A continuous-time dynamic model of a network of N nonlinear elements interacting via random asymmetric couplings is studied. A self-consistent mean-field theory, exact in the N~limit, predicts a transition from a stationary phase to a chaotic phase occurring at a critical value of the gain parameter. The autocorrelations of the chaotic flow as well as the maximal Lyapunov exponent are calculated.

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Abstract. This review reports on the research done during the past years on violations of the fluctuation-dissipation theorem (FDT) in glassy systems. It is focused on the existence of a quasi-fluctuation-dissipation theorem (QFDT) in glassy systems and the currently supporting knowledge gained from numerical simulation studies.It covers a broad range of non-stationary aging and stationary driven systems such as structural-glasses, spin-glasses, coarsening systems, ferromagnetic models at criticality, trap models, models with entropy barriers, kinetically constrained models, sheared systems and granular media. The review is divided into four main parts: 1) An introductory section explaining basic notions related to the existence of the FDT in equilibrium and its possible extension to the glassy regime (QFDT), 2) A description of the basic analytical tools and results derived in the framework of some exactly solvable models, 3) A detailed report of the current evidence in favour of the QFDT and 4) A brief digression on the experimental evidence in its favour. This review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the QFDT as well as for the more expert readers who may be interested in more specific results.

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The average eigenvalue distribution p(X) of N*N real random asymmetric matrices /,, (//, ?*,/,,) is calculated in the limit of TV-* «>. It is found that piX) is uniform in an ellipse, in the complex plane, whose real and imaginary axes are 1 + r and 1 -r, respectively. The parameter r is given by z=N [JijJji]j and N[jjj]j is normalized to 1. In the T = 1 limit, Wigner's semicircle law is recovered. The results are extended to complex asymmetric matrices.

We analyze the Thouless-Anderson-Palmer (TAP) approach to the spherical p-spin spin glass model in zero external field. The TAP free energy is derived by summing up all the relevant diagrams for $N\to \infty$ of a diagrammatic expansion of the free energy. We find that if the multiplicity of the TAP solutions is taken into account, there is a first order transition in the order parameter at the critical temperature $T_{\rm c}$ higher than that predicted by the replica solution $T_{\rm RSB}$, but in agreement with the results of dynamics. The transition is of "geometrical" nature since the new state has larger free energy but occupies the largest volume in phase space. The transition predicted by the replica calculation is also of "geometrical" nature since it corresponds to the states with smallest free energy with positive complexity

We investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales. In particular, we study the case of 3D turbulence at high Reynolds numbers by introducing a finite-size Lyapunov exponent which measures the growth rate of finite-size perturbations. For sufficiently small perturbations this quantity coincides with the usual Lyapunov exponent. When the perturbation is still small compared to large-scale fluctuations, but large compared to fluctuations at the smallest dynamically active scales, the finite-size Lyapunov exponent is inversely proportional to the square of the perturbation size. Our results are supported by numerical experiments on shell models. We find that intermittency corrections do not change the scaling law of predictability. We also discuss the relation between finite-size Lyapunov exponent and information entropy. PACS numbers 47.27.Gs, 05.45.+b, 47.27.Jv