree well with those predicted by the Hopf-bifurcation theory.

4. Mechanistic Interpretation of Reentrant Synchrony

The reentrance of synchronization is a direct consequence of the collective interplay between active and inactive subpopulations. In a homogeneous active network, increasing frequency detuning ($|\Delta\omega|$) always impairs synchrony. With a significant damped population ("inactive reservoir"), moderate detuning enables the two active subgroups to entrain the inactives, destroying global coherence via their competing pulls. However, as detuning grows further, each active subgroup interacts increasingly only with itself, and the influence of the damped population weakens; eventually, the active subpopulations can directly synchronize with each other through the global coupling, restoring synchrony.

Mathematically, this corresponds to the real part of the relevant Hopf eigenvalue $\mathrm{Re}\,\lambda$ becoming negative again for large $|\Delta\omega|$, as precisely captured by the phase boundary curve $p_{\rm ii}(\Delta\Omega)$.

5. Generality and Implications for Dynamical Networks

Because the Stuart–Landau oscillator represents the universal (normal) form of a supercritical Hopf bifurcation, the reentrance mechanism is expected in systems where: (i) a fraction of units can become excitable or damped; (ii) the network topology is not necessarily globally coupled; and (iii) the underlying oscillatory unit dynamics is higher-dimensional or more complex. Thus, reentrant synchronization should manifest broadly in coupled oscillator systems with inactive reservoirs, not just in the specific model.

A plausible implication is that in engineered and natural oscillator arrays—such as lasers, electrochemical systems, or neural networks—introducing or considering inactive elements fundamentally alters the synchrony–detuning relation, with potential impacts on the design of robust synchronized states or on the emergent failure modes of such systems.

6. Analytical Phase Boundaries and Persistence of the Phenomenon

Closed-form analytic boundaries $p_{\rm i}(\Delta\Omega)$ and $p_{\rm ii}(\Delta\Omega)$ provide explicit criteria for the onset and restoration (reentrance) of synchrony. The fixed-point linearization suffices to predict both the initial loss of synchrony (first Hopf threshold) and its subsequent recovery (second Hopf), preceding amplitude death at extreme detuning or damping fractions.

The agreement between analytical theory and numerical simulations validates the approach, indicating that the phenomenon is quantitatively robust. In summary, the addition of damped oscillators transforms the classic synchronization–detuning paradigm into a rich, nonmonotonic structure with practical and theoretical ramifications for oscillator network dynamics (Inagawa et al., 16 Nov 2025).