Study on Evaluation of Order Degree of Water Resources Coupling System Considering Time Series Characteristics—Take Jiangxi Province as an Example

Abstract

In recent years, the order evaluation method of a coupled system based on synergetics has been successfully applied in the field of water resources evaluation and management. The evaluation of system order degree needs to comprehensively consider the simple giant system entropy of multiple order parameters. At this stage, the calculation of simple giant system entropy ignores the time series characteristics of order parameters, which makes the evaluation results of system order degree deviate from the actual changes of the system. Therefore, considering the time factor in the entropy calculation of a simple giant system, this paper proposes a synergetic order evaluation method considering the characteristics of time series. Then, taking Jiangxi Province as the research object, an example of a comprehensive evaluation of the order degree of the water resources coupling system is carried out. The relevant experimental results show that: The evaluation results of the synergetic order degree method considering the characteristics of time series presented in this paper are closer to the actual evolution situation of the system than the evaluation results without considering the characteristics of the time series, and maintaining high consistency with the actual evolution situation; the order degree of the water resources subsystem is greatly affected by the changes of water resources; and the entropy change of the water resources coupling system in Jiangxi Province is greatly affected by the entropy change of the water resources subsystem. The research result enhances the universality of the application of the synergetic order evaluation method in the related fields of a comprehensive evaluation of water resources systems and has a certain practical significance.

1. Introduction

The history of humanity can be written in terms of human interactions and interrelationships with water [1]. During the Anthropocene, humans exerted an unprecedented influence on the natural environment [2], and human–environment interactions are occurring at an unprecedented scale [3]. Human activities, such as the construction of dams and reservoirs, land use changes, and pollutant discharges, have resulted in watershed-scale [4] to significant changes in water systems. The fact that humans, water, and the natural environment are linked through interactions [5] is now widely recognized and accepted [6]. Therefore, it has become a common concern for water scientists and policy makers [7] to consider these interactions in water resource evaluation models to improve water management evaluation methods.

How to scientifically evaluate the complex water resources system with high social, economic, and ecological coupling is one of the important means to promote the realization of sustainable development and utilization of water resources and guarantee the sustainable and high-quality development of the national economy. Relevant scholars at home and abroad have conducted a lot of research in this field [8,9,10,11], and the comprehensive evaluation methods of water resources systems mainly include fuzzy evaluation methods [12,13,14], principal component analysis methods [15,16,17], system dynamics methods [18,19], multi-objective decision methods [20], entropy weight methods [21,22], artificial neural network methods [23], matter-element methods [24], and so on. Different methods can make a difference in assessment results. For example, the analytic hierarchy process and fuzzy comprehensive evaluation method focus on the judgments of experts, which are based on personal biases and often have significant differences in their experiences. The material element analysis method is suitable for multi-factor assessment, but the issue of incompatible indicators has not yet been fully addressed. The TOPSIS method and grey correlation analysis method are suitable for comparing multiple schemes. The TOPSIS method, due to its own limitations, can lead to changes in positive and negative ideal solutions, thus lacking stability [25]. The gray correlation analysis method has a limited scope of application and requires querying sample data with time series characteristics. Based on the analytic hierarchy process, the fuzzy comprehensive evaluation of the water resources carrying capacity of Shanxi Province was carried out [26]; Wang et al. used a theoretical assessment method based on a binary water cycle model in the early 21st century [27] to measure the impact of anthropogenic disturbances on natural circulation systems; Liu [28] analyzed the water resources supply, demand, and exploitation potential of the Aegean Islands in Greece; Ni’s research [29] includes two aspects: Water quantity and water quality. Water quantity evaluation includes surface water, groundwater, and total water resources, while water quality evaluation involves surface water and groundwater; Shivanita [30] evaluated water resources, precipitation, surface water, groundwater, the total amount of water resources, as well as the combination and transformation relationships between various components and their impacts; Huang [31] established the conceptual hydrogeological model of the groundwater system and the corresponding mathematical model and calculated the mining output of groundwater under different conditions using the numerical simulation method based on a finite difference. Duan Xinguang et al. [32] evaluated the current situation of water resource carrying capacity in Xinjiang by using a fuzzy comprehensive evaluation model. Cao Lijuan et al. [33] conducted a comprehensive evaluation and research on the water resource carrying capacity of some counties and cities in Gansu Province based on the principal component analysis method. Dai Tao et al. [34] established a water resources carrying capacity evaluation model based on the set-pair analysis method.

