Complex-Frequency Synchronization

• Complex-frequency synchronization is a multi-domain approach that synchronizes amplitude, phase, and spectral properties beyond simple scalar phase differences.
• It employs techniques such as PLL-based overlays, distributed LO locking, and multiharmonic estimation to achieve robust coordination across systems.
• Practical implementations in wireless, power grids, and laser networks demonstrate low BER, sub-dB losses, and improved stability under varying operational conditions.

Complex-Frequency Synchronization is a cross-domain term rather than a single standardized construct. In the surveyed literature, it can denote a secondary signaling mechanism in which a primary phase-locked loop synchronizes to a composite phase while revealing an embedded frequency modulation, the joint alignment of the phasors $e^{j(\omega t+\phi)}$ across distributed radio nodes, the use of the complex quantity $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$ to couple voltage-amplitude and phase dynamics in power systems, and multiharmonic or spectral formulations of synchronization in heterogeneous oscillator networks (Montierth et al., 18 Jun 2026, Mghabghab et al., 2020, Milano, 2021, Gao et al., 2019). Across these usages, the common theme is that synchronization is characterized through a quantity richer than a scalar phase difference: carrier-frequency offsets, complex phasors, voltage-amplitude rates, higher harmonics, spectral gaps, or multimodal frequency statistics.

1. Terminological scope and core meanings

The term appears in several technical lineages, and the intended meaning depends on the underlying state variable and application domain. In wireless signaling, “Complex-Frequency Synchronization” refers to the primary PLL continuously synchronizing to the composite phase $\theta(t)$ while its loop-filter output exposes the embedded frequency modulation as a secondary signal. In distributed RF systems, it refers to alignment of both the carrier frequency $\omega$ and the instantaneous phase $\phi$ of node phasors $e^{j(\omega t+\phi)}$. In power systems, it refers to synchronizing the complex frequency $\lambda=\varepsilon+j\omega$, where $\varepsilon$ is the normalized rate of change of voltage magnitude and $\omega$ is the angular frequency. In oscillator-network theory, the phrase is used for synchronization phenomena governed by multiple frequency channels, complex spectral operators, higher harmonics, or coexisting modal frequencies (Montierth et al., 18 Jun 2026, Mghabghab et al., 2020, Wei et al., 15 Aug 2025, Gao et al., 2019).

Domain	Meaning of synchronization	Representative paper
Wireless overlay signaling	PLL tracks composite phase while loop-filter output reveals embedded FM	(Montierth et al., 18 Jun 2026)
Distributed RF arrays	Joint alignment of $\omega$ and $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$0 across transceivers	(Mghabghab et al., 2020)
Power systems	Convergence of $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$1 across buses	(Wei et al., 15 Aug 2025)
Statistical synchronization	Cross-frequency consistency over multiple harmonic channels	(Gao et al., 2019)
Repulsive laser networks	Onset predicted by a complex spectral gap of $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$2	(Honari-Latifpour et al., 2023)

A recurring distinction is between phase synchronization and frequency synchronization. In grid-oriented Kuramoto formulations, phase locking is characterized by $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$3, whereas frequency synchronization is characterized by convergence of all instantaneous frequencies to a common asymptotic frequency and by $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$4, where $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$5 is the time-averaged variance of instantaneous frequencies around the instantaneous network average. The literature explicitly states that phase locking is not required for grid operation; grid stability requires frequency synchronization (Kim et al., 16 Jan 2026).

2. Wireless and RF realizations

In “Frequency Lock Encoding,” the secondary layer is embedded by introducing small, controlled variations in the carrier frequency of an existing single-carrier transmission. The transmitted lowpass-equivalent signal is

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$6

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$7

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$8

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$9

The modulation index $\theta(t)$0 is chosen “small enough to allow the PLL at the detector to track out the phase variations.” At the primary receiver, a decision-directed second-order PLL absorbs the imposed offsets as if they were ordinary CFO or phase noise. When in lock, the loop-filter output satisfies

$\theta(t)$1

and this approximation holds “only when the bandwidth of the secondary pulse train is less than the closed-loop bandwidth of the PLL” (Montierth et al., 18 Jun 2026).

