# Entropy and Stability Analysis of Delayed Energy Supply–Demand Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Local Stability and the Existence of Hopf Bifurcation

#### 2.1. Case 1. ${\tau}_{1}>0,{\tau}_{2}=0$

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**1.**

#### 2.2. Case 2. ${\tau}_{1}>0,{\tau}_{2}>0$

**Lemma**

**3.**

**Lemma**

**4.**

**Theorem**

**2.**

## 3. Numerical Simulation and Analysis

#### 3.1. The Influence of ${\tau}_{1}$ on the Stability of Equation (22)

#### 3.2. The Influence of ${\tau}_{1}$ on the Entropy of Model (22)

#### 3.3. The Influence of ${\tau}_{1},{a}_{1},{d}_{3}$ on the Stability of Model (22)

^{13}for $({\tau}_{1},{a}_{1})=(1,0.05)$, and the maximum value of $x$ is 8.701 × 10

^{11}for $({\tau}_{1},{a}_{1})=(0.25,0.65)$. The analysis shows that the energy consumption coefficient ${a}_{1}$ has the greatest impact on the energy demand when other parameters are fixed. Therefore, the unscientific energy consumption coefficient will affect the accuracy of the energy demand forecast and further impact the social and economic development.

^{12}for $({\tau}_{1},{d}_{3})=(0.85,1)$, and the maximum value of $x$ is 4.387 × 10

^{11}for $({\tau}_{1},{d}_{3})=(1,1)$. The $x$ is approximately 0.85 in stable regions. If the other parameters are unchanged, we must ensure ${\tau}_{1}<0.5$ or ${d}_{3}<0.85$ in order to maintain the stability of energy demand.

#### 3.4. The Influence of ${\tau}_{2}$ on the Stability of Model (22)

#### 3.5. The Influence of ${\tau}_{2},{b}_{3},{b}_{4}$ on the Stability of Model (22)

^{11}for $({\tau}_{2},{b}_{2})=(0.09,0.2)$ and the minimum value of $x$ is −6.97 × 10

^{12}for $({\tau}_{2},{b}_{2})=(0.04,0.2)$. Most regions of $({\tau}_{2},{b}_{2})$ plane are stable at approximately 6 × 10

^{12}. The above features are shown in Figure 14.

#### 3.6. The Influence of ${\tau}_{1},{\tau}_{2}$ on the Stability of Model (22)

## 4. Bifurcation Control

#### 4.1. Bifurcation Value

#### 4.2. Model (23) Is Unstable

#### 4.3. Model (23) Is Stable

#### 4.4. The Influence of Control Parameter $k$ on Entropy

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The impact of ${\tau}_{1}$ on the stability of Model (22) when ${\tau}_{2}=0$: (

**a**) bifurcation diagram; and (

**b**) the largest Lyapunov exponent plot.

**Figure 2.**(

**a**) Time-domain plot; and (

**b**) Poincare section when ${\tau}_{1}=0.1<{\tau}_{10}=0.2208$, ${\tau}_{2}=0$.

**Figure 3.**The attractor of Model (22) when ${\tau}_{1}=0.1<{\tau}_{10}=0.2208$, ${\tau}_{2}=0$: (

**a**) $x(t),y(t),z(t)$; and (

**b**) $x(t),z(t),w(t)$.

**Figure 4.**(

**a**) Time-domain plot; and (

**b**) Poincare section when ${\tau}_{1}=0.4>{\tau}_{10}=0.2208$, ${\tau}_{2}=0$.

**Figure 5.**The attractor when ${\tau}_{1}=0.4>{\tau}_{10}=0.2208$, ${\tau}_{2}=0$: (

**a**) $x(t),y(t),z(t)$; and (

**b**) $x(t),z(t),w(t)$.

**Figure 10.**The influence of ${\tau}_{2}$ on the stability of Model (22) when ${\tau}_{1}=0.1$: (

**a**) bifurcation diagram; and (

**b**) the largest Lyapunov exponent plot.

**Figure 11.**The influence of ${\tau}_{2}$ on the stability of Model (22) when ${\tau}_{1}=0.1,{\tau}_{2}=0.05<{\tau}_{20}=0.1429$: (

**a**) Time-domain plot; (

**b**) energy supply–demand attractor $(x(t),y(t),z(t))$; and (

**c**) energy supply–demand attractor $(x(t),z(t),w(t))$.

**Figure 12.**The influence of ${\tau}_{2}$ on the stability of Model (22) when ${\tau}_{1}=0.1,{\tau}_{2}=0.35>{\tau}_{20}=0.1429$: (

**a**) Time-domain plot; (

**b**) energy supply–demand attractor $(x(t),y(t),z(t))$; and (

**c**) energy supply–demand attractor $(x(t),z(t),w(t))$.

**Figure 17.**The influence of $k$ on the stability of Model (23) when ${\tau}_{1}=0.4,{\tau}_{2}=0$: (

**a**) bifurcation diagram; and (

**b**) the largest Lyapunov exponent plot.

**Figure 18.**Model (23) is unstable when $k=0.6<0.7013$ for ${\tau}_{1}=0.4,{\tau}_{2}=0$: (

**a**) time-domain plot; (

**b**) energy supply–demand attractor $(x(t),y(t),z(t))$; and (

**c**) energy supply–demand attractor $(x(t),z(t),w(t))$.

**Figure 19.**Model (23) is stable when $k=0.8>0.7013$ for ${\tau}_{1}=0.4,{\tau}_{2}=0$: (

**a**) time-domain plot; (

**b**) energy supply–demand attractor $(x(t),y(t),z(t))$; and (

**c**) energy supply–demand attractor $(x(t),z(t),w(t))$.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Wang, J.; Si, F.; Wang, Y.; Duan, S.
Entropy and Stability Analysis of Delayed Energy Supply–Demand Model. *Entropy* **2016**, *18*, 434.
https://doi.org/10.3390/e18120434

**AMA Style**

Wang J, Si F, Wang Y, Duan S.
Entropy and Stability Analysis of Delayed Energy Supply–Demand Model. *Entropy*. 2016; 18(12):434.
https://doi.org/10.3390/e18120434

**Chicago/Turabian Style**

Wang, Jing, Fengshan Si, Yuling Wang, and Shide Duan.
2016. "Entropy and Stability Analysis of Delayed Energy Supply–Demand Model" *Entropy* 18, no. 12: 434.
https://doi.org/10.3390/e18120434