The purpose of the comprehensive evaluation of water resources system research is to address the coordinated development of regional economic and social, ecological, and water resources and other subsystems. The development and evolution of subsystems often have interactive relationships [35], thus constituting a coupled system of water resources. How to coordinate the interaction of social, economic, ecological, and energy subsystems in the water resources coupling system and maintain the dynamic balance among the systems, so as to realize the benign cycle and evolution of the water resources coupling system, is the main purpose of water resources evaluation and management [36]. Therefore, it is necessary to carry out research related to the comprehensive evaluation of water resource coupling systems from the perspective of system science and to expand and improve the theoretical and methodological systems in the field of water resources evaluation and management research. Synergetics is a science of systems across natural and social disciplines, proposed by German physicist Hermann Hacken in the 1970s [37]. Synergetics focuses on the interaction of intrinsic factors in complex coupled systems and analyzes the laws and characteristics of systems that exhibit a non-equilibrium ordered structure and transition from disorder to order under specific conditions. By carrying out system analysis through synergetics, we can identify the key control factors driving the evolution of the system, build an orderly spatio-temporal structure of the coupled system by analyzing the synergy of each subsystem, and analyze the interaction mechanism among the subsystems in the coupled system [38].

At present, synergetics has been introduced into the field of water resources evaluation and management research, and some research results have been obtained. The concept of the water–food–energy nexus has been developed and intensively studied internationally in response to the enormous pressure on water, food, and energy from global climate change and socio-hydrological changes [39,40,41]. Sepehri et al. [42] mapped food susceptibility in Hamadan city based on the principle of synergetics. After selecting and normalizing the valid indices related to food, food susceptibility in Pingtung City was assessed using the synergistic theory [43]. The existing methods for calculating the order degree of synergetics are mainly based on simple giant system entropy and normalization treatment considering the regulatory threshold boundary. Among them, the normalization process considering the regulation threshold boundary needs to understand the quantitative and qualitative evaluation basis of the order parameters and the regulation threshold, while the simple giant system entropy mainly relies on the quantitative relationship of the order parameters in the system for evaluation, which is simple and easy to implement [44,45]. At this stage, the simple giant system entropy calculation process in the order degree of synergetics ignores the time series characteristics of the order parameters. The entropy value is calculated based on the information of different order parameters in the system, and in some scenarios (see the comparative analysis of examples in subSection 3.3 later and the application examples in reference [46]), the evaluation results of the order degree of the system and the evolutionary dynamics of order parameters are not consistent, which makes the ordered degree of the system deviate from the actual evolutionary dynamics of the system. To this end, this article proposes an evaluation method of synergy degree that takes into account the characteristics of time series in order to improve the theoretical foundation of the application of the evaluation method of synergy degree in the field of water resource evaluation and management research.

2. Research Methods

2.1. Order Degree of Synergetics Evaluation Considering Time Series Characteristics of Order Parameters

The order parameters are important fundamental concepts in covariance, used to represent the ordered structure and type of the system and also characterize the contribution of subsystems to the interaction of the coupled system. The order parameter reflects the interaction between the subsystems in the coupling system and also reflects the state of different observations of the subsystems. The correct selection of the system order parameter is the key to realizing the comprehensive evaluation of the orderliness of the water resources coupling system [47].

Order degree is defined as the state function of order parameters in synergetics, which is used to measure the degree of order or chaos of the system. It is assumed that there are N subsystems in the coupled system, and the j-th subsystem S_j evolution process needs to consider N_j sequential parameters, and the measurement value of order parameters is denoted as $e_j=(e_{j,1},e_{j,2},\dots ,e_{j,i},\dots ,e_{j,N_j})$, where e_j,i denotes the i-th order parameter in the j-th subsystem S_j. The measurement values are converted into measurement degrees by methods such as fuzzy mathematical affiliation principle [12,13,14] and normalization processing method [43], and then the measurement degree of subsystem S_j is noted as ${\xi }_j=({\xi }_{j,1},{\xi }_{j,2},\dots ,{\xi }_{j,i},\dots ,{\xi }_{j,N_j}) \mathrm{and} {\xi }_{j,i}\in \left[0,1\right]$. Then, the simple giant system entropy H(S_j) of subsystem S_j is calculated [42,43] as:

$$H(S_j)=-{\displaystyle \sum _{i=1}^{N_j}{P}_{j,i}\cdot \log P_{j,i}}$$

(1)

$$P_{j,i}=\frac{{\xi }_{j,i}}{{\displaystyle \sum _{i=1}^{N_j}{\xi }_{j,i}}}$$

(2)

