This paper reports a new four-dimensional energy-saving and emission-reduction chaotic system. The system is obtained in accordance with the complicated relationship between energy saving and emission reduction, carbon emission, economic growth, and new energy development. The dynamics behavior of the system will be analyzed by means of Lyapunov exponents and equilibrium points. Linear feedback control methods are used to suppress chaos to unstable equilibrium. Numerical simulations are presented to show these results.

Since energy saving and emission reduction is the most effective way to control carbon emissions, how to promote energy saving and emission reduction is becoming the hot topic of academic research. Calculation and control analysis of carbon emissions have attracted a great deal of attention from various fields of researchers. Feng et al. [

From the above analysis, we can see that previous researches mainly explored the influence of energy intensity by energy structures, technical change and management level, energy consumption structure, economic structure, energy prices, and so on [

Chaos analysis and applications in dynamical systems are observed in many practical applications in engineering, biology, and economics [

In the three-dimensional energy-saving and emission-reduction system, the authors have not considered clean energy development (including wind energy, solar energy, hydropower, geothermal, biomass energy, and so forth), but most of the countries are developing and making use of new energy resources. Therefore, it is necessary to add new energy resources to the three-dimensional energy-saving and emission-reduction system.

By adding a new variable (new energy development) to the three-dimensional energy-saving and emission-reduction system, a new four-dimensional energy-saving and emission-reduction system is obtained.

This paper establishes a new four-dimensional energy-saving and emission-reduction system. It is organized as follows: Section

By adding a new variable

The first formula in (

The second formula in (

The third formula in (

The fourth formula in (

We can obtain that the system (

For simplicity, we fix the following parameters:

We fix the following parameters:

Solving (

Let

We fix parameters as above, and then obtain the equilibrium point

Consider the following:

If

To study the long-term dynamical behavior of the system (

Let

The subsystem presented in (

Equation (

The characteristic polynomial is obtained as

Suppose

When parameters are fixed as above at

For the discussion of the homoclinic orbit of

Next, substitute (

The following equation can be obtained:

If

According to the above analysis, homoclinic orbit

The subsystem presented in (

The proof is similar to the proof of Theorem

Theorems

We chose a set of parameters as follows:

A four-dimensional energy-saving and emission-reduction chaotic attractor.

A limit cycle.

Lyapunov exponent spectrum.

Because an energy resource system in the chaotic state is very sensitive to its initial condition and chaos often causes irregular behavior, chaos is undesirable. In this section, linear feedback methods [

We guide the chaotic trajectory

The Jacobian matrix of the system (

Let

According to Routh-Hurwitz criteria, if

Numerical experiments are carried out to integrate the controlled system (

The stable equilibrium point

Similarly, we can prove that another three equilibrium points

The stable equilibrium point

We have established a four-dimensional nonlinear dynamics model for the energy-saving and emission-reduction system and have analyzed the dynamics behavior of the system. When some parameters are adjusted, the dynamic behavior of energy-saving and emission-reduction, carbon emissions, economic growth, and new energy development displays some regulated phenomena. By observing these phenomena, we can figure out the affecting factors for energy intensity and grasp the statistical results which meet the real situation. This four-dimensional energy-saving and emission-reduction system will be more satisfactory for actual energy saving and emission reduction and instructive for the energy saving and emission reduction of China. The research results provide a key to energy saving and emission reduction, that is, to develop energy-saving and emission reduction as soon as possible with proper strategies rather than simply increasing investment. The theoretical proof and the empirical study ensure the necessity and significance to carry out comprehensive energy saving and emission reduction.

The research was supported by Nanjing Normal University Taizhou College Project and Taizhou Natural Science and Technology Development Project 2012.