Two Phases, Two Faces: “Janus Oscillators” Undergo Explosive Synchronization

What do algae, grandfather clocks and a two-faced Roman god have in common? On the face of it, not much—but they all play a part in a recent paper out of Northwestern University.

This story begins in the mid-17th century with a Dutch physicist and mathematician named Christiaan Huygens. In addition to his talent for abstraction, Huygens was a prolific inventor. In fact, he both created the first accurate pendulum clock in 1656 and correctly summarized the mathematical theory driving it some 17 years later.

As Huygens realized, as long as the angle of oscillation is small (that is, the pendulum doesn’t swing terribly far to either side), a pendulum becomes what’s known in physics as a simple harmonic oscillator, or SHO. The SHO can take a variety of forms—clocks ticking, springs bouncing, strings vibrating—but all these are connected by the single mathematical expression that requires them to oscillate back and forth with a regular period, which is determined from the specifics of the system.

A mass suspended from a spring is one classic example of the SHO.
Image credit: Svjo via Wikimedia Commons

In the case of a clock, it’s important that the period be exactly one second, so that each swing of the pendulum can trigger a mechanism to move the second hand one tick further. Although Galileo had experimented with pendulums and timekeepers, it was Huygens who derived the equations governing their periods of oscillations—a critical moment in the history of horology.

Inspired by the success of his first pendulum clock, no doubt, Huygens soon built a second…and many more, all accurate timepieces. However, when he started mounting the clocks on the wall of his study he noticed something strange: after a while, the originally disparate pendulums appeared to sync up! Unwittingly, by connecting the clocks with a single wall beam, Huygens had discovered the coupled oscillator, a system of multiple interconnected SHOs.

Later studies have shed a little more light on the behavior of two or more oscillators, no longer simple harmonic but coupled through some physical means. These systems are inherently more complex than the SHO, as each component reacts to the other’s movement, while still feigning a respect for its own period of oscillation.

Two pendulums hanging from a string is one example of a coupled oscillator.
 Image Credit: Lucas V. Barbosa via Wikimedia Commons

To describe the state of an oscillating system, one almost always needs to use a phase. This variable just describes where the oscillator is in its back-and-forth motion—is it reversing direction, or whizzing past the center line? When we start talking about coupled oscillators, though, it gets a little more complicated.

Notice how, in the animation above, the amplitude (or size) of the oscillation varies as it appears to pass from one of the oscillators to the other. Suddenly this means that the system needs both a phase and an amplitude to describe what’s happening at any moment; this is what’s known as a phase-amplitude oscillator, and it has been studied fairly extensively. However, there are other, less familiar types of coupled oscillators, and one of these is the kind featured in the new Northwestern study.

Dr. Adilson Motter is a professor of physics and astronomy at the university, where he heads a lab dedicated to the study of complex systems and networks. He and two postdoctoral researchers in his group, Dr. Zachary Nicolaou and Dr. Deniz Eroglu, have recently investigated not a phase-amplitude oscillator, but a phase-phase oscillator.

To explain this somewhat abstract idea, the group points to celestial oscillators. For example, the Moon rotates around the Earth in a cyclical motion. Because the same side of the Moon always faces the Earth, this cycle can be mathematically described with a single phase. The distance between the two bodies also varies, however, so an amplitude is also needed to correctly describe the system at any time; this is a standard phase-amplitude coupled oscillator.

“On the other hand,” the group writes, “the Earth’s orbit around the sun is nearly circular, and there is no need for an amplitude variable. As the Earth rotates around the Sun, however, it also rotates around its own axis, and thus the Earth-Sun system has two phases, corresponding to the day cycle and the year cycle.” Another example of such a phase-phase coupled oscillator is the cardio-respiratory system, which has two distinct frequencies: the heart rate and the breathing rate. Nevertheless, these “oscillators” are coupled to each other, rising under physical stress and dropping when the body is at rest.

Taking their inspiration from the analogous particles known as Janus particles, the research team dubbed this type of coupled oscillator—described by multiple phases with different natural frequencies—a Janus oscillator. (As a happy aside for pun aficionados, the oscillator with two phases thus takes its name from the Roman god with two faces.) Although the mechanics of a single Janus oscillator are not too complicated, they were curious about what would happen if, say, fifty such oscillators were all coupled together in a ring formation.

This is a simple schematic of a ring of coupled Janus oscillators.
 Image Credit: Nicolaou et al. via Physical Review

Using sophisticated mathematical modeling, they were able to simulate the effect that each Janus oscillator would have on the others as time went on. In particular, they studied what happened as the coupling constant increased—that’s the value that determines how strongly the oscillators are connected.

One reason this question is interesting, beyond the many real-world applications, is simply that it’s impossible to visualize without running serious simulations. “Salt is very different from the sodium and chlorine that it is composed of,” the researchers comment. Similarly, they found that this huge ring of Janus oscillator units behaved very differently than rings of identical phase oscillators.

For a weak coupling constant—the equivalent of tying each oscillator together extremely loosely—no real pattern emerged since each oscillator was free to bounce back and forth with roughly its natural frequency. As the coupling constant increased, however, strange patterns appear suddenly, in a so-called “explosive” transition toward synchrony. Small groups of oscillators would spontaneously synchronize, then just as suddenly drift apart again. Regions of synchronization would move along the ring, oscillator to oscillator, while the rest remained in complete incoherence—an effect known as a “chimera state” for its seeming impossibility. Finally, the group found that for certain conditions, it was necessary for the Janus oscillators themselves to not be identical in order for the system to settle into a phase-locked state, with each Janus oscillator bound to its neighbors and confined in its movement.

Each of these effects has been observed in the past, but historically they have each been relegated to a specific type of system. “Nobody expected to see chimera states and explosive synchronization, let alone all the other new behaviors we observed, occurring together in such a simple system,” the researchers say. These “new behaviors” included inverted synchronization transitions, in which an increase in the coupling between the Janus oscillators leads to an unexpected decrease in the synchrony among them. And as much as it might seem to the contrary, this system is simple thanks to its perfect symmetry. Although this study was purely theoretical, it wouldn’t be too difficult to construct a physical apparatus for further study.

Aside from pure academic interest, large systems of Janus oscillators could actually help us understand the world a little better. For example, Chlamydomonas, a type of algae, each have two flagella that operate much like a Janus oscillator. At the same time, each alga communicates with its neighbors via physical and possibly chemical signals, effectively coupling them. By developing a model of such systems, it could be possible to understand how large communities interact and, at times, synchronize.

One can only imagine what Christiaan Huygens might think.

—Eleanor Hook

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