Principle Emerges Function: Structure-Independent Synchronization Admissibility Boundary

Introduction

What makes the synchronization critical state so special^5,12,13? To answer this question, we conducted an in-depth analysis of the simulation data from the standard power grid test system. We recorded the time series data of oscillator amplitude and angular frequency and classified the outcome as stable or unstable on the basis of time stability (see S1 in the supplemental material). In these cases, a striking regularity emerged: When the system is in a critical state, the amplitude and angular frequency of the oscillator pair exhibit a specific geometric relationship. This relationship can be expressed by an elegant equation (see S1 in the supplemental material for details). This equation is used to describe the synchronization admissibility boundary:

(1)

denotes the oscillation amplitude of the Kth oscillator, and denotes the angle difference between and . denotes the angle of angular frequency of the Kth oscillator (not the phase angle), and synchronization corresponds to . denotes the angular frequency of Kth oscillator and is the common angular frequency of the system.

Any test can only indicate whether the oscillators are in sync, but cannot directly identify the boundary, since the boundary is not a single state but a set of critical points. Therefore, we locate it by simulating the stability of the synchronous cluster: by gradually increasing the disturbance intensity, we observe how the system transitions from a stable state to a split state in the form of a synchronous cluster. Here, “admissibility” refers to whether it is physically permissible for two oscillators to belong to the same synchronization cluster: when a point lies inside the boundary, the two oscillators K and L belong to the same cluster; outside the boundary, they do not. Global synchronization corresponds to a special type of synchronization cluster in which all oscillators belong to the same cluster, while the moment of cluster splitting and its division are marked by points crossing the boundary.

Eq. (1) presents a boundary for a pair of oscillators. Directly applying Eq. (1) to multibody problems introduces a fundamental challenge: evaluating pairs of oscillators simultaneously is necessaty. This characteristic leads to the curse of dimensionality and renders visual verification of the synchronization admissibility boundary impossible.

As shown in Figure 1a, Eq. (1) can be visualized as a surface in a three-dimensional (3D) space, dividing that space into two regions. When the system corresponds to the inner region, synchronization is admissible. Otherwise, it is inadmissible. Therefore, the surface provides a definitive geometric criterion for oscillator system admissibility. On the other hand, because the variables , , and serve as the axes of the coordinate system, the surface defined by Eq. (1) is geometrically fixed. In other words, changes to the structure do not alter this surface. This finding intuitively demonstrates that the boundary is independent of the structure. The central thesis of this paper thus reduces to a testable question: does this fixed surface effectively determine admissibility across different structures?

To test the boundary, a test system (10-oscillator) was used. A strong disturbance occurred at node 18. The data in Figure 1 were obtained from numerical simulation experiments designed to validate Eq. (1). represents the disturbance duration, and denotes the step size of the variation in .

(a). Visualization of the admissibility boundary. The admissibility boundary in Eq. (1) (blue surface) and the pink planes , and collectively define the boundaries and enclose the admissibility domain. The identical boundary form enables a unified admissibility assessment.

(b). Determination of synchronization admissibility and partial synchronization. represents the three-dimensional coordinate point formed by the Kth and Lth meta-oscillators. are calculated via Eqs. (S6) and (S7). At the disturbance duration , all coordinate points clustered near the boundary (cyan). At , three coordinate points [, , and ] deviated outside the boundary, whereas the others remained clustered near it (magenta). The position of the point relative to the boundary directly determines the synchronization state.

(c). Real-time determination of synchronization admissibility. Observe the trajectory of in (b) separately. The temporal trajectory of coordinate point is shown for the unstable case (). The calculation range was . Each cyan point represents the mean position of for a period of 1 second. The time T and the time interval are defined in Eq. (S8). The trajectory crossed the admissibility boundary outwardly [(7 s, 8 s)], preceding a rapid increase in the angle difference [(9 s,10 s)]. The outward crossing of a trajectory across the boundary indicates the onset of synchronization loss.

(d) and (e). Experimental validation of the admissibility ( and ). The horizontal axis represents the time. The vertical axis represents the value of . The maximum value is represented by (magenta), and the minimum value is represented by (cyan). In the stable case (), , indicating sustained synchronization. In the unstable case (), increased sharply after 9 s (), confirming system desynchronization.

(f). Experimental verification of partial synchronization. . Meta-oscillator 1 (cyan line) desynchronized after ~9 s, followed by meta-oscillators 2 (orange dashed line) and 10 (magenta line). Meta-oscillators 3–9 form a synchronized cluster (green lines). This pattern matches the cluster prediction from the spatial distribution in (b).

