Experimental Evidence of a Relationship between Permutation Entropy and the Largest Lyapunov Exponent in a Glow Discharge Plasma

Abstract

Complexity-entropy (CH) analysis has been applied to time-series data from fluctuating glow-discharge plasmas, and the permutation entropy (H) and statistical complexity (C) of emission-intensity signals have been evaluated as the discharge current is varied. Permutation entropy provides information distinct from the standard deviation of time-series data and enables characterization of the power spectrum distribution. The dynamical properties embedded in the time-series data are visualized and quantified using the CH diagram. Direct evidence of a relationship between permutation entropy and the largest Lyapunov exponent is reported here for the first time in a plasma experiment. These results are expected to contribute to the further development and application of CH analysis in plasma research.

1.  Introduction

Plasmas are well known for exhibiting a range of nonlinear behaviors, including intermittency, chaos, turbulence, and transitions among dynamical states. Characterizing these dynamics is crucial not only for advancing fundamental plasma physics but also for practical applications such as magnetically confined fusion and industrial plasma processing. Accordingly, developing easily and reliably measurable indicators of dynamical properties has become an important objective in plasma diagnostics and in achieving more flexible plasma control.

The complexity-entropy (CH) analysis, which uses two statistical measures—permutation entropy ($H$) and statistical complexity ($C$)—has been recognized as an effective method for measuring the complexity of time-series data. Permutation entropy relies on the data’s order rather than its amplitudes, and the CH analysis offers strong noise resistance and computational efficiency [1]. These benefits make the CH analysis especially useful for analyzing experimental data. By classifying these two statistics on the CH plane, it becomes possible to discuss the dynamic properties of the underlying system both intuitively and quantitatively [2].

Although several applications of the CH analysis to plasma experiments have been reported [3–9], it has not yet been widely adopted as a primary analytical method. There are very few examples in which experimental conditions have been systematically varied—from periodic states to stochastic states—to comprehensively examine the system’s behavior. Moreover, numerical studies have shown that the permutation entropy correlates with the largest Lyapunov exponent in the logistic map [1]. However, to the best of the author’s knowledge, no direct confirmation of the relationship between permutation entropy and the Lyapunov exponent has been reported in plasma experiments. As a result, the applicability and validity of the CH analysis to the fluctuation analysis in plasmas have not yet been fully established. Experiments using a simple system that reproduces a wide range of fluctuation behaviors and enables comprehensive characterization of their properties—specifically, the accumulation of experimental data on the correlation between the permutation entropy and other conventionally used quantities—are valuable for expanding the scope of the CH analysis.

In this paper, we report experimental results from applying the CH analysis to time-series data of fluctuations excited in a DC glow-discharge plasma. We demonstrate, for the first time, a clear positive correlation between the permutation entropy and the largest Lyapunov exponent in plasma experiments. The findings provide evidence that the CH analysis is an effective method for quantifying fluctuation characteristics and confirm that the permutation entropy is a practical diagnostic indicator.

2.  Complexity-entropy Analysis

We provide brief explanations of the statistical quantities used in the CH analysis. Their definitions are identical to those commonly used; for a detailed explanation, please refer to the relevant literature [2, 10].

Let us now consider a time-series data consisting of $N$ data points, as $\{x_t\}=\{x_1,x_2,\mathrm{\dots },x_N\}$, where $x$ and $t$ are the measured quantity and the time index, respectively. The time series is transformed into a sequence of ordinal patterns using an embedding procedure characterized by an embedding dimension ($d$) and an embedding delay ($\tau$ = 1, 2, …). For each time step, a d-dimensional vector is constructed by ${\mathit{X}}_t=(x_t,x_{t+\tau },x_{t+2\tau },\mathrm{\dots },x_{t+(d-1)\tau })$. Then, the elements of each ${\mathit{X}}_{t=1,2,\mathrm{\dots }}$ are arranged in order according to their relative magnitudes, forming a permutation. Since there are $d!$ possible permutation patterns, the probability distribution $P={\{p_i\}}_{i=1}^{d!}$ can be calculated, where $p_i$ ($i$ = 1, 2, … $d!$), is the probability satisfying ${\sum }_i^{d!}{p}_i$ = 1. Here, in the presence of equal values within a vector, the element corresponding to the earlier time is defined to be smaller. However, such cases are very rare in our experiments and do not affect the results presented below.

