Open AccessArticle

Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays

Jing Wang

^1,* and

Yuling Wang

Department of Mathematics and Physics, Bengbu University, Bengbu 233030, China

School of Economics, South-Central University for Nationalities, Wuhan 430074, China

Author to whom correspondence should be addressed.

Entropy 2016, 18(10), 371; https://doi.org/10.3390/e18100371

Submission received: 2 August 2016 / Revised: 13 October 2016 / Accepted: 13 October 2016 / Published: 19 October 2016

(This article belongs to the Section Complexity)

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Abstract

In this paper, we build a model of energy-savings and emission-reductions with two delays. In this model, it is assumed that the interaction between energy-savings and emission-reduction and that between carbon emissions and economic growth are delayed. We examine the local stability and the existence of a Hopf bifurcation at the equilibrium point of the system. By employing System Complexity Theory, we also analyze the impact of delays and the feedback control on stability and entropy of the system are analyzed from two aspects: single delay and double delays. In numerical simulation section, we test the theoretical analysis by using means bifurcation diagram, the largest Lyapunov exponent diagrams, attractor, time-domain plot, Poincare section plot, power spectrum, entropy diagram, 3-D surface chart and 4-D graph, the simulation results demonstrating that the inappropriate changes of delays and the feedback control will result in instability and fluctuation of carbon emissions. Finally, the bifurcation control is achieved by using the method of variable feedback control. Hence, we conclude that the greater the value of the control parameter, the better the effect of the bifurcation control. The results will provide for the development of energy-saving and emission-reduction policies.

Keywords:

energy-savings; emission-reduction; entropy; stability; two delays; Hopf bifurcation

1. Introduction

Energy resources are the backbone of any national economy. Scholars worldwide have been studying energy prices, the low-carbon economy and energy-savings and emission-reduction. Lv and Zhou [1] studied the dynamic behavior of the energy price model with time delay. They explained the reasons why the energy price model was developed and maintained periodically by the bifurcation theory. Tian et al. [2] developed a nonlinear model for the development of oil, gas and other energy resources against the backdrop of the energy structure in Jiangsu, China, one that predominantly relies on the coal consumption, and evidenced empirically the feasibility of the corresponding energy countermeasures.

Despite the fact that energy promotes economic development and improves people’s living standards, excessive energy consumption undoubtedly damages the environment, so it is necessary to adopt a low-carbon economy and reduce the possible environmental damage. In fact, the energy-saving and emission-reduction system is a complex nonlinear system, which involves the interactions between a variety of factors such as energy-saving and emission-reduction, carbon emissions, economic growth, energy efficiency, carbon tax and energy intensity, and so forth [3,4,5]. In view of this, a novel three-dimensional energy-saving and emission-reduction chaotic system is proposed [6]. The system is established in accordance with the complicated relationship among energy-savings and emission-reduction, carbon emissions and economic growth. This system displays a very complex phenomenon by including a special chaotic attractor named the energy-saving and emission-reduction attractor (the EE attractor for short), which is different from the previous chaotic attractor, such as Lorenz attractor [7], Chen attractor [8], Lü attractor [9], Energy resource attractor [10,11,12] and so on. Moreover, drawing up rational and effective policies and laws will ensure a better progress in energy-saving and emission-reduction [13,14]. Wang and Xu [15] reported a new four-dimensional energy-saving and emission-reduction chaotic system. The system is obtained in accordance with the complicated relationship among energy-saving and emission-reduction, carbon emission, economic growth and new energy development. The model is described as follows:

{\begin{cases} \dot{x} (t) = a_{1} x (t) (\frac{y (t)}{M} - 1) - a_{2} y (t) + a_{3} z (t), \\ \dot{y} (t) = - b_{1} x (t) + b_{2} y (t) (1 - \frac{y (t)}{C}) + b_{3} z (t) (1 - \frac{z (t)}{E}) - b_{4} u (t), \\ \dot{z} (t) = c_{1} x (t) (\frac{x (t)}{N} - 1) - c_{2} y (t) - c_{3} z (t) + c_{4} u (t) (\frac{u (t)}{L} - 1), \\ \dot{u} (t) = d_{1} y (t) + d_{2} z (t) (\frac{z (t)}{K} - 1) - d_{3} u (t), \end{cases}

(1)

where

x (t)

is the time-dependent variable of energy-saving and emission-reduction;

y (t)

the time-dependent variable of carbon emissions;

z (t)

the time-dependent variable of economic growth (GDP) and

u (t)

the time-dependent variable of new energy development.

a_{i}, d_{i}, b_{j}, c_{j}, (i = 1, 2, 3; j = 1, 2, 3, 4)

are coefficients and

M, N, L, K, C, E

positive constants [15]. Model (1) stands for the development and utilization of new energy sources. Therefore, the four-dimensional model of energy-saving and emission-reduction is a step closer to the actual conditions.

In view of the current mode of economic development and technological conditions, economic development will increase the consumption of coal and oil, and in turn this will cause an increase in carbon emissions. However, there are many other approaches to promote economic growth, such as those in the modern service sector that are low energy-consuming but high value-added, so economic development may not immediately cause a substantial energy consumption load. In other words, it might not lead to an obvious increase in carbon emissions in the short term. In this essay, we have included the delay between economic development and the significant growth of carbon emission in this kind of economic model.

Moreover, energy-savings and emission-reduction reduce carbon emissions by saving or reducing energy consumption. However, due to the complexity of the economic system, the implementation of energy-saving and emission-reduction measures normally will not lead to a significant reduction, showing a certain lag. Meanwhile, energy-saving and emission-reduction will slow down the pace of economic development. This will force enterprises to change their mode of economic development, strengthen energy conservation and efficient use, and actively comply with a recycling economy and low carbon economy, so energy-savings and emission-reduction will promote the sound and fast development of economy, but it still demonstrates an obvious delay. Therefore, the impact of energy-saving and emission-reduction on carbon emission and economic development is estimated as delayed. On the basis of Model (1), we propose a two-delay model as follows:

{\begin{cases} \dot{x} (t) = a_{1} x (t) (\frac{y (t)}{M} - 1) - a_{2} y (t) + a_{3} z (t - τ_{1}), \\ \dot{y} (t) = - b_{1} x (t - τ_{2}) + b_{2} y (t) (1 - \frac{y (t)}{C}) + b_{3} z (t - τ_{1}) (1 - \frac{z (t - τ_{1})}{E}) - b_{4} u (t), \\ \dot{z} (t) = c_{1} x (t - τ_{2}) (\frac{x (t - τ_{2})}{N} - 1) - c_{2} y (t) - c_{3} z (t) + c_{4} u (t) (\frac{u (t)}{L} - 1), \\ \dot{u} (t) = d_{1} y (t) + d_{2} z (t) (\frac{z (t)}{K} - 1) - d_{3} u (t), \end{cases}

(2)

where

τ_{1}

represents the delay time between economic development and carbon emission, and

τ_{2}

represents the delay time among energy-saving and emission-reduction, economic development and carbon emission.

The rest of this paper is organized as follows: in Section 2, we focus on the local stability and the existence of Hopf bifurcation at the equilibrium point. In Section 3, we study the effects of delays and the feedback control on the stability of the system. In Section 4, the effective Hopf bifurcation control of the system is examined. Finally, conclusions are drawn in Section 5.

2. Equilibrium Points and Local Stability

The calculation formulas of equilibrium points of Equation (2) are fairly complicated, so the equilibrium points can be calculated directly by using specific parameters value in numerical simulation section. We assume that Equation (2) has an equilibrium point, called hereafter by

E (x^{*}, y^{*}, z^{*}, u^{*})

. It means that

x (t)

y (t)

z (t)

and

u (t)

can be in an equilibrium state through dynamic game. In this paper, we focus on the influence of

τ_{1}

τ_{2}

and the feedback control on the stability of system (2) at the equilibrium point.

