research article modeling and dynamical analysis of the...

Research ArticleModeling and Dynamical Analysis of the Water ResourcesSupply-Demand System A Case Study in Haihe River Basin

Chongli Di and Xiaohua Yang

State Key Laboratory of Water Environment Simulation School of Environment Beijing Normal University Beijing 100875 China

Correspondence should be addressed to Xiaohua Yang xiaohuayangbnueducn

Received 15 March 2014 Accepted 8 May 2014 Published 26 May 2014

Academic Editor Song Cen

Copyright copy 2014 C Di and X Yang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The relationship between water resources supply and demand is very complex and exhibits nonlinear characteristics which leadsto fewer models that can adequately manage the dynamic evolution process of the water resources supply-demand system In thispaper we propose a new four-dimensional dynamical model to simulate the internal dynamic evolution process and predict futuretrends of water supply and demand At the beginning a new four-dimensional dynamical model with uncertain parameters isestablished Then the gray code hybrid accelerating genetic algorithm (GHAGA) is adopted to identify the unknown parametersof the system based on the statistic data (1998ndash2009) Finally the dynamical analysis of the system is further studied by Lyapunov-exponent phase portraits and Lyapunov exponent theory Numerical simulation results demonstrate that the proposed waterresources supply-demand system is in a steady state and is suitable for simulating the dynamical characteristics of a complex watersupply and demand system According to the trends of the water supply and demand of several nonlinear simulation cases thecorresponding measures can be proposed to improve the steady development of the water resources supply-demand system

1 Introduction

Water is indispensable to human survival agriculture andindustry Recently there is an increasing shortage of water inmost river basins mainly due to human activity and climatechange [1 2] With the population growth degradation ofwater quality and loss of potential sources of freshwaterwater demand is becoming increasing greater and watersupply has been increasing scarce in many developing coun-tries [3] The internal dynamic evolution process and therelationship between water resources supply and demandare very complex and exhibiting nonlinear characteristics[4] Water management is rapidly becoming more and moredifficult due to the complex water resources system Senge [5]discussed two types of complexity (1) detail and (2) dynamicDetail complexity is associated with systems with manycomponents Dynamic complexity however is in contactwith effects that are separated by time or space It is thedynamic complexity found in the water resources system thatpresents great difficulties for water management Thus howto analyse the internal dynamic evolved mechanism of the

water resources supply-demand system and solve the issueof imbalanced water resources supply-demand has an impor-tant practical significance to the achievement of sustainabledevelopment and management of water resources

The problem of water resources supply and demandhas attracted much attention and a considerable amount ofresearches Rayan [6] studied the water supply and demandof Sinai Egypt and introduced desalination options Someprevious works applied artificial neural networks (ANNs)to predict and forecast water resource system [7ndash9] Georgeet al [10] proposed a city water balance model to analysealternative strategies to manage the supply of water in orderto come to terms with the water scarcity problems facingpolicy makers in Hyderabad Qi and Chang [4] used systemdynamics model to estimate domestic water demand underuncertain economic impacts from 2003 to 2009 for ManateeCountry Based on the data for meteorology hydrologysoil planting vegetation and socioeconomic developmentof the irrigation region in the middle reaches of the HeiheRiver basin Ji et al [11] established the model of balanceof water supply and demand and assessed the security of

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 171793 10 pageshttpdxdoiorg1011552014171793

2 Journal of Applied Mathematics

water resources development A probabilistic forecast modelof future water demand for the city of Mecca in Saudi Arabiawas proposed in [12] Liu et al [13] adopted WDF-ANNmodel to forecast water demand in Weinan city Yurdusevet al [14] predicted monthly water consumption by usingfeed-forward and radial-basis neural networks However thenonlinear dynamic behavior of the water resources supply-demand system is seldom considered by these methods

Any very complicated dynamical system can be solvedby a simple mathematical model [15] The dynamical systemmethod is one approach to nonlinear modeling It hasbeen applied increasingly in water environments and waterresources system such as hydrological time series prediction[16ndash18] and the dynamical analysis of runoff flows [19 20]In this paper according to the current situation of the waterresources supply and demand we establish a new dynamicalmodel to express the complex relationship between waterresources supply and demand by using dynamical systemtheory This model can be used to analyse the internaldynamic evolved mechanism of the water resources supply-demand system of a river basin or a city