Denote H_m(S_j) as the maximum value of the entropy of the simple giant system of subsystem S_j. It is used to indicate the most disordered or chaotic state of the subsystem. Then the order degree μ_j(t) of subsystem S_j is expressed as

$${\mu }_j(t)=1-\frac{H_{}(S_j)}{H_m(S_j)}$$

(3)

In the formula, μ_j(t)∈[0,1], the greater the μ_j(t), the greater the stability of the subsystem. And the order degree μ(t) of the coupling system can be weighted by the order degree μ_j(t) of each subsystem and the corresponding weight coefficient λ_i. The formula is:

$$\mu (t)={\displaystyle \sum _{j=1}^{N_j}{\lambda }_j{\mu }_j(t)}\hspace{1em}{\lambda }_j\ge 0\hspace{1em}\mathrm{also}\hspace{1em}{\displaystyle \sum _{j=1}^{N_j}{\lambda }_j}=1\hspace{0.17em}$$

(4)

The above-mentioned evaluation method based on the order degree of the synergetics of complex giant systems is simple to calculate and easy to implement, and it can reflect the evolution law of multi-indicator complex coupled systems to a certain extent. However, in Equation (2), which does not consider the time series characteristics of the order parameters and only considers the information of different order parameters in the subsystem to calculate the entropy H(S_j) of the subsystem, this method ignores the changes in the order parameters of the subsystem at the time level. This makes the entropy H(S_j) of the simple giant system unchanged or changed very little (see the comparative analysis of examples in Section 3.3 later) when the multiple order parameters in the subsystem maintain similar rates of change (including the direction and magnitude of change), and the entropy of the simple giant system is calculated by applying Equation (2), even though the order parameters continue to grow or decrease with time. This leads to inconsistency in the evolutionary dynamics of the order degree and order parameters of the subsystem over time, which affects the evaluation effectiveness of the method. To this end, the study redetermines the calculation of the entropy of this simple giant system by considering the time series characteristics of the order parameters and proposes an orderliness evaluation method that considers the time series characteristics of the order parameters.

A coupled system with N subsystems exists, and the evolution of the j-th subsystem S_j needs to consider N_j order parameters. The order parameter measurements considering the time series characteristics are denoted as $e_j\left(t\right)=(e_{j,1,t},e_{j,2,t},\dots ,e_{j,i,t},\dots ,e_{j,N_j,t})$, where e_j,i,t denotes the value of the i-th order parameter in the j-th subsystem at the t-th time period, t = 1,2,…,T, and T denotes the number of time periods of the time series. If there is a clear intermediate or interval type indicator characteristic of the order parameters, the normalization process is carried out by the fuzzy mathematical affiliation principle; the rest of the very small or very large indicators can be normalized by using the great value and the very small value or other normalization methods to convert the measurement value e_j into the measurement degree ${\xi }_j\left(t\right)=({\xi }_{j,1,t},{\xi }_{j,2,t},\dots ,{\xi }_{j,i,t},\dots ,{\xi }_{j,N_j,t}) \mathrm{and} {\xi }_{j,i,t}\in \left[0,1\right]$. The great value and the very small value in the time series of the order parameters need to be considered in the normalization process here. In particular, it should be noted that it is generally believed that the smaller the entropy, the smaller the uncertainty of the system, and the smoother the system. Therefore, when normalizing the transformed metrics, a smaller measurement of the order parameters indicates a better characterization of the order parameter metrics and vice versa.

