Analysis of chaotic climatic process in the Tarim River Basin(I)

ZuHan Liu

1.Key Laboratory of the Education Ministry for Poyang Lake Wetland and Watershed Research,Jiangxi Normal University,Nanchang,Jiangxi 330022,China

2.Jiangxi Province Key Laboratory for Water Information Cooperative Sensing and Intelligent Processing,Nanchang Institute of Technology,Nanchang,Jiangxi 330099,China

ABSTRACT Based on observational data obtained from 1961 to 2011 in the Tarim River Basin,China,we investigated the chaotic dynamics of temperature,precipitation,relative humidity,and evaporation.The main findings are as follow:(1)The four data series have significant chaotic and fractal behaviors,which are the result of the evolution of a nonlinear chaotic dynamic system.The climatic process in the Tarim River Basin also has deterministic and stochastic characteristics.(2)To describe the temperature,precipitation,relative humidity,and evaporation dynamics,at least three independent variables at daily scale are required;in terms of complexity,their order is evaporation ＞temperature ＞precipitation ＞relative humidity.(3)Their respective largest Lyapunov exponent λ1 shows the order of their degree of complexity is relative humidity ＞temperature ＞precipitation ≈evaporation;the maximum time scales for which the four systems can be predicted are 17 days,17 days,16 days,and 16 days,if calculated separately.(4)The Kolmogorov entropy K illustrates that the complexity of the nonlinear precipitation system is much greater than that of the other three systems.Both temperature and evaporation systems exhibit weaker chaotic behavior,their predictability is better,and the degree of complexity is less than that of the other two factors.

Keywords:chaos;climatic process;largest Lyapunov exponent;Kolmogorov entropy;correlation dimension

1 Introduction

With the rapid development of meteorological science and technology,there is growing concern about climate change,which to a certain degree threatens the further development of human society.Currently,the challenge for meteorological workers facing this important issue is how to produce the mutation phenomenon of climate change,or rather the climatic process change or jump from one steady state into another,and to understand that the process of weather is chaotic after the climate abruptly changes.

As a complex nonlinear dynamic behavior,chaos is a general phenomenon in nature and is one of the most important discoveries of the later 20th century.Chaos is the regressive nonperiodic behavior state caused by the nonlinear system and having sensitive dependence on the initial conditions,which represents the inner randomicity of chaos time-series in a dynamic system,is pseudorandom in nature,and certainty is one of the basic properties of the dynamic system of chaos action.Chaos theory is only recently breaking down the myth and misreading that determinism and forecast must be connected.The chaos theory negates Laplace's mechanic determinism and enriches and develops the theory of scientific determinism.

Qian(1994)first proposed to construct a"geospherology"focusing attention on the Earth's surface layer with an open complex giant system(OCGS)approach.Chaos science is facing many uncertainties brought about by singularity,diversity,and evolvability.Chaos is a certainty,as inherent randomness of a deterministic system is a certainty;but in instability physics,it reveals the probability of intrinsic kinetics of significance,strongly suggesting that the probability of objectivity is different from the traditional sense of full determinacy.Chaotic systems from the time-series is an irregular system of randomness and faint fluctuations(Wernitz and Hoffmann,2012).The precision of the instantaneous state and a description of its physical laws are among many uncertainty factors;these two are the main ones influencing the complex system prediction(Palmeret al.,1994).

In recent years,research on the temporal evolution of the climatic process and its impacts on ecosystems has become a key issue in hydrology and ecology.These theories and perspectives adopt a wide purview,including the wavelet and neuro-fuzzy conjunction model(Honoratoet al.,2019),the regional atmospheric modeling system(RAMS)(Liston and Pielke,2000;Wen and Duan,2019),a latent Gaussian Markov random-field model(Zheng and Yao,2019),and the generalized linear model(Segondet al.,2006;Nguelifack and Kemajou-Brown,2019).However,climatic process has been identified as a set of atmospheric states of a dynamical,chaotic system showing deterministic variability(Lorenz,1962).The interpretation of climate as a complex intervariable organization is a key issue for understanding temporal evolution of dynamic systems(Millánet al.,2009;Liuet al., 2014a). Subsequently, valuable applications of chaos and nonlinear dynamical systems theory in atmospheric and climate science have been developed(Tsonis,1996).The idea of climate's evolving on a strange attractor is often invoked,sometimes implicitly,to illustrate qualitative aspects of theories and conjectures regarding the climate system's behavior(e.g.,Palmer,1999;Branstator and Teng,2010).

