*5.2. Simulation Results*

We select the above group of parameters given in Section 5.1, that is,

$$a\_1 = 0.14, \ a\_2 = 0.2, \ k = 0.5 \text{ S}\_1 = 2 \text{ S}\_2 = 0.5 \text{ N}\_1 = 276.27, \ \text{N}\_2 = 151.9485, \ m = 138.135.$$

After calculation, we find that the group of parameters satisfies hypothesis conditions (*H*1) and (*H*3), the hypothesis condition (*H*2) is not satisfied, that is, *E*<sup>1</sup> and *E*<sup>3</sup> exist, but *E*<sup>2</sup> does not exist. Since *E*<sup>1</sup> is a boundary equilibrium, we should actually discuss the stability of the non-trivial equilibrium, so we will discuss the stability of the equilibrium *E*<sup>3</sup> in the following part, and *E*<sup>3</sup> = (119.81, 119.82); from Equation (17), we calculate that *T*<sup>2</sup> = 0.7365 > 0, *D*<sup>2</sup> = 0.0815 > 0. Thus, equilibrium *E*<sup>3</sup> is locally asymptotically stable when *τ* = 0. We select initial value (132, 50), and the simulation result of the stable equilibrium *E*<sup>3</sup> is shown in Figure 3.

**Figure 3.** Equilibrium *E*<sup>3</sup> of system (3) is locally asymptotically stable when *τ* = 0.

**Remark 2.** *From Figure 3, we take τ* = 0*, which indicates that the impact of technological innovation on carbon emissions is instantaneous. We conclude that China will reach the carbon peak in 2026, and the peak value is about 14.5 billion tons. With time going by, the carbon emissions will become smaller and smaller, and gradually reach stability in 2067. The stable value is at about 11.981 billion tons, carbon absorption will increase with time and stabilize at 11.982 billion tons around 2067. For this reason, if China implements the current policy, it will achieve peak carbon dioxide emissions by 2030, but not carbon neutrality by 2060. Therefore, China should adopt stronger policies to lay the foundation for carbon neutrality. In Figure 1, we can see that China's peak carbon dioxide emissions time is 2029, with a peak value of 13.81 billion tons, while in Figure 3, the simulation results show that China has completed peak carbon dioxide emissions in an earlier time and the peak value will increase. As the coefficients in our model are constant, but in real life, the coefficients of the model may change with time. Another reason is that our model does not consider too many factors, such as the influence of construction industry and carbon sink, which leads to a slight deviation in our simulation results. However, our simulation results are at least consistent with the realization of peak carbon dioxide emissions before 2030.*

After that, we consider the time of carbon peak at the difference of the influence factors of industrial structure and energy structure on carbon emissions (see Figure 4).

**Figure 4.** Change of carbon emission under different values for the influence coefficient of energy structure reform on carbon emissions amount *k*.

**Remark 3.** *From Figure 4, when the influence coefficient of energy structure reform on carbon emissions amount k increases, that is, with the optimization of industrial structure and the improvement of energy structure, the time for China to reach the carbon peak will become shorter and shorter. When k* = 0.4*, we predict that the peak value of carbon will reach 14.573 billion tons in April 2026. When k* = 0.6*, China will reach the peak of carbon in April 2025, with a peak of 14.43 billion tons. When k* = 0.8*, we predict that China will reach peak carbon dioxide emissions in 2025, with a peak of 14.34 billion tons. This is reasonable because when China's emission reduction policy is effectively implemented, corresponding policies are introduced, new development of new energy exploration and application technologies is achieved, and industrial reform is conducted in depth. China's secondary industry with high carbon emissions will be transformed into a green and sustainable tertiary industry in an all-round way. The proportion of clean energy, mainly natural gas, will be greatly increased, and the peak value of carbon emissions will be reduced while reaching the maximum value ahead of time.*

Later, we select the previous data, and when the natural growth rate of carbon absorption increases, we arrive at the following conclusions: (see Figure 5).

