Research Article Stability and Complexity Analysis of Temperature Index Model Considering Stochastic Perturbation

Hindawi Advances in Mathematical Physics Volume 2018, Article ID 2789412, 18 pages https://doi.org/10.1155/2018/2789412

Jing Wang

Faculty of Science, Bengbu University, Bengbu 233030, China

Correspondence should be addressed to Jing Wang; [email protected]

Received 20 October 2017; Accepted 12 December 2017; Published 1 January 2018

Academic Editor: Giampaolo Cristadoro

Copyright © 2018 Jing Wang. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A temperature index model with delay and stochastic perturbation is constructed in this paper. It explores the infuence of parameters and stochastic factors on the stability and complexity of the model. Based on historical temperature data of four cities of Anhui Province in China, the temperature periodic variation trends of approximately sinusoidal curves of four cities are given, respectively. In addition, we analyze the existence conditions of the local stability of the temperature index model without stochastic term and estimate its parameters by using the same historical data of the four cities, respectively. Te numerical simulation results of the four cities are basically consistent with the descriptions of their historical temperature data, which proves that the temperature index model constructed has good ftting degree. It also shows that unreasonable delay parameter can make the model lose stability and improve the complexity. Stochastic factors do not usually change the trend in temperature, but they can cause high frequency fuctuations in the process of temperature evolution. Stability control is successfully realized for unstable systems by the variable feedback control method. Te trend of temperature changes in Anhui Province is deduced by analyzing four typical cities.

1. Introduction daily simulation approach, which we found in the scientifc literature, is explored by Dischel [6, 7] and is refned by Weather derivatives trading carried out in the world to Dornier and Queruel [8]. Tereafer, Torro´ et al. [9], Alaton actively respond to climate change and the development and et al. [10], Brody et al. [11], and Benth et al. [1, 12, 13] prosperity of fnancial market, which has experienced great extend the well-known fnancial difusion processes in order development since its inception, has made a positive contri- to incorporate the basic statistical features of temperature bution to the world economy to prevent the efect of climate models, combining both the time continuous approach and change and to bufer weather disaster loss. Temperature is an the econometric approach. In addition, more research has important indicator of climate change but can also be used focused on derivatives pricing. Jewson and Brix [14] distin- as the underlying asset of weather derivatives. In 2007-2008, guish between three diferent approaches for the valuation of 70% of all weather derivatives contracts traded were based on weather derivatives. Schiller et al. [15] propose a temperature temperature as the underlying factor. According to Benth and model which uses splines to remove trend and seasonality Saltyte-Benth [1] and Fujita and Mori [2], 98-99% of weather efects from the temperature time series in a fexible way. derivatives traded are based on temperature. Te majority of Fortuna et al. [16] consider the solar radiation daily patterns, traded weather contracts are currently being written based divided into clear sky, intermittent clear sky, completely cloud on temperature. For this reason, the dynamic analysis of sky, and intermittent cloud sky, according to the original temperature processes is necessary. It helps to study climate features-based classifcation strategy. Te results show that change and the development of weather derivatives trading. features-based classifcation is superior to the traditional neu- Previous studies focus on two diferent categories of ral network classifer when operating on high-dimensional ftting daily temperature variation processes correctly. Cao solar radiation patterns. and Wei [3], Campbell and Diebold [4], and Jewson and However, few papers focus on the complexity and stability Caballero [5] rely on a time series approach. Instead, a of the temperature process or climate change. Balint et al. [17] 2 Advances in Mathematical Physics investigate the micro- and macroeconomics of climate perturbation on the dynamic behavior of the model and change from the perspective of complexity science, and they evaluate the ftting degree of temperature index model. discuss the challenges faced by this series of studies. Jacobson et al. [18] report that students use agent-based computer Defnition 1 (daily average temperature). Daily average tem- models to learn and understand the complex system of cli- perature is defned as the average of the maximum and mate change. Te experimental conditions and experimental minimum temperature for that day [10]; that is, if the maximum temperature for any day is �max and the minimum procedures for obtaining the results are explained in detail. � � Te paper points out that the research conclusion is consistent for the same day is min, then daily average temperature ave with theory and practice. Mihailovic´ et al. [19] investigate the is given by fuxaggregationefectinthesubgridscaleparameterization (� +� ) � = max min . (1) basedonthedynamicalsystemapproach.Testabilityof ave 2 turbulentheatexchangeovertheheterogeneousenvironmen- tal interface in climate models is discussed with emphasis. Historical temperature data usually exhibits a trend. Te Sunetal.[20]analyzethestabilityofannualtemperature reason may be attributed not only to the efects of global between winter and summer in the urban functional zone. warming, but also to urbanization efects that have led to local warming. We establish three assumptions: Te research conclusion of this article can be applied in urban (i) Te interference of noise and other factors is taken into planning to improve climate adaptability. consideration, thus assuming the temperature model consists Diferent from previous studies, this paper discusses the of deterministic part and stochastic part. stability and the complexity of temperature index model (ii) Te period of oscillations is one year, but leap years under intervene of delay and stochastic factors based on have 366 days. In order to allow for assumption (iii) our entropy theory and nonlinear dynamics theory. Te work modelwillneglecttheefectofleapyears. mainlyincludesthefollowing:(1)thehistoricaltemperature (iii) Te seasonality of the temperatures equals one year changes trends of four cities of Anhui Province in China and the temperatures are modeled on a daily basis, thus are given; (2) a temperature index model with delay and setting � = 2�/365. stochastic term is constructed; (3) the parameters of the model are estimated by using the historical temperature data Modeling daily average temperature is a difcult task of four cities; (4) the stability and the complexity of temper- because of the existence of multiple variables that govern ature index model in four cities are analyzed, respectively, weather. Looking to the past, we will attempt to obtain under the infuence of delay and stochastic term; (5) the precious information about the behavior of temperature. ftting degree of the constructed temperature index model 2.1. Data Description. Te dataset investigated includes tem- is explored by comparison with the temperature changes ∘ trend obtained from the historical temperature data; (6) perature observations measured in centigrade degrees (± C) thevariablefeedbackcontrolmethodisusedtocontrolthe on four measurement stations of Anhui Province: Bozhou, unstable system; (7) the overall temperature evolution trend Bengbu, Hefei, and Anqing. Data consist of observations on of Anhui Province is discussed. the daily maximum and minimum temperature from China’s Te present paper is organized as follows. We take a meteorological data network. Te sample period extends panoramic view of the situation to generate the temperature fromJanuary1,2010,toMay31,2016,withatotalof2349 index model and simulate daily average temperature data observations per each measuring station. fromJanuary1,2010,toMay31,2016,inAnhuiProvince To assist in fnding a good model, we use a database with in Section 2. Based on the work of Alaton et al. [10], temperatures to analyze the trend of temperature change in we give a general parameter estimation of the model in the last several years from diferent cities in Anhui Province, Section 3. In Section 4, the local stability condition of the China. Te temperature data consists of daily average temper- delayed temperature index model without stochastic term is atures, computed according to (1). In Figure 1, we have plotted discussed. In Section 5, the temperature evolution trends of the daily average temperatures of the above-mentioned four four cities with or without delay and stochastic perturbation cities in Anhui Province for six consecutive years. are investigated through numerical simulation, respectively. From the temperature data in Figure 1, we clearly see that In Section 6, the stability control is realized by applying there is a strong seasonal variation in the temperature and thevariablefeedbackcontrolmethod.Briefsummaryand ftting with the sine function. We will build a temperature concluding remark are given in Section 7. model below and show the temperature trends of four cities during the same time. Te temperature index model is 2. Temperature Index Model evaluated by comparing it with Figure 1. Since we have decided to focus only on the temperature 2.2. Modeling Temperature. Te underlying weather indices index as the underlying variable in the fnance market, we of the derivatives traded at the CME for US cities are will try to fnd a model that describes the temperature in directly derived from the daily average temperature, which this section. Te goal is to fnd a dynamic process describing is defned as the average of the minimum and the maximum the movements of temperature. Te latter part of this paper temperature at a weather station on a day �, �∈�. will analyze the infuence of delay parameter and stochastic According to the mean reversion model by Alaton et al. [10] Advances in Mathematical Physics 3

