Symbolic diffusion entropy rate of chaotic time series as a surrogate measure for the largest Lyapunov exponent

Existing methods for estimating the largest Lyapunov exponent from a time series rely on the rate of separation of initially nearby trajectories reconstructed from the time series in phase space. According to Ueda, chaotic dynamical behavior is viewed as a manifestation of random transitions between unstable periodic orbits in a chaotic attractor, which are triggered by perturbations due to experimental observation or the roundoff error characteristic of the computing machine, and consequently consists of a sequence of piecewise deterministic processes instead of an entirely deterministic process. Chaotic trajectories might have no physical reality. Here, we propose a mathematical method for estimating a surrogate measure for the largest Lyapunov exponent on the basis of the random diffusion of the symbols generated from a time series in a chaotic attractor, without resorting to initially nearby trajectories. We apply the proposed method to numerical time series generated by chaotic flow models and verify its validity.

Figure 1

Two-dimensional data points ($•$) occupying each sector partitioned by a diagonal plane (dashed line) on an attractor (elliptical circle, solid curve) and their diffusion to other sectors over a time of $\tau DT\mathrm{\Delta }t$.

Figure 2

Estimates of $\alpha$ as a function of ${\lambda }_1$ for the time series of the variable $x$ of the Rössler model. $D=5$.

Figure 3

Estimates of $\alpha$ as a function of ${\lambda }_1$ for the time series of the variable $z$ of the Rössler model. $D=5$.

Figure 4

Estimates of $\alpha$ as a function of ${\lambda }_1$ for the variable $x$ of the Lorenz model. $D=5$.

Figure 5

Estimates of $\alpha$ as a function of ${\lambda }_1$ for the variable $z$ of the Lorenz model. $D=5$.

Figure 6

Estimates of $\alpha$ as a function of ${\lambda }_1$ for the time series of the variable $x$ of the forced Duffing oscillator. $D=5$.

Figure 7

Estimates of $\alpha$ as a function of ${\lambda }_1$ for the time series of the variable $y$ of the forced Duffing oscillator. $D=5$.

Figure 8

Estimates of $\alpha$ as a function of ${\lambda }_1$ for the time series of the variable $x$ of the augmented Lorenz model. $D=5$.

Figure 9

Estimates of $h\left(\tau DT\mathrm{\Delta }t\right)=H\left(\tau DT\mathrm{\Delta }t\right)/{log}_2(D!)$ [denoted as $h\left(\tau \right)$ for simplicity] as a function of $\tau$ for the time series of the Ornstein-Uhlenbeck process. $T=2612$ and $D=5$. The initial alphabet is “12345”.