Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
How is it that two fireflies, each blinking to its own rhythm, can come into sync with each other, flashing at the same time? How do we even begin to represent this situation mathematically?
A single firefly, if left to its own devices, will flash with some regularity. To model this situation mathematically requires a function that has periodicity, which simply means that it returns to the same value at regular intervals. As we saw in our unit on the connections between music and mathematics, a good mathematical function that models periodicity is a sinusoid. A sine wave oscillates smoothly between one value and another. For the firefly, these two values would be the states "on" and "off."
It would be reasonable to model the flashing of a single firefly by looking at the sine of theta, where theta represents where the firefly is in its flashing cycle. The firefly flashes when θ equals zero.
Another way to think about this is to imagine a runner on a circular track. Picture the runner traveling at a constant speed, corresponding to how quickly the firefly charges up its flash. The flash itself corresponds to the runner crossing the start/finish line. The angle theta then represents where the runner is on the track in relation to the start/finish line.
So, if theta represents where the firefly or the runner is in his cycle, the derivative of this will indicate how fast that position is changing.
= the rate at which θ changes.
This value is intuitively related to the frequency of oscillation—the more quickly θ changes, the more cycles the runner, or the firefly, will complete. Let's call the frequency that the runner or firefly would have alone, without any influence from others, the natural frequency, denoted by ω.
Things get interesting when we introduce another oscillator and consider two fireflies, or two runners, that interact with one another. We can model each one as an oscillator, just as we did in the single case, but because they interact with each other, the expression is somewhat more complicated.
Because we now have two oscillators, we will have two phases (θ1 and θ2) and two natural frequencies (ω1 and ω2) to account for. If we assume that the natural frequencies are fixed, then we will need two equations for the two unknowns θ1 and θ2.
The first firefly has phase θ1 and frequency ω1. The second firefly has phase θ2 and frequency ω2. For both fireflies to flash in sync with one another, the two thetas must be equal to one another. Mathematically, θ1 – θ2 must equal zero.
The phase difference, θ1 – θ2, determines the extent of "correction" each firefly needs to make to synchronize with the other one. The necessary adjustment varies depending on how far apart the two fireflies are in their cycles. If the two fireflies are very far apart in their cycles, a large correction is needed. If they are only slightly out of sync, only a slight nudge is required. However, the situation is a bit more complex than this.
The adjustment each firefly makes can be either to slow down or to speed up its flashes. How does it determine which to do? Consider the case of perfect alternation, with one firefly flashing and then the other flashing at perfectly spaced intervals. Should the one slow down or speed up to match the other? It can speed up, basically doubling its frequency temporarily so that its next flash coincides with the other, or it can slow down, halving its frequency, skipping the next flash in the attempt to synchronize with the other.
If the flash of the first firefly occurs at a point in time that is less than half the firefly's cycle time from the second firefly's next flash, it makes sense to speed up. On the other hand, if it is more than half way through its cycle, it is better to slow down and wait for the other firefly to catch up. The difference in θ is what influences the firefly as to what to do. A function capable of modeling either a speed up or a slow down must be able to periodically take on positive or negative values, depending on the difference in θ. Once again this is ideally a sinusoid. So, our mathematical model of how a firefly adjusts its flashing cycle to achieve synchronization with another should look something like this:
The sine terms should be mediated by a constant that represents how strongly the two fireflies interact with each other. This constant can take into account things such as distance and ambient light levels that affect a firefly's perception. Let's designate this constant K1 for the first firefly and K2 for the second firefly. Incorporating these factors yields these modified expressions:
Finally, we shouldn't forget the influence of each firefly's natural rhythm, ω1 and ω2 respectively.
These two equations represent the changes that each firefly should make, based on what the other is doing, in order to achieve synchronization. Mathematically, these are the equations of coupled oscillators. In our study of sync, we need to analyze the behavior of these equations to find out the various conditions under which spontaneous synchronization can occur. This is a simple, standard model that can be applied to many different situations in which synchronization is observed.
Recall that synchronization is defined to be the condition in which both oscillators are in phase. Mathematically, this occurs when:
θ1 = θ2 or θ1 – θ2 =0
We can let φ = θ1 – θ2 to introduce a single, convenient variable to represent the phase difference. The change in φ, representing how the phase difference changes, would then be:
Using our equations for the derivatives of the flashing cycle equations of the two fireflies from above, we can get:
What this equation tells us, via φ and , is that the fireflies' synchronization with one another is based on the difference in their natural frequencies, ω1 - ω2, and how that difference compares to the strength of the signals they send and receive from each other, K1 +K2, also called the coupling strength. If the difference in frequency is less than the coupling strength, the fireflies will spontaneously synchronize. If the difference is too great, they will go on flashing at their individual rates.
This is a relatively straightforward model of potentially synchronous behavior with two oscillators. Real-world systems, however, are often made up of many oscillators. In the next section, we will explore how to expand our model to deal with more-complicated systems such as these.