e trend of temperature change in Bozhou e trend of temperature change in Bengbu 35 35 30 30 25 25 20 20 15 15 10 10 5 5 0 0 −5 −5 −10 −10 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500

Daily mean temperature Daily mean temperature Seasonal trend Seasonal trend (a) (b) e trend of temperature change in Hefei e trend of temperature change in Anqing 35 35 30 30 25 25 20 20 15 15 10 10 5 0 5 −5 0 −10 −5 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500

Daily mean temperature Daily mean temperature Seasonal trend Seasonal trend (c) (d)

Figure 1: Daily average temperatures of four cities in Anhui Province during 2010–2016. (a) Bozhou; (b) Bengbu; (c) Hefei; (d) Anqing. and the temperature long-term change trend, we assume the years. Te greenhouse efect brought by global warming or temperature index satisfes the following equation: city urbanization causes the temperature to show a slightly rising trend, so �� captures the trend of temperature change. �� =��(�) +�(�(�) −�)��+�� , � � � � (2) From (2) and (3) we can defnitely say that the daily average temperature can be broken into the time term, where � determines the speed of the mean reversion, �� is seasonal term, and stochastic term. Te next step will be to the daily volatility of temperature variation, and �� is a determine the unknown parameter of the so-specifed model Brownian increment. Te daily average temperature for day through estimation based on historical temperatures. � is �(�)(� = 1,2,...,�), which represents the changes of seasonal. AscanbeseenfromFigure1,thedailymeantemperature 3. Parameter Estimation changes process of four cities shows an obvious seasonal variation trend and exhibits periodicity. Te seasonality of the For (3), we performed the following transformation: temperature is modeled with a sine curve; it can be expressed as � (�) =�+��+�sin (�� + �) � (�) =�+��+� (�� + �) , (4) sin (3) =�+��+�(sin (��) cos �+cos (��) sin �) . where � is the phase angle and the parameters A, B, C,and � need to be estimated. In addition, the oscillation period of Let us consider �=�⋅cos �, �=�⋅sin �, �1 = sin(��), daily mean temperature is one year, so we ignore the efect and �2 = cos(��). Terefore, (4) is equivalent to the linear of the leap year (excluding the temperature data on February regression equation: 29 of the leap years) and assume � = 2�/365.Teunknown parameters are estimated by using historical temperature data � (�) =�+��+�� +�� . measured in four cities of Anhui Province in the last few 1 2 (5) 4 Advances in Mathematical Physics

Table 1: Parameters estimation. Arguments City �� 15.3292 −1.8561� − 004 −3.629285 −12.75431 13.2606 −1.8480 0.109500 0.967525 Bozhou (0.129957) (0.000096) (0.091505) (0.092323) (0.0013) 15.6110 −1.9633� − 005 −3.893407 −12.25677 12.8603 −1.8784 0.125200 1.000233 Bengbu (0.128585) (0.000095) (0.090539) (0.09138) (0.0044) 16.3834 2.7225� − 004 −3.908884 −11.96285 12.5853 −1.8866 0.086997 0.95595 Hefei (0.13199) (0.0000975) (0.092937) (0.093767) (0.0057) 17.1427 −2.2414� − 004 −0.443545 −11.91876 11.9270 −1.6080 0.079950 1.02330 Anqing (0.156511) (0.000124) (0.110663) (0.110126) (0.0029) Data sources: China Meteorological Data Service Center (CMDC) http://data.cma.cn/site/index.html.