Contemporary China is implementing the One Belt and One Road open strategy and strategic programs of reconstructing the beautiful landscape.Given this background and facing the frequent extreme weather events of climate change(Wanget al.,2019),how to understand the complexity of the climate system has important theoretical and practical significance.Here lies the advantage of chaos theory.The sensitivity and randomness of the system to the initial conditions,the density of the periodic points,the Lyapunov exponent,and the fractal dimension are the objects of chaos discussion (Rocha and Varandas,2018).The chaotic characteristics of an unknown system are often more accurately identified by numerical methods. The reconstruction parameters are determined by phase-space reconstruction;and the correlation dimension,maximum Lyapunov exponent,and Kolmogorov entropy are calculated,respectively.It can be shown whether the time-series has chaotic attractors,so as to enable study of the dynamic variation law of chaotic dynamics of the system.

The purpose of this study was to further understanding of the temporal complexity of climatic processes,based on observed average daily temperature,precipitation,relative humidity,and evaporation at 23 meteorological stations in the Tarim River Basin,China,from January 1,1961,to December 31,2010.Following chaos theory,we used a combination of quantitative and qualitative,microscopic analyses to explore the chaotic characteristics and complexity of the combination of four climate-factor time-series.The quantitative degree of system chaos was acquired,and the average time scale that could forecast weather was found.More specifically,the reconstruction parameters(namely,the embedding dimensionmand delay timeτ)were determined by application of mutual information and the Cao method;and the characteristic quantities as the largest Lyapunov exponentλ1,Kolmogorov entropyK,and the correlation dimensionDwere calculated.All the results indicated an obvious chaotic characteristic in the four climatic-factor timeseries in the Tarim River Basin,all of which resulted from evolution of the nonlinear chaotic dynamic system.Finally,their chaotic behavior and some influencing factors are discussed.

2 Study area and data

The Tarim River Basin(34°20'N-43°39'N,71°39'E-93°45'E)is the hinterland of Eurasia;high mountains and areas of barren soil,covering almost the entire southern part of Xinjiang,including 42 cities or counties and 55 collective farms managed by the Xinjiang Production and Construction Corporation.Its area is 1.02×106km2,and over 97%of the area belongs to a drainage basin internal to China.This area has a typical desert climate,with an average annual temperature of 10.6°C to 11.5°C.Monthly mean temperature ranges from 20°C to 30°C in July and-10°C to-20°C in January.The highest and lowest temperatures are 43.6°C and-27.5°C,respectively.The accumulative temperature of ＞10°C ranges from 4,100°C to 4,300°C.Average annual precipitation is 116.8 mm in the basin,ranging from 200 to 500 mm in the mountainous area,50 to 80 mm at the edges of the basin,and only 17.4 to 25.0 mm in the central area of the basin.Within any given year,the distribution of precipitation is quite uneven.More than 80%of the total annual precipitation falls between May and September,and less than 20%falls from October to next April.

The records used in this paper were obtained from high-quality data of three climatic factors—temperature,precipitation,and relative humidity—from January 1,1961,to December 31,2011,at 23 meteorological stations in the Tarim River Basin and processed by the Information Center of Meteorological Office in the Xinjiang Uygur Autonomous Region.Figure 1 shows locations of the 23 stations,and more detailed information can be found in the Liuet al.(2014a)reference.According to Chenet al.(2002),scaling of correlated data series was not affected by randomly cutting out segments and stitching together the remaining parts,even when 50%of the points were removed.Therefore,we removed the missing values from the raw data over some stations,as they are only a small part of the series.Furthermore,to overcome the natural nonstationarity of the data due to season trends,we removed the annual cycle from the raw dataeby computing the anomaly seriese'=e-＜e＞dfor relative humidity,where ＜＞ddenotes the long-time average value for the given calendar day.We applied the standard normal homogeneity test(SNHT),using the Buishand and Pettit homogeneity test method,to check these data(Pettit,1979;Buishand,1982;Alexandersson,1986).The stepwise multiple linear-regression method was employed to revise the inhomogeneity of the time-series.

