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Characterizing weak chaos in nonintegrable Hamiltonian systems: the fundamental role of stickiness and initial conditions

Weak chaos in high-dimensional conservative systems can be characterized through sticky effect induced by invariant structures on chaotic trajectories. Suitable quantities for this characterization are the higher cummulants of the finite time Lyapunov exponents (FTLEs) distribution. They gather the {\it whole} phase space relevant dynamics in {\it one} quantity and give informations about ordered and random states. This is analyzed here for discrete Hamiltonian systems with local and global couplings. It is also shown that FTLEs plotted {\it versus} initial condition (IC) and the nonlinear parameter is essential to understand the fundamental role of ICs in the dynamics of weakly chaotic Hamiltonian systems.