# Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems

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## Abstract

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## 1. Introduction

## 2. Designing the Complexity Measuring Algorithms

#### 2.1. Data Processing and Quantification

- Step 1:
- Normalization of time series. For given time series $\{{x}_{j}\left(n\right),n=1,2,3,\cdots ,N,j=1,2,\cdots ,d\}$, where d is the number of time series or the dimension of the chaotic system. Since amplitudes of different time series are different, normalization processing is necessary. The normalization function is given by:$${\tilde{x}}_{j}\left(n\right)=\frac{{x}_{j}\left(n\right)-min\left({x}_{j}\right)}{max\left({x}_{j}\right)-min\left({x}_{j}\right)}.$$
- Step 2:
- Coarse graining. To design multiscale complexity measuring algorithms, the multiscale coarse-grained processing should be carried out firstly. For the j-th time series, its consecutive coarse-grained time series is constructed by [23]:$${y}_{j}^{\tau}\left(k\right)=\frac{1}{s}\sum _{i=(k-1)s+1}^{ks}{\tilde{x}}_{j}\left(i\right),$$
- Step 3:
- Data quantification. For the given k and scale factor $\tau $, [${y}_{1}^{\tau}\left(k\right)$, ${y}_{2}^{\tau}\left(k\right)$, ⋯, ${y}_{d}^{\tau}\left(k\right)$] can be modeled as a pattern by introducing the idea of the Bandt–Pompe pattern [26]. Obviously, there are $d!$ possible patterns. Let the pattern space be given by $\Lambda =\left\{{\pi}_{1},{\pi}_{2},\cdots ,{\pi}_{d!}\right\}$, and thus, a pattern series $\left\{{\mathsf{\Psi}}^{\tau}\left(k\right):{\mathsf{\Psi}}^{\tau}\left(k\right)\in \Lambda ,k=1,2,\cdots ,\lfloor N/\tau \rfloor \right\}$ can be obtained. Moreover, let ${\pi}_{l}=l\left(l=1,2,\cdots ,d!\right)$; we can get a quantification pattern series, which is given by $\left\{{\Phi}^{\tau}\left(k\right):{\Phi}^{\tau}\left(k\right)\in \mathrm{N},k=1,2,\cdots ,\lfloor N/\tau \rfloor \right\}$.

#### 2.2. Complexity Measuring Algorithms

#### 2.2.1. Multiscale Multivariate Permutation Entropy

#### 2.2.2. Multiscale Multivariate Lempel–Ziv Complexity

- Step 1:
- Suppose that the quantification pattern series is $\left\{{\Phi}^{\tau}\left(k\right)={s}_{1},{s}_{2},{s}_{3},\cdots ,{s}_{N}\right\}$. Let S and Q be two character strings.
- Step 2:
- For the step n $(n=1,2,3,\cdots ,N)$, let $S=\left({s}_{1},{s}_{2},{s}_{3},\cdots ,{s}_{n}\right)$, and $Q={s}_{n+1}$ or $Q=\left({s}_{n+1},{s}_{n+2},\cdots ,{s}_{n+k}\right)$, then we get:$$SQ=\left({s}_{1},{s}_{2},{s}_{3},\cdots ,{s}_{n},{s}_{n+1}\right),$$$$SQ=\left({s}_{1},{s}_{2},{s}_{3},\cdots ,{s}_{n},{s}_{n+1},{s}_{n+2},\cdots ,{s}_{n+k}\right).$$Define:$$S{Q}_{v}=\left({s}_{1},{s}_{2},{s}_{3},\cdots ,{s}_{n}\right).$$$$S{Q}_{v}=\left({s}_{1},{s}_{2},{s}_{3},\cdots ,{s}_{n},{s}_{n+1},{s}_{n+2},\cdots ,{s}_{n+k-1}\right).$$If there exist an $i\phantom{\rule{4pt}{0ex}}(1\le i\le n)$ and the following relationship is satisfied:$$\left({s}_{n+1},{s}_{n+2},\cdots ,{s}_{n+k}\right)=\left({s}_{i},{s}_{i+1},\cdots ,{s}_{i+k-1}\right).$$
- Step 3:
- In Step 2, we obtained a series of dots; thus, we can calculate the number of dots and denote the complexity as $c\left(n\right)$.
- Step 4:
- According to [27], Lempel–Ziv complexity will reach a stable value, which is given by:$$L{Z}_{Stable}=\underset{n\to \infty}{lim}c\left(n\right)=\frac{n}{{log}_{2}\left(n\right)}.$$$$MMLZC=\frac{c\left(n\right)}{L{Z}_{stable}},\left(0\le MMLZC\le 1\right).$$