The outline of this paper is organized as follows Section 2introduces the study area and data In Section 3 the waterresources supply demand model is established and themethod of parameter identification of the proposed modelis introduced Section 4 provides the results and discussionson the real water resources supply-demand system in HaiheRiver Basin The conclusions of the paper are summarized inSection 5

2 Study Area and Data

The HRB is the political economic and cultural center ofChina However water resources shortage in the HRB hasbecome a serious problem in recent years due to the rapideconomic development and climate change [21]The locationof the HRB is shown in Figure 1 The basin covers an areaof 318200 km2 with 60 being mountains in the westernand northern part and 40 being plains in the eastern andsouthern parts It belongs to continental monsoon climatezone and locates in a semihumid and semiarid region Theaverage annual precipitation is 52469mm and the totalamount of water resources is 37439times10

8m3 (1956ndash2008) thewater volume per capita is 272m3 merely 17 of the nationalaverage and 124 of the world average [22]

Thewater resources demand and supply data from 1998 to2009 are obtained from the water resources bulletin of HRBTable 1 is the water resources statistic data (1998ndash2009) ofHRB where 119883 is water demand 119884 is surface water supply 119885is groundwater supply and119882 is other kinds of water supply

3 Modeling and Methods

31 Establishment of the Water Resources Supply-DemandModel Let 119883(119905) be the demand for water resources 119884(119905)

the surface water supply (including diversion water supply)119885(119905) the groundwater supply and 119882(119905) other kinds of watersupply (mainly including wastewater reuse direct seawater

China

RiverHaihe River Basin

Haihe River Basin

0 80 160 320

(km)

42∘N

40∘N

38∘N

36∘N

112∘E 114

∘E 116∘E 118

∘E 120∘E

N

Figure 1 The location of the Haihe River Basin

utilization and collection of rainwater) The model for thewater resource system is established as follows

119889119883

119889119905= 1198861119883(1 minus

119883

119872) minus 1198862119884 minus 1198863119885 minus 1198864119882

119889119884

119889119905= minus 119887

1119884 minus 1198872119885 + 1198873119883 [119873 minus (119883 minus 119885 minus 119882)]

119889119885

119889119905= 1198881119883 minus 1198882119884 minus 1198883119885 minus 1198884119882

119889119882

119889119905= 1198891119882(119883 minus 119889

2) minus 1198893119884

(1)

where 119886119894 119887119894 119888119894 119889119894119872119873119870 are constants and119872 gt 0119873 gt 0

Additionally 1198861is the elastic coefficient of the water

resources demand 1198862 1198863 and 119886

4are respectively the influ-

ence coefficients of the surface water supply the groundwatersupply and other kinds of water supply on thewater resourcesdemand 119887

1is the influence coefficient of the surface water

supply on the change rate of the surface water supply 1198872is the

influence coefficient of the groundwater supply on the surfacewater supply 119887

3is the influence coefficient of water resources

demand on the surface water supply 1198881 1198882 1198883 and 119888

4are

respectively the influence coefficients of the water resourcesdemand the surface water supply the groundwater supplyand other kinds of water supply on the groundwater supply1198891is the velocity constant of the recycled water supply

1198892is the cost of recycled water supply 119889

3is the influence

coefficient of the surface water supply on the other kinds ofwater supply 119872 is the maximum value of water resourcesdemand119873 is the valve value of diversion water supply

In system (1) 1198861119883(1 minus (119883119872)) denotes that the rate of

water resources demand 119889119883119889119905 is in direct ratio to waterresources demand when the current water resources demand

Journal of Applied Mathematics 3

Table 1 Water resources statistic data in Haihe River Basin (Unit billion m3)

Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009119883 42400 43150 40060 39200 39983 37705 36801 38046 39285 38452 37339 37002119884 16070 16268 13660 12270 12794 11364 11823 12335 13415 12864 12310 12550119885 26190 26750 26279 26774 27029 26138 24697 25294 25185 25001 24060 23588119882 140 129 121 156 160 203 283 417 685 583 969 864

of the study area is less than 119872 and 119889119883119889119905 is inverselyproportional to water resources demand when the waterresources demand of the study area is larger than 119872 minus119886

