# Information Transfer Among the Components in Multi-Dimensional Complex Dynamical Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Two-Dimensional Formalism of Information Transfer (the LK2005 Formalism [6])

#### 2.1. Continuous Flows

#### 2.2. Discrete Mappings

## 3. n-Dimensional Formalism of Information Transfer

#### 3.1. Continuous Flows

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1**

#### 3.2. Discrete Mappings

## 4. The Application of Multi-Dimensional Formalism of Information Transfer

#### 4.1. The Lorenz System

- Initialize the joint density $\rho ({x}_{1},{x}_{2},{x}_{3})$ with a preset distribution ${\rho}_{0}$, then generate an ensemble through drawing samples randomly according to the initial distribution ${\rho}_{0}$.
- Partition the sample space $\mathsf{\Omega}$ into “bins”.
- Obtain an ensemble prediction for the Lorenz system at every time step.
- Estimate the three-variable joint probability density function $\rho $ via bin counting at every time step.

#### 4.2. The Chua’s System

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Discrete Mappings

**Theorem**

**A1.**

#### Appendix A.1.

**Proof**

**of**

**Theorem**

**A1**

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**Figure 2.**Left panel: a sample data (${X}_{1},{X}_{2}$ and ${X}_{3}$) of the Lorenz system generated by a fourth order Runge–Kutta method with $\Delta t=0.01$. Right panel: an estimated marginal density of ${x}_{1},{x}_{2}$ and ${x}_{3}$ via counting the bins and initializing with a Gaussian distribution, respectively.

**Figure 3.**Left panel: the multivariate information flow of the Lorenz system: blue dot-dash line: ${T}_{2,3\to 1}$; green star line: ${T}_{1,3\to 2}$; red solid line: ${T}_{1,2\to 3}$ (in nats per unit time); Right panel: the information strength of transfer in the Lorenz system: blue dot-dash line: ${S}_{2,3\to 1}$; green star line: ${S}_{1,3\to 2}$; red solid line: ${S}_{1,2\to 3}$ (arbitrary unit).

**Figure 5.**${T}_{1\to 2}$, ${T}_{3\to 2}$ and ${T}_{1,3\to 2}$ in the Lorenz system (in nats per unit time).

**Figure 6.**Left panel: an estimated marginal density of $x,y,z$ and w via counting the bins and initializing with a Gaussian distribution, respectively; Right panel: the multivariate information flow over time of a 4D dynamical system.

**Figure 7.**The attractor of Chua’s system with $x\left(0\right)=-3,y\left(0\right)=2,z\left(0\right)=1.$ The former three trajectories are $x,z$-plane,$x,y$-plane and $y,z$-plane, respectively. The last trajectory is a 3D plot of $x,y$ and $z.$

**Figure 8.**Left panel: a sample data ($X,Y$ and Z) of the Chua’s system generated by a fourth order Runge–Kutta method with $\Delta t=0.01$; Right panel: the purple line, black line, and blue line represent an estimated marginal density of $x,y,z$ by counting bins, respectively.

**Figure 9.**Left panel: the multivariate information flow of the Chua’s system: green dot-dash line: ${T}_{y,z\to x}$; red dot-dash line: ${T}_{x,z\to y}$; blue dot-dash line: ${T}_{x,y\to z}$ (in nats per unit time); Right panel: the information strength of transfer in the Chua’s system: green dot-dash line: ${S}_{y,z\to x}$; red dot-dash line: ${S}_{x,z\to y}$; blue dot-dash line: ${S}_{x,y\to z}$ (arbitrary unit).

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Yin, Y.; Duan, X.
Information Transfer Among the Components in Multi-Dimensional Complex Dynamical Systems. *Entropy* **2018**, *20*, 774.
https://doi.org/10.3390/e20100774

**AMA Style**

Yin Y, Duan X.
Information Transfer Among the Components in Multi-Dimensional Complex Dynamical Systems. *Entropy*. 2018; 20(10):774.
https://doi.org/10.3390/e20100774

**Chicago/Turabian Style**

Yin, Yimin, and Xiaojun Duan.
2018. "Information Transfer Among the Components in Multi-Dimensional Complex Dynamical Systems" *Entropy* 20, no. 10: 774.
https://doi.org/10.3390/e20100774