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We investigate the connection between local and global dynamics of twoN-degree of freedom Hamiltonian systems with different origins describingone-dimensional nonlinear lattices: The Fermi-Pasta-Ulam (FPU) model and adiscretized version of the nonlinear Schrodinger equation related toBose-Einstein Condensation (BEC). We study solutions starting in the vicinityof simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phasemotion (OPM), which are known in closed form and whose linear stability can beanalyzed exactly. Our results verify that as the energy E increases for fixedN, beyond the destabilization threshold of these orbits, all positive Lyapunovexponents exhibit a transition between two power laws, occurring at the samevalue of E. The destabilization energy E_c per particle goes to zero as N goesto infinity following a simple power-law. However, using SALI, a very efficientindicator we have recently introduced for distinguishing order from chaos, wefind that the two Hamiltonians have very different dynamics near their stableSPOs: For example, in the case of the FPU system, as the energy increases forfixed N, the islands of stability around the OPM decrease in size, the orbitdestabilizes through period-doubling bifurcation and its eigenvalues movesteadily away from -1, while for the BEC model the OPM has islands around itwhich grow in size before it bifurcates through symmetry breaking, while itsreal eigenvalues return to +1 at very high energies. Still, when calculatingLyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunovexponents decrease following an exponential law and yield extensiveKolmogorov--Sinai entropies per particle, in the thermodynamic limit of fixedenergy density E/N with E and N arbitrarily large.Comment: 29 pages, 10 figures, published at International Journal of  Bifurcation and Chaos (IJBC

CHAOTIC DYNAMICS OF N-DEGREE OF FREEDOM HAMILTONIAN SYSTEMS