Next, Equation (2) is linearized at the equilibrium point

E (x^{*}, y^{*}, z^{*}, u^{*})

by Jacobian matrix as follows:

{\begin{cases} \dot{x} (t) = (a_{1} \frac{y^{*}}{M} - a_{1}) x (t) + (a_{1} \frac{x^{*}}{M} - a_{2}) y (t) + a_{3} z (t - τ_{1}), \\ \dot{y} (t) = - b_{1} x (t - τ_{2}) + (b_{2} - \frac{2 b_{2} y^{*}}{C}) y (t) + (b_{3} - \frac{2 b_{3} z^{*}}{E}) z (t - τ_{1}) - b_{4} u (t), \\ \dot{z} (t) = (\frac{2 c_{1} x^{*}}{N} - c_{1}) x (t - τ_{2}) - c_{2} y (t) - c_{3} z (t) + (\frac{2 c_{4} u^{*}}{L} - c_{4}) u (t), \\ \dot{u} (t) = d_{1} y (t) + (\frac{2 d_{2} z^{*}}{K} - d_{2}) z (t) - d_{3} u (t), \end{cases}

(3)

The characteristic equation of Equation (3) is:

| \begin{matrix} λ - J_{11} & - J_{12} & - J_{13} e^{- λ τ_{1}} & - J_{14} \\ - J_{21} e^{- λ τ_{2}} & λ - J_{22} & - J_{23} e^{- λ τ_{1}} & - J_{24} \\ - J_{31} e^{- λ τ_{2}} & - J_{32} & λ - J_{33} & - J_{34} \\ - J_{41} & - J_{42} & - J_{43} & λ - J_{44} \end{matrix} | = 0

where:

J_{11} = (a_{1} \frac{y^{*}}{M} - a_{1}), J_{12} = a_{1} \frac{x^{*}}{M} - a_{2}, J_{13} = a_{3}, J_{14} = 0,

J_{21} = - b_{1}, J_{22} = b_{2} - \frac{2 b_{2} y^{*}}{C}, J_{23} = (b_{3} - \frac{2 b_{3} z^{*}}{E}), J_{24} = - b_{4},

J_{31} = (\frac{2 c_{1} x^{*}}{N} - c_{1}), J_{32} = - c_{2}, J_{33} = - c_{3}, J_{34} = \frac{2 c_{4} u^{*}}{L} - c_{4},

J_{41} = 0, J_{42} = d_{1}, J_{43} = \frac{2 d_{2} z^{*}}{K} - d_{2}, J_{44} = - d_{3} .

We further get:

\begin{array}{c} λ^{4} + A_{3} λ^{3} + A_{2} λ^{2} + A_{1} λ + A_{0} + (B_{2} λ^{2} + B_{1} λ + B_{0}) e^{- λ τ_{1}} + \\ (C_{2} λ^{2} + C_{1} λ + C_{0}) e^{- λ τ_{2}} + (D_{2} λ^{2} + D_{1} λ + D_{0}) e^{- λ (τ_{1} + τ_{2})} = 0 \end{array}

(4)

where:

A_{3} = - J_{11} - J_{22} - J_{33} - J_{44},

A_{2} = J_{11} J_{22} + J_{11} J_{33} + J_{11} J_{44} - J_{14} J_{41} + J_{22} J_{33} + J_{22} J_{44} - J_{24} J_{42} + J_{33} J_{44} - J_{34} J_{43},

\begin{array}{c} A_{1} = - J_{11} J_{22} J_{33} - J_{11} J_{22} J_{44} + J_{11} J_{24} J_{42} - J_{12} J_{24} J_{41} + J_{14} J_{22} J_{41} - J_{11} J_{33} J_{44} \\ + J_{11} J_{34} J_{43} + J_{14} J_{33} J_{41} - J_{22} J_{33} J_{44} + J_{22} J_{34} J_{43} - J_{24} J_{32} J_{43} + J_{24} J_{33} J_{42} \end{array},

A_{0} = J_{11} J_{22} J_{33} J_{44} - J_{11} J_{22} J_{34} J_{43} + J_{11} J_{24} J_{32} J_{43} - J_{11} J_{24} J_{33} J_{42} + J_{12} J_{24} J_{33} J_{41} - J_{14} J_{22} J_{33} J_{41},

B_{2} = J_{23} J_{32}, B_{1} = J_{11} J_{23} J_{32} - J_{13} J_{34} J_{41} + J_{23} J_{32} J_{44} - J_{23} J_{34} J_{42},

B_{0} = - J_{11} J_{23} J_{32} J_{44} + J_{11} J_{23} J_{34} J_{42} - J_{12} J_{23} J_{34} J_{41} + J_{13} J_{22} J_{34} J_{41} - J_{13} J_{24} J_{32} J_{41} + J_{14} J_{23} J_{32} J_{41},

C_{2} = J_{12}^{2}, C_{1} = J_{12}^{2} J_{33} + J_{12}^{2} J_{44} - J_{12} J_{14} J_{42} - J_{14} J_{31} J_{43},

C_{0} = J_{12}^{2} J_{33} J_{44} + J_{12}^{2} J_{34} J_{43} - J_{12} J_{14} J_{32} J_{43} + J_{12} J_{14} J_{33} J_{42} - J_{12} J_{24} J_{31} J_{43} + J_{14} J_{22} J_{31} J_{43},

D_{2} = J_{13} J_{31}, D_{1} = J_{12} J_{13} J_{32} - J_{12} J_{23} J_{31} + J_{13} J_{22} J_{31} + J_{13} J_{31} J_{44},

D_{0} = J_{12} J_{13} J_{32} J_{44} - J_{12} J_{13} J_{34} J_{42} + J_{12} J_{23} J_{31} J_{44} - J_{13} J_{22} J_{31} J_{44} + J_{13} J_{24} J_{31} J_{42} - J_{14} J_{23} J_{31} J_{42} .

2.1. Case 1 $τ_{1} > 0, τ_{2} = 0$

For

τ_{2} = 0

, Equation (4) can be simplified as follows:

λ^{4} + M_{3} λ^{3} + M_{2} λ^{2} + M_{1} λ + M_{0} + (N_{2} λ^{2} + N_{1} λ + N_{0}) e^{- λ τ_{1}} = 0,

(5)

where:

M_{3} = A_{3}, M_{2} = A_{2} + C_{2}, M_{1} = A_{1} + C_{1}, M_{0} = A_{0} + C_{0},

N_{2} = B_{2} + D_{2}, N_{1} = B_{1} + D_{1}, N_{0} = B_{0} + D_{0},

Let

λ = i ω_{1} (ω_{1} > 0)

be the root of Equation (5). Separating the real and imaginary parts, we obtain the following:

{\begin{cases} N_{1} ω_{1} \cos ω_{1} τ_{1} + (N_{2} ω_{1}^{2} - N_{0}) \sin ω_{1} τ_{1} = M_{3} ω_{1}^{3} - M_{1} ω_{1} \\ N_{1} ω_{1} \sin ω_{1} τ_{1} - (N_{2} ω_{1}^{2} - N_{0}) \cos ω_{1} τ_{1} = M_{2} ω_{1}^{2} - ω_{1}^{4} - M_{0} \end{cases} .

(6)

From Equation (6), we can get:

\cos ω_{1} τ_{1} = \frac{N_{2} ω_{1}^{6} + (N_{1} M_{3} - N_{2} M_{2} - N_{0}) ω_{1}^{4} + (N_{2} M_{0} + N_{0} M_{2} - N_{1} M_{1}) ω_{1}^{2} - N_{0} M_{0}}{N_{1}^{2} ω_{1}^{2} + {(N_{2} ω_{1}^{2} - N_{0})}^{2}} .

(7)

Squaring both sides, adding both equations and regrouping by powers of

ω_{1}

, we obtain that

ω_{1}

(

ω_{1} > 0

) satisfies the following polynomial:

ω_{1}^{8} + (M_{3}^{6} - 2 M_{2}) ω_{1}^{6} + (M_{2}^{2} - 2 M_{1} M_{3} + 2 M_{0} - N_{2}^{2}) ω_{1}^{4} + (M_{1}^{2} - 2 M_{0} M_{2} - N_{1}^{2} + 2 N_{0} N_{2}) ω_{1}^{2} + M_{0}^{2} - N_{0}^{2} = 0 .

(8)

Let

r_{1} = ω_{1}^{2}

, Equation (8) transformed into:

r_{1}^{4} + q_{3} r_{1}^{3} + q_{2} r_{1}^{2} + q_{1} r_{1} + q_{0} = 0,

(9)

where:

q_{3} = M_{3}^{6} - 2 M_{2}, q_{2} = M_{2}^{2} - 2 M_{1} M_{3} + 2 M_{0} - N_{2}^{2},

q_{1} = M_{1}^{2} - 2 M_{0} M_{2} - N_{1}^{2} + 2 N_{0} N_{2}, q_{0} = M_{0}^{2} - N_{0}^{2} .

In the following, we need to seek conditions under which Equation (9) has at least one positive root. Denote:

h (r_{1}) = r_{1}^{4} + q_{3} r_{1}^{3} + q_{2} r_{1}^{2} + q_{1} r_{1} + q_{0} .

Since

\lim_{r_{1} \to \infty} h (r_{1}) = \infty

, we conclude that if

q_{0} < 0

, then Equation (9) has at least one positive root [16].