After determining the measurement degree of each order parameter ${\xi }_j=\left\{{\xi }_j\left(1\right),{\xi }_j\left(2\right),\dots ,{\xi }_j\left(t\right),\dots ,{\xi }_j\left(T\right)\right\}$ in the subsystem S_j, the entropy of the simple giant system of the subsystem considering the time series characteristics is calculated as

$$H\left(S_j\left(t\right)\right)=-{\displaystyle \sum _{i=1}^{N_j}{P}_{j,i,t}\cdot \log P_{j,i,t}}$$

(5)

$$P_{j,i,t}=\frac{{\xi }_{j,i,t}}{{\displaystyle \sum _{t=1}^T{\xi }_{j,i,t}}}$$

(6)

The biggest difference between the evaluation method of order degree of synergetics considering the time series characteristics of order parameters and that without considering the time series characteristics lies in Formulas (2) and (6). Formula (2) calculates P_j,i in different order parameters of subsystem S_j in the same period, while Formula 6 calculates P_j,i,t in the same order parameters of subsystem S_j in different periods, so that the calculation of measurement degree can better consider the evolution trend and law characteristics of order parameters as time series. The remaining order degree calculation is the same as Formulas (3) and (4).

2.2. Entropy Change Value

To discern the direction of evolution of the order degree of a subsystem or a coupled system of water resources [47], the entropy variant of the system is also defined in synergetics.

$$\Delta \mu (t+1)=\mu (t+1)\hspace{0.17em}-\mu (t)$$

(7)

In the formula, μ(t) and μ(t + 1) are the order degree of the system in time period t and t + 1, respectively, and Δμ(t + 1) is the change value of the order degree from time period t to t + 1. According to the magnitude of the change value of the order degree Δμ(t + 1), the evolution direction and the internal stability of the system at time slot t + 1 can be judged. When Δμ(t + 1) > 0, it means that the order degree of the system increases, the system evolves in an orderly direction, and the system behaves as “normal state”. When Δμ(t + 1) < 0, the subsystem or coupled system has at least one order parameter or subsystem is not transformed to the order direction, and the water resources system is “abnormal state”. When Δμ(t + 1) = 0, that the water resources system is in a “steady state”, the order degree does not increase or decrease.

2.3. Entropy Weight Method

In the calculation of the order degree of the coupled system, the above order degree evaluation needs to set the weights of different subsystems. The study uses the entropy method of objective assignment to calculate the weights of each subsystem λ_i. The core idea is to calculate the objective weights by the degree of variability of indicators. The formula for calculating the information entropy of the l-th indicator in the comprehensive evaluation considering L indicators is:

$$E_l=-\frac{1}{\ln N_m}{\displaystyle \sum _{m=1}^{N_m}{p}_{l,m}·\ln \left(p_{l,m}\right)}\hspace{0.17em}$$

(8)

$$p_{l,m}=\frac{X_{l,m}}{{\displaystyle \sum _{m=1}^{N_m}{X}_{l,m}}}$$

(9)

In the formula, E_l denotes the information entropy of the l-th indicator; N_m denotes the length of the sequence of the l-th data; and X_l,m denotes the m-th data in the sequence of the l-th indicator after normalization. In addition, considering that p_l,m may be zero, therefore define $\underset{p_{l,m}\to 0}{\lim }{p}_{l,m}·\ln (p_{l,m})=0$. Further, the weight can be calculated from the information entropy of each indicator, and the weight calculation formula is:

$${\lambda }_l=\frac{1-E_l}{L-{\displaystyle \sum _{l=1}^L{E}_l}}$$

(10)

3. Examples of Applications

3.1. Research Area

This paper takes the water resources coupling system in Jiangxi Province as the research object to carry out an example application study. Jiangxi Province is located on the south bank of the Yangtze River, where the middle and lower reaches meet, and has a well-developed water system and many rivers. Jiangxi Province as a whole is in a water-abundant area, but the spatial and temporal distribution of water resources is uneven, and the timing of water supply and water use is not synchronized. There are different degrees of resource water shortage, engineering water shortage, and water quality water shortage. Therefore, it is of great practical significance to carry out a comprehensive evaluation study of the coupled water resources system in Jiangxi Province based on the order degree of synergetics for the socio-economic development and sustainable utilization of water resources in Jiangxi Province. The geographical location map of Jiangxi Province is shown in Figure 1.

3.2. Selection and Normalization of Order Parameters

To construct an evaluation model of synergy degree, the first step is to determine the order parameters of a coupled water resources system. The water resources coupling system in Jiangxi Province mainly includes water resources, social, economic, energy, and ecological subsystems. The selection of the order parameters should take into account the current situation of water resource development and utilization, the law of the hydrological cycle, the influence of regional social and economic factors, as well as the difficulty of obtaining digital data. Taking the above factors into account, the order parameters of the water resources coupling system in Jiangxi Province were selected as listed in

Table 1. Order parameter index of water resources coupling system in Jiangxi.