First, we tested the ability of boundaries to distinguish between admissibility and inadmissibility. We increased the disturbance duration in small increments and observed the system's response. As shown in Figure 1b, a minute increase in disturbance duration from to , a mere 0.001 s, caused a dramatic shift in system behavior. Under admissible conditions (, cyan), all points clustered at the boundary, whereas at the inadmissible threshold (, magenta), specific points [,, and ] deviated outside it. The significant difference indicates that the boundary distinguishes between admissibility and inadmissibility. The correspondence between a point's position relative to the boundary and the synchronization was validated by time-domain simulations: The clustering of cyan points coincides with a bounded difference in in Figure 1d, whereas the deviation of magenta points from the boundary indicates the large, growing desynchronization evident in Figure 1e. This consistency reveals the capability of the boundary to discriminate synchronization admissibility. To more thoroughly validate the discriminative capability of the boundary equation, we conducted systematic tests across all 39 nodes of the 10-oscillator system, generating a total of 78 distinct scenarios (39 admissible and 39 inadmissible cases, as detailed in Figure S2 and Table S1 in the supplemental material). In each scenario, the geometric criterion provided by Eq. (1) correctly matched the stability state (see S1 in the supplementary material) determined by time-domain simulations.

Moreover, partial synchronization can also be diagnosed geometrically using the same criterion. As shown in Figure 1b, the positions of the 9 coordinate points reveal distinct synchronization clusters. Specifically, points ,, and residing outside the boundary indicate that the corresponding meta-oscillators (1st, 2nd, and 10th, respectively) lost synchronization admissibility. This result effectively divided the system into four synchronization groups: three desynchronized individual units (1st, 2nd and 10th) and one synchronized cluster comprising meta-oscillators 3–9. The time-domain results in Figure 1f are consistent with this diagnosis, confirming the manifestation of cluster synchronization¹⁶. Our approach thus offers a geometric interpretation for this phenomenon: Cluster synchronization arises from the localized loss of synchronization admissibility between oscillators when their representative coordinate points lie outside the boundary. This result, which simplifies the interpretation of partial synchronization^15,17, was further corroborated by additional data (Table S1 in the supplemental material).

Typically, we need to not only know whether a system is ultimately admissible, but also obtain this knowledge in real time. The timing of cluster splitting is well-suited for such real-time verification, as it involves multiple degrees of freedom, nonlinear transients, and multiple time scales^18–21. The real-time transition of the system from admissibility to inadmissibility is shown in Figure 1c, where the coordinate point crossed the boundary outwardly during the time interval (8 s,9 s). This event is the direct precursor to the subsequent physical response: A dramatic increase in the angle difference during (9 s,10 s). This finding indicates that the desynchronization is caused by crossing the boundary. The results of the time series in Figure 1e confirm that (9 s,10 s) marked the beginning of the loss of admissibility. This consistency indicates that the boundary is valid. This finding also demonstrates the capability of the boundary to distinguish complex admissibility phenomena in multiple oscillators¹¹.

(a). Synchronization stabilization discrimination and partial synchronization. Two coordinate points clustered at the boundary when stable (, cyan dots). was outside the boundary and away from when unstable (, magenta dots).

(b). The real-time transition in admissibility for multiple oscillators. . Mirroring the dynamics in Figure 1c, crossed the boundary outwardly in the time interval (2 s, 3 s), and rapidly increased in the time interval (4 s, 5 s). These findings are in good agreement with the results presented in Figure 2d.

(c) and (d) show the experimental validation of the synchronization admissibility and the real-time transition in admissibility for multiple oscillators. The time-series data corroborate the state predictions in (a), showing maintained synchrony at () and loss of synchrony at ().

(e). The phenomenon of partial synchronization. . After approximately 4 s, the system split into two synchronized clusters. Meta-oscillator 1 disengaged from the cluster (black line). Meta-oscillators 2 and 3 formed a synchronized cluster (red line and blue line).

The validity of Eq. (1) across fundamentally different systems indicates a deeper insight: it may stem from a more fundamental physical origin than structure. This origin is ultimately rooted in a cornerstone of physics: For measurable physical quantities, their measured values must be real numbers. This cornerstone is a foundational premise of physics, prior to and independent of any assumptions about the type of oscillator or the structure of the interactions. Therefore, the angular frequency of any oscillator is a real number. The common angular frequency emerges from the frequencies of these oscillators. That is, when , we obtain . Thus, for any physically realizable synchronous state in oscillator systems, the common angular frequency must be a real number. This is the real number constraint, a direct corollary of that cornerstone. It is a tacit assumption so fundamental that it is never questioned in physical measurements. However, this seemingly trivial requirement, when applied to the synchrony boundary, has profound consequences: It imposes a condition on the relationship between oscillator amplitudes and frequencies.

