Abstract:

An example of an oscillatory system with a time-reversible nonanalytic nonlinearity is shown to be a pendulum with a flexible cord sandwiched between two identical circular disks, in contrast to the analytic case of a pendulum interrupted by a single circular disk. The amplitude-dependent frequencies of both cases are perturbatively calculated, and are compared to numerical simulations. The nonanalyticity causes the unusual effect of the frequency to vary linearly with amplitude for small amplitudes, which has also been observed in compressional standing waves in sandstone. The general condition for a nonanalytic nonlinearity to yield this behavior is presented. The amplitude-dependent frequency for the double-interrupted pendulum allows an explanation for Huygens' observation that circular interrupters were as effective as cycloidal interrupters in achieving isochronous motion. A lattice of linearly coupled double-interrupted pendulums is described near the lower and upper cutoff modes by quadratic nonlinear Schrodinger (NLS) equations, in contrast to cubic NLS equations which arise for analytic pendulum lattices as well as typical acoustic and surface wave guides. Solitary breather and kink solutions to the quadratic NLS equations are presented, and the quadratic NLS breather is predicted to occur in a bar of sandstone. [Work supported by ONR.]