**Remark 4.** *With the increase of natural growth rate of carbon absorption, the peak value of carbon emissions becomes smaller and smaller, and the time for carbon to reach the peak value becomes shorter and shorter, and the time for carbon neutrality becomes shorter. The red line in Figure 5 shows the apparent trend of carbon emissions and carbon absorption at a*<sup>2</sup> = 0.1*. China will reach peak carbon dioxide emissions around 2028, but it will take nearly one hundred years to achieve carbon neutrality. The blue curve shows the change trend of carbon emissions and carbon absorption at a*<sup>2</sup> = 0.2*, and it is predicted that China will reach peak carbon dioxide emissions around 2026 and be carbon neutral in 2072. The black curve shows the change trend of carbon emissions and carbon absorption at a*<sup>2</sup> = 0.3*. It can be seen that China will reach peak carbon dioxide emissions* *around 2026 and become carbon neutral in 2057. Although China can achieve peak carbon dioxide emissions by 2030 at different natural growth rates of carbon absorption, China cannot achieve carbon neutrality by 2060 at a lower natural growth rate of carbon absorption. It also shows that with the country's emphasis on ecological protection and green development, people have a clearer understanding of green development, saving energy, planting trees, increasing forest vegetation coverage and increasing urban green space, which leads to an increase in the natural growth rate of carbon absorption. The smaller the carbon peak, the shorter the time to achieve carbon neutrality. In Figure 5, we can see that when the natural growth rate of carbon absorption reaches 0.3, it is possible for China to achieve carbon neutrality by 2060. As the natural growth rate of carbon absorption is relatively high, in order to achieve China's goal of carbon neutrality by 2060, we should not only consider increasing the natural growth rate of carbon absorption to achieve carbon neutrality, but also consider optimizing the industrial structure and energy structure to reduce the natural growth rate of carbon emissions. Therefore, we also need to deepen the industrial reform and optimize the energy structure to reduce the natural growth rate of China's carbon emissions.*

**Figure 5.** Analysis of the influence of natural growth rate of carbon absorption on carbon absorption and emission.

Further on, we find that Formula (22) has only one positive root, so we calculate that *τ*(0) <sup>1</sup> <sup>=</sup> 3.606. From Theorem 3, we know that when *<sup>τ</sup>* <sup>∈</sup> [0, *<sup>τ</sup>*(0) <sup>1</sup> ), the equilibrium *E*<sup>3</sup> is locally asymptotically stable and unstable when *τ* > *τ*(0) <sup>1</sup> . From Equations (37) and (41), we have

$$\text{Re}(M) = 0.1955 > 0; \text{Re}(H) = -3.51 \ast 10^{-7} < 0; \ r = \sqrt{-\frac{\text{Re}(M)\mu}{\text{Re}(H)}} > 0. \tag{44}$$

When *τ* = 0.5 < *τ*(0) <sup>1</sup> = 3.606, the simulation result of the stable equilibrium *E*<sup>3</sup> is as shown in Figure 6.

**Remark 5.** *From Figure 6, carbon emissions x reached their peak around 2060, with a peak of about 14.6 billion tons, and then decreased year by year and gradually stabilized, and they became stable* *around 2063. Carbon absorption y tends to be stable around 2067. Once carbon emissions and carbon absorption are stable, the value of carbon emissions is about 11.9 billion tons, and the value of carbon absorption is about 12 billion tons. If this continues, it will be difficult for China to achieve carbon neutrality before 2060. Therefore, China needs to take some measures, such as improving the level of carbon emission reduction technology to reduce the carbon emissions of the steel industry.*

**Figure 6.** When *τ* = 0.5, the system (3) is locally asymptotically stable at the equilibrium point *E*3.

When *τ* = 3.61 > *τ*(0) <sup>1</sup> = 3.606, from Theorem 4, we learn that the system (3) has a stable periodic solution at the at the equilibrium *E*3, as shown in Figure 7.

**Figure 7.** When *τ* = 3.61, the system (3) has a stable periodic solution near the equilibrium *E*3.

**Remark 6.** *From Figure 7, when τ* = 3.61*, the system* (3) *is locally asymptotically stable at the equilibrium E*3*, which is consistent with* (4)*. τ* = 3.61*, that is, when the technical level is applied to the actual production of carbon emission reduction, the time required becomes longer, we can see that carbon emissions and carbon absorption fluctuate periodically, and the period is about 13 years. As a result of the long delay time, it is inconsistent with the actual production level at present, so we do not consider the practical application of this situation.*