Subsystems	Serial Number	Order Parameters	Property Types
Social subsystem	1	Year-end resident population (million people)	Max-type
	2	Natural population growth rate (‰)	Max-type
	3	Per capita disposable income of residents (yuan)	Max-type
	4	Population urbanization rate (%)	Max-type
	5	Per capita consumption expenditure of residents (yuan)	Max-type
Economic subsystem	6	Gross regional product (billion yuan)	Max-type
	7	Gross regional product per capita (yuan per person)	Max-type
	8	Local fiscal general budget expenditure (billion yuan)	Max-type
Energy subsystem	9	Total energy consumption (million tons of standard coal)	Min-type
	10	Energy consumption of million yuan GDP (tons of standard coal)	Min-type
	11	Energy self-sufficiency rate (%)	Max-type
Water resources subsystem	12	Total water resources (billion m3)	Max-type
	13	Water resources per capita (m3)	Max-type
	14	Total reservoir group capacity (billion m3)	Max-type
	15	Total water consumption (billion m3)	Min-type
	16	Water consumption per million yuan GDP (m3)	Min-type
	17	Water consumption per capita (m3)	Min-type
	18	Water resource extraction rate (%)	Min-type
	19	Percentage of groundwater supply (%)	Min-type
Ecological subsystem	20	Sulfur dioxide emissions (million tons)	Min-type
	21	Chemical oxygen demand emissions (million tons)	Min-type
	22	Forest cover (%)	Max-type
	23	Soil erosion control area (thousand m2)	Max-type
	24	Ecosystem water use efficiency (%)	Max-type

. The data for this study were mainly obtained from the official website of the National Bureau of Statistics of China (https://data.stats.gov.cn/ [accessed on 3 July 2023]) and the Jiangxi Provincial Statistical Yearbook (http://tjj.jiangxi.gov.cn/col/col38595/index.html [accessed on 3 July 2023]) of the Jiangxi Provincial Bureau of Statistics.

After determining the order parameters of the water resources-coupled system, the measured values of the order parameters were normalized to obtain the measured degrees, and the process of measuring the measured degrees of different order parameters over time is given in Figure 2. In the social subsystem (Figure 2a), all the order parameters maintain a good development except for order parameter 2, which has a large fluctuation and an overall deterioration trend between 2005 and 2021. Although the implementation of the Amendment to the Population and Family Planning Law (Draft) and the official opening of the two-child policy in Jiangxi Province in January 2016 could make the natural population growth rate in Jiangxi Province maintain a good state, the sudden outbreak of the new crown pneumonia epidemic in 2020 led to a sharp rise in order parameter 2 and a sharp decline in the natural population growth rate in Jiangxi Province. In the economic subsystem (Figure 2b), all the order parameters show a good development trend. In the energy subsystem (Figure 2b), except for order parameter 10, which shows a good development trend with the development of science and technology and the implementation of energy conservation and emission reduction policies, the local primary energy in Jiangxi Province is less and the energy demand is rising year by year, which makes order parameters 9 and 11 show a deteriorating trend. In the ecological subsystem (Figure 2d), except for the order parameter 21, which has a large fluctuation and shows a deterioration trend, all the order parameters show an improvement. In the water resources subsystem (Figure 2e), except for order parameters 14, 16, and 19, which are influenced by the development of science and technology and the support of national policies and continue to show a good development trend, the other order parameters 13, 15, 17 and 18 are influenced by the abundance and depletion of water resources in different years of order parameter 12 and show large fluctuations.

3.3. Comparison of Two Evaluation Methods of Synergetic Order Degree

In order to test the effectiveness of the evaluation method of synergy degree that considers the time series characteristics of the order parameters proposed in the study, the economic subsystem with the most obvious trend of system evolution was selected for the example analysis, and the evaluation results of the two synergetic order degrees calculated according to Equations (1)–(7) are shown in Figure 3 and listed in

Table 2. Evaluation results of ordering degree of economic subsystem with and without time series attributes.

