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.

Figure 1

(a) Infinite Bernoulli scheme. Lines represent the transformation. (b) Ant-lion map. The dashed lines represent the envelopes of the map.

Figure 2

Bifurcation diagram and basins of attraction for the ant-lion map (10). (a) Bifurcation diagram. This diagram is drawn as follows: initial points $x_0^i$ are randomly chosen $(i=1,\cdots ,100)$. After ${10}^4$ iterations, we plot the values of $x_{{10}^4}^i$ $(i=1,\cdots ,100)$ for each parameter $A$. (b) Basins of attraction. Vertical axis represents initial point. The figure is drawn as follows: initial points are chosen as $i/100$ for each parameter $A=i/100$ ($i=1,\cdots ,100$), and then we plot the values of $x_{{10}^3}$ by their colors.

Figure 3

The generalized Lyapunov exponent. Symbols represent the results of numerical simulations for finite lengths $N$ of sums. The solid curve is the theoretical curve (23) without fitting parameter.

Figure 4

The generalized Lyapunov exponent of the log-Weibull map. The generalized Lyapunov exponent ${\Lambda }_0$ defined by Eq. (40) converges to $A_{LW}\cong 1.43$. Inset figure shows that $\langle S_n\rangle /lnn$ increases with $lnlnn$.