Edge of chaos as critical local symmetry breaking in dissipative nonautonomous systems

The fully nonlinear notion of resonance$-$$\mathit{\text{geometrical}}\mathit{\text{resonance}}$$-$in the general context of dissipative systems subjected to spatially periodic $\mathit{\text{phase-modulated}}$ 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 $\mathit{\text{local}}\mathit{\text{symmetry}}$ 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.