Year	No Consideration	Consideration	Year	No Consideration	Consideration
2005	0.00	0.00	2014	2.47 × 10−3	0.33
2006	7.04 × 10−6	0.02	2015	7.35 × 10−3	0.38
2007	3.21 × 10−5	0.04	2016	5.42 × 10−3	0.43
2008	1.95 × 10−5	0.07	2017	9.89 × 10−3	0.50
2009	4.59 × 10−5	0.09	2018	0.02	0.61
2010	1.24 × 10−5	0.13	2019	0.09	0.73
2011	1.02 × 10−4	0.19	2020	0.22	0.80
2012	6.93 × 10−4	0.23	2021	0.00	1.00
2013	1.39 × 10−3	0.28

. The analysis results show that the gross regional product, per capita gross regional product, and local fiscal general budget expenditure all continue to grow, and the measurement degrees of order parameters 6, 7, and 8 all gradually decrease, and the system should develop in an orderly direction. And in the evaluation results of the order degree without considering the time series characteristics, the order degree of the economic subsystem fluctuates close to zero. This is due to the fact that the population of Jiangxi Province has changed little in recent years, and the growth trends of the indicators of gross regional product, per capita gross regional product, and local fiscal general budget expenditure are basically the same, resulting in little change in the entropy of the simple giant system calculated according to Equations (1) and (2), resulting in the calculation results of the orderliness of the subsystem being close to zero and the evaluation results deviating from the actual situation. In contrast, the evaluation results of the order degree calculated according to Equations (5) and (6) show that the order degree of the economic subsystem gradually increases and the system develops in the direction of order, which is consistent with the actual situation that the economic subsystem of Jiangxi Province maintains a good development trend from 2005 to 2021, which indicates that the evaluation results of the order degree considering the characteristics of time series are more reasonable.

In addition, the study also takes the energy subsystem, considering only ordinal covariates 9 and 11 as an example, and the evaluation results of the two synergetic order degrees calculated according to Equations (1)–(7) are shown in Figure 4 and listed in

Table 3. Evaluation results of ordering degree of energy subsystem with and without time series attributes.

Year	No Consideration	Consideration	Year	No Consideration	Consideration
2005	0.00	1.00	2014	8.97 × 10−3	0.29
2006	0.07	0.89	2015	6.25 × 10−3	0.23
2007	0.01	0.77	2016	2.03 × 10−4	0.16
2008	0.06	0.73	2017	2.80 × 10−3	0.09
2009	0.08	0.67	2018	6.63 × 10−3	0.04
2010	2.39 × 10−4	0.50	2019	1.42 × 10−3	0.03
2011	0.02	0.47	2020	1.37 × 10−3	0.02
2012	0.02	0.43	2021	1.40 × 10−4	0.00
2013	0.02	0.36

. The energy subsystem, considering only order parameters 9 and 11, develops in an unfavorable direction with the increase in total energy consumption and the decrease in energy self-sufficiency. The order degree of the energy subsystem considering time series characteristics shows a decreasing trend, while the order degree of the energy subsystem fluctuates near zero in the evaluation results of the order degree without considering time series characteristics, which again indicates that the evaluation results of the order degree considering time series characteristics are more reasonable.

3.4. Evaluation of Order Degree of Water Resources Coupling System and Analysis of Change Value of Order Degree

On the premise of considering the time series characteristics of the order parameters, the evaluation of the order degree of the water resources coupling system in Jiangxi Province was carried out. After determining the measurement degree of the order parameters of each subsystem, the order degree of each subsystem in different time periods was calculated, and the weights were determined by the entropy weight method to obtain the order degree of the water resources coupling system. The evaluation results are shown in

Table 4. Evaluation results of order degree of water resources coupling system in Jiangxi Province.

Year	Social Subsystem	Economic Subsystem	Energy Subsystem	Ecological Subsystem	Water Resources Subsystem	Water Coupling System
2005	0.0000	0.0000	0.4002	0.0083	0.3001	0.1414
2006	0.0236	0.0150	0.4110	0.000	0.3823	0.1605
2007	0.0826	0.0377	0.3150	0.0511	0.0317	0.1155
2008	0.1458	0.0653	0.3282	0.0676	0.1295	0.1520
2009	0.1992	0.0880	0.2754	0.3566	0.0615	0.2079
2010	0.2153	0.1295	0.1820	0.2188	0.4905	0.2243
2011	0.2430	0.1912	0.2101	0.1052	0.0000	0.1589
2012	0.2525	0.2340	0.1976	0.0860	0.4542	0.2245
2013	0.2394	0.2832	0.1588	0.1343	0.1376	0.1928
2014	0.2899	0.3310	0.1057	0.3254	0.2296	0.2540
2015	0.3417	0.3824	0.0698	0.3410	0.4262	0.2954
2016	0.4450	0.4300	0.0313	0.5122	0.5504	0.3681
2017	0.6041	0.5034	0.0033	0.6326	0.3417	0.4021
2018	0.6192	0.6082	0.000	0.7062	0.2142	0.4281
2019	0.6253	0.7281	0.0144	0.6909	0.4894	0.4951
2020	0.5384	0.7976	0.0198	0.5431	0.4559	0.4622
2021	0.6456	1.000	0.0717	0.6422	0.4596	0.5629

