High-order curvatures and harmonicity regression

We consider curvatures kappa(i) of all orders, as defined by the generalized Frenet-Serret formulae, along the trajectories of a classical Hamiltonian system with N degrees of freedom. In the spirit of previous experiments on the first two of them, time averages are numerically computed for the curvatures up to fifth order and for the microcanonical density in a typical anharmonic system (the FPU quartic chain), with checks in other models. Neat breakdowns of harmonic-like behaviour define thresholds to anharmonicity for every kappa(i) at distinct values (u) over tilde(i) of the order parameter (the energy density u). The threshold (u) over tilde(i) at fixed order i is independent of the total N, and it rapidly decreases as i grows. However, all curvatures are simultaneously sensitive or not to the initial conditions, for u < (u) over tilde(1) or u > (u) over tilde(1) respectively, confirming the previous identification of (u) over tilde(1) as an efficient indicator of the strong stochasticity transition. This phenomenology, which is discussed within the weak/strong stochasticity problem, gives a new insight into the progressive enforcement of a harmonic-like structure as u decreases.