Ten we have analyze the relative stability of the temperature change trend �=� by observing the state of the temperature index model. We assume that the daily average temperature before �(� = �=� 0, 1, 2, . . .) days can be used instead of the current daily average temperature. At this time, if the temperature index model �=√�2 +�2 (6) is stable, the diference between the historical temperature and the current temperature is small, and the temperature � change is relatively stable. Otherwise, the unstable model �=arctan ( ). � means that the temperature changes greatly fuctuate, and the relative steady change trend of temperature has changed. It We made parameter estimation rely on the history data of is necessary to explain that we ignore the individual daily the daily average temperature in four cities of Anhui Province temperature mutation in the relative stable interval in the with the least squares method by using Matlab sofware process of analysis. Tat is to say, it is a macro discussion and (Matrix Laboratory, MathWorks Company, America). analysis. For this reason, on the basis of (7), we can get the Te second estimator is derived by (2) and considers the following model with delay parameter: discretized equation as a regression equation, which can be seen as a regression of today’s temperature on yesterday’s �� =[�+��cos (�� + �) temperature. It is natural to estimate the unknowns by � ̂ � ̂ +�(�+��+� (�� + �) − � (�−�))] �� minimizing the least squares criterion ∑�=1(��−��) (��−��), sin (9) ̂ with �� being the estimate for � in the �th simulation runs +��, with �=1,2,...,�. Based on above description and discussion, the param- where � represents the delay parameter, which means that the eters estimation results of four cities in Anhui Province are average temperature � days before can be approximated as the shown in Table 1. current average temperature. Table 1 presents the estimates of the unknown parameters Now, the deterministic part of (9) is as follows: driving the daily average temperature of each city. Te standard errors are reported in parentheses to measure the �� =[�+��cos (�� + �) signifcance of parameters. (10) Tus far, substituting (3) into (2), the following equation +�(�+��+�sin (�� + �) − � (�−�))] ��. of temperature index model is obtained: In order to analyze the infuence of delay parameter �� =[�+��cos (�� + �) and stochastic factors on the stability and complexity of (7) models, some numerical simulations are given in Section 5. +�(�+��+� (�� + �) − � )] �� + � �� . sin � � � In addition, Bozhou, Bengbu, Hefei, and Anqing are located in the northern, north central, south central, and southern If stochastic factors are not taken into account, the determin- part of Anhui Province, respectively, and they are four rep- istic part of (7) is as follows: resentative cities in the temperature changes. Tus, we adopt the parameter estimation of these four cities for numerical �� =[�+��cos (�� + �) (8) simulation, respectively. On the one hand, we fnd the trend +�(�+��+�sin (�� + �) − ��)] ��. of temperature changes in Anhui Province and observe the efects of delay parameter and stochastic factors on the trend Te daily average temperature in Anhui varies greatly in of temperature changes. On the other hand, the ftting degree diferent periods, but the temperature in Anhui will have a of the temperature index model is measured, that is, whether short relatively stable state during the periodic change. We it refects the changes trend of the real temperature. Advances in Mathematical Physics 5

Tis paper discusses four cases: the stochastic diferential and then ±��1 is a pair of purely imaginary roots of (13) (�) equation without delay (i.e., (7)), the deterministic diferen- when �=� . tial equation without delay (i.e., (8)), the stochastic difer- Defning ential equation with delay (i.e., (9)), and the deterministic diferential equation with delay (i.e., (10)). Next, the local �10 =�1, stability conditions of (10) are analyzed. (�) (20) �0 = min {� } , � = 0, 1, 2, . . . , 4. Local Stability Analysis of (10) let �(�) = �(�) + ��1(�) be the root of (13) near �=�0 �(� )=0 � (� )=0 Now we investigate the stability of (10). It can be rewritten as which satisfes 0 , 1 0 . Diferentiating both sides of (13) with respect to �,wecan ̇ �� =�+��cos (�� + �) obtain (11) +�(�+��+� (�� + �) − � (�−�)). �� −�� sin = . (21) �� 1−��−�� For convenience, we assume that the equilibrium point of ∗ (11) is �(� ). Next, (11) is linearized through Jacobian matrix Accordingly, as follows: �� −1 1 � ̇ [ ] = − . (22) �� = −�� (�−�) . (12) �� −��

Te characteristic equation of (12) is Substituting �=��10 (�10 >0)into (22), we have �+��−�� =0. −1 (13) �� (�0) 1 Re [ ] = Re [ ] −��0 �� =��10 4.1. �=0. For �=0, (10) becomes (8). Te characteristic (23) �� equation (13) can be simplifed to = 10 sin 10 0 . 2 2 � �10 �+�=0. (14) Since According to Routh-Hurwitz, we can draw the following −1 conclusions. �� (�0) �(Re �(�0)) [ ]= [ ] , (24) sign Re �� sign �� Teorem 2. If �>0, the root of (14) will have negative real ∗ parts. Ten, the equilibrium point �(� ) of (10) is locally stable � � =0̸ [��(� )/��]−1 =0̸ without delay parameter. if sin 10 0 ,thenRe 0 . According to the corollary in [21–24], we have the following conclusions. �>0 4.2. . In this case, the characteristic equation of (10) is Teorem 3. If �=0 ̸ and sin �10�0 =0̸ , then the equilibrium �=�� (� >0) ∗ still (13). Let 1 1 be the root of (13). Ten, we point �(� ) of (10) is asymptotically stable for �∈[0,�0),andit can obtain is unstable when �>�0. In addition, (10) begins to lose stability near �=�0. ��1 =−�(cos �1�−�sin �1�) . (15) Separating the real and imaginary parts, we have 5. Simulate and Analyze

� sin �1�=�1 We consider the infuence of delay parameter and stochastic (16) factors on the temperature index model. Te sequence of � � �=0. cos 1 random numbers in (7) and (9) obeys a normal distribution with mean being 0 and variance being 1. Further, we can obtain 2 2 �1 =�. (17) 5.1. Te Stability of (7) and (8) when �=0

