Universal finite-sample effect on the perturbation growth in chaotic dynamical systems

The finite-sample effect on the growth of moments of the perturbation observed in numerical simulations of chaotic dynamical systems is studied. To numerically estimate the moments, only a limited number of sample trajectories can be utilized, and therefore the moments exhibit pure exponential growth only initially, and give way to relaxed growth thereafter. Such transition is a consequence of the unobservability of rare events in finite sample sets. Using the large-deviation formalism for chaotic time series, we estimate the relaxation time and derive the post-relaxation growth law. We demonstrate that even after the relaxation, each moment still obeys a universal growth law of different type, which reflects physical information on the statistics of chaotic expansion rates.

Figure 1

(Color online) Results for the modified Bernoulli map. The number of samples is $N=1000$, and the number of averages is $M=1000$. (a),(b) Growth of $\overline{\ln \left[q\right(t{)}^k]}$ and $\overline{\ln \left\{\right[q\left(t{)}^k\right]∕\left[q\right(t\left){]}^k\right\}}$ at several values of $k$ (the bottom curve corresponds to $k=0.5$, and the top curve to $5.0$, with curves in intervals of $0.5$ in between). (c),(d),(e) Growth rate $\lambda \left(k\right)$, intermittency exponent $\Lambda \left(k\right)$, and generalized Lyapunov exponent $h\left(k\right)$. Values measured in the initial period and those measured after the onset of relaxation are shown. Theoretical curves are also plotted for comparison. In (e), the ordinary Lyapunov exponent $h$ corresponding to the $k\to 0$ limit is also shown. (f) Rate function $S\left(u\right)$ of the finite-time Lyapunov exponent $u$. The theoretical curve, the Legendre transform of the numerically obtained $\lambda \left(k\right)$, direct numerical calculations using time intervals of $t=20$ and $t=50$, and the quadratic fit around the minimum are plotted.

Figure 2

(Color online) Results for the logistic map. The number of samples is $N=1000$, and the number of averages is $M=1000$. (a),(b) Growth of $\overline{\ln \left[q\right(t{)}^k]}$ and $\overline{\ln \left\{\right[q\left(t{)}^k\right]∕\left[q\right(t\left){]}^k\right\}}$ at several values of $k$ (the bottom curve corresponds to $k=0.5$, and the top curve to $5.0$, with curves in intervals of $0.5$ in between). (c),(d),(e) Growth rate $\lambda \left(k\right)$, intermittency exponent $\Lambda \left(k\right)$, and generalized Lyapunov exponent $h\left(k\right)$. Values measured in the initial period and those measured after the onset of relaxation are shown. Theoretical curves are also plotted. (f) Rate function $S\left(u\right)$ of the finite-time Lyapunov exponent. Theoretical curve and the results of direct numerical calculations using time intervals of $t=50$ and $t=100$ are plotted.

Figure 3

(Color online) Results of the modified Bernoulli map. (a) Growth of $\overline{\ln \left[q\right(t{)}^k]∕\{\left[q\right(t\left)\right]{\}}^k}$ obtained numerically at several values of $k$ (the bottom curve corresponds to $k=0.5$, and the top curve to $5.0$, with curves in intervals of $0.5$ in between) using $N=1000$ samples. (b) Rescaled relative moments obtained using $N=100$ and $N=1000$ samples at $2.0\le k\le 5.0$ with intervals of $0.5$. (c) Rescaled moments obtained using $N=100$ and $N=1000$ samples at $2.0\le k\le 5.0$ with intervals of $0.5$. (d) Double-logarithmic plot of the rescaled moments shown in (c). Theoretically expected stretched-exponential law with $t^{1∕2}$ is also shown. The number of averages is $M=1000$.

Figure 4

(Color online) Results of the Logistic map. (a) Rescaled numerical moments at $2.0\le k\le 5.0$ with curves in intervals of $0.5$ in between obtained using $N=100$ and $N=1000$. (b) Double-logarithmic plot of the rescaled numerical moments shown in (a). The number of averages is $M=50$.

Figure 5

(Color online) Results for the Duffing model. (a),(b) Growth of $\overline{\ln \left[q\right(t{)}^k]}$, and $\overline{\ln \left\{\right[q\left(t{)}^k\right]∕\mathbf{\left(}\right[q\left(t\right)\left]{\mathbf{)}}^k\right\}}$. (c) Rate function $S\left(u\right)$ obtained by direct numerical simulations using time intervals of $t=50$ and $t=100$. Quadratic fit around the infimum is also plotted. (d) Rescaled moments shown in double-logarithmic scales. The number of samples is $N=1000$ and the number of averages is $M=100$.

Figure 6

(Color online) Results for the Lorenz model. (a),(b) Growth of $\overline{\ln \left[q\right(t{)}^k]}$, and $\overline{\ln \left\{\right[q\left(t{)}^k\right]∕\mathbf{\left(}\right[q\left(t\right)\left]{\mathbf{)}}^k\right\}}$. (c) Rate function $S\left(u\right)$ obtained by direct numerical simulations using time intervals of $t=20$, $30$, and $40$. Linear fit around the infimum is also plotted. (d) Rescaled moments shown in double-logarithmic scales. The sample number is $N=100$ or $N=1000$, and the number of averages is $M=1000$.