Applying this constraint to Eq. (1) yields Eq. (2) (see S3.1 in the supplementary material for details). Starting from Eq. (1) and treating the common angular frequency (before disturbance), the real number constraint requires that the resulting quadratic equation in must have real roots (Eq. (S12)):

This requirement yields a discriminant condition , and its boundary simplifies to an equation containing no , an algebraic criterion depending only on observable states:

(2)

Where . Eq. (2) defines the exact mathematical boundary for the existence of a real synchronous angular frequency, thereby defining the condition for the existence of synchronization itself between the pair of oscillators K, L. Remarkably, it depends only on the oscillator states , confirming its independence from network topology or coupling parameters. In fact, Eq. (2) is mathematically equivalent to the real number constraint (see S3.1 in the supplementary material). This equivalence explains our initial question: critical points are so unique because they correspond to boundary between real and complex numbers.

Upon further scrutiny (see S3.2 in the supplementary material), we found that Eq. (1) is an approximation. The exact boundary equation is shown below:

(3)

Unlike Eq. (1), is defined as . Although Eq. (1) is an approximation, the discovery and validation sections of this paper follow the true sequence of scientific inquiry: we first identified this concise form of Eq. (1) from the data and then conducted all validation experiments based on it. To preserve the authentic narrative sequence of scientific discovery and maintain consistency in the visual analysis throughout the main text, we continue to use Eq. (1) in the discovery and validation sections. We provide a direct comparison of the results from Eqs. (1) and (3) (Figure S3 in the supplemental material), demonstrating that Eq. (1) is a reliable and valid approximate expression. Therefore, the core conclusions of this paper are not affected by this approximation.

Eq. (2) and Eq. (3) are mathematically equivalent (see S3.2 in the supplementary material): Eq. (3) is a parametric projection of Eq. (2) onto the observable state space via the real number . The equivalence between Eq. (2) and Eq. (3), together with the equivalence between Eq. (2) and the real number constraint, establishes the equivalence: the synchronization admissibility boundary is an equivalent representation of the real number constraint. Thus, our interpretation reveals a profound fact: When applied to the synchronization of coupled oscillators, the requirement that physical quantities be real numbers emerges as the synchronization admissibility boundary. This key function is ultimately rooted in a cornerstone of physics.

The disturbance duration increased from 0.140 s to 0.155 s. A strong external disturbance occurred at the node 18. .

(a). Spontaneous synchronization of metaoscillators. The horizontal axis represents the disturbance duration. The vertical axis represents the standard deviation of . The yellow area indicates the thin layer where spontaneous synchronization occurs. Here, the standard deviation began to decrease at 0.147 s (magenta) and increased by 1300% at 0.155 s (cyan) when the system was unstable. can be calculated via Eq. (S9).

(b). Phenomenon of the bottleneck. increased from 0.140 s to 0.154 s. The black arrow indicates the direction in which increased. From (the magenta dots) onward, the distance between neighboring points decreased in the direction of the black arrow (elliptical shaded area).

(c). Starting points of spontaneous synchronization. The horizontal axis represents the disturbance duration. The amplified points indicate the starting points of spontaneous synchronization. All the starting points appeared at . The trends of all were almost identical. Combining (b) and (c) reveals that in this case, the structures in (b) exist between any two meta-oscillators. To clearly present the trend of , the linear scale on the vertical axis has been omitted; the corresponding data can be found in Table S2 in the supplemental material.

We report a phenomenon of spontaneous synchronization that occurs as the system approaches the admissibility boundary. This phenomenon is evidenced by a decrease in the standard deviation (Figure 3a), indicating that the subsystems are spontaneously evolving toward a common angular frequency (not necessarily the one prior to disturbance). This evolution is a hallmark of spontaneous synchronization. However, this evolution does not restore the system to the complete synchronous state. Crucially, this self-organization is confined to a thin layer (yellow region in Figure 3a) immediately preceding the eventual loss of admissibility at . This geometric region may reflect collective self-organization near the admissibility threshold²³. The phenomenon of spontaneous synchronization has been observed across different disturbance sites (Figure S4 in the supplemental material) and in distinct test systems (Figure S5 in the supplemental material), confirming its universality as an intrinsic characteristic of systems approaching the admissibility boundary. Furthermore, the reduction in the value of value within the thin layer reveals a change in the system's symmetry. If the system's state was merely a single-valued function of the control parameter , then its forward and backward variations should follow identical trajectories. The observed reduction in directly breaks this directional symmetry: During backward variation, the system cannot retrace the non-monotonic process of “first decreasing and then increasing” seen in the forward path because an increase in contradicts the definition of “synchronization” when a disordered state evolves into an ordered state. Consequently, the system state depends not only on but also on the manner in which the critical point is approached.