Namely, 5.1.1. Te Stability of (8). AsshowninTable1,theestimations of parameter � of Bozhou, Bengbu, Hefei, and Anqing are √ 2 � = 0.109500 � = 0.125200 � = 0.086997 �1 = � . (18) �� , �� , ℎ� ,and �� = 0.079950,respectively.Tesubscripts��, ��, ℎ�,and From (16), if �=0 ̸ ,weobtain �� are the abbreviations of the above-mentioned four cities. � >0�=�� ℎ� �� 1 Because � , , , , , according to Teorem 2, �(�) = ( (0) +2��) , �=0,1,2,..., the temperature index models (8) of four cities are stable. � arccos (19) 1 Taking the parameters estimation values of four cities into 6 Advances in Mathematical Physics

e trend of temperature change in Bozhou e trend of temperature change in Bengbu 30 30

25 25

20 20 t t 15 15 X X

10 10

5 5

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b) e trend of temperature change in Hefei e trend of temperature change in Anqing 30 30

25 25

20 20 t 15 t 15 X X

10 10

5 5

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (c) (d)

Figure 2: Daily average temperatures of four cities for (8) during 2010–2016. (a) Bozhou; (b) Bengbu; (c) Hefei; (d) Anqing.

(8), respectively, then their time series diagrams are shown � will lead to the growth of �� when � ≥ 0.005;theefect in Figure 2. on �� by � is more signifcant. Te growing process can be We observed that the temperature changes in all four divided into four stages, which are represented by the light cities show stable periodic characteristics, similar to those blue, green, yellow, and red, respectively. It is a slow process of sinusoidal curves. Te temperature changes trends of the for � to cause a small increase in ��. In Figure 3(b), the value four cities vary slightly. What is most obvious is that the of �� iscloseto20when�≥0.005. minimum temperature in Anqing is greater than or equal In Figure 3(c), �� increases gradually with the increase ∘ to 5 C,whichishigherthanthatoftheotherthreecities. of � when � ≥ 0.005.Techangeprocessof�� can be Tis is because Anqing is located in the south of Anhui; divided into fve stages, with denoted dark blue, light blue, the daily average temperature is on the high side. Compared green, yellow, and red, respectively. Figure 3(d) shows the with Figure 1, the temperature changes of the four cities relationships among �, �,and�.Wenoticethat� and � have showninFigure2arebasicallyconsistentwithanalysis little efect on ��,andthereisagradualincreasein�� with the results of historical temperature data. Terefore, (8) has good growth of �. What is interesting is that the trend is basically ftting degree without considering the delay parameter and thesameasFigure3(c). stochastic factors, and it can better show the temperature From the above analysis, we can draw the following changestrendsoffourcitiesinalongerperiod. conclusion: (1) the parameter � has a signifcant efect on ��, whereas �� is almost unafected by � and �; (2) the efect of 5.1.2. Te Efect of �, �, �, � on the Stability of (8) in Anqing. � on �� is divided into two stages: �� does not change with Taking the parameter estimation of Anqing as an example, the theincreaseof� when � ≥ 0.005,and�� decreases with the infuence of �, �, �,and� on the stability of (8) is analyzed. decrease of � when � < 0.005. Terefore, the reasonable range As can be seen from Figures 3(a)–3(c), no matter how �, � of parameters can maintain the stability of the system. change, �� remains at a low level when � is very small (the critical value is about 0.005). At this point, �� will rise with 5.1.3. Te Stability of (7). Te parameters estimate values of an increase of �. However, in the remaining cases, � has little four cities are taken into (7). Teir corresponding time series efect on ��. Figure 3(a) shows that the increase of � and diagrams are shown in Figure 4. Compared with Figure 2, Advances in Mathematical Physics 7

20 20

0.1 0.1 15 0.09 15 0.09 0.08 0.08 10 0.07 10 0.07 0.06 0.06 5 k 0.05 5 k 0.05 0.04 0.04 0 0.03 0 0.03 0.02 0.02 0.01 0.01 −5 20 −5 1 0 15 0 20 20 15 10 15 0 −10 10 A −10 10 ×10 5 5 5 B C 0 0 C 0 −1 (a) (b) 22 20 0.1 20 0.09 18 20 0.08 16 0.07 15 15 0.06 14 k 0.05 12 0.04 10 C 10 10 0.03 8 0.02 5 6 0.01 5 0 4 0 20 1 15 20 2 1 0.5 10 −3 0 15 0 ×10 10 −3 0 5 ×10 −0.5 5 A B −1 0 A B −1 0 (c) (d)

Figure 3: Te efect of �, �, �,and� on (8) in Anqing; (a) � = −0.00022414;(b)� = 17.1427;(c)� = 11.927;(d)� = 0.07995.