Suppose that Equation (9) has positive roots. Without loss of generality, we assume that it has four positive roots, defined by

r_{11}

r_{12}

r_{13}

and

r_{14}

, respectively. Then Equation (8) has four positive roots:

ω_{11} = \sqrt{r_{11}}, ω_{12} = \sqrt{r_{12}}, ω_{13} = \sqrt{r_{13}}, ω_{14} = \sqrt{r_{14}} .

For each fixed

ω_{1 k} (k = 1, 2, 3, 4)

, there exists a sequence

{τ_{1 k}^{(j)} | k = 1, 2, 3, 4; j = 0, 1, 2, ...}

that satisfies Equation (6). According to Equation (7), we can get:

\begin{array}{l} τ_{1 k}^{(j)} = \frac{1}{w_{1 k}} \arccos \frac{N_{2} ω_{1 k}^{6} + (N_{1} M_{3} - N_{2} M_{2} - N_{0}) ω_{1 k}^{4} + (N_{2} M_{0} + N_{0} M_{2} - N_{1} M_{1}) ω_{1 k}^{2} - N_{0} M_{0}}{N_{1}^{2} ω_{1 k}^{2} + {(N_{2} ω_{1 k}^{2} - N_{0})}^{2}} + \frac{2 j π}{ω_{1 k}} \\ k = 1, 2, 3, 4; j = 0, 1, 2, 3, ... \end{array}

(10)

Let

τ_{10} = \min {τ_{1 k}^{(j)} | k = 1, 2, 3, 4; j = 0, 1, 2, ...} = \min_{k \in {1, 2, 3, 4}} {τ_{1 k}^{(0)}} = τ_{1 k_{0}}

ω_{10} = ω_{1 k_{0}}

. So the

τ_{10}

is:

τ_{10} = \frac{1}{ω_{10}} \arccos \frac{N_{2} ω_{10}^{6} + (N_{1} M_{3} - N_{2} M_{2} - N_{0}) ω_{10}^{4} + (N_{2} M_{0} + N_{0} M_{2} - N_{1} M_{1}) ω_{10}^{2} - N_{0} M_{0}}{N_{1}^{2} ω_{10}^{2} + {(N_{2} ω_{10}^{2} - N_{0})}^{2}}

(11)

Based on the above analysis, the main results are presented as below:

Lemma 1.

q_{0} < 0

, Equation (5) has a pair of pure imaginary roots

\pm i ω_{10}

when

τ_{1} = τ_{10}

. Next, take the derivative with respect to

τ_{1}

in Equation (5) and we can obtain:

{[\frac{d λ}{d τ_{1}}]}^{- 1} = \frac{(4 λ^{3} + 3 M_{3} λ^{2} + 2 M_{2} λ + M_{1}) e^{λ τ_{1}} + (2 N_{2} λ + N_{1})}{(N_{2} λ^{3} + N_{1} λ^{2} + N_{0} λ)} - \frac{τ_{1}}{λ}

Re {[\frac{d λ (τ_{10})}{d τ_{1}}]}_{λ = i ω_{10}}^{- 1} = \frac{P_{1} P_{3} + P_{2} P_{4}}{P_{1}^{2} + P_{2}^{2}}

where:

P_{1} = - N_{1} ω_{10}^{2}, P_{2} = N_{0} ω_{10} - N_{2} ω_{10}^{3},

P_{3} = 4 ω_{10}^{3} \sin ω_{10} τ_{10} - 3 M_{3} ω_{10}^{2} \cos ω_{10} τ_{10} - 2 M_{2} ω_{10} \sin ω_{10} τ_{10} + M_{1} \cos ω_{10} τ_{10},

P_{4} = - 4 ω_{10}^{3} \cos ω_{10} τ_{10} - 3 M_{3} ω_{10}^{2} \sin ω_{10} τ_{10} + 2 M_{2} ω_{10} \cos ω_{10} τ_{10} + M_{1} \sin ω_{10} τ_{10} + 2 N_{2} ω_{10} + N_{1},

Lemma 2.

Suppose that

P_{1} P_{3} + P_{2} P_{4} \neq 0

, then

{\frac{d Re λ (τ_{10})}{d τ_{1}} |}_{λ = i ω_{10}} = Re {[\frac{d λ (τ_{10})}{d τ_{1}}]}_{λ = i ω_{10}}^{- 1} \neq 0

. So it satisfies the transversality condition.

According to the Lemmas 1–2 and the Hopf bifurcation theorem in [17], we obtain the following results:

Theorem 1:

The equilibrium point

E (x^{*}, y^{*}, z^{*}, u^{*})

of Equation (2) is asymptotically stable for

τ_{1} \in [0, τ_{10})

and unstable for

τ_{1} > τ_{10}

; Equation (2) undergoes a Hopf bifurcation when

τ_{1} = τ_{10}

2.2. Case 2 $τ_{1} > 0, τ_{2} > 0$

For

τ_{1} > 0, τ_{2} > 0

, we consider the characteristic Equation (4) with

τ_{1}

in its stable intervals (the range of

τ_{1}

when the stability of the system is not affected), that is to say,

τ_{1} \in [0, τ_{10})

[18]. We study the influence of

τ_{2}

on the stability of the system when

τ_{1}

is fixed.

Proposition 1.

Suppose that

L_{0} < 0

, then the characteristic Equation (4) has a pair of pure imaginary roots

\pm i ω_{20}

for

τ_{2} = τ_{20}

. where:

τ_{20} = \frac{1}{ω_{20}} \arccos \frac{h_{6} ω_{20}^{6} + h_{5} ω_{20}^{5} + h_{4} ω_{20}^{4} + h_{3} ω_{20}^{3} + h_{2} ω_{20}^{2} + h_{1} ω_{20} + h_{0}}{f_{4} ω_{20}^{4} + f_{3} ω_{20}^{3} + f_{2} ω_{20}^{2} + f_{1} ω_{20} + f_{0}}

(12)

Proof.

Let

λ = i ω_{2} (ω_{2} > 0)

be a root of Equation (4). Then we get:

{\begin{cases} (C_{2} ω_{2}^{2} - C_{0} + D_{2} ω_{2}^{2} \cos ω_{2} τ_{1} - D_{1} ω_{2} \sin ω_{2} τ_{1} - D_{0} \cos ω_{2} τ_{1}) \sin ω_{2} τ_{2} \\ + (C_{1} ω_{2} + D_{2} ω_{2}^{2} \sin ω_{2} τ_{1} + D_{1} ω_{2} \cos ω_{2} τ_{1} - D_{0} \sin ω_{2} τ_{1}) \cos ω_{2} τ_{2} \\ = A_{3} ω_{2}^{3} - A_{1} ω_{2} - B_{2} ω_{2}^{2} \sin ω_{2} τ_{1} - B_{1} ω_{2} \cos ω_{2} τ_{1} + B_{0} \sin ω_{2} τ_{1}, \\ - (C_{2} ω_{2}^{2} - C_{0} + D_{2} ω_{2}^{2} \cos ω_{2} τ_{1} - D_{1} ω_{2} \sin ω_{2} τ_{1} - D_{0} \cos ω_{2} τ_{1}) \cos ω_{2} τ_{2} \\ + (C_{1} ω_{2} + D_{2} ω_{2}^{2} \sin ω_{2} τ_{1} + D_{1} ω_{2} \cos ω_{2} τ_{1} - D_{0} \sin ω_{2} τ_{1}) \sin ω_{2} τ_{2} \\ = A_{2} ω_{2}^{2} - ω_{2}^{4} - A_{0} + B_{2} ω_{2}^{2} \cos ω_{2} τ_{1} - B_{1} ω_{2} \sin ω_{2} τ_{1} - B_{0} \cos ω_{2} τ_{1} \end{cases}

(13)

From Equation (13), we can obtain:

\cos ω_{2} τ_{2} = \frac{h_{6} ω_{2}^{6} + h_{5} ω_{2}^{5} + h_{4} ω_{2}^{4} + h_{3} ω_{2}^{3} + h_{2} ω_{2}^{2} + h_{1} ω_{2} + h_{0}}{f_{4} ω_{2}^{4} + f_{3} ω_{2}^{3} + f_{2} ω_{2}^{2} + f_{1} ω_{2} + f_{0}}