The secondary receiver uses the primary loop-filter output as its observable, followed by DC blocking when needed, matched filtering to $\theta(t)$2, secondary timing recovery, downsampling by $\theta(t)$3, and slicing to the secondary alphabet. For the implementation with $\theta(t)$4 cycles/symbol, the measured transparency envelope is

$\theta(t)$5

Within this region, the paper reports primary BER near baseline over a wide operating region, with only $\theta(t)$6 dB implementation loss at more extreme settings such as $\theta(t)$7, $\theta(t)$8. Over the air, using QPSK primary, binary PAM overlay, $\theta(t)$9, and $\omega$0 in the $\omega$1 MHz ISM band, both layers achieved $\omega$2, and secondary data rates exceeded $\omega$3 kbps (Montierth et al., 18 Jun 2026).

Related RF synchronization systems use explicit frequency transfer and ranging rather than an overlay. In a $\omega$4 m microwave wireless link, a self-mixing circuit extracts a $\omega$5 MHz reference from two tones at $\omega$6 and $\omega$7 MHz,

$\omega$8

and this reference locks the secondary node’s LO. Phase alignment is then obtained from range through

$\omega$9

An adaptive PI loop adjusts the two-tone separation to maintain $\phi$0 mm. The system demonstrated continuous high-accuracy links over a $\phi$1 m distance for durations up to seven days, with coherent operation at $\phi$2, $\phi$3, and $\phi$4 GHz attainable for probabilities $\phi$5, $\phi$6, and $\phi$7, respectively, of achieving at least $\phi$8 coherent gain (Mghabghab et al., 2020).

A distinct remote-carrier formulation for distributed sensing uses full-duplex dual-carrier synchronization around an application central frequency $\phi$9, with outbound tones at $e^{j(\omega t+\phi)}$0 and return tones at $e^{j(\omega t+\phi)}$1. Averaging the two received phases suppresses symmetric channel phase and Doppler skew, and the follower beamforming phase satisfies

$e^{j(\omega t+\phi)}$2

with the closed-loop objective $e^{j(\omega t+\phi)}$3 inside the loop bandwidth. The paper reports residual phase error of a few degrees at $e^{j(\omega t+\phi)}$4–$e^{j(\omega t+\phi)}$5 dB and loop bandwidth $e^{j(\omega t+\phi)}$6–$e^{j(\omega t+\phi)}$7 Hz, but also notes that delay margin constrains $e^{j(\omega t+\phi)}$8 as the inter-node separation increases (Duncan et al., 2020).

3. Power-system formulations

In power-system studies, the defining object is the bus-voltage phasor

$e^{j(\omega t+\phi)}$9

and the complex frequency is

$\lambda=\varepsilon+j\omega$0

The real part $\lambda=\varepsilon+j\omega$1 is the normalized rate of change of voltage magnitude, and the imaginary part $\lambda=\varepsilon+j\omega$2 is the instantaneous angular frequency. In this formulation, complex-frequency synchronization means that all buses converge to a common $\lambda=\varepsilon+j\omega$3, so that both voltage-amplitude rates and phase-angle rates are synchronized (Milano, 2021, Wei et al., 15 Aug 2025).

The paper “Complex-Frequency Synchronization of Converter-Based Power Systems” uses this viewpoint to analyze dispatchable virtual oscillator-controlled converters. In complex-angle coordinates, the measurement-filtered dVOC law is

$\lambda=\varepsilon+j\omega$4

The dVOC core is equivalent to a complex-power–frequency droop law,

$\lambda=\varepsilon+j\omega$5

and the analysis separates the dynamics into a fast linear subsystem that synchronizes complex frequency and a slower, approximately linear subsystem that stabilizes voltage amplitude. A sufficient parametric condition is

$\lambda=\varepsilon+j\omega$6

which implies the spectral condition for fast synchronization. The slow subsystem is globally asymptotically stable under a quadratic Lyapunov function (He et al., 2022).

The later system-level formulation extends the same idea to region-level assessment. It defines local convergence times $\lambda=\varepsilon+j\omega$7 and $\lambda=\varepsilon+j\omega$8, synchronization errors $\lambda=\varepsilon+j\omega$9, disturbance impact sets $\varepsilon$0 and $\varepsilon$1, and a generalized inertia

$\varepsilon$2

Here $\varepsilon$3 unifies voltage-support capability with the conventional inertia term $\varepsilon$4. On the IEEE $\varepsilon$5-bus system, the paper reports that $\varepsilon$6 synchronizes fastest, $\varepsilon$7 converges more slowly and constrains the global synchronization speed, and increasing $\varepsilon$8 markedly reduces the magnitude of $\varepsilon$9 (Wei et al., 15 Aug 2025).