and Figure 5, and then the weights of each subsystem calculated by the entropy weight method are shown in

Table 5. Subsystem weight calculated by entropy weight method.

	Social Subsystem	Economic Subsystem	Energy Subsystem	Ecological Subsystem	Water Resources Subsystem
Weight	0.1541	0.2404	0.2477	0.2231	0.1347

. Combined with the good development trend of the order parameters of the economic subsystem in Figure 2b, the order degree of the economic subsystem in Figure 5 increases year by year. While the measurement degree of order parameters 9 and 11 of the energy subsystem in Figure 2c continues to deteriorate, even though order parameter 10 shows a good development trend, the overall level of order degree of the energy subsystem in Figure 5 is low and continues to deteriorate. In contrast, the social subsystem in Figure 2a, except for order parameter 2, shows a deteriorating development, and the remaining order parameters show a continuous good development, so the order degree of the social subsystem in Figure 5 shows a yearly increase from the overall point of view. While the order parameter 21 of the ecological subsystem in Figure 2d shows a deterioration trend, the order parameter 24 shows a good trend, but its volatility and overall level are low, so the order degree of the ecological subsystem in Figure 5 shows a good trend, but there are fluctuations. Combined with the change process of the order parameter 12 of the water resources subsystem in Figure 2e, it can be seen that the total amount of water resources in 2007, 2009, 2011, 2013, and 2018 is less, which makes the order degree of the water resources subsystem in that year appear to have very small values, which indicates that the order degree of the water resources subsystem is influenced by the order parameter 12 (total water resources).

Further, to analyze the influence of different subsystems on the change trend of the order degree of the water resources coupled system, this paper calculates the entropy variation value of the water resources coupled system and the entropy variation value of each subsystem in Jiangxi Province (see Figure 6) according to Equation (7). The correlation results show that (1) in Figure 2e, the water resources in 2007, 2009, 2011, 2013, 2017, and 2018 are depleted, and the entropy variation value of the water resources subsystem is negative, which indicates that the water resources subsystem is more influenced by the change of water resources’ abundance and depletion, and the frequency and magnitude of its entropy variation value fluctuations in Figure 6 are higher than those of other subsystems; (2) although the weight of the water resources subsystem in Table 5 is only 0.1347, when the entropy variation value of the water resources subsystem is negative (For example, 2007, 2011 and 2013 are dry years), the entropy variation value of the water resources coupled system is also negative, and the orderliness of the corresponding coupled system is reduced. In summary, the entropy variation value of the water resources coupling system in Jiangxi Province is influenced by the entropy variation value of the water resources subsystem. Similarly, in the evolution of the orderliness of the water resources coupling system, the evolutionary trend of the orderliness of the water resources subsystem is more critical. After the implementation of the strictest water resources management system in Jiangxi Province in 2012, the order degree of the water resources subsystem in the remaining years showed an obvious increasing trend, except for 2013, 2017, and 2018, which suffered from dry water years, resulting in a decrease in the order degree of the water resources subsystem. With the continuous improvement of the order degree of social, economic, and ecological subsystems and the fluctuation of the order degree of the water resources subsystem but the overall growth trend, the order degree of the water resources coupled system in Jiangxi Province shows a continuous increasing trend.

4. Discussions

The order evaluation model of the water resources coupled system based on the principle of synergetics, considering time series characteristics, first needs to determine the order parameters of each subsystem and normalize the measured value of the order parameters to obtain the measurement degree, then establish the order model of each subsystem and the coupled system, and then establish the order evaluation model based on the principle of synergetics, and then carry out the comparison of the two synergetic order evaluation methods. Based on the synergistic principle of considering the time series characteristics of the coupled water resources system, the modeling steps of the coupled water resources system are as simple as those of the traditional method, and the evaluation results are more reasonable, which can open up a new way of evaluating the coupled water resources system and provide a reference for the scientific allocation of water resources.