2119884

minus1198863119885 and minus119886

4119882 denote that the increase in surface water

supply groundwater supply and other kinds of water supplywill reduce the demand of the water resources respectivelyminus1198871119884 minus 119887

2119885 denotes that the rate of surface water supply

will decrease with the surface water supply and groundwatersupply 119887

3119883[119873minus(119883minus119885minus119882)] denotes that when the diversion

water supply is less than the threshold value (ie 119883(119905) minus

119885(119905) minus 119882(119905) lt 119873) the rate of diversion water supply willincrease with the increase of 119883(119905) and when diversion watersupply is large enough (or 119883(119905) minus 119885(119905) minus 119882(119905) gt 119873) witheconomic development the people and enterprises aroundthe Yellow River and Yangtze River Basin will need a lot ofwater which results in the rate of diversion water supplydecreasing with the increase rate of 119883(119905) 119888

1119883 denotes that

the increase in water resources demand will increase thegroundwater supply minus119888

2119884 minus 119888

3119885 minus 119888

4119882 denotes that the

decrease in surface water supply the groundwater supplyand other kinds of water supply will decrease the rate ofgroundwater supply 119889

1119882(119883 minus 119889

2) denotes that the rate of

reused water is in direct ratio to the amount of recycled waterwhich increases with the increase of water resources demandand decrease with the increase of the cost for reused water

System (1) is an uncertainty system with unknownparameters and it has different characteristics with differentparameters When a set of parameters are set as follows 119886

1=

006 1198862

= 02 1198863

= 009 1198864

= 015 1198871

= 022 1198872

= 021198873

= 02 1198881

= 013 1198882

= 014 1198883

= 012 1198884

= 021198891= 016 119889

2= 02 119889

3= 0001 119872 = 11 and 119873 = 06 and

the initial condition is set as [06 03 02 01] the Lyapunovexponents of the system (1) are 119871

1= minus00156 119871

2= minus00371

1198713

= minus00853 and 1198714

= minus01801 Because these Lyapunovexponents are all less than 0 the orbit of the system (1) is alimit cycle as shown in Figure 2 When 119886

1= 006 119886

2= 008

1198863= 03 119886

4= 02 119887

1= 005 119887

2= 004 119887

3= 004 119888

1= 01

1198882= 02 119888

3= 005 119888

4= 004 119889

1= 02 119889

2= 03 119889

3= 0001

119872 = 11 and 119873 = 05 and the initial conditions are setas [07 02 01 01] the Lyapunov exponents of the system(1) are 119871

1= minus00020 119871

2= minus00134 119871

3= minus00169 and

1198714= minus00592The systemhas a stable focus point and its orbit

is shown in Figure 3 When 1198861= 008 119886

2= 013 119886

3= 006

1198864

= 008 1198871

= 015 1198872

= 015 1198873

= 014 1198881

= 0131198882

= 015 1198883

= 019 1198884

= 018 1198891

= 016 1198892

= 0121198893= 0002 119872 = 25 and119873 = 08 and initial condition is set

as [08 01 03 02] the Lyapunov exponents of the system(1) are 119871

1= 00421 119871

2= minus00104 119871

3= minus01144 and

1198714= minus02631 Since there are two Lyapunov exponents that

are larger than zero therefore the system is hyperchaotic inthis situation System (1) has a chaotic attractor as shown inFigure 4

32 Parameter Identification To illustrate the behavior ofwater resources supply-demand system in a specified areathe unknown parameters should be identified first Parameteridentification can be seen as a global optimization problemMany methods for the global optimization problem havebeen proposed which can be roughly divided into two cat-egories deterministic optimization methods and stochasticoptimization methods The deterministic approaches takeadvantage of the analytical properties of the problem togenerate a sequence of points that converge to a global opti-mal solution and sometimes can give a guaranteed solution[23 24] However the deterministic optimization may betrapped in the local extreme point when the dimensionis high and there are numerous local optima Sometimestraditional deterministic methods cannot obtain the globaloptimization efficiently due to its complex computationand strict application conditions [25] Additionally otherresearch results suggest that the deterministic global algo-rithms have lower efficiency for the optimization problemswith a large number of parameters [26] Therefore the tradi-tional deterministic global algorithms are not considered tosolve our problemwhich not only contains many parametersbut also shows high nonlinearity and complex dynamiccharacteristics

