Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.

The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential chaos. Using one-dimensional maps, we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In subexponential chaos, we show superweak chaos, which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth. The scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process, which is related to infinite ergodic theory.

Subexponential instability in one-dimensional maps implies infinite invariant measure.

We characterize dynamical instability of weak chaos as subexponential instability. We show that a one-dimensional, conservative, ergodic measure preserving map with subexponential instability has an infinite invariant measure, and then we present a generalized Lyapunov exponent to characterize subexponential instability.

Intermittency route to strange nonchaotic attractors in a non-skew-product map.

Whether strange nonchaotic attractors (SNAs) can typically arise in non-skew-product maps has been a crucial question for more than two decades. Recently, it was shown that SNAs arise in a particular non-skew-product map related to quasiperiodically driven continuous dynamical systems [R. Badard, Chaos, Solitons Fractals 28, 1327 (2006); Chaos 18, 023127 (2008)]. In the present paper, we derive Badard's non-skew-product map from a periodically driven continuous dynamical system with spatially quasiperiodic potential and investigate onset mechanisms of SNAs in the map. In particular, we focus on a transition route to intermittent SNAs, where SNAs appear after pair annihilations of stable and unstable fixed points located on a ring-shaped invariant curve. Then the mean residence time and rotation numbers have a logarithmic singularity. Finally, we discuss the existence of SNAs in a special class of non-skew-product maps.

Characterization of intermittency in renewal processes: application to earthquakes.

We construct a one-dimensional piecewise linear intermittent map from the interevent time distribution for a given renewal process. Then, we characterize intermittency by the asymptotic behavior near the indifferent fixed point in the piecewise linear intermittent map. Thus, we provide a framework to understand a unified characterization of intermittency and also present the Lyapunov exponent for renewal processes. This method is applied to the occurrence of earthquakes using the Japan Meteorological Agency and the National Earthquake Information Center catalog. By analyzing the return map of interevent times, we find that interevent times are not independent and identically distributed random variables but that the conditional probability distribution functions in the tail obey the Weibull distribution.

Spectral analysis and an area-preserving extension of a piecewise linear intermittent map.

We investigate the spectral properties of a one-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius-Perron operator has two simple real eigenvalues 1 and lambda(d) Epsilon (-1,0) and a continuous spectrum on the real line [0,1]. From these spectral properties, we also found that this system exhibits a power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.