Linear response spectra of a driven intrinsic localized mode in a micromechanical array are measured as it approaches two fundamentally different kinds of bifurcation points. A linear phase mode associated with this autoresonant state softens in frequency and its amplitude grows as the upper frequency bifurcation point is approached, similar to the soft-mode kinetic transition for a single driven Duffing resonator. A lower frequency bifurcation point occurs when the four-wave-mixing partner of this same phase mode intercepts the top of the extended wave branch, initiating a second kinetic transition process.

An intrinsic localized mode (ILM) represents a localized vibrational excitation in a nonlinear lattice. Such a mode will stay in resonance as the driver frequency is changed adiabatically until a bifurcation point is reached, at which point the ILM switches and disappears. The dynamics behind switching in such a many body system is examined here through experimental measurements and numerical simulations. Linear response spectra of a driven micromechanical array containing an ILM were measured in the frequency region between two fundamentally different kinds of bifurcation points that separate the large amplitude ILM state from the two low amplitude vibrational states. Just as a natural frequency can be associated with a driven harmonic oscillator, a similar natural frequency has been found for a driven ILM via the beat frequency between it and a weak, tunable probe. This finding has been confirmed using numerical simulations. The behavior of this nonlinear natural frequency plays important but different roles as the two bifurcation points are approached. At the upper transition its frequency coalesces with the driver and the resulting bifurcation is very similar to the saddle-node bifurcation of a single driven Duffing oscillator, which is treated in an Appendix. The lower transition occurs when the four-wave mixing partner of the natural frequency of the ILM intersects the topmost extended band mode of the same symmetry. The properties of linear local modes associated with the driven ILM are also identified experimentally for the first time and numerically but play no role in these transitions.

The speed of a traveling intrinsic localized mode (ILM) in the acoustic spectrum of a micromechanical cantilever array is experimentally measured at high resolution as a function of the driving frequency. A repeating speed pattern is observed for chaotic and regular traveling ILMs between adjacent extended wave normal mode frequencies. The speed of a regular traveling ILM is almost the same as the plane wave dispersion group velocity at that frequency. Since ILM amplification only occurs during reflections at the ends of the array the phase matching condition for long time stability is greatly relaxed. A double humped distribution of speeds, found for chaotic ILMs, is shifted from the regular ILM nearly monochromatic speed value due to the modulational instability. Numerical simulations reproduce many of the experimental observations, demonstrating that intrinsic dynamical properties of the small array are being measured.

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Abstract:The experimental linear response spectrum of an auto-resonant intrinsic localized mode (ILM) in a driven 1-D cantilever array is composed of several resonances including a phase mode of the ILM. This autoresonant state is stable in a finite frequency range between the upper and lower bifurcation frequencies. Here we examine the robustness of the lower frequency transition point to an added lattice perturbation. In the unperturbed ILM state an even linear localized mode crosses the phase mode as the transition is approached and bifurcation occurs when the phase mode intersects the highest-frequency odd-symmetry band mode of the lattice. When an impurity mode is introduced into the lattice near the even linear local mode it breaks the local symmetry so that the lower bifurcation frequency of the ILM is now shifted to the point where the even mode and the odd phase mode frequencies coalesce.