Genetic algorithm (GA) is a type of stochastic optimiza-tion method based on the mechanics of natural selection andnatural genetics It can provide a robust procedure not onlyto explore broad and promising regions of solutions but alsoto avoid being trapped in the local optimization [27] Manyresearchers have applied the GA techniques to identify linearand nonlinear systems Based on many improved geneticalgorithms Yang et al [28] proposed a gray-encoded hybridaccelerating genetic algorithm (GHAGA) with Nelder-Meadsimplex searching operator and simplex algorithm for theglobal optimization of dynamical systems The GHAGAmethod is an improved genetic algorithm Compared withtraditional genetic algorithm and pure random search algo-rithm the GHAGA method has the advantage of higher cal-culation precision and efficiency rapider convergent speedand wider scope of application It has been proved asan effective method to be used in various complex waterenvironment optimization problems [28 29] Therefore theGHAGA method is chosen to identify the parameters of thewater resources supply-demand system (1)


minus4minus2

02

minus6minus4

minus20

2minus2

minus1

0

1

2

3

XY

Z

(a)

minus4minus2

02

minus6minus4

minus20

20

005

01

015

02

XY

W

(b)

0 200 400 600 800 1000minus5

05

0 200 400 600 800 1000minus10

010

0 200 400 600 800 1000minus5

05

0 200 400 600 800 10000

01

02

X

Y

Z

t

t

t

t

W

(c)

Figure 2 A limit cycle of system (1) (a) a 3D view (119883-119884-119885) (b) a 3D view (119883-119884-119882) and (c) the time series of119883(119905) 119884(119905) 119885(119905)119882(119905)

321 Discretization Before determining the parameters sys-tem (1) needs to be discretized Using the discrete transformthe water resources system (1) is discretized into system (2)

119883 (119896 + 1)

= 119883 (119896) + 119905 [1198861119883 (119896) (1 minus

119883 (119896)

119872) minus 1198862119884 (119896)

minus1198863119885 (119896) minus 119886

4119882(119896) ]

119884 (119896 + 1)

= 119884 (119896) + 119905 [ minus 1198871119884 (119896) minus 119887

2119885 (119896)

+1198873119883 (119896) [119873 minus (119883 (119896) minus 119885 (119896) minus 119882 (119896))]]

119885 (119896 + 1)

= 119885 (119896) + 119905 [1198881119883 (119896) minus 119888

2119884 (119896) minus 119888

3119885 (119896) minus 119888

4119882(119896)]

119882 (119896 + 1) = 119882 (119896) + 119905 [1198891119882(119896) (119883 (119896) minus 119889

2) minus 1198893119884 (119896)]

(2)

322 The Algorithm of GHAGA The GHAGA with Nelder-Mead simplex searching operator and simplex algorithm isused for the parameter identification A flowchart for theidentification of the system parameters by GHAGA is givenin Figure 5 and the detailed steps of the algorithm are givenin the following

Consider the parameter optimization of the waterresources model as

min 119891 (1199091 1199092 119909

119899)

st 119886119894le 119909119894le 119887119894 for 119894 = 1 2 119899

(3)

where 119909 = 119909119894 119894 = 1 2 119899 119909

119894is a parameter to be

optimized 119891 is an objective function and 119891 ge 0 for system(1) 119899 = 16

Step 1 (gray encoding) Suppose gray encoding length is 119890

in every parameter the 119894th parameter range is the interval


minus020

0204

0608

minus010

0102

03minus02

minus01

0

01

02

XY

Z

(a)

minus020

0204

0608

minus02minus01

001

02minus005

0

005

01

015

XY

Z

(b)

0 200 400 600 800 1000minus1

01

0 200 400 600 800 1000minus02

002

0 200 400 600 800 1000minus02

002

0 200 400 600 800 10000

01

02

X

Y

t

t

t

t

Z

W

(c)

Figure 3 A stable focus point of system (1) (a) a 3D view (119883-119884-119885) (b) a 3D view (119883-119885-119882) and (c) the time series of119883(119905) 119884(119905) 119885(119905)119882(119905)

