Lyapunov exponents in high-dimensional systems

Although Lyapunov exponents are well established tools, some features of the tangent-space dynamics and even basic properties of the Lyapunov spectra have remained unexplored for a long time. One of the novel directions is the extension of the multifractal formalism to space-time chaos to quantify the effective interactions among the various degrees of freedom. This is done by studying the Lyapunov-exponents fluctuations and their scaling behaviour. Moreover, fluctuations, as well as the structure of the associated covariant vectors, can help to quantify the hyperbolicity of the system dynamics. In globally coupled systems, even the Lyapunov spectrum must be carefully analysed, as it may reveal peculiar features, such as the presence of a non-extensive component. A prominent such example is the Hamiltonian mean field, in which case, even the question whether the maximum Lypaunov exponent is strictly larger than zero in the thermodynamic limit remained open for a long time. Another important question is whether the standard (microscopic) Lyapunov exponents can capture the presence of collective motion. This issue is discussed with reference to an ensemble of rotators.