Figure 1 Locations of meteorological stations in the Tarim River Basin

3 Methodology

Chaos theory can be traced to 1963,when the U.S.meteorologist Lorenz simulated atmospheric turbulence between two infinite planes(Lorenz,1963).

For a climatic-factor time-seriesx(t),we suppose it is generated by a nonlinear dynamic system withmdegrees of freedom.To restore the dynamic characteristic of the original system,the first step is to construct an appropriate series of state vectors,X(m)(t),with delay coordinates in them-dimensional phase space according to the phase-space reconstruction(Grassberger and Procaccia,1983b):X(m)(t)={x(t),x(t+τ),x(t+2τ),…,x(t+(m-1)τ)},wheremis the embedding dimension andτis an appropriate delay time(Xuet al.,2016).To reconstruct an appropriate phase space,τandmshould be determined(they are the two most basic important parameters).It must be said that the choice ofτ,which can usually be obtained by the autocorrelation function and C-C method (Kolokolov and Lyubinskii,2019)or the mutual information method(Fraser and Swinney,1986),plays an important role.As for the embedding dimension(m),it should not be too small or too large.If it is too small or too large,it can cause noise interference due to the extra dimension.Takens(1981)theoretically proved that whenm＞2D+1,an embedding dimension of the attractor could be obtained,whereDis the correlation dimension.

In addition,the three parameters,the correlation dimension(Grassberger and Procaccia,1983a),the largest Lyapunov exponent(Wolfet al.,1985),and the Kolmogorov entropy(Grassberger and Procaccia,1983b)are available in the literature to identify the existence of chaos in a time-series.

The first parameter is the largest Lyapunov exponentλ1,which is is an important parameter in detecting and characterizing chaos produced from a dynamical system.The slope of the"average"liney(i)behaves as a function of the value of time steps forward in the trajectoryi(Wanet al.,2018).We can draw they(i)-icurve from the natural logarithm and fit the curve by the least-squares method(LSM),then the slope of the largest Lyapunov exponent isλ1.It defines the average rate of divergence or convergence of two neighbouring trajectories in the state-space.Specifically,the bigger theλ1,the longer the time scale is and the stronger the degree of complexity of the system.The predictable maximum time scale is the reciprocal of the largest Lyapunov exponentλ1,namely 1/λ1.

The second parameter is the correlation dimension.Its benefits lie in determining the embedding dimension for the phase-space reconstruction of the climatic-factor time-series,which is of vital importance in describing geometric features of the strange attractor of a climatic system and is usually applied to analyze a time-series and determine if it exhibits a chaotic dynamic characteristic(Eggleston,2018).The most widely used method is the G-P algorithm proposed by Grassberger and Procaccia(1983a).The correlation integralC(ε)behaves as a power-law function of the surveyor's rod for distanceε,data present scaling:C(ε)∝εD.The correlation dimension(D)is defined as the slope of the regression line for all points[log(ε),log(C(ε))].

The third parameter is the Kolmogorov entropy,which is the infinite resolution limit of block entropies and characterizes the degree of dynamical randomness of system.Asm→∞andε→0,K2=()converges to a steady value,which is the Kolmogorov entropyK.For a regular motion,the series corresponds toK=0 andK→∞,and the system indicates random motion;for 0 ＜K＜∞,the system embodies deterministic chaotic motions.The bigger theKvalue,the more serious the degree of chaos of the system,the higher the complexity and the loss rate of information.

For further detailed computation about chaos theory,see Grassberger and Procaccia(1983),Liuet al.(2006),and Eggleston(2018),etc..

4 Results and discussion

4.1 Delay time and embedding dimension

The delay time and embedding dimension are calculated by using the mutual information method and the Cao method(Cao,1997).Figure 2 shows the change of delay timeτwith the mutual information of the four climatic-factor time-series.When theirτ=8,5,11,and 13,respectively,their mutual information reaches the lowest point for the first time,soτ=8,5,11,and 13 are selected for their optimum delay time,respectively.Figure 3 showsE1andE2change with the embedding dimension of the four climaticfactor time-series.When embedding dimensionsmrespectively are less than 15,3,3,and 6,E1andE2changes are increased significantly and gradually become steady untilm=15, 3, 3, and 6. In this case,the standard deviation ofE2fluctuation(E2=0.2391,0.2035,0.2196,and 0.2252,respectively)is much larger than(E1=0.1099,0.0315,0.0755,and 0.0755,respectively).With further increasing of their embedding dimensionsm,the fourE1values tend to l,the four climatic factors all exist determined parts.As mentioned,m=15,3,3,and 6 are selected as the saturated embedding dimension of their reconstructing-phase space.