(14)

where:

h_{6} = C_{2} + D_{2} \cos ω_{2} τ_{1},

h_{5} = D_{2} A_{3} \sin ω_{2} τ_{1} - D_{1} \sin ω_{2} τ_{1}

h_{4} = C_{1} A_{3} - D_{2} B_{2} + D_{1} A_{3} \cos ω_{2} τ_{1} - C_{2} A_{2} - C_{2} B_{2} \cos ω_{2} τ_{1} - C_{0} - D_{2} A_{2} \cos ω_{2} τ_{1} - D_{0} \cos ω_{2} τ_{1},

h_{3} = C_{2} B_{1} \sin ω_{2} τ_{1} - C_{1} B_{2} \sin ω_{2} τ_{1} - D_{2} A_{1} \sin ω_{2} τ_{1} - D_{0} A_{3} \sin ω_{2} τ_{1} + D_{1} A_{2} \sin ω_{2} τ_{1},

\begin{array}{c} h_{2} = - C_{1} A_{1} - C_{1} B_{1} \cos ω_{2} τ_{1} + D_{2} B_{0} - D_{1} A_{1} \cos ω_{2} τ_{1} - D_{1} B_{1} + D_{0} B_{2} + C_{2} A_{0} \\ + C_{2} B_{0} \cos ω_{2} τ_{1} + C_{0} A_{2} + C_{0} B_{2} \cos ω_{2} τ_{1} + D_{2} A_{0} \cos ω_{2} τ_{1} + D_{0} A_{2} \cos ω_{2} τ_{1} \end{array},

\begin{array}{c} h_{1} = C_{1} B_{0} \sin ω_{2} τ_{1} + D_{1} B_{0} \cos ω_{2} τ_{1} \sin ω_{2} τ_{1} + D_{0} A_{1} \sin ω_{2} τ_{1} \\ - C_{0} B_{1} \sin ω_{2} τ_{1} - D_{1} A_{0} \sin ω_{2} τ_{1} - D_{1} B_{0} \sin ω_{2} τ_{1} \cos ω_{2} τ_{1} \end{array},

h_{0} = - D_{0} B_{0} - C_{0} A_{0} - C_{0} B_{0} \cos ω_{2} τ_{1} - D_{0} A_{0} \cos ω_{2} τ_{1},

f_{4} = C_{2}^{2} + D_{2}^{2} + 2 D_{2} C_{2} \cos ω_{2} τ_{1},

f_{3} = - 2 D_{1} C_{2} \sin ω_{2} τ_{1} + 2 C_{1} D_{2} \sin ω_{2} τ_{1},

f_{2} = 2 D_{2} C_{0} \cos ω_{2} τ_{1} - 2 C_{0} C_{2} + D_{1}^{2} - 2 D_{0} C_{2} \cos ω_{2} τ_{1} - 2 D_{0} D_{2} + C_{1}^{2} + 2 D_{1} C_{1} \cos ω_{2} τ_{1},

f_{1} = 2 D_{1} C_{0} \sin ω_{2} τ_{1} - 2 D_{0} C_{1} \sin ω_{2} τ_{1},

f_{0} = C_{0}^{2} + D_{0}^{2} + 2 C_{0} D_{0} \cos ω_{2} τ_{1} .

Similar to Case 1, from Equation (13), we have:

ω_{2}^{8} + L_{7} ω_{2}^{7} + L_{6} ω_{2}^{6} + L_{5} ω_{2}^{5} + L_{4} ω_{2}^{4} + L_{3} ω_{2}^{3} + L_{2} ω_{2}^{2} + L_{1} ω_{2} + L_{0} = 0

(15)

where

L_{7} = 0, L_{6} = A_{3}^{2} - 2 A_{2} - 2 B_{2} \cos ω_{2} τ_{1}, L_{5} = 2 B_{2} A_{3} \sin ω_{2} τ_{1} - 2 B_{1} \sin ω_{2} τ_{1},

L_{4} = B_{2}^{2} - 2 A_{1} A_{3} - 2 A_{3} B_{1} \cos ω_{2} τ_{1} + A_{2}^{2} + 2 A_{0} + 2 A_{2} B_{2} \cos ω_{2} τ_{1} + 2 B_{0} \cos ω_{2} τ_{1} - C_{2}^{2} - D_{2}^{2} - 2 D_{2} C_{2} \cos ω_{2} τ_{1},

L_{3} = 2 B_{2} A_{1} \sin ω_{2} τ_{1} + 2 B_{0} A_{3} \sin ω_{2} τ_{1} - 2 B_{1} A_{2} \sin ω_{2} τ_{1} + 2 D_{1} C_{2} \sin ω_{2} τ_{1} - 2 C_{1} D_{2} \sin ω_{2} τ_{1},

\begin{array}{c} L_{2} = A_{1}^{2} + B_{1}^{2} + 2 A_{1} B_{1} \cos ω_{2} τ_{1} - 2 B_{0} B_{2} - 2 A_{0} A_{2} - 2 B_{2} A_{0} \cos ω_{2} τ_{1} - 2 B_{0} A_{2} \cos ω_{2} τ_{1} \\ - 2 D_{2} C_{0} \cos ω_{2} τ_{1} + 2 C_{0} C_{2} + D_{1}^{2} + 2 D_{0} C_{2} \cos ω_{2} τ_{1} + 2 D_{0} D_{2} - C_{1}^{2} - 2 D_{1} C_{1} \cos ω_{2} τ_{1} \end{array},

L_{1} = - 2 B_{0} A_{1} \sin ω_{2} τ_{1} + 2 B_{1} A_{0} \sin ω_{2} τ_{1} - 2 D_{1} C_{0} \sin ω_{2} τ_{1} + 2 D_{0} C_{1} \sin ω_{2} τ_{1},

L_{0} = A_{0}^{2} + 2 B_{0} A_{0} \cos ω_{2} τ_{1} - C_{0}^{2} - D_{0}^{2} - 2 D_{0} C_{0} \cos ω_{2} τ_{1},

We turn Equation (15) into the following form:

f (ω_{2}) = ω_{2}^{8} + L_{7} ω_{2}^{7} + L_{6} ω_{2}^{6} + L_{5} ω_{2}^{5} + L_{4} ω_{2}^{4} + L_{3} ω_{2}^{3} + L_{2} ω_{2}^{2} + L_{1} ω_{2} + L_{0}

(16)

Since

\lim_{ω_{2} \to \infty} f (ω_{2}) = \infty

, we conclude that if

L_{0} < 0

, then Equation (15) has at least one positive root. Without loss of generality, we assume that Equation (15) has a finite number of positive roots defined by

{ω_{21}, ω_{22}, ..., ω_{2 s}}

From Equation (14), we denote:

τ_{2 k}^{(j)} = \frac{1}{ω_{2 k}} \arccos \frac{h_{6} ω_{2 k}^{6} + h_{5} ω_{2 k}^{5} + h_{4} ω_{2 k}^{4} + h_{3} ω_{2 k}^{3} + h_{2} ω_{2 k}^{2} + h_{1} ω_{2 k} + h_{0}}{f_{4} ω_{2 k}^{4} + f_{3} ω_{2 k}^{3} + f_{2} ω_{2 k}^{2} + f_{1} ω_{2 k} + f_{0}} + \frac{2 j π}{ω_{2 k}}, k = 1, 2, ..., s; j = 0, 1, 2, ...

Then

(τ_{2 k}^{(j)}, ω_{2 k})

be a root of Equation (13). Therefore, when

τ_{2} = τ_{2 k}^{(j)}

, the characteristic Equation (4) has a pair of pure imaginary roots

\pm i ω_{2 k}

. Define:

τ_{20} = \min {τ_{2 k}^{(j)} | k = 1, 2, ..., s; j = 0, 1, 2, ...} = \min_{k \in {1, 2, ..., s}} {τ_{2 k}^{(0)}} = τ_{2 k_{0}}, ω_{20} = ω_{2 k_{0}} .

Let

λ (τ_{2}) = α (τ_{2}) + i ω (τ_{2})

be the root of the characteristic Equation (4) near

τ_{2} = τ_{2 k}^{(j)}

satisfying:

α (τ_{2 k}^{(j)}) = 0, ω (τ_{2 k}^{(j)}) = ω_{2 k} .

Then on the basis of above analysis, we can get the following conclusion:

τ_{20} = \frac{1}{ω_{20}} \arccos \frac{h_{6} ω_{20}^{6} + h_{5} ω_{20}^{5} + h_{4} ω_{20}^{4} + h_{3} ω_{20}^{3} + h_{2} ω_{20}^{2} + h_{1} ω_{20} + h_{0}}{f_{4} ω_{20}^{4} + f_{3} ω_{20}^{3} + f_{2} ω_{20}^{2} + f_{1} ω_{20} + f_{0}}

(17)

The characteristic Equation (4) has a pair of pure imaginary roots

\pm i ω_{20}

when

τ_{2} = τ_{20}

. ☐

Proposition 2.