the stochastic factors do not change the overall trends of 5.2.1. Te Efect of �ℎ� on (10) in Hefei. Figure 5 shows that temperatureevolutioninthefourcities,buttheregularity the system moves from stable to unstable state as the �ℎ� of periodic change is obviously afected. High frequency increases, which also improves the complexity of the system. fuctuations accompany the whole process of temperature As you can see from Figure 5(a), there is a large fuctuation evolution, which seriously interfere with the accuracy of when �ℎ� > 17.44; this unstable state is obviously not temperature changes. By comparing Figures 4 with 1, we can conducive to our analysis of temperature. We do a further conclude that (7) is not ideal under stochastic factors inter- analysis of (10) with the largest Lyapunov exponent (LLE). ference, but it can refect more realistic trends of historical LLEmeansthatthestabilityofsystemisjudgedaccordingto temperature changes in four cities. the relation between the exponent value (LLE-ev) and zero. If LLE-ev is less than zero, the system is stable. It is unstable 5.2. Te Stability and Complexity of (9) and (10) when �>0. when LLE-ev is greater than zero. When LLE-ev is equal to According to (18), (19), and (20), we get ��,��10,and��0 zero, it represents the system losing stability. From Figure 5(b) (� = ��, ��, ℎ�,��) by using the parameters estimation values we fnd that the critical value is �ℎ�0 = 17.44;thatistosay,the of four cities, respectively, in Table 1. Te subscript 0 indicates system is stable on the lef of the critical value, but on the right the critical value of system stability. By combining Teorems it is unstable. Tus, Figures 5(a) and 5(b) describe the efect 2 and 3, the stability of temperature index model of each city of �ℎ� on the stability of the system from diferent angles, but can be determined. Te analysis results are shown in Table 2. they express the same meanings. As a provincial capital city, Hefei is located in the central Complexity is another important factor to consider in and southern part of Anhui Province, which is representative systemanalysis.Temorecomplexthesystemis,theharderit of the temperature change. Terefore, we take Hefei city as an is to understand, and vice versa. Entropy indicates the degree example to explore the infuence of delay parameter on the of randomness of a system. Te same applies to measuring the stability and complexity of temperature index model. complexity of a system. Kolmogorov entropy (K entropy) is 8 Advances in Mathematical Physics

e trend of temperature change in Bozhou e trend of temperature change in Bengbu 35 35

30 30

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20 20 t t X 15 X 15

10 10

5 5

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b) e trend of temperature change in Hefei e trend of temperature change in Anqing 35 40 30 35 25 30 25 20 t t 20 X X 15 15 10 10 5 5 0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (c) (d)

Figure 4: Daily average temperatures of four cities for (7) during 2010–2016. (a) Bozhou; (b) Bengbu; (c) Hefei; (d) Anqing.

Table 2: Calculation results and stability conditions of (10) for each city.

Bozhou Bengbu Hefei Anqing � 0.109500 0.125200 0.086997 0.079950 sin �10�0 1111 ∗ ��(� ) 15.3244 15.6105 16.3769 17.1420

��10 0.109500 0.125200 0.086997 0.079950

��0 14.3452 12.5463 17.4400 19.6472

Stable �� ∈[0,��0)�� ∈[0,��0)�ℎ� ∈[0,�ℎ�0 )�� ∈[0,��0)

Losing stability �� =��0 �� =��0 �ℎ� =�ℎ�0 �� =��0

Unstable �� >��0 �� >��0 �ℎ� >�ℎ�0 �� >��0

one of the most commonly used entropy. Tus, this paper uses value is greater than zero as �ℎ� > 17.44.Itcanbeobserved K entropy to measure the complexity of the system by using intuitively that the entropy value increases continuously with Matlab sofware. Te rules of K entropy for judging system the increase of �ℎ� . In other words, the excessive increase complexity are as follows: If the system is stable or regular of �ℎ� will make the system more complex. From what is movement, the entropy value equals zero. Tat is to say, the mentioned above, we may come to the conclusion that �ℎ� system retains the original complexity. On the other hand, if must be controlled within a certain range in order to keep the system is unstable, the entropy value is greater than zero, the system stable. Otherwise, the system will lose stability and which means the system is more complex than the original increase the complexity. one. Terefore, it must be clear that the complexity of the From the above analysis, it is known that the system has system is positively related to the entropy value. Figure 5(c) diferent states on both sides of �ℎ�0 .Now,let�ℎ� =16< provides an entropy diagram to describe the impact of �ℎ� on �ℎ�0 and �ℎ� =19>�ℎ�0 , their corresponding time series the complexity of the system. We noticed that the entropy diagrams are shown in Figures 6(a) and 6(b), respectively. value is equal to zero when �ℎ� < 17.44, while entropy Figure 6(a) shows the historical temperature changes trend of Advances in Mathematical Physics 9

200 0.02

0.01 100 X: 17.44 X: 17.44 Y: 17.94 Y: 0 0 0 −0.01 −100 −0.02 t −200 X LLE −0.03 −300 −0.04 −400 −0.05 − 500 −0.06

−600 −0.07 5 10 15 20 5 10 15 20   (a) (b) 3.5

2.5

Entropic 1.5

0.5 X: 17.44 Y: 0 0 5 10 15 20  (c)

Figure 5: Te efect of � on the stability and complexity of (10) in Hefei. (a) Change trend diagram; (b) LLE diagram; (c) entropy diagram.

(10) for Hefei. Te temperature exhibits a periodic variation In addition, the smaller the �ℎ� is, the better the ftting with when �ℎ� =16. Figures 6(a) and 2(c) are basically the same, the sine curve is, and vice versa. Tat is, the daily average so we can infer that as long as the delay parameter �ℎ� is temperature near the current date is a better choice. within a reasonable range, the daily average temperature of history can replace the present daily average temperature, and 5.2.2. Te Efect of �ℎ� on (9) in Hefei. Te parameters itcanalsoshowthecharacteristicsofperiodicchangesin estimation values of Hefei are taken into (9) and the delay temperature. By comparing Figures 6(a) and 1(c), we fnd that parameters are the same as those shown in Figure 6. Te theresultsarebasicallythesamebasedonthesamehistorical dynamical characteristics are shown in Figure 7. temperature data. So (10) is an ideal model �ℎ� =16<�ℎ�0 . Comparing Figures 7(a) and 5(a), it can be found that However, if the system is unstable, the temperature the stochastic factors do not change the trend of the system evolution is irregular and disorganized. Te instability of the evolution, but the system has smaller fuctuations in the system is described in Figure 6(b). Tus, we should carefully process of change. However, there is a larger diference in select �ℎ� to avoid adverse changes in system stability and the complexity of the system. From Figures 7(b) and 5(c) we complexity. fnd that stochastic disturbance leads to premature loss of Trough the simulation of (10) in Hefei, it is shown that stability and complexity of the system. Under the infuence the stability analysis is correct. In other words, when �ℎ� of stochastic factors, the complexity of the system has a increases to more than �ℎ�0 , (10) changes from a steady state to reciprocating change in the process of increase, and the trend an unstable state, and the complexity of (10) will be improved. of overall increase has not changed. 10 Advances in Mathematical Physics

e trend of temperature change in Hefei e trend of temperature change in Hefei 30 300

25 200

20 100 t t 15 0 X X

10 −100

5 −200

0 −300 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b)

Figure 6: Time series diagram for (10) in Hefei. (a) �ℎ� =16;(b)�ℎ� =19.

