Estimation of the degree of dynamical instability from the information entropy of symbolic dynamics

A positive Lyapunov exponent is the most convincing signature of chaos. However, existing methods for estimating the Lyapunov exponent from a time series often give unreliable estimates because they trace the time evolution of the distance between a pair of initially neighboring trajectories in phase space. Here, we propose a mathematical method for estimating the degree of dynamical instability, as a surrogate for the Lyapunov exponent, without tracing initially neighboring trajectories on the basis of the information entropy from a symbolic time series. We apply the proposed method to numerical time series generated by well-known chaotic systems and experimental time series and verify its validity.

Figure 1

Histograms of alphabets for the time series of the variable $x$ of the Lorenz model. $T=0.01$ ($k=1$) and $D=4$. (a) PE and (b) SEG (closed triangles) and SEL (open circles).

Figure 2

Histograms of alphabets for the time series of the variable $x$ of the Lorenz model. $T=3$ ($k=300$) and $D=4$. (a) PE and (b) SEG (closed triangles) and SEL (open circles).

Figure 3

Estimates of $h$ as a function of $T$ for the time series of the variable $x$ of the Lorenz model. $D=4$. (a) PE, (b) SEG, and (c) SEL. Dashed curves indicate the approximate functions defined by Eqs. (5) and (8), where the coefficient $c$ was determined using the estimates of $\alpha$, $\lambda$, and $h\left(0\right)$. The largest Lyapunov exponent $\lambda$ is cited from Ref. [19].

Figure 4

Plots of $\alpha$ versus the largest Lyapunov exponent $\lambda$ for (a) PE, (b) SEG, and (c) SEL.

Figure 5

Histograms of alphabets for the time series of the variable $x$ of the Lorenz model for SEG (closed triangles) and SEL (open circles) at $T=0.01$. (a) $D=5$ and (b) $D=6$.

Figure 6

Flame front location of an unstable swirling premixed flame as a function of time [20].

Figure 7

Estimates of $h$ as a function of $T$ for the time series of the flame front location of an unstable swirling premixed flame. $D=4$. (a) PE, (b) SEG, and (c) SEL. Dashed curves indicate the approximate functions defined by Eqs. (5) and (8).

Figure 8

Largest Lyapunov exponent $\lambda$ as a function of the embedding dimension for the flame front location of an unstable swirling premixed flame.

Figure 9

Flame front location of buoyancy-induced flame oscillations as a function of time [21].

Figure 10

Estimates of $h$ as a function of $T$ for the time series of the flame front location of buoyancy-induced flame oscillations. $D=4$. (a) PE, (b) SEG, and (c) SEL. Dashed curves indicate the approximate functions defined by Eqs. (5) and (8).

Figure 11

Largest Lyapunov exponent $\lambda$ as a function of the embedding dimension for the flame front location of buoyancy-induced flame oscillations.