  We report on low temperature measurements performed on
micro-electro-mechanical systems (MEMS) driven deeply into the non-linear
regime. The materials are kept in their elastic domain, while the observed
non-linearity is purely of geometrical origin. Two techniques are used,
harmonic drive and free decay. For each case, we present an analytic theory
fitting the data. The harmonic drive is fit with a Lorentz-like lineshape
obtained from an extended version of Landau and Lifshitz's non-linear theory.
The evolution in the time domain is fit with an amplitude-dependent frequency
decaying function derived from the Lindstedt-Poincar'e theory of non-linear
differential equations. The technique is perfectly generic and can be
straightforwardly adapted to any mechanical device made of ideally elastic
constituents, and which can be reduced to a single degree of freedom, for an
experimental definition of its non-linear dynamics equation.
