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

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