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Oseledets' Splitting in Large Hamiltonian Systems: the HMF Phase Transition

We consider the covariant Lyapunov vectors (CLV) of a high-dimensional Hamiltonian flow in the case of long range potential, namely the Hamiltonian Mean Field (HMF) problem, by studying the behavior of the Lyapunov spectra and the Oseledets' splitting principal angles (the mutual orientation between stable and unstable subspaces) when a phase transition takes place. Motivated by several results connecting the dynamical properties of systems to their Lyapunov exponents and vectors, we first find confirmation of an explicit sensitivity of such quantities to the transition: our main finding is that the mutual orientation between stable and unstable subspaces fluctuates in time around specific constant values. Moreover, the fluctuations statistics are different above and below the critical threshold, and thus intimately connected to the transition. Pictorially, the evolution of the stability subspaces along a reference orbits turns out to happen only through rigid global rotations plus weak internal deformations between the two subspaces; we are not aware of any theoretical explanation for such a non- trivial setting but, through analogies with integrable systems, we provide some little insight on 'how much' the system resembles an integrable one.