The system's trajectory in state space reveals a unique structural signature, a ‘bottleneck’, that, to our knowledge, has not been previously reported. For a constant step size , the distance between successive states decreases within the shaded area of Figure 3b, forming this bottleneck structure that exhibits a collective slowing-down. This spatial manifestation of deceleration emerges precisely at the temporal onset of spontaneous synchronization, i.e., when begins to decrease at (the magenta dots correspond exactly to in Figure 3b). A comparison between Figure 3b and S6a in the supplemental material confirms that this bottleneck structure is not a misleading result introduced by Eq. (S4), as it is also present in the oscillator's dynamic data. We hypothesize that the formation of a “bottleneck” may be related to emergent long-range coherence near critical points. In our visualization, this coherence coincides with trajectory clustering and deceleration (Figure 3b).

An accurate timeline of spontaneous synchronization is delineated by a distinct transition in the indicator . As shown in Figure 3(c), the value of for all meta-oscillators undergoes a simultaneous and coherent shift at (amplified points). This timing coincides exactly with the moment when the standard deviation begins to decrease in Figure 3a, unambiguously linking the reversal in to the onset of spontaneous synchronization. Prior to this moment, remains positive; immediately afterward, becomes negative and remains so. This system-wide reversal indicates that marks the starting point of the spontaneous synchronization stage. The stage end is clearly indicated by the system's loss of synchronization at , a state change that is demonstrated by a dramatic increase in the standard deviation in Figure 3a and the corresponding time-domain simulation results in Figure 1e. Thus, the layer of spontaneous synchronization is bounded between and .

This newly identified spontaneous synchronization constitutes a distinct phenomenon, separate from those previously reported in power grids²⁴. Existing phenomena typically describe how systems self-organize into synchronized states. In contrast, the novel spontaneous synchronization is uniquely characterized by its occurrence during a forward, disturbance-driven process toward inadmissibility and its association with the novel “bottleneck” structure. Moreover, its emergence is confined to the immediate vicinity of the synchronization admissibility boundary, as evidenced by experimental results under multiple disturbance scenarios (Figure 3a and S4 in the supplemental material). The intimate link between spontaneous synchronization and the admissibility boundary has a profound effect: The emergent layer of spontaneous synchronization is independent of the structure. More specifically, this independence is invariant across different structures: Throughout spontaneous synchronization, the coordinate point consistently resides near the boundary in the 3D coordinate system . More direct evidence comes from experiments using a separate test system (3-oscillator), where the phenomenon was also observed (Figure S5a in the supplemental material). Our observations provide direct evidence for a type of structure-independent spontaneous synchronization. First, the fact that spontaneous synchronization is independent of structure and occurs only near the boundary supports the view that the boundary is independent of structure. Second, spontaneous synchronization itself is a dynamic phenomenon, and the location where it emerges in state space is typically considered to be determined by structure, thus serving as an example of the “structure determines function” view⁴. However, our observations contradict this view and support the independence of the boundary. Moreover, the co-emergence of spontaneous synchronization and the boundary may suggest an intrinsic and close connection between emergent critical phenomena and admissibility conditions.

In the case of the “bottleneck” phenomenon, the first derivative of the the disturbed trajectory is continuous (see Figure 3c). Moreover, we observed a different phenomenon that occurred more frequently near the onset of spontaneous synchronization. The slope abruptly changes from greater than 1 to less than 1, causing the curve in Figure 3b to kink. is defined by . As shown in Figure 3b, when the state is admissible, and satisfy a relationship of the form . Thus, . Owing to the monotonicity of with respect to , i.e., . Therefore, and occur simultaneously (see Figure S5b,c in the supplemental material). These results reveal the complexity of spontaneous synchronization.

Conclusion

Methods

was set to 314.16 rad/s. All external control mechanisms were turned off. We calculated the angular frequency and the terminal amplitude of the ith oscillator for various disturbance durations over a time window T of 20 seconds, starting from disturbance inception. The disturbance scenario (the site, the type, and the duration ) was defined. We set all the disturbances as three-phase short circuits to ground. The simulation process was repeated until the system transitioned from a stable state to a loss of synchronization state, as confirmed by the simulation results.

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