# Symplectic Entropy as a Novel Measure for Complex Systems

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Symplectic Entropy

**X**

_{m×d}in phase space can be reconstructed by time delay coordinates method:

**X**reflects the characteristics of the system. Then, a Hamilton matrix

**M**of the system $\mathit{f}(x)$ can be built from

**X**:

**Theorem**

**1.**

**Theorem**

**2.**

**H**is:

**H**is a symplectic unitary matrix. ${\varpi}^{*}$ is $\varpi $ conjugate transposition.

**Theorem**

**3.**

**S**can be decomposed as $\mathit{S}=\mathit{Q}\mathit{R}$, where

**Q**is a symplectic unitary matrix and

**R**is an upper triangle matrix.

**H**can be used as the matrix

**Q**. Then, the Hamilton matrix

**M**can be transformed into an upper Hessenberg matrix by the Household matrix

**H**,

**B**is upper Hessenberg matrix (

**b**

_{ij}= 0, i > j + 1). Meanwhile,

**H**can be obtained from the matrix

**A**[4]. The eigenvalues of matrix

**B**are given as $\lambda (\mathit{B})=\left\{{\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{d}\right\}$. The eigenvalues

**λ**(

**A**) of matrix

**A**are equal to those of matrix

**B**; i.e., $\mathit{\mu}=\lambda (\mathit{B})=\lambda (\mathit{A})$. The eigenvalues $\mathit{\mu}=\left\{{\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{d}\right\}$ are sorted in descending order; that is,

**A**with relevant symplectic orthonormal bases. They reflect the distribution of the energy of the system in different symplectic directions. The probability of the energy distribution in each direction can be defined, respectively, as ${\mathit{p}}_{1},{\mathit{p}}_{2},\cdots ,{\mathit{p}}_{d}$, and the probability space of the equation can be expressed as:

#### 2.2. Materials

#### 2.2.1. Synthetic Time Series

- Lorenz chaotic system:$$\begin{array}{l}\dot{x}=10(y-x)\\ \dot{y}=28x-y-xz\\ \dot{z}=xy-\frac{8}{3}z\end{array}$$
- Rössler chaotic system:$$\begin{array}{l}\dot{x}=-y-z\\ \dot{y}=x+0.15y\\ \dot{z}=0.2+z(x-10)\end{array}$$
- Van der Pol chaotic system:$$\ddot{x}-5(1-{x}^{2})\dot{x}+x=5\mathrm{cos}(2.466t)$$

#### 2.2.2. Real Time Series

#### 2.2.3. Surrogate Data and Null Hypothesis

**z**value is given:

_{orig}is the test statistic value for the original data, and $\langle {Q}_{s}\rangle $ is the mean of the statistic values for the surrogates. σ

_{s}is the standard deviation (SD) of the statistic values for the surrogates. z > 1.96 means that the null hypothesis can be rejected for two-sided testing at a 95% (α = 0.05) confidence level. For α = 0.05, the number of surrogates is

**B**= 2/α − 1 = 39 [36,38,45]. For this, 39 sets of surrogate data are generated for each analyzed data in this study.

## 3. Results

#### 3.1. Applicantion to Synthetic Time Series

#### 3.1.1. Tests on Gaussian White Noise Process

#### 3.1.2. Tests on Chaotic Dynamical Systems

#### 3.2. Application to Real Time Series

#### 3.2.1. The EEG for ASD and Healthy Subjects

#### 3.2.2. The Time Series for Diesel Engine and Air Compressor

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The probability values of the Gaussian white noise in different directions in dimension d = 3:5:28.

**Figure 2.**The characteristic test of the Gaussian white noise process: (

**a**) the symplectic entropy (SymEn) values of noise and its surrogate data; (

**b**) the approximate entropy (ApEn) values of noise and its surrogate data; (

**c**) the sample entropy (SampEn) values of noise and its surrogate data; (

**d**) the fuzzy entropy (FuzzyEn) values of noise and its surrogate data.

**Figure 3.**Comparison of nonlinear tests on four entropy methods for Lorenz chaotic dynamical system.

**Figure 4.**Comparison of nonlinear tests on four entropy methods for Rössler chaotic dynamical system.

**Figure 5.**Comparison of nonlinear tests on four entropy methods for Van de Pol chaotic dynamical system.

**Figure 6.**The nonlinearity test on the electroencephalogram (EEG) signal of ASD subject based on the SymEn method by comparison with the ApEn, SampEn, and FuzzyEn methods: (

**a**) The raw EEG data in the right O2; (

**b**) The SymEn functions of the raw data and its surrogates in the embedding dimension d = 2:25; (

**c**) The SymEn values in d = 25 for the raw data and its surrogates, * for the raw data; (

**d**) The ApEn values of the raw data and its surrogates in the embedding dimension d = 2:25; (

**e**) The SampEn values of the raw data and its surrogates in the embedding dimension d = 2:25; (

**f**) The FuzzyEn values of the raw data and its surrogates in the embedding dimension d = 2:25.

**Figure 7.**The nonlinearity test on the EEG signal of healthy subject: (

**a**) The raw EEG data in the right O2; (

**b**) In the range of 2 to 25 dimension, the SymEn functions of the raw data and its surrogates; (

**c**) The SymEn measures in d = 25 for the raw data and its surrogates; (

**d**) The ApEn values of the raw data and its surrogates in d = 2:25; (

**e**) the SampEn values of the raw data and its surrogates in d = 2:25; (

**f**) The FuzzyEn values of the raw data and its surrogates in d = 2:25.

**Figure 8.**The test on the abnormal sound in the diesel engine: (

**a**) The raw 1000-point sound time series; (

**b**) The SymEn functions of the raw data and its surrogates in d = 2:25; (

**c**) In d = 25, the SymEn measures of the raw data and its surrogates, * for the raw data, histogram for its surrogates; (

**d**) The ApEn values of the raw data and its surrogates in d = 2:25; (

**e**) The SampEn values of the raw data and its surrogates in d = 2:25; (

**f**) The FuzzyEn values of the raw data and its surrogates in d = 2:25.

**Figure 9.**The test on the acceleration time series in the vibration of the air compressor: (

**a**) The raw acceleration signal; (

**b**) The SymEn measures in d = 2:25 for the raw data and its surrogates; (

**c**) The SymEn measure (*) of the raw data in d = 25, and the histogram of the SymEn values for its surrogates; (

**d**) The ApEn values of the raw data and its surrogates in d = 2:25; (

**e**) The SampEn values of the raw data and its surrogates in d = 2:25; (

**f**) The FuzzyEn values of the raw data and its surrogates in d = 2:25.

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Lei, M.; Meng, G.; Zhang, W.; Wade, J.; Sarkar, N.
Symplectic Entropy as a Novel Measure for Complex Systems. *Entropy* **2016**, *18*, 412.
https://doi.org/10.3390/e18110412

**AMA Style**

Lei M, Meng G, Zhang W, Wade J, Sarkar N.
Symplectic Entropy as a Novel Measure for Complex Systems. *Entropy*. 2016; 18(11):412.
https://doi.org/10.3390/e18110412

**Chicago/Turabian Style**

Lei, Min, Guang Meng, Wenming Zhang, Joshua Wade, and Nilanjan Sarkar.
2016. "Symplectic Entropy as a Novel Measure for Complex Systems" *Entropy* 18, no. 11: 412.
https://doi.org/10.3390/e18110412