Grid-oriented oscillator models also stress that frequency synchronization and phase synchronization are not identical. For the second-order Kuramoto swing equation

$\omega$0

frequency synchronization is measured by

$\omega$1

With inertia and damping scaling as

$\omega$2

decentralization can induce a non-monotonic dependence of the critical coupling and a double phase transition. For $\omega$3, $\omega$4, and $\omega$5, the paper finds an abrupt transition near $\omega$6 followed by a continuous transition near $\omega$7, and notes that increasing damping to $\omega$8 suppresses the multi-stage transition behavior (Kim et al., 16 Jan 2026).

4. Multiharmonic, structured, and optimization-based network theories

One line of work formulates synchronization directly as a multi-frequency estimation problem. In “Multi-Frequency Phase Synchronization,” the unknown phases $\omega$9 are estimated by enforcing cross-frequency consistency across harmonic channels: $\omega$0 The two-stage algorithm uses PPE-SPC for initialization and MFGPM for refinement. PPE-SPC computes leading eigenvectors $\omega$1, forms $\omega$2, estimates pairwise angles through a periodogram, and performs a final spectral synchronization. Under a sub-Gaussian/Wigner model, the paper proves

$\omega$3

and extends the framework to synchronization over compact Lie groups such as $\omega$4 (Gao et al., 2019).

A related network-theoretic literature studies how structural heterogeneity and phase frustration shape frequency synchronization thresholds. For the Sakaguchi–Kuramoto model

$\omega$5

with partial degree–frequency correlation, the onset equations couple the critical coupling $\omega$6 and the group angular velocity $\omega$7. On scale-free networks, explosive synchronization appears over broad $\omega$8 intervals for $\omega$9, whereas on Erdős–Rényi networks it is observed only in narrow $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$00 windows when $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$01 (Kundu et al., 2018).

Higher-order interactions generate another formal meaning of the term. For Sakaguchi–Kuramoto oscillators on simplicial complexes,

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$02

the synchrony alignment function is

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$03

Setting $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$04 yields the analytically derived frequency assignment

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$05

which guarantees $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$06 at the targeted parameter point. The paper verifies this on Erdős–Rényi, scale-free, and small-world networks and finds both first-order and second-order transitions depending on topology and phase frustration (Dutta et al., 2023).

Weighted complex networks provide yet another route. In repulsive or mismatch-weighted settings,

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$07

and explosive synchronization can be induced or enhanced by combining local frequency mismatch with edge betweenness. In the fully connected analytical limit, the hysteretic interval is approximately $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$08 for the normalized coupling threshold, while in heterogeneous networks moderate $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$09 in

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$10

restores explosive synchronization that pure mismatch weighting may not produce (Leyva et al., 2013).

5. Detuning, spectral alignment, and multimode oscillator platforms

Several experimental platforms use the phrase for synchronization that is mediated by detuning, spectral alignment, or multimodal frequency structure rather than simple phase locking. In a silicon-nitride optomechanical oscillator, synchronization was demonstrated up to the fourth harmonic of the $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$11 MHz fundamental frequency, together with purely optomechanical frequency division up to $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$12, from $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$13 MHz to $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$14 MHz. The effective slow-phase dynamics follows

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$15

and the measured Arnold tongues show that $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$16 and $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$17 tongues are wider than $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$18, while $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$19 is conspicuously narrow. At $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$20 dBm injection, the divide-by-$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$21 output shows phase noise better than $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$22 dBc/Hz (Rodrigues et al., 2021).

In multimode chaotic BA-VCSELs under weak unidirectional injection, synchronization appears when a dominant transverse mode of the master becomes spectrally aligned with a slave mode, without requiring matching between the spatial field profiles. The paper identifies fast-chaotic synchronization up to $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$23 GHz and low-frequency polarization hopping synchronization at $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$24 MHz. Correlations up to $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$25 are obtained at $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$26 MHz cutoff for suitable detunings, and both normal and inverse synchronization appear depending on whether the injected master mode aligns with a slave mode of the same or orthogonal polarization (Mercadier et al., 18 Jun 2025).