Suppose that

Δ \neq 0

, then

Re {[\frac{d λ (τ_{20})}{d τ_{2}}]}_{λ = i ω_{20}}^{- 1} \neq 0

Re {[\frac{d λ (τ_{20})}{d τ_{2}}]}_{λ = i ω_{20}}^{- 1}

and

Δ

have the same sign, where:

Δ = Q_{1} Q_{3} + Q_{2} Q_{4}

Proof.

Substituting

λ (τ_{2})

(the root of the characteristic Equation (4), which is defined above) into Equation (4) and taking the derivative with respect to

τ_{2}

, we obtain:

{[\frac{d λ}{d τ_{2}}]}^{- 1} = \frac{Q_{10} + Q_{20} + Q_{30} + Q_{40}}{λ e^{- λ (τ_{1} + τ_{2})} (D_{2} λ^{2} + D_{1} λ + D_{0}) + λ e^{- λ τ_{2}} (C_{2} λ^{2} + C_{1} λ + C_{0})} - \frac{τ_{2}}{λ}

(18)

Q_{10} = 4 λ^{3} + 3 A_{3} λ^{2} + 2 A_{2} λ + A_{1}, Q_{20} = (2 B_{2} λ + B_{1} - τ_{1} B_{2} λ^{2} - τ_{1} B_{1} λ - τ_{1} B_{0}) e^{- λ τ_{1}},

Q_{30} = (2 C_{2} λ + C_{1}) e^{- λ τ_{2}}, Q_{40} = (2 D_{2} λ + D_{1} - τ_{1} D_{2} λ^{2} - τ_{1} D_{1} λ - τ_{1} D_{0}) e^{- λ (τ_{1} + τ_{2})},

From Equation (18), we have:

Re {[\frac{d λ (τ_{20})}{d τ_{2}}]}_{λ = i ω_{20}}^{- 1} = \frac{Q_{1} Q_{3} + Q_{2} Q_{4}}{Q_{1}^{2} + Q_{2}^{2}} = \frac{Δ}{Q_{1}^{2} + Q_{2}^{2}}

where:

\begin{array}{c} Q_{1} = - D_{1} ω_{20}^{2} \cos (τ_{1} + τ_{20}) - D_{2} ω_{20}^{3} \sin (τ_{1} + τ_{20}) + D_{0} ω_{20} \sin (τ_{1} + τ_{20}) \\ - C_{1} ω_{20}^{2} \cos ω_{20} τ_{20} - C_{2} ω_{20}^{3} \sin ω_{20} τ_{20} + C_{0} ω_{20} \sin ω_{20} τ_{20} \end{array}

\begin{array}{c} Q_{2} = - D_{2} ω_{20}^{3} \cos (τ_{1} + τ_{20}) + D_{0} ω_{20} \cos (τ_{1} + τ_{20}) + D_{1} ω_{20}^{2} \sin (τ_{1} + τ_{20}) \\ - C_{2} ω_{20}^{3} \cos ω_{20} τ_{20} + C_{0} ω_{20} \cos ω_{20} τ_{20} + C_{1} ω_{20}^{2} \sin ω_{20} τ_{20} \end{array}

\begin{array}{c} Q_{3} = - 3 A_{3} ω_{20}^{2} + A_{1} + 2 B_{2} ω_{20} \sin ω_{20} τ_{1} + B_{1} \cos ω_{20} τ_{1} + τ_{1} B_{2} ω_{20}^{2} \cos ω_{20} τ_{1} - τ_{1} B_{1} ω_{20} \sin ω_{20} τ_{1} \\ - τ_{1} B_{0} \cos ω_{20} τ_{1} + 2 C_{2} ω_{20} \sin ω_{20} τ_{20} + C_{1} \cos ω_{20} τ_{20} + 2 D_{2} ω_{20} \sin (τ_{1} + τ_{20}) \\ + D_{1} \cos (τ_{1} + τ_{20}) + τ_{1} D_{2} ω_{20}^{2} \cos (τ_{1} + τ_{20}) - τ_{1} D_{1} ω_{20} \sin (τ_{1} + τ_{20}) - τ_{1} D_{0} \cos (τ_{1} + τ_{20}) \end{array}

\begin{array}{c} Q_{4} = - 4 ω_{20}^{3} + 2 A_{2} ω_{20} + 2 B_{2} ω_{20} \cos ω_{20} τ_{1} - B_{1} \sin ω_{20} τ_{1} - τ_{1} B_{2} ω_{20}^{2} \sin ω_{20} τ_{1} - τ_{1} B_{1} ω_{20} \cos ω_{20} τ_{1} \\ + τ_{1} B_{0} \sin ω_{20} τ_{1} + 2 C_{2} ω_{20} \cos ω_{20} τ_{20} - C_{1} \sin ω_{20} τ_{20} + 2 D_{2} ω_{20} \cos (τ_{1} + τ_{20}) \\ - D_{1} \sin (τ_{1} + τ_{20}) - τ_{1} D_{2} ω_{20}^{2} \sin (τ_{1} + τ_{20}) - τ_{1} D_{1} ω_{20} \cos (τ_{1} + τ_{20}) + τ_{1} D_{0} \sin (τ_{1} + τ_{20}) \end{array}

Since

Q_{1}^{2} + Q_{2}^{2} > 0

, thus

Re {[\frac{d λ (τ_{20})}{d τ_{2}}]}_{λ = i ω_{20}}^{- 1} \neq 0

. We conclude that

Re {[\frac{d λ (τ_{20})}{d τ_{2}}]}_{λ = i ω_{20}}^{- 1}

and

Δ

have the same sign. ☐

Thus, according to propositions 1 and 2 and the Hopf bifurcation theorem in [17], we have the following theorem.

Theorem 2:

For

τ_{1} \in [0, τ_{10})

τ_{10}

is defined by Equation (11). The equilibrium point

E (x^{*}, y^{*}, z^{*}, u^{*})

of Equation (2) is asymptotically stable for

τ_{2} \in [0, τ_{20})

and unstable when

τ_{2} > τ_{20}

. Equation (2) has a Hopf bifurcation at

τ_{2} = τ_{20}

3. Numerical Simulation and Analysis

In this section, the numerical simulation is carried out in order to support the theoretical analysis. Let

x (0) = 0.3

;

y (0) = 0.5

;

z (0) = 0.7

;

u (0) = 0.9

;

a_{1} = 4.5

;

a_{2} = 1.3

;

a_{3} = 20

;

b_{1} = 0.25

;

b_{2} = 0.85

;

b_{3} = 0.35

;

b_{4} = 0.04

;

c_{1} = 0.5

;

c_{2} = 0.3

;

c_{3} = 0.2

;

c_{4} = 0.1

;

d_{1} = 0.01

;

d_{2} = 0.02

;

d_{3} = 0.06

;

M = 10

;

C = 1

;

E = 8

;

N = 12

;

L = 2

;

K = 2

. We consider the following system with given parameter values:

{\begin{cases} \dot{x} (t) = 4.5 x (t) (\frac{y (t)}{10} - 1) - 1.3 y (t) + 20 z (t - τ_{1}), \\ \dot{y} (t) = - 0.25 x (t - τ_{2}) + 0.85 y (t) (1 - y (t)) + 0.35 z (t - τ_{1}) (1 - \frac{z (t - τ_{1})}{8}) - 0.04 u (t), \\ \dot{z} (t) = 0.5 x (t - τ_{2}) (\frac{x (t - τ_{2})}{12} - 1) - 0.3 y (t) - 0.2 z (t) + 0.1 u (t) (\frac{u (t)}{2} - 1), \\ \dot{u} (t) = 0.01 y (t) + 0.02 z (t) (\frac{z (t)}{2} - 1) - 0.06 u (t), \end{cases}

(19)

The following equilibrium points can be obtained by simple calculation:

E_{0} (0, 0, 0, 0), E_{1} (- 0.6625, 1.1410, - 0.0578, 0.2100),

E_{2} (- 5.2219 \pm 54.1330 i, - 3.8795 \pm 0.7677 i, - 0.9477 \pm 17.0453 i, - 48.6048 \pm 10.9389 i),

E_{3} (13.3977 \pm 0.7680 i, 0.4248 \pm 1.7180 i, 2.9437 \pm 0.2407 i, 0.5241 \pm 0.1303 i),

E_{4} (- 251.5640 \pm 371.2492 i, 14.7184 \pm 1.4202 i, 15.8006 \pm 47.5443 i, - 337.9472 \pm 234.7986 i),

