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

Abstract

The fully nonlinear notion of resonance -$\textit{geometrical resonance}$- in the general context of dissipative systems subjected to $\textit{nonsteady}$ 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 steady. The geometrical resonance solutions represent a \textit{local symmetry} whose critical breaking leads to a new analytical criterion for the order-chaos threshold. This physical criterion is deduced in the co-moving frame from the local energy conservation over the shortest significant timescale. Remarkably, the new criterion for the onset of chaos is shown to be valid over large regions of parameter space, thus being useful beyond the perturbative regime and the scope of current mathematical techniques