The permutation entropy is obtained by calculating the Shannon entropy of $P$, and the normalized permutation entropy ($H$) is defined as

  

$$H[P]=-\frac{1}{\log d!}\sum _{i=1}^{d!}{p}_i\log p_i,$$(1)

which takes values in the range 0 ≤ $H$ ≤ 1; for a monotonic signal, the value is close to zero ($H$ = 0), and it becomes unity for a completely random signal ($H$ = 1) [1].

Statistical complexity refers to the statistical distance between the probability distribution ($P$) and a uniform probability distribution ($P_e={\{\frac{1}{d!}\}}_{i=1}^{d!}$). The Jensen–Shannon statistical complexity ($C$) [11, 12] is defined as

  

$$C[P]=Q_J[P,P_e]H[P].$$(2)

A quantity $Q_J$ indicates the Jensen-Shannon divergence defined by

  

$$Q_J[P,P_e]=Q_{J,\text{max}}^{\mathrm{-1}}\left(H\left[\frac{P+P_e}{2}\right]-\frac{H[P]+H[P_e]}{2}\right)\log d!,$$(3)

where

  

$$Q_{J,\text{max}}=-\frac{1}{2}\left[\frac{d!+1}{d!}\log (d!+1)-2\log (2d!)+\log d!\right].$$(4)

This formulation ensures that the statistical complexity vanishes in both the monotonically varying time-series case ($H$ = 0) and the completely random case ($H$ = 1), taking intermediate values when structured patterns are present.

A pair of ($H$, $C$) values derived from the time-series data is mapped to a point on the CH plane, providing an intuitive understanding of the dynamical properties inherent in the data. Monotonically varying signals typically appear in regions of low entropy and low complexity, while stochastic signals tend to cluster in regions of high entropy and low complexity. Chaotic systems, on the other hand, are found in an intermediate region characterized by moderate entropy and high complexity, in which more disordered chaotic states have higher permutation entropy.

Although chaotic and stochastic signals may both exhibit irregular and seemingly unpredictable fluctuations, their underlying generation mechanisms are fundamentally different. Chaotic signals arise from deterministic nonlinear dynamics and exhibit sensitivity to initial conditions, whereas stochastic signals originate from random processes without deterministic evolution. As a result, chaotic dynamics preserve some temporal correlations in the sequence of ordinal patterns, while stochastic signals exhibit a more random distribution of these patterns. This difference in ordinal organization leads to higher statistical complexity for chaotic signals than for stochastic signals with comparable entropy values. Therefore, the trajectories of data points on the CH plane provide information about changes in the dynamical states [2, 12].

3.  Experimental Setup

In this work, time-series data are obtained from a DC glow-discharge plasma. A schematic of the experiment is shown in Fig. 1(a), and the fundamental experimental settings are the same as those in Ref. [13]. A neon (Ne) plasma is generated in a sealed cylindrical glass tube with an axial length of 300 mm and a diameter of 10 mm. In the plasma produced within this simple device—consisting of the discharge tube, a stabilizing resistor (25 kΩ), and a DC power supply—kHz-range fluctuations occur spontaneously, and their characteristics vary with the discharge current ($I_{\text{d}}$). A photodiode is placed at the center of the discharge tube to measure the emission intensity of Ne-I, and the typical signals are shown in Figs. 1(b) and (c). The spatial resolution of the emission-intensity measurements is 1.5 mm, and the upper frequency limit of the measurement system in this experimental setup is 50 kHz.