E_{5} (- 48.6048 \pm 10.9389 i, 6.9068 \pm 6.8940 i, - 28.3090 \pm 13.8694 i, 112.0947 \pm 134.3508 i),

E_{6} (131.6126 \pm 3.3325 i, 3.3618 \pm 6.3095 i, 19.4028 \pm 18.7720 i, - 1.8941 \pm 114.1013 i),

According to actual economic meaning, only

E_{1}

is Nash Equilibrium point. So we examine the system stability of the system at the equilibrium point

E_{1} (- 0.6625, 1.1410, - 0.0578, 0.2100)

For Case 1, from Equations (8) and (11), we can obtain

ω_{10} = 4.5329

τ_{10} = 0.4618

, so Equation (5) has a pair of pure imaginary roots

\pm i ω_{10}

when

τ_{1} = τ_{10}

and

τ_{2} = 0

. We also get

P_{1} P_{3} + P_{2} P_{4} = 37.4226 \neq 0

. By theorem 1, the equilibrium point

E_{1}

of Equation (2) is asymptotically stable when

τ_{1} \in [0, 0.4618)

and unstable when

τ_{1} > 0.4618

. It has a Hopf bifurcation at

τ_{1} = 0.4618

For Case 2, Let

τ_{1} = 0.4 \in [0, τ_{10})

, from Equations (16) and (17), we can obtain

ω_{20} = 39.6736

τ_{20} = 0.0622

, so Equation (4) has a pair of pure imaginary roots

\pm i ω_{20}

when

τ_{1} = 0.4

and

τ_{2} = τ_{20}

. At this point,

Q_{1} Q_{3} + Q_{2} Q_{4} = 23.8153 \neq 0

. By theorem 2, the equilibrium point

E_{1}

of Equation (2) is asymptotically stable when

τ_{2} \in [0, 0.0622)

for

τ_{1} = 0.4

and unstable when

τ_{2} > 0.0622

for

τ_{1} = 0.4

. Equation (2) undergoes a Hopf bifurcation when

τ_{2} = 0.0622

for

τ_{1} = 0.4

3.1. The Influence of $τ_{1}$ on the Stability of Equation (19)

Equation (19) moves from stable to unstable with the increase in

τ_{1}

and undergoes Hopf bifurcation at

τ_{1} = 0.4618

when

τ_{2} = 0

. Figure 1a shows the dynamic evolution process of Equation (19). In this paper, we calculate the largest Lyapunov exponent (LLE) by the method of Wolf reconstruction [19]. We judge whether the system is stable according to the exponent value. If it is less than 0, the system is stable. If it is greater than 0, the system is unstable. If it is equal to 0, the system appears bifurcation. In Figure 1b, we can know that the system has a bifurcation at

τ_{1} = 0.4618

. This is consistent with the conclusion of theoretical analysis.

According to Theorem 1, Equation (19) is stable when

τ_{1} = 0.4 < τ_{10} = 0.4618

and

τ_{2} = 0

. In this case,

x (t)

y (t)

z (t)

and

u (t)

will converge to the equilibrium point

E_{1}

through the game. Figure 2 and Figure 3 show the above properties.

Equation (19) is unstable when

τ_{1} = 0.5 > τ_{10} = 0.4618

and

τ_{2} = 0

by Theorem 1. At this point, the frequency spectrum plot is discrete, which means that the system has a periodic solution. The

x (t)

y (t)

z (t)

and

u (t)

will lie on the basin of attraction [20] through the game. These analyses can be illustrated by Figure 4 and Figure 5.

We discover that economic growth (slowdown) will lead to an increase (reduction) in carbon emissions when other parameters are fixed. The two variables share a consistent pattern, but with a certain delay in time. Based on the above simulation, we find out that when the delay is greater than the bifurcation value, the carbon emissions pattern will not be consistent with the economic growth trend and volatility surfaces. Therefore, we should guarantee the timing of achieving the goal of reducing emissions and make it less than the bifurcation value. Only in such a stable environment, the implementation of energy-saving and emission-reduction policies will intend a favorable effect.

3.2. The Influence of $τ_{1}$ on the Entropy of Equation (19)

Kolmogorov entropy can be used to measure the degree of complexity of the system. Let

k

be the value of Kolmogorov entropy. If the system is stable and in regular motion, then

k = 0

, otherwise

k > 0

. The greater the value of

k

is, the more complex the system is. We have already found that Equation (19) displays bifurcation at

τ_{1} = 0.4618

, as is shown in Figure 1. Based on the above analysis, we can get that if

τ_{1} < 0.4618

, then

k = 0

, otherwise

k > 0

. The more complex the system is, the longer and the more difficult it will be for the system to return to stability. The entropy properties are displayed in Figure 6.

3.3. The Influence of $τ_{1}, b_{3}, b_{4}$ on the Stability of Equation (19)

The influence of

τ_{1}

and

b_{3}

y

is shown in Figure 7. The system shows a sign of instability at

τ_{1} = 0.4618

. However,

b_{3}

has no obvious impact on

y

when

τ_{1} < 0.55

. With the increase of

b_{3}

y

becomes larger and then decreases when

τ_{1} > 0.55

. The minimum value of

y

is −1.448 for

(τ_{1}, b_{3}) = (0.5, 0.85)

while the maximum value of

y

is 5.124 for

(τ_{1}, b_{3}) = (0.6, 0.55)

. The value of

y

is −0.66 approximately when

τ_{1} < 0.4618

. Thus, the impact of

b_{3}

y

can be ignored when

τ_{1} < 0.4618

Figure 8 shows that

b_{4}

has a greater impact on

y

. With an increase in

b_{4}

, the system shifts from unstable to stable. When

b_{4} = 0.2

, the system is beginning to become stable. The minimum value of

y

is −1.838 for

(τ_{1}, b_{4}) = (0.5, 0.15)

and the maximum value of

y

is 4.207 for

(τ_{1}, b_{4}) = (0.6, 0)

. The value of

y

is stable at 2.889, but

τ_{1}

has little effect on

y

. Therefore, we should consider the effect of

b_{4}

on the stability of the system.

We focus on the influence of

τ_{1}, b_{3}

and

b_{4}

y

in Figure 9. The system is stable when

τ_{1} < 0.4618

, and unstable when

τ_{1} > 0.4618

. The change in

b_{3}

and

b_{4}

leads only to a larger fluctuation but when

τ_{1} > 0.4618

, the fluctuation trend is basically symmetrical. However,

b_{3}

and

b_{4}

have no effect on

y

when

τ_{1} < 0.4618

. We conclude that the stability of energy-saving and emission-reduction system can be maintained, only controlling for the value of delay parameter

τ_{1}

3.4. The Influence of $τ_{2}$ on the Stability of Equation (19)

In this part, we study the effect of

τ_{2}

on the stability of the system when

τ_{1}

is fixed. With an increase in

τ_{2}

, Equation (19) will lose stability. Figure 10 displays that Equation (19) undergoes bifurcation at

τ_{2} = 0.0622

for

τ_{1} = 0.4

. This is consistent with the theoretical analysis.

Equation (19) is stable when

τ_{1} = 0.4, τ_{2} = 0.04 < τ_{20} = 0.0622

by Theorem 2.

x (t)

y (t)

z (t)

and

u (t)

will converge to the equilibrium point

E_{1}

through the game. The properties are shown in Figure 11 and Figure 12.

On the basis of theorem 2, Equation (19) is unstable when

τ_{1} = 0.4, τ_{2} = 0.08 > τ_{20} = 0.0622

. Meanwhile, the Poincare plot has five discrete points, which means that the system has a periodic solution.

x (t)

y (t)

z (t)

and

u (t)

will lie on the basin of attraction through the game. Figure 13 and Figure 14 show these characteristics.

3.5. The Influence of $τ_{2}$ on the Entropy of Equation (19)

From Figure 10, we know that Equation (19) has a bifurcation when

τ_{1} = 0.4, τ_{2} = 0.0622

. Equation (19) is unstable for

τ_{2} > 0.0622

, then the entropy value is more than 0. As in Section 3.2, the growth in the value of entropy shares the same pattern with that in

τ_{2}

. The change of entropy is shown in Figure 15. With the increase of entropy value, the system will be more unstable. In this case, the implementation effect of energy-saving and emission-reduction is poorly maintained.

