  The fully nonlinear notion of resonance$-$\textit{geometrical resonance}$-$in
the general context of dissipative systems subjected to spatially periodic
\textit{phase-modulated} potentials is discussed. It is demonstrated that there
is an exact local invariant associated with each geometrical resonance solution
which reduces to the system's energy when the potential is stationary. The
geometrical resonance solutions represent a \textit{local symmetry} whose
critical breaking leads to a new analytical criterion for the onset of chaotic
instabilities. This physical criterion is deduced in the co-moving frame from
the local energy conservation over the shortest significant timescale.
Remarkably, the new physical criterion for the onset of chaotic instabilities
is shown to be valid over large regions of parameter space, thus being useful
beyond the scope of current mathematical techniques. More importantly, the
present theory helps to understand the unreasonable effectiveness of the
Melnikov's method beyond the perturbative regime.