Fig. 1.  (a) A schematic diagram of the experimental setup. The typical emission intensity signals are shown in the bottom: $I_{\text{d}}$ = (b) 5, (c) 8.5 mA.

Although we do not discuss the spatial characteristics or dispersion relations of the fluctuations, we have confirmed that the observed fluctuations, as shown in Fig. 1, are backward waves that exhibit characteristics of ionization waves [14, 15]. In this paper, we focus on the characteristics of time-series emission-intensity data as the discharge current varies, using CH, Fourier-based, and Lyapunov analyses.

4.  Results and Discussions

Figure 2 shows the discharge current dependence of (a) the power spectrum, (b) standard deviation ($\mathit{\sigma }$), and (c) $H$ at $(d,\tau )$ = (5, 10), evaluated from time-series data consisting of $N$ = 2 × 10⁴ data points. In the present calculation of the CH statistics, the embedding delay is fixed at $\tau$ = 10, which is close to the value at which the mutual information reaches its first local minimum [16]. The temporal resolution of the measurements is $\mathrm{\Delta }t$ = 5 μs; therefore, the time t and the embedding delay $\tau$ on the real-time scale (in seconds) are obtained by multiplying their indices by $\mathrm{\Delta }t$. The standard deviation, which reflects the magnitude of fluctuations, increases monotonically with the discharge current. By contrast, the shape of the power spectrum does not vary monotonically with the discharge current, a feature of DC glow-discharge plasmas associated with mode transitions [17–19]. The characteristics of the power spectrum vary between coherent and broadband regimes depending on the discharge current range. Notably, the permutation entropy appears to qualitatively capture the shape of the power spectrum: it takes larger values as the power spectrum broadens, and vice versa. This provides evidence that permutation entropy is an effective statistical measure for extracting and quantifying fluctuations independently of their amplitude.

Fig. 2.  Dependence of (a) the power spectrum of the emission intensity, (b) the standard deviation, $\mathit{\sigma }$, and (c) the permutation entropy, $H$ ($d$ = 5), of the spontaneous emission intensity on the discharge current.

As shown in Fig. 3, the dynamical properties under different discharge conditions are depicted on the CH plane, which is bounded by $C_{\text{min}}$ and $C_{\text{max}}$. Two computationally generated data sets are also shown in this figure: one is a sine wave with a typical dominant frequency observed in the experiments, represented by diamonds, and the other is fractional Brownian motion (fBm) [20], represented by a dashed line. The fBm data is calculated by varying the Hurst exponent from 0 to 1. All the experimental data are in a relatively high-complexity region compared with the fBm process and appear to lie on a specific curve. These observations suggest that the measured time series exhibit structured ordinal organization that is less likely to arise in purely stochastic signals. As the discharge conditions vary, the corresponding trajectories on the CH plane indicate changes in the structural organization of the time series, while the time series remains in a relatively high-complexity region. Therefore, the CH plane provides a useful means of visualizing and comparing changes in the structural properties of the time series, even when the permutation entropy alone exhibits complex behavior under varying experimental conditions.

Fig. 3.  CH diagram of experimental data for each discharge current (circles); $d$ = 5, $\tau$ = 10. A sine wave with a frequency of 8.3 kHz is represented by diamonds. The dashed line indicates the fractional Brownian motion (fBm) with the Hurst exponent ranging from 0 to 1.

The results presented so far demonstrate that CH analysis effectively extracts structural properties from experimentally obtained time-series data. To further investigate the characteristics of the permutation entropy, we present its relationship with the largest Lyapunov exponent ($\lambda$) in Fig. 4, using the same dataset as in Figs. 2 and 3. The value of $\lambda$ is calculated using the standard procedure [21]; for a sine wave (or a limit cycle), $\lambda$ is zero, whereas for chaotic dynamics, it is positive. Furthermore, in the figure, $\lambda$ is normalized by the fundamental frequency ($f_0$) for each discharge current.

