ILMs in a Coupled Pendulum Array Imagine a line of pendulums whose motion is rotationally coupled via springs between them. This system is perhaps the simplest model for coupled oscillators. Each oscillator, here pendulum, has an intrinsic frequency which is dependent on its amplitude of oscillation. In addition, each oscillator is also sensitive to the state of its neighbor, here due to the springs between the pendulums. The motion of each pendulum is goverened by the discrete sine-Gordon equation,
This system of nonlinear differential equation permits localized solutions, aka ILMs, discrete breathers or discrete solitons. But what if an external driver is added to the equations as well as dissipation? Interestingly, ILMs can still exist over a range of driver amplitudes and frequencies. We do, however, now observe non-negligible motion of pendulums far away from the ILM center. The following movies -taken by an overhead (wide-angle) webcam- show the motion of pendulums in response of a driver shaking them. In many instances a stable localized response is observed to a non-local driver. The first movie demonstrates the modulational instability of the uniform pendulum mode, and that is can give rise to ILMs in the long run. Note that the movie is spiced from three separate clips of the same run. The second movie shows two stable ILMs oscillating side by side. And the third movie shows what happens when the driver frequency is increased above the stability domain for the ILM. If the (torsional) spring stiffness is increased, the ILMs broaden and more closely approximate the shape of a classical soliton. Here is one example with a center amplitude of >180 degrees.
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