Delay-induced multistability in one-dimensional swarmalators with common intrinsic frequency

Imagine a world where tiny, self-propelled units move through space while simultaneously ticking like clocks. In physics, these are called swarmalators. They are not just particles that drift, nor are they just oscillators that beat in time; they are a hybrid. Their movement through space influences how they synchronize their internal rhythms, and conversely, their synchronization dictates how they move. This coupling creates a feedback loop where assembly and timing feed into one another, a phenomenon seen in everything from the collective motion of robotic swarms to the behavior of biological cells. For years, scientists have studied these systems to understand how order emerges from chaos. A particularly simple version of this system, where the units move along a single ring, was already known to settle into three distinct, predictable patterns: a chaotic drift where everyone is out of step, a synchronized state where everyone moves and ticks together, and a "phase wave" where the units form a traveling wave of synchronization.

However, real-world signals rarely travel instantly. When a unit at one end of the ring sends a signal to another, there is a delay, a lag caused by the finite speed of information. Previous studies looked at what happens when these units have no natural rhythm of their own, or when that rhythm is zero. But what if every unit has a steady, non-zero internal beat? This is the question tackled by Rommel Tchinda Djeudjo, Timoteo Carletti, and K. P. O'Keeffe in their recent study. They asked how a shared, constant internal frequency changes the behavior of these delayed swarmalators. The answer turned out to be far more complex and surprising than anyone expected. Instead of simply shifting the existing patterns, the combination of a shared rhythm and a time delay completely reorganizes the system, creating new zones of stability and, most remarkably, a rare state where three completely different behaviors can exist at the exact same time.

To understand the researchers' discovery, one must first grasp how time delay interacts with a steady rhythm. In a system without delay, if every unit is spinning at the same speed, you can simply change your point of view to a rotating frame of reference, effectively canceling out that speed and making the math look like the zero-frequency case. But time delay breaks this trick. Because the units are reacting to where they were a moment ago, the delay accumulates a phase shift that cannot be erased. This delay-induced lag acts as a hidden force that reshapes the entire landscape of possible behaviors. The researchers modeled a ring of these units, tracking both their position and their phase, and introduced a fixed time lag in how they influenced one another. They then ran thousands of computer simulations, testing different strengths of attraction and repulsion between the units, and varying the length of the time delay.

The results revealed a dramatic restructuring of the system's stability. In the absence of delay, the "asynchronous" state—where units drift apart and lose all coordination—is usually stable only when the units repel each other. With a shared frequency and a delay, this state becomes stable even when the units are attracted to one another, but only within specific, narrow windows of time delay. These windows appear as distinct lobes on a stability map, regions where the delay paradoxically stabilizes the chaos. Similarly, the "synchronized" state, where everything locks together, undergoes a transformation. It is no longer a static, locked branch. Instead, it becomes a rotating branch that can remain stable even when the physical forces between the units are repulsive. The delay accumulates a phase that effectively turns the repulsion into attraction, allowing the swarm to synchronize against the odds.

Perhaps the most intricate change occurred in the "phase wave" state. In the zero-frequency case, this state is a simple, uniform branch. With a shared frequency and delay, it fractures into a series of narrow, resonant stability bands. These bands appear at specific combinations of delay and coupling strength, creating a complex, striped pattern of stability that was absent in previous models. The researchers found that these new stability regions for the asynchronous, synchronized, and phase wave states do not just exist side by side; they overlap. In certain regions of the parameter space, the system is linearly stable in all three forms simultaneously. This means that depending entirely on how the system started—its initial history—it can settle into one of three completely different outcomes.

The most striking instance of this phenomenon occurs in a very narrow window near a specific balance point between the coupling forces. Here, the researchers found a genuine tristability. If they initialized the simulation with a history resembling the asynchronous state, the system settled into incoherence. If they started it with a history resembling the synchronized state, it locked into perfect unison. If they started it with a history resembling the phase wave, it formed a traveling wave. All three outcomes were stable attractors for the exact same set of physical parameters. This is a level of multistability that does not exist in models without delay or in models where the units have no intrinsic frequency. It suggests that in systems with shared rhythms and signal lags, the past history of the system can permanently determine its future state, locking it into one of three distinct realities.

In the gaps between these stable regions, where no single pattern is favored, the system does not simply collapse into randomness. Instead, the researchers observed persistent, regular oscillations in the order parameters. The system would not settle into a static cluster or a fixed pattern; instead, the degree of synchronization and the spatial arrangement would cycle continuously. These oscillations were not chaotic or unpredictable; they were rhythmic, with sharp peaks in their frequency spectra, indicating a form of macroscopic dynamics that is neither static nor turbulent. The researchers confirmed this behavior through extensive basin surveys, testing hundreds of random starting conditions at these points, and found that every single one led to these persistent oscillations.

This work fundamentally changes our understanding of how delay and intrinsic frequency interact in coupled systems. It demonstrates that a shared rhythm, when combined with a time lag, is not a minor perturbation but a powerful organizer that creates new states of matter. The discovery of a tristable point, where incoherence, full synchrony, and a phase wave can all coexist as stable options for the same group of units, highlights the profound role of history in determining collective behavior. While the study was conducted using computer simulations of a simplified one-dimensional model, the principles it uncovers—delay-selected stability windows, the restoration of synchrony through repulsive forces, and the emergence of genuine tristability—offer a new lens for understanding complex systems where timing and movement are inextricably linked. The findings suggest that in the real world, from biological swarms to robotic collectives, the interplay of a shared internal clock and the finite speed of communication could allow a single group to exist in multiple, mutually exclusive states of order, depending entirely on how it began.