1) Stable damped-driven ILMs: direct, horizontal shaking
2) Sub-harmonic shaking
3) Edge breathers
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. This is seen in the experimental data shown below.
The experimental pendulum chain used in these experiments is monitored from directly above (as shown by the web-cam image below). More recently, we have shown that these ILMs can also be excited via sub-harmonic resonance at a driver frequency in a band around three-times the linear resonance frequency. In this case, it is fairly easy to get ILM amplitudes in excess of 180 degrees (see the image from the overhead web-cam below).
Furthermore, we can stabilize so-called edge-breathers - nonlinearly localized modes that are trapped at the edges of the lattice:
In this figure, the circles are experimental measurements and the blue stars are the corresponding numerical results. A Floquet perturbation analysis reveals these states to be dynamically stable (right panel). Their amplitude is somewhat smaller than the corresponding bulk mode (see dotted trace), but it can persist over a slightly wider driving-frequency window. The edge topology seems to slightly stabilize these localized modes.
Figures from Phys. Lett. A 380, 402 (2015).