Entropy diagram of Hefei 200 3.5

100 X: 17.44 3 Y: 18.82 0 2.5 −100 2 t −200 X

Entropic 1.5 −300 1 −400

−500 0.5

−600 0 5 10 15 20 5 10 15 20   (a) (b) e trend of temperature change in Hefei e trend of temperature change in Hefei 60 140

50 120

100 40 80 t t 30 X X 60 20 40

10 20

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (c) (d)

Figure 7: Te efect of �ℎ� on the stability and complexity of (9) in Hefei. (a) Change trend diagram; (b) entropy diagram; (c) �ℎ� =16;(d) �ℎ� =19. Advances in Mathematical Physics 11

e trend of temperature change in Bozhou e trend of temperature change in Bengbu 30 30

25 25

20 20 t t 15 15 X X

10 10

5 5

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b) e trend of temperature change in Anqing 35 30 25 20 t X 15 10 5 0 0 500 1000 1500 2000 2500 t (c)

Figure 8: Daily average temperatures of (10) during 2010–2016. (a) Bozhou, �� =13<��0;(b)Bengbu,�� =11<��0;(c)Anqing, �� =18<��0.

Figures 7(c) and 7(d) correspond to the two states. Table 3: Delay parameters selectionof (10) for three cities. RelativetoFigure6,itisclearthattheperiodicvariationofthe temperature index model is not very regular and fuctuates Bozhou Bengbu Anqing � =13<� � =11<� � =18<� greatly. It is worth noting that the random factors lead to Stable �� 0 �� 0 �� 0 � =15>� � =13>� � =20>� temperatures greater than 0 except the initial temperature in Unstable �� 0 �� 0 �� 0 Figure 7(d). In addition, we can deduce that the infuence of stochastic factors on the stable system is more obvious than the unstable system. Tis is obviously not desirable, so we shouldminimizetheimpactofrandomfactorsonthesystem Sections 5.2.1 and 5.2.2, so the temperature evolution of the to ensure better system stability. other three cities is comprehensively analyzed. In order to In a word, the efects of stochastic perturbations are more supportthetheoreticalanalysisofTable2,wesetthedelay obvious, which accelerate the instability and complexity of parameters values for three cities as shown in Table 3. the system. Te trends of temperature changes of three cities from In short, the delay diferential equation (10) without January 1, 2010, to May 31, 2016, are shown in Figures 8 and stochastic term is stable when �ℎ� ∈[0,�ℎ�0 ),andthe 9. Te entropy diagrams corresponding to them are shown complexity of the system does not change. Te complexity in Figure 10. As you can see from Figure 8, the systems are increases and the system loses stability only when �ℎ� > stable when �� <��0 (� = ��, ��, ��),andthetemperature �ℎ�0 . However, the temperature variation shown by the delay changes trends of the three cities are similar to that of the sine diferential equations (9) with stochastic term is seriously curve. Figure 9 shows that the systems are unstable as �� > distorted with high frequency fuctuations. Tis corresponds ��0 (� = ��, ��, ��);atthistimethetemperaturechanges to the increase in complexity to a certain extent, and the com- show chaotic and irregular trends. Tose are consistent with plexity of the system increases more clearly when �ℎ� >�ℎ�0 . the results of theoretical analysis. By comparing Figures 8 and Terefore, the model with stochastic perturbation basically 1, we fnd that (10) has a high ftting degree. refects the overall trend of temperature changes but has a In addition, Figure 10 shows the state transition of the negative impact on the stability and complexity of the system. system from the point of view of system complexity. Te critical points of system complexity and stability are the same. 5.2.3. Te Stability and Complexity of (10) in Bozhou, Bengbu, When the system is unstable, the complexity of the system and Anqing. Te analysis processes are similar to those in rises with the increase of �� (� = ��, ��, ��). 12 Advances in Mathematical Physics

e trend of temperature change in Bozhou e trend of temperature change in Bengbu 300 300

200 200

100 100 t 0 t

X 0 X

−100 −100

−200 −200

−300 −300 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b) e trend of temperature change in Anqing 1500

1000

500 t X 0

−500

−1000 0 500 1000 1500 2000 2500 t (c)

Figure 9: Daily average temperatures of (10) during 2010–2016. (a) Bozhou, �� =15>��0;(b)Bengbu,�� =13>��0;(c)Anqing, �� =20>��0.