#### 2.2.3. Process for Complexity Measuring

- Step 1:
- Figure 1a. Solve the chaotic system and observe the state of the system based on the phase diagrams, preliminarily.
- Step 2:
- Figure 1b. Cut three segments of chaotic time series, which are the three state variables of the 3D chaotic system. Data processing and coarse graining are carried out by employing the method given in the Section 2.1.
- Step 3:
- Figure 1c. Quantize the scaled time series using the Bandt–Pompe approach; thus, a symbol time series is obtained.
- Step 4:
- Figure 1d. Estimate the MMLZC and MMPE according to the obtained sequence, where the steps of MMPE and MMLZC are shown in Section 2.2.1 and Section 2.2.2, respectively.
- Step 5:
- Figure 1e. Illustrate the complexity measuring results with different figures. Here, the two measures are shown in the MMPLC-MMPE plane.Note that we also illustrate the complexity with MMLZC and MMPE as shown by the curve and surfaces for comparison.

## 3. Complexity Analysis of Self-Reproducing Chaotic Systems

#### 3.1. Case A: One-Directional Self-Reproducing System

#### 3.2. Case B: Two-Directional Self-Reproducing System

## 4. Discussion

#### 4.1. Comparison with the Corresponding Original Systems

#### 4.2. Comparison of MMPE and MMLZC

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Steps to analyze the complexity of a chaotic system through the multiscale multivariate Lempel–Ziv complexity (MMLZC) and multiscale multivariate permutation entropy (MMPE) and algorithms. (

**a**) Chaotic systems; (

**b**) Chaotic time series; (

**c**) Permutation Vector Discretization; (

**d**) Measuring complexity based on $\Phi $; (

**e**) Show the results.

**Figure 2.**Coexisting attractors and complexity analysis results of System (13). (

**a**) Coexisting attractors; (

**b**) MMLZC; (

**c**) MMPE; (

**d**) MMLZC-MMPE plot.

**Figure 4.**Coexisting attractors and complexity analysis results of System (14). (

**a**) Coexisting attractors; (

**b**) MMLZC; (

**c**) MMPE; (

**d**) MMLZC-MMPE plot.

**Figure 5.**Complexity analysis results of System (14) with simultaneous variations of ${x}_{0}$ and ${y}_{0}$. (

**a**) MMPE analysis result; (

**b**) MMLZC analysis result.

**Figure 7.**Complexity analysis results of System (15). (

**a**) MMPE with ${y}_{0}$ and b; (

**b**) MMPE with ${y}_{0}$ and b; (

**c**) MMPE with ${y}_{0}$ and b; (

**d**) MMLZC with ${z}_{0}$ and b.

**Figure 9.**Dynamics of System (15) with $({x}_{0},{y}_{0},{z}_{0})=(0,0.1-82\pi ,-82\pi )$. (

**a**) Phase diagram; (

**b**) time series x; (

**c**) time series y; (

**d**) time series z.

**Figure 10.**States of System (15) under different scale factors. (

**a**) Phase diagram under $\tau =1$; (

**b**) quantification pattern series under $\tau =1$; (

**c**) probability distribution under $\tau =1$; (

**d**) phase diagram under $\tau =50$; (

**e**) quantification pattern series under $\tau =50$; (

**f**) probability distribution under $\tau =50$; (

**g**) phase diagram under $\tau =100$; (

**h**) quantification pattern series under $\tau =100$; (

**i**) probability distribution under $\tau =100$.

(Original, New) | ${\mathit{L}\mathit{E}}_{\mathit{m}\mathit{a}\mathit{x}}$ | MMPE | MMLZC |
---|---|---|---|

(Sys(16), Sys(13)) | (0.0363, 0.0285) | (1.3805, 1.5579) | (0.5402, 0.5376) |

(Sys(17), Sys(14)) | (0.1149, 0.1101) | (1.1610, 1.2657) | (0.5402, 0.5907) |

(Sys(18), Sys(15)) | (0.0938, 0.0890) | (1.7435, 1.7414) | (0.6483, 0.6976) |

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**MDPI and ACS Style**

He, S.; Li, C.; Sun, K.; Jafari, S.
Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems. *Entropy* **2018**, *20*, 556.
https://doi.org/10.3390/e20080556

**AMA Style**

He S, Li C, Sun K, Jafari S.
Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems. *Entropy*. 2018; 20(8):556.
https://doi.org/10.3390/e20080556

**Chicago/Turabian Style**

He, Shaobo, Chunbiao Li, Kehui Sun, and Sajad Jafari.
2018. "Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems" *Entropy* 20, no. 8: 556.
https://doi.org/10.3390/e20080556