Number of first-passage times as a measurement of information for weakly chaotic systems

We consider a general class of maps of the interval having Lyapunov subexponential instability $|\delta x_t|\sim |\delta x_0|exp\left[{\Lambda }_t\left(x_0\right)\zeta \left(t\right)\right]$, where $\zeta \left(t\right)$ grows sublinearly as $t\to \infty$. We outline here a scheme [J. Stat. Phys. 154, 988 (2014)] whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate $\zeta \left(t\right)$. This matching approach is based on the infinite measure property of such systems. We show that the average information that is necessary to record without ambiguity a trajectory of the system tends to $\langle \Lambda \rangle \zeta \left(t\right)$, suitably extending the Kolmogorov-Sinai entropy and Pesin's identity. For such systems, information behaves like a random variable for random initial conditions, its statistics obeying a universal Mittag-Leffler law. We show that, for individual trajectories, information can be accurately inferred by the number of first-passage times through a given turbulent phase-space cell. This enables us to calculate far more efficiently Lyapunov exponents for such systems. Lastly, we also show that the usual renewal description of jumps to the turbulent cell, usually employed in the literature, does not provide the real number of entrances there. Our results are supported by exhaustive numerical simulations.

Figure 1

Schematic calculation of $\int \chi d\mu$ for a typical two branches map. For the horizontal line starting from $x_{<}$ $\left(x_{>}\right)$, blue upper and red lower stripes denote intervals for which $\sigma \left(x\right)\sigma \left[T\right(x\left)\right]$ and $\sigma \left[T\right(x\left)\right]$ have nonzero measurements, respectively. Crosshatched stripes depict differences between corresponding intervals, i.e., nonzero measurement of $\chi$ according to Eq. (29).

Figure 2

Graphics of the algorithm complexity $C_t$, calculated by Eq. (18), as a function of $N_t$, the number of entrances of a given trajectory into the cell $A_1=(x_{*},1]$ during $t={10}^6$ iterations of Thaler's map with $\alpha =1/2$. The slopes $\gamma$ for each straight line were calculated according to Eqs. (32) and (33) for the same map. Each straight line is matched by $4\times {10}^3$ numerical points obtained from initial conditions uniformly distributed on the interval $[0,1]$. Solid red (dashed blue) lines depict partitions with $x_{*}>c$ $\left(x_{*}<c\right)$, $x_{*}$ increasing in the counterclockwise (clockwise) direction. Critical line comes from the standard partition $x_{*}=c$. In all cases, the error in $\gamma$ is less than $2\%$ of the corresponding theoretical value.

Figure 3

Solid lines from top $\left(\alpha =0.3\right)$ to bottom $\left(\alpha =0.8\right)$ represent the semi-log plot of $\gamma$ according to Eqs. (32) and (33) for Thaler's map. Each numerical point was obtained from the map iterations and by least squares fitting of $C_t\to \gamma N_t$ over ${10}^3$ initial conditions uniformly distributed on the interval $[0,1]$.

Figure 4

Distribution of the random variable ${\xi }_{\alpha }=N_t/\langle N_t\rangle$ for Thaler's map with $\alpha =2/3,t={10}^8$, and standard partition of phase space. Both histograms were built from the same $2.5\times {10}^5$ initial conditions uniformly distributed on the interval $[0,1]$. The background and the inset plots correspond, respectively, to rates ${\xi }_{\alpha }$ whereby $\langle N_t\rangle$ was employed according to Eqs. (35) and (43). For both cases, solid line is the Mittag-Leffler probability density with $\alpha =2/3$ and unit expected value. As one can see, the background distribution behaves according to our prediction in Sec. 5, contrary to the renewal approach provides (inset).