Comparison between Figures 10 and 5(c) shows that Equation (9) disturbed by stochastic factors cannot ft the ��0 is the largest, ��0 is the smallest, �ℎ�0 and ��0 are in trends of the historical temperature changes well. the middle. Tat is to say, the daily average temperature in Terefore, if stochastic factors play an important role, Anqing is not very diferent over a period of time, and the the trajectory of temperature changes will be unstable, which average temperature of current day can be replaced by the is not conducive to the study of the historical temperature daily average temperature of ��0 days ago at most. However, and the future temperature. As compared with Figure 9, the the daily average temperature varies greatly in Bengbu. Te unstable equation (9) shown in Figure 12 is even more chaotic. daily average temperature within ��0 days of the current date In other words, stochastic factors exacerbate the system’s will be of reference value. Te similar conclusions can be dramatic fuctuations and irregular motion. In a word, the interference of stochastic factors enhances drawn from Hefei and Bozhou. Te diferences of ��0 (� = ��, ��, ��) are closely related to the geographical position and the instability and complexity of the system. Whether the surrounding environment of the four cities and conform to system is stable or not, there is a lot of unstable factors in it, and the system becomes complex. Te entropy diagrams the actual situation. describe the evolution process of the system complexity under stochastic perturbation in Figure 13. Compared with 5.2.4. Te Stability and Complexity of (9) in Bozhou, Bengbu, Figure 10, the interference of stochastic factors improves the and Anqing. Te temperature evolution trends of three cities complexity of the stable system and changes the trajectory of under stochastic perturbation are shown in Figures 11 and complexity in unstable system. 12. It can be seen that the impacts of stochastic factors on the system are very obvious whether the systems are stable 6. Stability Control or not. Te temperature changes are always accompanied by high frequency fuctuations. Comparing Figures 11 and Basedonabovedescriptionanddiscussion,weknowthat 8, the stochastic factors seriously destroy the accuracy of an unstable system may not function properly and cannot temperature periodic variations. However, this can describe achieve its intended goal. Terefore, we will take efective the periodic phenomenon of temperature changes macro- measures to ensure that the system is transformed from scopically. Te diferences between Figures 11 and 1 are large. instability to stability. In other words, the unstable system Advances in Mathematical Physics 13

Entropy diagram of Bozhou Entropy diagram of Bengbu 3 3

2.5 2.5

2 2

1.5 1.5 Entropic Entropic 1 1

0.5 0.5 X: 14.35 X: 12.55 Y: 0 Y: 0 0 0 5 10 15 20 5 6 7 8 9 10 11 12 13 14 15   (a) (b) Entropy diagram of Anqing 3.5

2.5

1.5 Entropic

0.5 X: 19.65 Y: 0 0 5 10 15 20 25  (c)

Figure 10: Entropy diagrams of (10). (a) Bozhou; (b) Bengbu; (c) Anqing. is controlled by variable feedback method stabilized by Figure 14(b) describes the impact of � on the system from adjusting the control parameters. a complexity perspective. Figures 14(a) and 14(b) express the same meaning. In addition, the smaller the � is, the stronger 6.1.TeControlof(10)inHefei. Adding the control term in thesysteminstabilityisandthemorecomplexitis,andvice (10) based on parameter estimation of Hefei as follows: versa. Terefore, the system returns to stable state only when � > 0.01382. �̇ = 2.7225� − 004 + 0.2169 (0.0172� − 1.8866) � cos Next, let � = 0.005 < �0, according to the previous anal- + 0.090069 (16.3834 + 2.7225� − 004� ysis, (25) is unstable. Figure 15(a) confrms this conclusion. � = 0.02 > � (25) Ten set 0; as can be seen in Figure 15(b), + 12.5853 sin (0.0172� − 1.8866) −�(�−�)) (25) is stable and the temperature varies periodically. Tis is consistent with the above discussion. Terefore, it can be −��, concluded that the unstable systems achieve stability control by adjusting control parameter �. where � is a control parameters, the purpose of the following discussion is to achieve system control by adjusting �. According to the analysis of Section 5.2.1, the system is 6.2. Te Control of (9) in Hefei. Afer adding the control item, (9) based on parameter estimation of Hefei can be changed to unstable when �ℎ� =19, and the corresponding time series diagram is shown in Figure 6(b). Now, we study the fact �� = [2.7225� − 004 + 0.2169 (0.0172� − 1.8866) � � cos that the efect of on the stability and complexity of (25) + 0.090069 (16.3834 + 2.7225� − 004� under the other parameters is the same. Figure 14(a) shows (26) that (25) is unstable when � < 0.01382,anditisstable + 12.5853 sin (0.0172� − 1.8866) −�(�−�))] �� when � > 0.01382.Tecriticalvalueis�0 = 0.01382. + 0.95595�� −��, 14 Advances in Mathematical Physics

e trend of temperature change in Bozhou e trend of temperature change in Bengbu 45 45 40 40 35 35 30 30 25 25 t t X X 20 20 15 15 10 10 5 5 0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b) e trend of temperature change in Anqing 50 45 40 35 30 t 25 X 20 15 10 5 0 0 500 1000 1500 2000 2500 t (c)

Figure 11: Daily average temperatures of (9) during 2010–2016. (a) Bozhou, �� =13<��0;(b)Bengbu,�� =11<��0;(c)Anqing, �� =18<��0.

e trend of temperature change in Bozhou e trend of temperature change in Bengbu 80 140 70 120 60 100 50 80 t t 40 X X 60 30 20 40 10 20 0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b) e trend of temperature change in Anqing 120

100

t 60 X

0 0 500 1000 1500 2000 2500 t (c)

Figure 12: Daily average temperatures of (9) during 2010–2016. (a) Bozhou, �� =15>��0;(b)Bengbu,�� =13>��0;(c)Anqing, �� =20>��0. Advances in Mathematical Physics 15

Entropy diagram of Bozhou Entropy diagram of Bengbu 4 3 3.5 2.5 3 2 2.5 2 1.5 Entropic 1.5 Entropic 1 1 0.5 0.5 0 0 5 10 15 20 5 6 7 8 9 10 11 12 13 14 15   (a) (b) Entropy diagram of Anqing 3.5 3 2.5 2 1.5 Entropic 1 0.5 0 5 10 15 20 25  (c)

Figure 13: Entropy diagrams of (9) during 2010–2016. (a) Bozhou; (b) Bengbu; (c) Anqing.