3.6. The Influence of $τ_{2}, b_{3}, b_{4}$ on the Stability of Equation (19)

Figure 16 shows that the influence of

τ_{2}

y

is greater. With an increase in

τ_{2}

, the system moves from stable to unstable. The system is becoming unstable at

τ_{2} = 0.06

. The minimum value of

y

is 1.048 for

(τ_{2}, b_{3}) = (0.1, 0.1)

and the maximum value of

y

is 1.212 for

(τ_{2}, b_{3}) = (0.09, 0.1)

. The value of

y

is stable in the vicinity of 1.1. But

b_{3}

has little effect on

y

. Only when

τ_{2} > 0.06

, the value of

y

will turn from large to small with the increase of

b_{3}

. Therefore, we have to make

τ_{2} < 0.06

, so as to guarantee the effective implementation of energy-savings and emission-reduction.

When

τ_{2}

and

b_{3}

are larger and

b_{4}

is smaller, the value of

y

is closer to 1. Otherwise,

y

approximates −1. The change in the value of the parameters will cause the transformation of

y

between two states. These properties are shown in Figure 17. Therefore, we have to ensure that

τ_{2}

and

b_{3}

are as small as possible, so as to reduce carbon emissions.

3.7. The Influence of $τ_{1}, τ_{2}$ on the Stability of Equation (19)

It can be seen from Figure 18 that the system loses stability if

τ_{1}

shows an increase. In that case,

x

demonstrates a large range of ups and downs. On the other hand,

τ_{2}

has no impact on

x

3.8. The Influence of $τ_{1}, τ_{2}$ on the Entropy of Equation (19)

Figure 19 shows that the entropy becomes greater than 0 with an increase in

τ_{1}

. Meanwhile, the system is unstable. According to Section 3.4,

(τ_{1}, τ_{2}) = (0.4, 0.0622)

is a bifurcation point, so we can infer that the point

(0.4, 0.0622)

is on the boundary of the entropy change.

τ_{2}

still has no effect on entropy. So keeping the other parameters constant, the system is stable when

τ_{1} < 0.4

4. Bifurcation Control

The fact that the system loses stability at the bifurcation will seriously impact the effectiveness of the implementation of the policy. Thus we have to take measures to ensure the stability of the system. The system can be controlled by some approaches, for example, modified straight-line stabilization method [21], pole placement method [22], the OGY method (a control method of chaos proposed by Ott, Grebogi and Yorke) [23], time-delayed feedback method [24], and so forth. Here, we adopt the method of variable feedback control (see, e.g., [25], and references therein) to control for the Hopf bifurcation. The controlled system is given as follows:

{\begin{cases} \dot{x} (t) = a_{1} x (t) (\frac{y (t)}{M} - 1) - a_{2} y (t) + a_{3} z (t - τ_{1}) - k x (t), \\ \dot{y} (t) = - b_{1} x (t - τ_{2}) + b_{2} y (t) (1 - \frac{y (t)}{C}) + b_{3} z (t - τ_{1}) (1 - \frac{z (t - τ_{1})}{E}) - b_{4} u (t) - k y (t), \\ \dot{z} (t) = c_{1} x (t - τ_{2}) (\frac{x (t - τ_{2})}{N} - 1) - c_{2} y (t) - c_{3} z (t) + c_{4} u (t) (\frac{u (t)}{L} - 1), \\ \dot{u} (t) = d_{1} y (t) + d_{2} z (t) (\frac{z (t)}{K} - 1) - d_{3} u (t), \end{cases}

(20)

where

k

is a feedback control parameter. The state of Equation (20) can be changed by adjusting the value of

k

. If

k

is greater than the critical value, then Equation (20) returns to the stable state, otherwise the system is still in an unstable state.

4.1. Bifurcation Value of Equation (20) to $k$

In Case 1, we know that Equation (2) demonstrates the bifurcation when

τ_{1} = 0.4618, τ_{2} = 0

and it is unstable for

τ_{1} = 0.5, τ_{2} = 0

. It can be seen from Figure 4 and Figure 5.

Figure 20 shows the dynamic evolution process of Equation (20) to

k

for

(τ_{1}, τ_{2}) = (0.5, 0)

. We find out that Equation (20) undergoes bifurcation at

k = 0.1689

. In other words, Equation (20) is unstable when

k < 0.1689

and is stable when

k > 0.1689

4.2. Equation (20) is Unstable When $k < 0.1689$

Keep the values of other parameters unchanged and let

k = 0.05 < 0.1689

, and the time-domain plot and the EE attractor of Equation (20) are shown in Figure 21. We find out that the system (20) is unstable and has a basin of attraction. It fails to achieve bifurcation control.

4.3. Equation (20) is Stable When $k > 0.1689$

When the other parameters are consistent with the original, let

k = 0.2 > 0.1689

and the time-domain plot and the EE attractor of Equation (20) are displayed in Figure 22. We can see that Equation (20) maintains stable and

x

y

and

z

tend to equilibrium point. Therefore, bifurcation control is successful as

k > 0.1689

. We can conclude that the control effect will be larger if the control parameter

k

stands higher.

Based on the above analysis, we assume that the unstable system can return to stability if effective control is exerted. This means that when problems arise from energy-saving and emission-reduction policies and lead to market volatility, we can take control measures to ensure that the system returns to a stable state.

5. Conclusions

In this paper, we examine the stability of a four dimensional energy-saving and emission-reduction model with two delays. We focus on the impacts of delays and the feedback control on the stability of the system. The system shifts from stable to unstable when the delays are larger than the bifurcation values. In this case, a large fluctuation shows up in the system and affects the implementation of energy-saving and emission reduction policies. However, we can control the bifurcation of the system by adopting the variable feedback control approach and the system will return to a stable state when the control parameters are set.

The results show that time delays play an important role in the stability of the system. Only when the energy-savings and emission-reduction system is stable, can the purpose of reducing carbon emissions be achieved. Therefore, we must ensure that the delay parameters are in a stable region and then add other parameters to maintain a long-term stable system.

According to the above analysis, if we want to keep the energy-saving and emission-reduction system running in a stable state, we can take the following strategies when other things are equal. First, when

τ_{2} = 0

, we must make sure that

τ_{1} < 0.4618

b_{4} > 0.5

. Second, when

τ_{1} = 0.4

, we must keep

τ_{2} < 0.0622

Acknowledgments

The authors thank the reviewers for their careful reading and providing some pertinent suggestions. The research was supported by The Key Project of Natural Science Research of Anhui province (No. KJ2015A144); The Fundamental Research Funds for the Central Universities, South-Central University for Nationalities (CSY13011).

Author Contributions

Jing Wang performed mathematical derivation and numerical simulation; Yuling Wang built the economic model and provided economic interpretation of the conclusions; they wrote this research article together. Both authors have read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The influence of

τ_{1}

on the stability of Equation (19) when

τ_{2} = 0

. (a) bifurcation diagram; (b) the largest Lyapunov exponent plot.

Figure 1. The influence of

τ_{1}

on the stability of Equation (19) when

τ_{2} = 0

. (a) bifurcation diagram; (b) the largest Lyapunov exponent plot.

Figure 2. Time-domain plot when

τ_{1} = 0.4 < τ_{10} = 0.4618

τ_{2} = 0

Figure 2. Time-domain plot when

τ_{1} = 0.4 < τ_{10} = 0.4618

τ_{2} = 0

Figure 3. The EE attractor when

τ_{1} = 0.4 < τ_{10} = 0.4618

τ_{2} = 0

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 3. The EE attractor when

τ_{1} = 0.4 < τ_{10} = 0.4618

τ_{2} = 0

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 4. Equation (19) is unstable when

τ_{1} = 0.5 > τ_{10} = 0.4618, τ_{2} = 0

. (a) time-domain plot; (b) frequency spectrum plot.

Figure 4. Equation (19) is unstable when

τ_{1} = 0.5 > τ_{10} = 0.4618, τ_{2} = 0

. (a) time-domain plot; (b) frequency spectrum plot.

Figure 5. The EE attractor when

τ_{1} = 0.5 > τ_{10} = 0.4618, τ_{2} = 0

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 5. The EE attractor when

τ_{1} = 0.5 > τ_{10} = 0.4618, τ_{2} = 0

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 6. The entropy plot respect to

τ_{1}

when

τ_{2} = 0

Figure 6. The entropy plot respect to

τ_{1}

when

τ_{2} = 0

Figure 7. The influence of

τ_{1}

and

b_{3}

y

when

τ_{2} = 0

Figure 7. The influence of

τ_{1}

and

b_{3}

y

when

τ_{2} = 0

Figure 8. The influence of

τ_{1}

and

b_{4}

y

when

τ_{2} = 0

Figure 8. The influence of

τ_{1}

and

b_{4}

y

when

τ_{2} = 0

Figure 9. The influence of

τ_{1}, b_{3}

and

b_{4}

y

when

τ_{2} = 0

Figure 9. The influence of

τ_{1}, b_{3}

and

b_{4}

y

when

τ_{2} = 0

Figure 10. The influence of

τ_{2}

on the stability of Equation (19) when

τ_{1} = 0

. (a) bifurcation diagram; (b) the largest Lyapunov exponent plot.

