The problem of anomalous diffusion in the extended Sinai billiard or periodic Lorentz gas is investigated by means of power spectral analysis. A logarithmic divergence of the power spectrum is observed and explained by a detailed analytic calculation, which yields a t−1 tail for the velocity autocorrelation function and a t ln t divergence for the mean-square displacement.

We report a new type of anomalous diffusion in chaotic systems with periodic symmetry. It is characterized by mean square displacements diverging faster than linearly in time. By a comparison of power spectra we conclude that the phenomenon has occurred in a recent observation of $\frac{1}{f}$ noise in Josephson junctions.

Intermittent diffusion arises through tangent bifurcations from drifting periodic orbits in dynamical systems. We show the existence of infinite sequences of parameter values where intermittent diffusion sets in. These sequences are found to converge geometrically and their rate of convergence is determined. In continuous-time approximations we calculate the velocity autocorrelation function, its power spectrum, and the meansquare displacement. The spectrum exhibits excess noise (ω−2) at low frequencies. The mean-square displacement grows liket2 below a crossover time.

In unbounded systems with discrete translational symmetry the Pomeau-Manneville scenario turns into a scenario involving intermittent diffusion. The velocity autocorrelation function, its power spectrum $S(\ensuremath{\omega})$, and the mean-square displacements ${\ensuremath{\sigma}}^{2}(t)$ are calculated. We find excess noise ($S\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}2}$) at low frequencies and anomalous diffusion (${\ensuremath{\sigma}}^{2}\ensuremath{\sim}{t}^{2}$) of transient duration. We explain that the phenomenon can easily be observed in driven Josephson junctions.

This paper reviews the universal critical properties exhibited by some 1d discrete dynamical systems: period-doubling systems and systems generating diffusive motion. While the period-doubling bifurcations have the universal asymptotic bifurcation rate δ=4.6692..., the tangent bifurcations present within the chaotic region do not follow this rate. We show that the tangent bifurcations giving rise to a fine structure of periodic windows have bifurcation rates γk which can be calculated analytically. They converge to a universal constant γ=2.94805... We have found that a class of dynamical systems show the onset of a diffusive motion in addition to period-doubling. The diffusion is self-generated and does not rely on the presence of random external forces. The onset of diffusion has strong analogies with a phase-transition. The diffusion coefficient is the order parameter and has a universal critical exponent. The dependence on random external fluctuations is also universal and can be expressed in terms of a universal scaling function which is calculated analytically.

A new type of universality associated with the onset of a deterministic diffusion for systems described by iterative one-dimensional maps is reported. The diffusion coefficient $D$ plays the role of an order parameter with a universal critical exponent. For the presence of external noise, the existence of a universal scaling function $d$ is shown. An analytic expression is derived for $d$ which is in good agreement with results of a numerical experiment.

We report on universal scaling properties for the onset of chaos in Id discrete dynamical systems. For period doubling systems we show that the fine structure of the chaotic region is governed by bifurcation rates γk, which are determined and which converge to a universal constant γ = 2.94805… In certain discrete systems a self-generated diffusion is observed which has critical properties in analogy to phase transitions. The diffusion coefficient is the order parameter with a universal critical exponent. For the presence of external noise a universal scaling function is shown to exist and is calculated analytically.

A new relation is reported which quantitatively describes the fine structure of the chaotic region of period-doubling systems. The relation determines the onset of fundamental periods and of ergodic behavior. It involves bifurcation rates ${\ensuremath{\gamma}}_{k}$, which converge to a new universal constant $\ensuremath{\gamma}=2.94805\dots{}$. This theory is in agreement with numerical determinations of ${\ensuremath{\gamma}}_{k}$.

For period-doubling bifurcations of 1 d-maps the Lyapunov number λ is calculated explicitly using a renormalization procedure. We find that its slope diverges like (δ/2)k. In the chaotic regime the unstable cycles yield a continuous curve as an upper bound for the Lyapunov exponent of chaotic bands. The bound has a critical exponent t = 0.449 80…, which is the same as for the chaotic bands.