120 1.6

100 1.4 1.2 80 1

t 60 X 0.8 Entropic 0.6 40

X: 0.01382 0.4 20 Y: 15.91 0.2 X: 0.01382 Y: 0 0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05  

(a) (b)

Figure 14: Te efect of � on the stability and complexity of (25) in Hefei when �ℎ� =19. (a) Change trend diagram; (b) entropy diagram.

where � is a control parameter. Te instability of (9) is Compared to Figures 14 and 16, the evolution trends of the described in Figure 7(d) when �ℎ� =19. Te analysis process control parameters � and � arebasicallythesame,butonlya is similar to that in Section 6.1. Next we analyze the efect slight fuctuation occurs during the � change. Tat is to say, of the control parameter � on (26) under other unchanged the infuence of stochastic factors on the control parameter is parameters. limited. Further, the system is stable when � > 0.01382,and 16 Advances in Mathematical Physics

e trend of temperature change in Hefei 40 e trend of temperature change in Hefei 25 35

30 20 25

20 15 t

15 t X X 10 10 5

0 5 −5 − 10 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b)

Figure 15: Time series diagrams of (25) in Hefei when �ℎ� =19.(a)� = 0.005;(b)� = 0.02.

120 1.8

1.6 100 1.4

80 1.2

1 t 60 X 0.8 Entropic 40 0.6

X: 0.01382 0.4 20 Y: 16.37 0.2 X: 0.01382 Y: 0 0 0 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05  

(a) (b)

Figure 16: Te efect of � on the stability and complexity of (26) in Hefei when �ℎ� =19. (a) Change trend diagram; (b) entropy diagram. it is unstable when � < 0.01382.Tebiggerthe� is, the more can efectively avoid instability through the variable feed- stable the system is, so the unstable system can be returned to back control method. Te adjustment range of the control stable state by setting � > 0.01382. parameters directly afects the stability and complexity of Figures 17(a) and 17(b) show the control efect of (26) the system. Moreover, the variable feedback control method when � = 0.005 and � = 0.03, respectively. According can control the deterministic diferential equation and the totheanalysisabove,(26)isunstablewhen� = 0.005.In stochastic diferential equation well. However, the control comparison with Figures 17(b) and 7(d), we fnd that the efect of the deterministic diferential equation is better than controlled system exhibits the trend of periodic variation that of the stochastic diferential equation. when � = 0.03 > 0.01382. Because (26) is afected by random factors, compared with Figure 15(b), the periodic variation in 7. Conclusion Figure 17(b) is not very regular, but obvious control action has been shown with respect to Figure 7(d). In this paper, a temperature index model with delay and In brief, the unstable system is extremely likely to cause stochastic term is established. Te temperature variation temperature oscillations and cannot obtain accurate results. trends of four cities in Anhui Province are analyzed by However, adding control parameters in the original system using historical temperature data. Te results show that the Advances in Mathematical Physics 17

e trend of temperature change in Hefei e trend of temperature change in Hefei 70 40

60 35

30 50 25 40 t t 20 X X 30 15 20 10

10 5

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 t t (a) (b)

Figure 17: Time series diagram of (9) in Hefei when �ℎ� =19.(a)� = 0.005;(b)� = 0.03.

historical temperatures of all the four cities exhibit periodic comparing and analyzing simulation results, the model with- variation trends similar to that of sinusoidal curve. Te out stochastic term (delay parameter in reasonable range) local stability conditions of delay diferential equation are can better ft the trend of actual historical temperature; (7) explored. We analyze the efects of delay parameter and by analyzing the temperature evolution of four representative stochastic factors on the stability and complexity of the model cities, it can be inferred that the overall temperature in Anhui through temperature change trend diagram, the largest Lya- is also periodically changed; (8) parameters � and � have punov exponent diagrams, time series diagrams, entropy more obvious efect on the stability and complexity of the diagrams, and 4D diagrams. We carried out numerical system than parameters � and �; (9) the variable feedback simulation on the following models: the temperature index control method successfully achieves the stability control of model without delay and stochastic term, the temperature the deterministic diferential equation and stochastic difer- index model without delay under stochastic perturbation, ential equation. Tus, the unstable systems can be stabilized delay temperature index model without stochastic term, and by adjusting the control parameters. However, the control delay temperature index model with stochastic term. Te efect of deterministic diferential equations is better than that simulation results are compared with historical temperature of stochastic diferential equations. data to measure the ftting degree of the models. Te variable feedback control method is adopted to return the unstable Conflicts of Interest system to the stable state. Te conclusions of this paper are as follows: (1) the Te author declares no conficts of interest. historical temperature evolution trends of four cities in Anhui (2) Province periodically change; the temperature index Acknowledgments model without delay and stochastic term can better ft the temperature evolution process of four cities, similar to the Te author gratefully acknowledges fnancial support (3) sine curve; the temperature index model without delay from the National Natural Science Foundation of China under stochastic perturbation can show the trend of temper- (11626033); the Philosophy and Social Science Planning ature changes in four cities macroscopically, but the distur- ProjectofAnhui(AHSKQ2016D29);theUniversity bance of stochastic factors makes the temperature changes Outstanding Young Talent Support Project of Anhui (4) always accompanied by the high frequency fuctuations; (gxyq2017101);theHumanitiesandSocialScienceResearch delay temperature index model without stochastic term tells Project of Education Department of Anhui Province us that if the delay parameter is within a reasonable range, (SK2017A0434); the Social Science Planning Project of the model can refect the trend of temperature changes better; Bengbu (BB17C021); KYTD of BBC 2016-02. otherwise, the system will lose stability and increase the (5) complexity; because of the infuence of stochastic factors, References even if the delay parameter is within a reasonable range, delay temperature index model with stochastic term cannot [1] F. E. Benth and J. Saltyte-Benth, “Stochastic modelling of tem- accurately describe the changes of temperature, and it shows perature variations with a view towards weather derivatives,” the trend of periodic variation only macroscopically; (6) by Applied Mathematical Finance,vol.12,no.1,pp.53–85,2005. 18 Advances in Mathematical Physics

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