Figure 4 indicates the trend of a correlation integral spectrumεaccording to different embedding dimensionsm,and the change of correlation dimension and white noise withmvalues for the time-series and the white noise of using the G-P algorithm.Obviously,for each there exists a scale-free interval within a certain amount of four correlation integral spectrumε.Their slopes of scale-free interval increase and the distance between the curves decreases with increasingm.The distance has become a lot smaller when the embedding dimensionm≥17,18,17,and 17,which are the saturated values of the four attractors,respectively.Their corresponding correlation dimensionD,respectively,is 2.81,2.86,2.95,and 2.48;and everyDvalue is a non-integer.Therefore,the effective degree of freedom for the dynamical system described by the four time-series,respectively,is 17,18,17,and 17.Moreover,to describe the temperature,precipitation,relative humidity,and evaporation dynamics,at least three independent variables at the daily scale are required.

These results show the four attractors all possess a fractal structure,but their complexity is obviously different.To be specific,the highest factor is evaporation,next is temperature,the third is precipitation,and last is relative humidity.By comparison,the fourDvalues of the white noise(namely,a purely random system)do not converge with increasingm,so the dynamic behavior of the climatic process is different from that of white noise;therefore,the dynamic behavior in temporal variation has both obviously deterministic and random characteristics.It further illustrates that the four climatic-factor series all possess chaotic characteristics or chaotic attractors.

Figure 3 The change of E1 and E2 of average daily temperature(a),precipitation(b),relative humidity(c),and evaporation(d)in the Tarim River Basin

Figure 4 The trend of a correlation integral according to various embedding dimensions m,and the change of correlation dimension and white noise with m values for the time-series of average daily temperature(a1)and(a2),precipitation(b1)and(b2),relative humidity(c1)and(c2),and evaporation(d1)and(d2)

4.3 Largest Lyapunov exponent

For the average daily temperature,precipitation,relative humidity,and evaporation time-series,their Lyapunov exponentλ1can be calculated from the formula method mentioned.They(i)-icurves under cir-cumstances thatτ=8 andm=17,τ=5 andm=18,andτ=11 andm=17 are shown in Figure 5.Figure 5 shows slight differences among them;but overall,the slope of the"average"liney(i)values increases with increase in the value of time steps forward in the trajectoryi.Besides that,all the slope ranges ofy(i)-icurves become smaller and smaller,that is to say,they(i)change has a very good convergence.The largest Lyapunov exponentλ1can be obtained by the leastsquares method(LSM)to fit these curves;the fitting results are 0.0658,0.0594,0.0824,and 0.0744,respectively.Anyλ1value is greater than zero,which allows us to further contend that the four climatic-factor systems have chaotic characteristics,or a chaotic attractor.The order of the degree of complexity in the climatic-factor system is relative humidity ＞temperature ＞precipitation=evaporation.In other words,the divergent extent of the near orbit and the general chaotic level system of the four dynamic systems decrease in that order.Moreover,their maximum predictive time scales are 17 days,17 days,16 days,and 16 days,if considered separately.

In addition,the maximum time span of forecast of the four climatic-factor systems approximately equals the correlation dimension (attractor dimension)D.This aspect still needs further research.

