Magical Pendulums

The exploration of synchronization started during the Scientific Revolution. The renowned physicist Christiann Huygens observed a pair of pendulum clocks hanging from the same beam, and he noted their eventual tendency to swing in unison. Regardless of how he initiated their movements, they would ultimately couple their motions. This was quite perplexing! What could be the cause behind this behavior?

In mathematics, we refer to such a system as a coupled oscillator. This term describes two individual objects undergoing cycles that can influence one another. Often, a coupled oscillator system leads to a shared motion among the individual objects. Huygens termed this phenomenon "sympathy," likely inspiring the magical system in the acclaimed series The Kingkiller Chronicles. He conducted various experiments with coupled pendulums and kept detailed records but never fully unraveled the mystery.

Around the same period, naturalist Engelbert Kaempfer traveled to Southeast Asia, where he documented fireflies flashing in unison. This scenario is significantly more intricate and challenging to describe. Each firefly reacts solely to its neighbors, and somehow, they achieve synchronization. It took centuries for numerous synchronization studies to converge under a unified theory, as the necessary concepts had not yet been developed.

If you're interested in watching fascinating videos about coupled pendulums, I will provide links at the end of this article!

Finding Common Ground

While two coupled pendulums represent a relatively simple system, most synchronization examples become increasingly complex. The addition of more oscillators complicates matters significantly! This complexity is what makes describing natural synchronizing systems so challenging. How can we even begin to model such phenomena?

To start, let's establish a basic framework to describe a single oscillator. This oscillator has a period that denotes the time interval between each event. For fireflies, this corresponds to the time between flashes. We can succinctly represent this system with a sine wave!

We define the period as T. The oscillator progresses with time, indicated by the variable lowercase t. Another vital factor is A, which signifies the starting point of the oscillator's cycle, commonly referred to as the phase.

Now, let's adapt this to represent our fireflies. Each firefly initiates with a random phase that indicates its proximity to flashing. Additionally, each one begins with different periods, which will not be entirely random but will fall within a species-specific range. Using this information, we can create a straightforward model of a single firefly. Let's visualize this!

In the sine wave model, a firefly flashes at the peak of its oscillation. In this visual, I set the current time represented by a vertical line. Each time the wave reaches its peak value, I included an animation to show the firefly lighting up. It's astounding what can be achieved with Desmos!

Utilizing this equation, we can easily expand it to formulate a mathematical model of synchronization among multiple oscillators. One of the most recognized models was developed by Yoshiki Kuramoto in the 1980s. His aim was to create a simple equation to elucidate collective synchronization.

The equation may appear complex, but it's quite manageable. The left side describes the cycle's progression over time. The first term on the right, ω, represents the natural frequency. Each firefly has a preferred frequency at which it oscillates (or its period). However, this frequency is influenced by others, depicted in the subsequent term, which sums the behaviors of all other fireflies. A coupling constant, K, indicates the extent to which the fireflies desire synchronization. The equation then calculates the phase difference between each firefly and every other firefly, adjusting accordingly.

To summarize, this equation acts on each firefly. Every firefly harbors a frequency it aspires to oscillate at but experiences slight influences from each other firefly. This model assumes that every firefly is affected by all others, disregarding distance. The strength of these influences is described by the variable K. This can lead to intriguing behaviors, as demonstrated in the video below!

Three different Kuramoto systems with varying coupling constants depict this phenomenon. Each dot represents a firefly's movement around a circle, which serves as an alternative representation of the sine curve above. Just as the sine curve peaks, the firefly flashes when its dot reaches the circle's top. This video showcases three distinct coupling constants. On the left, with K = 1, there is no synchronization among the fireflies, and their flashes appear random. As we increase the coupling constant, they align more closely.

The yellow dot illustrates the overall phase of the entire system. When it is near the center, coherence is low, indicating that the fireflies are not synchronized. Conversely, high coherence signifies that the system is experiencing phase-locking. As K increases, the fireflies become more aligned.

This equation can be easily modified to represent other physical systems. As shown in the accompanying diagrams, applications use it to describe a 2D space of oscillators. In these images, colors denote the phase of each point. An additional factor is incorporated, allowing oscillators to be influenced solely by nearby oscillators. Depending on how distance is defined in this system, some stunning results emerge!

Another Simulation

As anticipated, scientists have devised numerous methods to describe the firefly system. The next model I'll introduce is much simpler. Each firefly possesses its own internal clock that begins at a random value. When it reaches a designated value, it flashes. However, if a sufficient number of neighboring fireflies flash, its clock resets. This model considers the distance between fireflies and does not necessitate complex differential equations for simulation.

A crucial aspect of this model is that every firefly starts with the same period of 10 seconds. The only randomized component is the starting phase. While this is an oversimplification, it remains illustrative. To demonstrate this model, I will use the incredible software NetLogo, which excels at modeling complex systems.

The image above showcases a screenshot from the NetLogo simulation described earlier. The yellow fireflies are flashing, while the grey ones are not. The image updates with each time step, and the fireflies move randomly across the screen. I appreciate this simulation because it allows us to observe synchronization developing over time.

This plot illustrates the number of fireflies flashing at each time step. Initially, the flashing appears random. However, as they adjust their clocks based on their neighbors' behavior, synchronization emerges! By the end of the simulation, the majority of fireflies are flashing together. This illustrates the power of simple models like this one.

Just for fun, I ran this model for over an hour, and by the conclusion, every single firefly was flashing in unison! While this may not accurately reflect nature—real firefly groups often maintain a few outliers that do not synchronize—it demonstrates the concept effectively.

Extending to Other Systems

As previously mentioned, fireflies are not the only natural systems that exhibit synchronization. Scientists have developed a variety of models to account for the diverse examples observed. One modeling approach is known as a limit cycle. A limit cycle is represented in a phase plot, where the oscillator moves around a circle, similar to the video of the spinning dots shown earlier.

Different equations governing limit cycles yield various types of synchronous behavior. Some allow synchronization only if the initial conditions are favorable. For example, an agent starting outside the circle will not fall into the oscillation pattern. Only those starting within the circle will exhibit regular behavior.

Other models utilize individual agents that modify their behavior based on their neighbors' actions. This resembles the second firefly model discussed earlier. Numerous adjustments can be made to account for the influence of neighbors and the initial conditions of each agent.

An additional example of synchronization in nature that I haven't yet mentioned is flocking. You've likely seen large flocks of birds flying together in the sky. Fish exhibit similar behavior! Scientists have created models to replicate this behavior akin to the Kuramoto model described earlier. However, instead of adjusting their phase, birds adjust their speed and direction based on their neighbors. This model can effectively replicate the beautiful patterns observed in flocking behavior.

Going Further

I hope you’ve gained some insights! The mathematics of synchronization is a rich and vibrant field of study. Many of the sources I referenced in this article comprise research conducted within the last thirty years! If you wish to delve deeper into synchronization and explore some captivating visuals, I have included additional resources below.

For those interested in more pendulum videos, I have plenty! Check out the following links for further exploration into this fascinating topic! If you're eager to see another simulation of fireflies, I recommend visiting this website for a fantastic visual representation of oscillating behavior.

For a deeper understanding of synchronization, consider reading Sync by Steven Strogatz. This more technical book is incredibly engaging and expands on the topics discussed here. Anything by Strogatz is worth exploring!

In case you've never witnessed a video of fireflies synchronizing, don't miss the one below!

Lastly, there are more ways to model flocking behavior than the one mentioned earlier. Be sure to check out this online simulation, which employs a different algorithm and allows you to adjust parameters!

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