Modulation instability in the region of the minimum group-velocity dispersion is analyzed by means of an extended nonlinear Schrodinger equation. It is shown that the critical modulation frequency saturates at a value determined by the fourth-order dispersion. Experimental results demonstrate the viability of generating a train of femtosecond pulses with repetition rates of a few terahertz in reasonable agreement with the theory.

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Low-dimensional non-linear maps are prototype models to study the emergence of complex behavior in nature. They may exhibit power-law sensitivity to initial conditions at the edge of chaos which can be naturally formulated within the generalized Tsallis statistics prescription which i s c haracterized by the entropic index q. General scaling arguments provide a direct relation between the entropic index q and the scaling exponents associated with the extremal sets of the multifractal critical attractor. The above result comes in favor of recent conjectures that Tsallis statistics is the natural frame for studying systems with a fractal-like structure in the phase-space. Power-law sensitivity i n high-dimensional dissipative and Hamiltonian systems are also discussed within the present picture. I IntroductionLow-dimensional non-linear maps are the prototype models to study the emergence of complex behavior in dynamical systems. Their typical behavior include the occurrence of bifurcation instabilities, long-range correlated sequences, fractal structures and chaos, which are commonly observed in a great variety of systems ranging from uids, magnetism, biology, social sciences and many others 1 .The study of the sensitivity to initial conditions of non-linear systems is one of the most important tools used to investigate the nature of the phase-space attractor. It is usually characterized by the Liapunov exponent , de ned for the simple case of a one-dimensional dynamical variable x as xt x0e t x0 ! 0; t ! 1 1 where x0 is the distance between two initially nearby orbits in an equivalent point of view, it is the uncertainty on the precise initial condition. If 0 the system is said to be strongly sensitive to the initial condition with the uncertainty on the dynamical variable growing exponentially in time and this characterizes a chaotic motion in the phase-space. On the other hand, if 0 the system becomes strongly insensitive to the initial condition which is expected for any state whose dynamical attractor is an orbit with a nite period. The problem of the sensitivity to initial conditions can be reformulated in an entropic language as a process of information loss in the case of chaotic behavior or recovery for periodic attractors. Within this context, it is useful to introduce the Kolmogorov-Sinai entropy K. It is basically the rate of variation of the Boltzmann- where S0 and SN are the entropies of the system evaluated at times t = 0 and t = N for maps = 1. With the simplifying assumption that at time t there are Wt occupied cells with the same occupation number, we h a ve from equation 2 that Wt = W 0e Kt ;3 which is equivalent to equation 1 for the sensitivity to initial condition and provides the well-known Pesin

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