Following the principles of scientificity, practicality, dynamism, and data availability [48], selecting the order parameters of the water resource coupling system reasonably can not only cover the relative situation of various elements of the regional resource environment and socio-economic system but also reflect the measurement degree changes of the order parameters in the time dimension, which is conducive to achieving objective evaluation of the water resource coupling system [46,47]. The spatial differences in the endowment of the domestic water resources coupled system are significant, and the geographical differences in economic and social development are obvious, so it is impossible to form a unified ordinal parameter system. Therefore, to evaluate the orderliness of a region, it is best to start from the exchange of material and energy between regional resources, environment, ecology, and socio-economic system, design the ordinal parameter system from the four dimensions of ecology, resources, environment, and socio-economy, and determine a number of ordinal parameter indexes based on the reality of regional development [49]. At the same time, policy-oriented principles can also be considered. Policy orientation reflects the problems that need to be further solved in social life and economic development and is the scientific basis for decision-makers. Then, correlation analysis, principal component analysis [50], and other methods are used to optimize and select the ordinal parameters, and finally design the ordinal parameter system of the water resources coupling system that meets the characteristics of the region.

In this paper, the improved synergistic orderliness evaluation method is initially applied to the evaluation of water resources coupled systems, and how to propose a detailed strategy for the water resources coupled system to be optimized year by year [51] for the study area according to the evaluation results is the place that deserves a deeper study in this paper. For example, the focus should be on improving the natural growth rate of the population while preventing the rediscovery of public health events and promoting the coordinated development of social subsystems, promoting the coordinated development of ecological subsystems by improving sewage treatment technology, and so on.

5. Conclusions

In this paper, we propose a method to evaluate the order degree of the water resources-coupled system in Jiangxi Province by considering the time series characteristics of the order parameters and carrying out a comprehensive evaluation of the order degree of the system. The results of the relevant example analysis show that:

(1)

In the economic subsystem, the gradual decrease in the measurements of order parameters 6, 7, and 8 indicates that the economic subsystem should develop in the direction of order; that is, the degree of orderliness of the economic subsystem should gradually increase. In the evaluation result of the orderliness without considering the time series characteristics, the orderliness of the economic subsystem fluctuates close to 0, which is seriously inconsistent with the actual situation mentioned above, and the evaluation result is unreasonable. In the evaluation results considering the order of time series characteristics, the order of the economic subsystem gradually increases, and the system develops in the direction of order, which is consistent with the actual situation that the economic subsystem of Jiangxi Province maintains a good development trend during the period from 2005 to 2021, which indicates that the evaluation results considering the order of time series characteristics are more reasonable. Similarly, in the evaluation results of the order of the energy subsystem, it is also confirmed that the evaluation results without considering the time series characteristics of the order deviate from the actual development of the system, while the evaluation method of the order of the time series characteristics of the sequential covariates proposed in this paper can make the order and the actual evolution of the sequential covariates consistent, so that the evaluation results are more scientific and accurate. The evaluation method provides a powerful tool for exploring the field of integrated evaluation and management of water resources systems.

(2)

In the comprehensive evaluation of the order degree of the water resources coupled system in Jiangxi Province, it is found that the water resources subsystem is affected by the changes in the abundance and depletion of water resources, and its entropy variation value fluctuates more frequently and with higher magnitude than other subsystems; the entropy variation value of the coupled water resources system in Jiangxi Province is strongly influenced by the entropy variation value of the water resources subsystem; after the implementation of the strictest water resources management system in Jiangxi Province in 2012, the order degree of the coupled water resources system in Jiangxi Province showed a continuous increasing trend as the order degree of the social, economic and ecological subsystems continued to improve, and the order degree of the water resources subsystem showed fluctuations but maintained an overall increasing trend.

It should be pointed out that the evaluation of water resource coupling systems involves multiple aspects such as society, economy, natural environment, and energy. Due to the impact of data availability, the order parameters selected in this article may not be comprehensive and complete, and the research area is only limited to Jiangxi Province. With the increase and enrichment of information, future research will further improve the comprehensive evaluation of water resources based on synergistic effects and combine other improvement methods to conduct more in-depth, comprehensive, and extensive research on the comprehensive evaluation and management of water resources.