Figure 10. The influence of

τ_{2}

on the stability of Equation (19) when

τ_{1} = 0

. (a) bifurcation diagram; (b) the largest Lyapunov exponent plot.

Figure 11. Time-domain plot when

τ_{1} = 0.4, τ_{2} = 0.04 < τ_{20} = 0.0622

Figure 11. Time-domain plot when

τ_{1} = 0.4, τ_{2} = 0.04 < τ_{20} = 0.0622

Figure 12. The EE attractor when

τ_{1} = 0.4, τ_{2} = 0.04 < τ_{20} = 0.0622

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 12. The EE attractor when

τ_{1} = 0.4, τ_{2} = 0.04 < τ_{20} = 0.0622

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 13. Equation (19) is unstable when

τ_{1} = 0.4, τ_{2} = 0.08 > τ_{20} = 0.0622

. (a) Time-domain plot; (b) Poincare plot.

Figure 13. Equation (19) is unstable when

τ_{1} = 0.4, τ_{2} = 0.08 > τ_{20} = 0.0622

. (a) Time-domain plot; (b) Poincare plot.

Figure 14. The EE attractor when

τ_{1} = 0.4, τ_{2} = 0.08 > τ_{20} = 0.0622

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 14. The EE attractor when

τ_{1} = 0.4, τ_{2} = 0.08 > τ_{20} = 0.0622

and

(x (0), y (0), z (0), u (0)) = (0.3, 0.5, 0.7, 0.9)

. (a)

x (t), y (t), z (t)

; (b)

x (t), y (t), u (t)

; (c)

x (t), z (t), u (t)

; (d)

y (t), z (t), u (t)

Figure 15. The entropy plot respect to

τ_{2}

when

τ_{1} = 0.4

Figure 15. The entropy plot respect to

τ_{2}

when

τ_{1} = 0.4

Figure 16. The influence of

τ_{2}

and

b_{3}

y

when

τ_{1} = 0.4

Figure 16. The influence of

τ_{2}

and

b_{3}

y

when

τ_{1} = 0.4

Figure 17. The influence of

τ_{2}, b_{3}

and

b_{4}

y

when

τ_{1} = 0.4

. (a,b) shown from different angles.

Figure 17. The influence of

τ_{2}, b_{3}

and

b_{4}

y

when

τ_{1} = 0.4

. (a,b) shown from different angles.

Figure 18. The influence of

τ_{1}

and

τ_{2}

x

when

τ_{1} \in [0.1, 0.6]

τ_{2} \in [0.01, 0.1]

Figure 18. The influence of

τ_{1}

and

τ_{2}

x

when

τ_{1} \in [0.1, 0.6]

τ_{2} \in [0.01, 0.1]

Figure 19. The influence of

τ_{1}

and

τ_{2}

on entropy of Equation (19) when

τ_{1} \in [0.1, 0.6]

τ_{2} \in [0.01, 0.1]

Figure 19. The influence of

τ_{1}

and

τ_{2}

on entropy of Equation (19) when

τ_{1} \in [0.1, 0.6]

τ_{2} \in [0.01, 0.1]

Figure 20. The influence of

k

on the stability of Equation (20) when

(τ_{1}, τ_{2}) = (0.5, 0)

. (a) bifurcation diagram; (b) the largest Lyapunov exponent plot.

Figure 20. The influence of

k

on the stability of Equation (20) when

(τ_{1}, τ_{2}) = (0.5, 0)

. (a) bifurcation diagram; (b) the largest Lyapunov exponent plot.

Figure 21. Equation (20) is unstable when

k = 0.05 < 0.1689

for

(τ_{1}, τ_{2}) = (0.5, 0)

. (a) time-domain plot; (b) the EE attractor.

Figure 21. Equation (20) is unstable when

k = 0.05 < 0.1689

for

(τ_{1}, τ_{2}) = (0.5, 0)

. (a) time-domain plot; (b) the EE attractor.

Figure 22. Equation (20) is stable when

k = 0.2 > 0.1689

for

(τ_{1}, τ_{2}) = (0.5, 0)

. (a) time-domain plot; (b) the EE attractor.

Figure 22. Equation (20) is stable when

k = 0.2 > 0.1689

for

(τ_{1}, τ_{2}) = (0.5, 0)

. (a) time-domain plot; (b) the EE attractor.

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Wang, J.; Wang, Y. Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays. Entropy 2016, 18, 371. https://doi.org/10.3390/e18100371

AMA Style

Wang J, Wang Y. Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays. Entropy. 2016; 18(10):371. https://doi.org/10.3390/e18100371

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Wang, Jing, and Yuling Wang. 2016. "Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays" Entropy 18, no. 10: 371. https://doi.org/10.3390/e18100371

APA Style

Wang, J., & Wang, Y. (2016). Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays. Entropy, 18(10), 371. https://doi.org/10.3390/e18100371

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Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays

Abstract

1. Introduction

2. Equilibrium Points and Local Stability

2.1. Case 1 $τ_{1} > 0, τ_{2} = 0$

2.2. Case 2 $τ_{1} > 0, τ_{2} > 0$

3. Numerical Simulation and Analysis

3.1. The Influence of $τ_{1}$ on the Stability of Equation (19)

3.2. The Influence of $τ_{1}$ on the Entropy of Equation (19)

3.3. The Influence of $τ_{1}, b_{3}, b_{4}$ on the Stability of Equation (19)

3.4. The Influence of $τ_{2}$ on the Stability of Equation (19)

3.5. The Influence of $τ_{2}$ on the Entropy of Equation (19)

3.6. The Influence of $τ_{2}, b_{3}, b_{4}$ on the Stability of Equation (19)

3.7. The Influence of $τ_{1}, τ_{2}$ on the Stability of Equation (19)

3.8. The Influence of $τ_{1}, τ_{2}$ on the Entropy of Equation (19)

4. Bifurcation Control

4.1. Bifurcation Value of Equation (20) to $k$

4.2. Equation (20) is Unstable When $k < 0.1689$

4.3. Equation (20) is Stable When $k > 0.1689$

5. Conclusions

Acknowledgments

Author Contributions

Conflicts of Interest

References

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Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays

Abstract

1. Introduction

2. Equilibrium Points and Local Stability

2.1. Case 1 τ 1 > 0 , τ 2 = 0

2.2. Case 2 τ 1 > 0 , τ 2 > 0

3. Numerical Simulation and Analysis

3.1. The Influence of τ 1 on the Stability of Equation (19)

3.2. The Influence of τ 1 on the Entropy of Equation (19)

3.3. The Influence of τ 1 , b 3 , b 4 on the Stability of Equation (19)

3.4. The Influence of τ 2 on the Stability of Equation (19)

3.5. The Influence of τ 2 on the Entropy of Equation (19)

3.6. The Influence of τ 2 , b 3 , b 4 on the Stability of Equation (19)

3.7. The Influence of τ 1 , τ 2 on the Stability of Equation (19)

3.8. The Influence of τ 1 , τ 2 on the Entropy of Equation (19)

4. Bifurcation Control

4.1. Bifurcation Value of Equation (20) to k

4.2. Equation (20) is Unstable When k < 0.1689

4.3. Equation (20) is Stable When k > 0.1689

5. Conclusions

Acknowledgments

Author Contributions

Conflicts of Interest

References

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2.1. Case 1 $τ_{1} > 0, τ_{2} = 0$

2.2. Case 2 $τ_{1} > 0, τ_{2} > 0$

3.1. The Influence of $τ_{1}$ on the Stability of Equation (19)

3.2. The Influence of $τ_{1}$ on the Entropy of Equation (19)

3.3. The Influence of $τ_{1}, b_{3}, b_{4}$ on the Stability of Equation (19)

3.4. The Influence of $τ_{2}$ on the Stability of Equation (19)

3.5. The Influence of $τ_{2}$ on the Entropy of Equation (19)

3.6. The Influence of $τ_{2}, b_{3}, b_{4}$ on the Stability of Equation (19)

3.7. The Influence of $τ_{1}, τ_{2}$ on the Stability of Equation (19)

3.8. The Influence of $τ_{1}, τ_{2}$ on the Entropy of Equation (19)

4.1. Bifurcation Value of Equation (20) to $k$

4.2. Equation (20) is Unstable When $k < 0.1689$

4.3. Equation (20) is Stable When $k > 0.1689$