Decay of the distance autocorrelation and Lyapunov exponents

This work presents numerical evidence that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent. Distinct decay laws for the distance autocorrelation are observed for different systems, namely, exponential decays for the quadratic map, logarithmic for the Hénon map, and power-law for the conservative standard map. In all these cases the decay exponent is close to the positive Lyapunov exponent. For hyperchaotic conservative systems the power-law decay of the distance autocorrelation is not directly related to any Lyapunov exponent.

Figure 1

$D_A$ as a function of $\mathrm{\Delta }t$ for the QM (8) considering two periodic windows: $r=1.47786$ (period-6) and $r=1.77043$ (period-3).

Figure 2

(a) Time evolution of $D_A$ for four parameters corresponding to the chaotic regime of the QM (8). The parameters are $r=1.6$, 1.7, 1.9, and 2.0. The IC used is $x_0=0.1$, and the displacement of the states between the samples is $\mathrm{\Delta }t=1$. (b) $D_A$ as a function of $\mathrm{\Delta }t$ for the same parameters from panel (a).

Figure 3

(a) $D_A$ for $\mathrm{\Delta }t=1$ (blue) and $\mathrm{\Delta }t=2$ (orange) and (b) LE, both as a function of $r$. The initial condition used is $x_0=0.1$, and the parameter $r$ is divided into ${10}^3$ equally spaced parts.

Figure 4

LE as a function of $D_A$ for the cases $\mathrm{\Delta }t=1$ and $\mathrm{\Delta }t=2$. The functions for the adjustments are given in the inset.

Figure 5

Curves plotted in semilog scale of the $\langle D_A\rangle$ as a function of $\mathrm{\Delta }t$ for (a) $r=1.7$, (b) $r=1.9$, and (c) $r=2.0$. The functions for the adjustments (dashed black lines) are given in the panels.

Figure 6

(a) LE and the adjustment parameter $\beta$ in function of $r$ and (b) the relationship between $\lambda$ and $\beta$.

Figure 7

(a) Finite-time LE and (b) the corresponding $D_A\left(\mathrm{\Delta }t\right)$ (green curve) in semilog scale for the HM (9) with parameters $r=1.4$ and $b=0.3$. In panels (c) and (e), the finite-time LEs for the SM (10) with mixed and chaotic phase space are displayed, respectively. Panels (d) and (f) show the corresponding $D_A\left(\mathrm{\Delta }t\right)$ (orange curve) in the log-log scale. The dashed black lines in panels (b), (d), and (f) are the associated adjustments.

Figure 8

$D_A\left(\mathrm{\Delta }t\right)$ for CSMs (11, 12) with $N=2$ (a) for the mixed (circles in cyan) and (b) for the chaotic (squares in magenta) cases. With $N=4$, the mixed (green diamonds) and the chaotic (orange triangles) cases are displayed in panels (c) and (d), respectively. The dashed-black lines are the associated adjustment. Results are summarized in Table 1.