Research
Synchronization in Coupled Nonlinear Oscillators
In recent years, studies on complex systems have often centered on emergent behavior: cooperative interactions among the parts of the system lead to organized (and sometimes unexpected) behavior of the whole. One simple example of emergent behavior is spontaneous mutual synchronization. Synchronization is ubiquituous in nature (it can be seen in the flashing of fireflies, the chirping of crickets, and the electrical activity of neurons) and is relevant to many applications. While significant progress has been made in understanding synchronization, many open questions remain even for relatively simple systems.
One such example is the case of two mechanical oscillators (e.g., pendulum clocks) that are coupled by placing them on a common platform. In 1665, Christiaan Huygens discovered that after some time had passed two pendulum clocks mounted on a common support would end up swinging exactly out of phase. If they were disturbed, the clocks would spontaneously resyncrhonize and always end up in the antiphase state. After careful study Huygens concluded that the key interaction was due to small movements of the supporting beam, but was unable provide any further analysis. This is forgivable given the fact that Newton's Principia would not be published for another 22 years!
In 2002, a group at Georgia Tech built an apparatus consisting of two clocks mounted on a common support, which was allowed to move in one dimension along an air track. They confirmed Huygens's observations and showed that such a system would always end up in the out of phase state. They also derived a mathematical model for the system that allowed them to predict the observed behavior.
Concurrently, James Pantaleone of the University of Alaska was carrying out experiments on a very similar system. In Pantaleone's experiments the clocks were replaced with mechanical metronomes and the airtrack was replaced by a common supporting platform rested on smooth cylinders to allow low-friction translation in one dimension. Pantaleone found that opposite to the clocks, the metronomes almost always synchronized in-phase, with antiphase motion observed “only under special circumstances.” Pantaleone's demonstration has inspired a host of YouTube videos such as the one below on Harvard's Natural Science Lecture Demonstrations channel.
In 2010, Prof. Kurt Wiesenfeld and I set out to explain why such different behaviors were observed in systems that should, in theory, be governed by very similar physics. While it was possible that the differences in observed behavior were the result of differences in the detailed physics (e.g., choice of escapement mechanism), we took the view that the systems are fundamentally the same, involving a single set of basic physical processes. We developed a model that allowed us to the differing behaviors in terms of the relative importance of these processes.
We developed an iterative map model that incorporates a few simple effects common to many coupled nonlinear oscillator systems. These include weak nonlinearity of the oscillators, which allows the phase of the oscillators to “slip,” and driving “kicks,” which model the energy input from the clock/metronome escapements. The map is simple enough that we were able to derive explicit expressions for the stability boundaries of the in-phase and antiphase modes and provide a straightforward explanation for the difference between the observations of the Georgia Tech group and Pantaleone. We concluded that the difference in observed behavior is attributable to differences in the relative importance of the main physical effects and were able to show that our results agreed with numerical simulations of the differential equations of motion. In particular, we discovered that the larger amplitude of oscillation of the metronomes provided a stabilizing effect for the in-phase mode, whereas the heavier platforms used by Huygens and the Georgia Tech group damped it resulting in anti-phase synchronization.
More recently, Prof. Wiesenfeld and I have been working on extending the iterative map model to include cases with large numbers of coupled oscillators and seeing whether it can be used to predict not just the asymptotic behavior of coupled oscillator systems, but also the dynamics of synchronization (e.g., predict the time to synchronization after a perturbation).
Related Publications
- K. Wiesenfeld and D. Borrero-Echeverry, “Huygens (and Others) Revisited,” Chaos 21, 047515 (2011).
© 2016 Daniel Borrero Echeverry | Last updated: 6-20-2016