Figure 5 The calculated result of largest Lyapunov exponent using the Rosenstein algorithm for the time-series of average daily temperature(a),precipitation(b),relative humidity(c),and evaporation(d)

Figure 6 indicatesK2versusm+1 of the four timeseries using G-P algorithms.The fourK2values all exhibit a decreasing trend,withm+1 increasing untilm+1=18,19,18,and 18(namelym=17,18,17,and 17),respectively.Kolmogorov entropy reaches saturation and converges toK.In this instance,Kseparately is equal to 0.091,2.184,0.518,and 0.092(＞0),further illustrating all four climatic processes are chaotic,that is,there is a chaos attractor in every nonlinear climatic system.What's more,the Kolmogorov entropyKof the precipitation time-series is much greater than that of the other three,which suggests the complexity of the precipitation system is much greater than the other three systems.The reason is that precipitation is affected by more external factors and defined by local influence of climate process.And it even is influenced by the common background of the global climate,which to a great extent affects temperature,humidity and evaporation.Moreover,the Kolmogorov entropyKof both the temperature and evaporation time-series is approximately equal to 0,but still greater than zero.The pattern indicates the chaotic characteristic is weaker in the nonlinear temperature and evaporation systems.And the predictability of these two systems is better and the degree of complexity less than those of the other two.That may be because their influencing factors are of relatively smaller magnitude,so the complexity of these two climatic factors is significantly lower.

Figure 6 The calculated results of Kolmogorov entropy K of the time-series of the average daily temperature(a),precipitation(b),relative humidity(c),and evaporation(d)

5 Conclusions

Based on the temperature,precipitation,relative humidity, and evaporation daily time-series in the Tarim River Basin from 1961 to 2011,their three characteristic quantities—the largest Lyapunov exponentλ1,Kolmogorov entropyK,and the correlation dimensionD—were analysed using chaos theory.The main findings are as follow:

(1)All results in the four series show obvious chaotic and fractal characteristics,which are the result of the evolution of nonlinear chaotic dynamic systems.The climatic process in the Tarim River Basin also has deterministic and stochastic characteristics.

(2)The correlation dimensionDof the four nonlinear climatic systems,respectively,is 2.81,2.86,2.95,and 2.48.To describe the temperature,precipitation,relative humidity,and evaporation dynamics,at least three independent variables at the daily scale are needed.Moreover,the four attractors all have fractal structures.The order of their complexity is evaporation＞temperature＞precipitation＞relative humidity.

(3)The largest Lyapunov exponentλ1of the four nonlinear climatic systems is 0.0658,0.0594,0.0824,and 0.0744(＞0),respectively.The finding allows us to further contend that the four climatic-factor systems all have chaotic characteristics,or chaotic attractors.The order of their degree of complexity is relative humidity＞temperature＞precipitation ≈evaporation.Their possible maximum prediction time scales are 17 days,17 days,16 days,and 16 days,if figured separately.

(4)The Kolmogorov entropyKof the four nonlinear climatic systems separately equals 0.091,2.184,0.518,and 0.092,which illustrates they all have chaotic characteristics and chaos attractors.What's more,the complexity of the nonlinear precipitation system is much larger than that of the other three systems,which may be explained by the local influence of climatological-hydrological processes in the Tarim River Basin,and even influenced by the common background of the global climate,which to a great extent affects temperature,humidity,and evaporation.Moreover,the temperature and evaporation systems exist weaker chaotic characteristics;and their predictability is better and the degree of complexity less than those of the other two.

The climatic process is an open,complex dynamic system with very strong nonlinear characteristics.Climate change is affected by many factors,such as geographical location,physics,chemistry,biology,meteorology,and human activities.Therefore,the system is one of the objects of fractal and chaos research.Chaos theory is applied to the one-dimensional time-series of a climatic factor,which can better reflect the intrinsic motion mechanism of the sequence and reveal the complex motion law and evolution process of the dynamic system.

According to chaos theory,the motion orbital divergence of the chaotic system is small in the short term,so the method realizes the possibility of shortterm prediction for climatic processes.At present,the forecast-measure problem of climate change needs further study,which is mainly reflected in the unpredictability in the long term and the improvement of forecasting accuracy in the short term.

Acknowledgments:

This work is supported by the China Postdoctoral Science Foundation(2016M600515),Postdoctoral Preferred Fund of Jiangxi Province(2017KY48),the Jiangxi Postdoctoral Daily Fund Project(2016RC25),the Opening Fund of Key Laboratory of Poyang Lake Wetland and Watershed Research (Jiangxi Normal University),Ministry of Education(PK2017002),the Open Research Fund of Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing(2016WICSIP012),the National Natural Science Foundation of China(61703199),and the Science and Technology Research Project of Jiangxi Provincial Education Department(GJJ180926).