Fig. 4.  Permutation entropy vs. the largest Lyapunov exponent. The Lyapunov exponent is normalized by the dominant frequency in each discharge current, $f_0$.

As shown in the figure, a positive correlation between $H$ and $\lambda$ is observed in the region of large Lyapunov exponents. In contrast, in the region where $\lambda \sim 0$, $H$ remains nearly constant. This implies that for time series exhibiting chaotic or weakly turbulent behavior, the permutation entropy can be used to estimate the largest Lyapunov exponent. Furthermore, this constitutes the first experimental demonstration of the positive correlation through direct comparison and provides evidence that this relationship holds in actual plasmas.

It should be noted that the positive correlation between the permutation entropy and the largest Lyapunov exponent does not hold universally for dynamical models in general. In this sense, the permutation entropy is not a quantity that can always be used to evaluate the largest Lyapunov exponent. Nevertheless, compared to methods that directly compute the largest Lyapunov exponent, the CH analysis offers greater robustness to noise and lower computational cost. The ability to estimate the largest Lyapunov exponent from the permutation entropy is a significant advantage for efficiently characterizing the underlying system. It is of interest whether the relationship between the permutation entropy and the largest Lyapunov exponent holds in other plasmas.

Finally, we show how permutation entropy varies with the embedding dimension. Figure 5 shows the discharge current dependence of the permutation entropy in the cases of $d$ = 3, 4, and 5, in which the data in $d$ = 5 is the same as that shown in Fig. 2(c). The permutation entropy exhibits qualitatively similar trends regardless of the embedding dimension, while its variation (i.e., the dynamic range) increases with embedding dimension. These results suggest that the insights obtained in the present study are not limited to any specific embedding dimension. The present conclusions are based on the robustness of the observed trends over the investigated embedding dimensions, while the asymptotic behavior for larger embedding dimensions remains an interesting subject for future investigation.

Fig. 5.  Discharge current dependence of the permutation entropy for each embedded dimension. The data for $d$ = 3, 4, and 5 are represented by triangles, squares, and circles, respectively.

In the CH analysis, the embedding dimension is a critical parameter that determines the required length of the time series, as the condition $N$ ≫ $d!$ must be satisfied for the probability distribution to converge sufficiently. A smaller value of $d$ reduces computational cost, but also lowers the resolution at which patterns inherent in the time-series data can be resolved. Hence, a larger embedding dimension is preferable for a more precise characterization of time-series data, whereas choosing the embedding dimension as small as possible enables computationally efficient analysis. The optimal choice of embedding dimension likely depends on the intended application—for example, whether the CH analysis is applied to tasks requiring high temporal responsiveness, such as feedback control, or to rigorous dynamical state analysis, such as comparison with theoretical models.

5.  Conclusions

The results demonstrate that permutation entropy reflects the structural properties of the power spectrum and is positively correlated with the largest Lyapunov exponent. These findings suggest that permutation entropy can serve as a fundamental statistical measure capable of capturing intrinsic dynamical features embedded in experimental data. Furthermore, unlike conventional statistical quantities such as the mean and standard deviation, permutation entropy provides complementary information related to the complexity and dynamical organization of fluctuations.

A particularly important outcome of this study is the first direct experimental confirmation of a positive correlation between permutation entropy and the largest Lyapunov exponent in a plasma system. By systematically varying discharge conditions and analyzing the resulting changes in fluctuation characteristics, this work extends and deepens previous experimental studies within the CH analysis framework.

Overall, the results demonstrate that CH analysis is a powerful and versatile tool for characterizing nonlinear plasma fluctuations exhibiting diverse dynamical behaviors and is expected to have broad applicability in plasma research.

Acknowledgements

This study was partially supported by the JSPS KAKENHI (grant number: JP23K03355) and the Collaborative Research Program of Research Institute for Applied Mechanics, Kyushu University (2026S2-CD-8).

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