[119886119894 119887119894] and then each interval is divided into 2

119890minus 1 subin-

tervals

119909119894= 119886119894+ 119868119894sdot 119888119894 (4)

where the length of subinterval of the 119894th parameter 119888119894

=

(119887119894minus 119886119894)(2119890minus 1) is constant The searching location 119868

119894is a

nonnegative integer and 0 le 119868119894lt 2119890 for 119894 = 1 2 119899 The

gray code array of the 119894th parameter is denoted by the gridpoints of 119889(119894 119896) | 119896 = 1 2 119890 for every individual

119868119894=

119890

sum

119895=1

(

119890

⨁

119896=119895

119889 (119894 119896)) sdot 2119895minus1

(5)

where⨁ denotes the operator of additionmodulo 2 on 0 1

[29] The GHAGA uses a code of parameters instead of theparameters and works on a population of points and not onone single point

Step 2 (randomly generating the initial population) Initiallythe chromosomes are generated at random in gray-encodedgenetic algorithm and p-chromosomes in father populationare

119868119894(119895) = int (119906 (119894 119895) sdot 2

119890)

119894 = 1 2 119899 119895 = 1 2 119901

(6)

where 119906(119894 119895) is uniform random numbers 119906(119894 119895) isin [0 1]119868119894(119895) is a searching location and int(sdot) is an integer function

From (3) the p-corresponding chromosomes are 119889(119894 119896 119895)for 119894 = 1 2 119899 119896 = 1 2 119890 119895 = 1 2 119901 To coverhomogeneously thewhole solution space and to avoid the riskof having too much individuals in the same region a largeuniformity random population is selected in this algorithm

Step 3 (decoding and fitness evaluation) Decoding of 119889(119894 119896 119895)for (119894 = 1 2 119899 119896 = 1 2 119890 119895 = 1 2 119901) worksthrough (4) and (5) and then corresponding parameter


minus20minus10

010

minus100minus50

050

minus10

0

10

20

30

X

Y

Z

(a)

minus20minus10

010

minus100minus50

0500

1

2

3

XY

W

(b)

0 500 1000 1500 2000 2500 3000minus20

020

t

t

t

t

0 500 1000 1500 2000 2500 3000minus100

0100

0 500 1000 1500 2000 2500 3000minus50

050

0 500 1000 1500 2000 2500 30000

2

4

X

Y

Z

W

(c)

Figure 4 A chaotic attractor of system (1) (a) a 3D view (119883-119884-119885) (b) a 3D view (119883-119884-119882) and (c) the time series of119883(119905) 119884(119905) 119885(119905)119882(119905)

119909119894(119895) is obtained Substitution of 119909

119894(119895) into (1) produces the

objective function 119891(119895) The smaller the value 119891(119895) is thehigher the fitness of its corresponding 119894th chromosome willbe Then the fitness function of 119894th chromosome is de-fined

119865 (119895) =1

[119891 (119895)]2

+ 01

(7)

Step 4 (selection) The chromosomes in the initial fatherpopulation are selected by a known probability 119875

119904(119895)

119875119904(119895) = 119865(119895)sum

119899

119894=1119865(119894) Such two groups of p-chromosomes

are selected by the above probabilities

Step 5 (two-point crossover) Perform crossover on eachchromosome pair according to probability 119875

119888to generate two

offsprings For two-point crossover two crossing points 1198681=

int(1198801sdot (119890 + 1)) and 119868

2= int(119880

2sdot (119890 + 1)) are randomly

chosen where1198801 1198802are uniformity randomnumbers where

1198801 1198802 isin [0 1] The two-point crossover between two indi-

viduals is performed through the crossing probability 119875119888

1198891015840

1(119894 119896 119895) =

1198892(119894 119896 119895) 119896 isin [119868

1 1198682]

1198891(119894 119896 119895) 119896 notin [119868

1 1198682]

(8)

1198891015840

2(119894 119896 119895) =

1198891(119894 119896 119895) 119896 isin [119868

1 1198682]

1198892(119894 119896 119895) 119896 notin [119868

1 1198682]

(9)

In order to enhance the diversity of population the crossingprobability is set as 119875