In repulsive laser networks, the onset of synchronization is predicted from the complex spectral matrix

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$27

where $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$28 is the signless Laplacian. If $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$29 are the two eigenvalues of $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$30 with the smallest real parts, the approximate threshold is

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$31

For identical oscillators this reduces to $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$32. The paper shows that local rings and all-to-all networks prevent frequency synchronization, whereas full bipartite networks have optimal synchronization properties (Honari-Latifpour et al., 2023).

Detuning can also promote synchronization in modular phase-oscillator networks that would otherwise sustain weak chimeras. For the interaction function

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$33

with $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$34, the equal-cluster entrainment condition is

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$35

and the weak-chimera locking window is approximately

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$36

Electrochemical-oscillator experiments implemented detuning by changing individual resistances; synchronization occurred robustly for $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$37 in the range $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$38–$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$39 ohm (Ocampo-Espindola et al., 7 May 2025).

A distinct neuronal usage emphasizes multimodal frequency statistics. In an all-to-all network of $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$40 Izhikevich neurons with $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$41 and $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$42, the synchronizing cluster develops matched mean frequencies while retaining bimodal most probable frequencies. When intrinsic frequencies are discretized, residence-time distributions for partially synchronized and unsynchronized states exhibit peaks at beat periods

$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$43

Using $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$44, $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$45, and $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$46, the paper obtains $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$47 s (Marghoti et al., 20 Nov 2025).

6. Limitations, distinctions, and recurrent misconceptions

A first misconception is that the term denotes a single universal mechanism. The surveyed papers show instead that it covers protocol-agnostic carrier overlays, distributed LO locking, power-system phase–amplitude dynamics, multimode spectral synchronization, and multiharmonic estimation. A plausible implication is that cross-paper comparisons require care: the shared phrase does not imply a shared state space, observable, or stability criterion (Montierth et al., 18 Jun 2026, Wei et al., 15 Aug 2025, Honari-Latifpour et al., 2023).

A second misconception is that phase locking and frequency synchronization are interchangeable. Grid papers distinguish them explicitly: $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$48 characterizes phase coherence, while $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$49 characterizes frequency synchronization, and the latter is the operational requirement for grid stability. Likewise, detuning-induced synchronization in modular oscillator networks yields a common average frequency $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$50 while allowing distinct cluster phases, and the power-system complex-frequency framework requires synchronization of both $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$51 and $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$52, not $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$53 alone (Kim et al., 16 Jan 2026, Ocampo-Espindola et al., 7 May 2025, Milano, 2021).

A third misconception is that structural or architectural changes automatically improve synchronization. The decentralization study shows that increasing decentralization can induce a non-monotonic dependence of the critical coupling strength and a double phase transition. The repulsive-laser study shows that complete graphs and local rings can prevent synchronization, while complete bipartite graphs are optimal. In wireless overlays, the method is protocol-agnostic only within the stated scope: the analysis and validation focus on single-carrier waveforms, and detailed treatment of OFDM or other multi-carrier schemes is beyond the scope of that paper (Kim et al., 16 Jan 2026, Honari-Latifpour et al., 2023, Montierth et al., 18 Jun 2026).

Across domains, the limiting factors are also specific. In Frequency Lock Encoding, exceeding PLL tracking capacity or choosing too small $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$54 causes failure, and many COTS radios do not expose loop-filter outputs. In the microwave distributed-array system, severe multipath, blockage, and insufficient SNR degrade $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$55, while large bandwidth reductions limit coherent operation. In the remote carrier synchronization loop, large separations constrain the allowable natural frequency because delay margin decreases as $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$56 increases. In optomechanics, many fractional tongues require $$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$57, beyond the semi-analytical small-$$\lambda(t)=\frac{1}{V(t)}\frac{dV(t)}{dt}+j\frac{d\theta(t)}{dt}$$58 regime. In the spectral network principle, the criterion is approximate and admits exceptions (Montierth et al., 18 Jun 2026, Mghabghab et al., 2020, Duncan et al., 2020, Rodrigues et al., 2021, Honari-Latifpour et al., 2023).

These limitations suggest that “complex-frequency synchronization” is best understood as a family of synchronization concepts for systems in which amplitude, phase, harmonic content, detuning structure, or spectral mode competition cannot be collapsed to a single scalar phase-difference criterion. In that narrower and technically precise sense, the term marks a shift from classical one-variable synchronization toward multivariable, multichannel, or multimode synchronization theory.