Investigation of an Unforced Duffing Equation having Fractional Power Damping

The Duffing ODE provides a standard model for nonlinear oscillations for a broad range of phenomena in the natural and engineering sciences. The effects of dissipation are generally included by adding a ``friction" force term, $f(v)$, proportional to an integer power of the velocity. Thus, oscillations take place, but with a decreasing amplitude, and which only decrease to zero in an infinite time interval. We examine the case where $f(v)=-av^p$, $a>0$ and $0 < p < 1$, and demonstrate that the amplitude of the oscillations become zero in a finite time [1]. This result may have relevance for the vibrations of carbon nanotubes and sheets of graphene sheets [2]. \\[4pt] [1] R. E. Mickens, Truly Nonlinear Oscillators (World Scientific, London, 2010). \\[0pt] [2] A. Eichler et al., Nature Nanotechnology, Vol. 6 (June 2011), 339--342.