  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.