119888ge 05

Step 6 (two-pointmutation) Theoperator of two-pointmuta-tion is for four random numbers 119881

1 1198812 1198813 1198814 isin [0 1]

If 1198811

le 05 the offspring is computed by (8) Other-wise the offspring 119889

1015840(119894 119896 119895) is computed by (9) Let the

mutating probability be 119901119898

isin [0 1] the two points


No

Variable initial interval

Gray encoding

Generating initial population

Decoding

Genetic operator(1) Fitness evaluation(2) Selection(3) Two-point crossover(4) Mutation

Simplex searching

Satisfy convergent condition

Satisfy evolution times

Variable new initial interval

Yes

Stop

Figure 5 The flowchart for the identification of the system param-eters by GHAGA

be 1198691= int(119881

2sdot (119890 + 1)) 119869

2= int(119881

3sdot (119890 + 1)) and then the

above offspring is mutated from

1198893(119894 119896 119895)

(119894119896119895)

=

0 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

0 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

1198891015840(119894 119896 119895)

(119894119896119895)otherwise

(10)

where a new offspring 1198893(119894 119896 119895) can be computed by a

mutating probability 119901119898

Step 7 (simplex evolution and alternating) The Nelder-Meadalgorithm is a useful local descent algorithm which does notmake use of the objective function derivatives [30] The bestpoint in the previous phase becomes a new initial solutionin the Nelder-Mead simplex algorithm and then a new bestpoint is obtained by thisNelder-Mead simplex algorithmThenew best point inside the offspring will be inserted to replacethe worst one in the previous phase Repeat Step 3 to Step 7until the evolution times 119876 is met

Step 8 (accelerating cycle) The parameter ranges of 119899119904-

excellent individuals obtained by 119876-times of the Nelder-Mead simplex evolution alternating become the new rangesof the parameters and then the whole process back tothe Gray-encoding The GHAGA computation is over untilthe algorithm running times get to the design 119879 times orthere exists a chromosome 119888fit whose fitness satisfies a givencriterion In the former case the 119888fit is the fittest chromosomeor the most excellent chromosome in the population Thechromosome 119888fit represents the solution

4 Results and Discussions

41 The Results of Parameters Identification The GHAGAis calculated by using Matlab Table 2 presents the variableinitial intervals of the 16 unknown parameters in system (1)which are given according to the meaning of the parametersand the empirical research The parameters of the GHAGAare selected as follows the length 119890 = 10 the population size119899 = 300 the number of excellent individuals 119899

119904= 10 the

times of simplex evolution alternating 119876 = 5 the crossoverprobability 119901

119888= 10 the mutation probability 119901

119898= 05 and

the times of simplex searching 119898 = 600 The computationalresults of parameter identification for system (1) are also listedin Table 2

42 The Dynamical Analysis of the Real System When theparameters values are taken as shown Table 2 system (1)has four equilibrium points 119874(0 0 0 0) 119878

1(09093 07669

08118 03969) 1198782(08856 04171 36269 minus09523) and 119878

3

(03586 minus28007 61825 00527) By calculation the eigen-values of the Jacobi matrix of the system (1) at119874(0 0 0 0) are1205821

= 06921 gt 0 1205822

= 00320 gt 0 12058223

= minus01155 gt 0and 120582

4= minus00876 gt 0 Because four eigenvalues are not all

negative therefore 119874(0 0 0 0) is an unstable point and thesystem is unstable at the point 119874(0 0 0 0)

The eigenvalues of the Jacobi matrix of the system (1)at 1198781(09093 07669 08118 03969) are 120582

1= minus00022 lt 0

1205822= minus05721 lt 0 and 120582

34= minus00519 plusmn 00677119894 Obviously

the real parts of all the four eigenvalues are negative sothe water resources supply-demand system is stable whenthe demand of water resources in Haihe River Basin thesurface water supply the groundwater supply and the otherkinds of water supply are in the local area of the equilib-riumpoint 119878

1(09093 07669 08118 03969) It indicates that

water resources supply-demand system has the capacity ofself-adjustment and demonstrates the status of steady devel-opment in the local area of 119878

1(09093 07669 08118 03969)

Similarly we can prove that the water resources supply-demand system is unstable when the demand of waterresources in Haihe River Basin the surface water supply thegroundwater supply and the other kinds of water supply arein the local area of the equilibrium points 119878

2and 1198783

43 Numerical Simulations In this section based on theresults of parameter identification by GHAGA the numericalsimulation results for the real water resources supply-demandsystem are obtained and corresponding measures are given


Table 2 The parametersrsquo initial intervals and identified values P stands for parameters II stands for initial intervals and IP stands foridentified parameters

P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)

Figure 6 The time series of119883(119905) 119884(119905) 119885(119905)119882(119905)

to ensure the steady development of the water resources sup-ply-demand system

Case 1 The result of parameter identification is used as thesystem parameters (Table 2) and the initial condition is setas [07 03 01 05] and the water resources supply-demandsystem is stable as shown in Figure 6 Under the currentinitial situation and reasonable water resources demand therelationship between water resourcesrsquo supply and demand ofthe HRB will be in a stable state From Figure 6 we can seethat the trend of the water resources demand will increasefirstly and then decrease the surface water supply will be in agrowth trend until steady state the groundwater supply willincrease firstly and then decrease until a steady state otherkinds of water supply will be in a growth state Accordingto the above analysis of the trend of the three kinds of watersupply sources we can obtain that in order to maintain thebalance between water resources supply and demand weneed to save water and improve water use efficiency givepriority to use of south-to-north water and improve the useof recycled water

0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)

Figure 7 The time series of119883(119905) 119884(119905) 119885(119905)119882(119905) when119872 gt 15

Case 2 Maintain other parameters constant and initial con-ditions are set as [07 03 01 05] when the maximum value119872 gt 15 the time series119883(119905) 119884(119905)119885(119905) and119882(119905) are shownin Figure 7 From Figure 7 we can obtain that the valuesof 119884(119905) 119885(119905) and 119882(119905) are too large and beyond the scopeof practical significance It indicates that if the maximumdemand of water resources is set to be very large that is to sayif the water resources demand is not controlled the systemwill be in an unstable stateTherefore wemust save water andcontrol water demand

Case 3 Set 1198892= 14 and maintain other parameters constant

and the initial conditions are set as [09 05 07 03] thesimulation result is shown in Figure 8 The figure shows thatthe trend of the other kinds of water resources supply willdecrease until zero the groundwater supply will increaseuntil larger than 1 and this will lead to overexploitation ofthe groundwater supply Therefore with the cost of recycledwater supply growing the other kinds of water supply willdecrease That is to say other types of water supply aredependent on the cost of the recycled water supply


0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)

Figure 8 The time series of119883(119905) 119884(119905) 119885(119905)119882(119905) when 1198892= 14

Case 4 Maintain other parameters constant and vary valvevalue 119873 the water resources supply-demand system stillmaintains a stable state It indicates that the valve value119873 hasno influence on the system However if the valve value 119873 isset to be very large it will cause water resources shortage inother regions and further affect the economic development ofthe other regionsTherefore an appropriate valve value needsto be selected

5 Conclusions

In this paper a novel model is proposed to analyse therelationship between water resources supply and demandThe main conclusions are as follows

(1) Using dynamical system theory we establish a newfour-dimensional water resources supply-demand systemwith unknown parameters This model can be used to an-alyse the complex nonlinear characteristics among the waterresources demand surface water supply groundwater supplyand other kinds of water supply of a river basin or a city

(2) Selecting the Haihe River Basin as a study areabased on its statistic data (1998ndash2009) we identify theunknown parameters of the systemwith the gray code hybridaccelerating genetic algorithm (GHAGA) Therefore a realwater resources supply-demand system of the Haihe RiverBasin is determined

(3) Numerical simulation results provide the trend of thewater resources demand and supply The results show thatthe obtained water resources supply-demand system has thecapacity of self-regulatory and demonstrates a steady stateSeveral cases are analysed and contribute to the formulation

of policy and the management of water resources Finally thecorresponding measures are proposed to ensure the steadydevelopment of the water resources supply-demand system

Our research provides a new method for water resourcessystem analysis based on dynamical system theory Thismethod can also be used to discuss the balance betweenwaterresources supply and demand in other regions or cities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (no 2010CB951104) the the Funds forCreative Research Groups of China (no 51121003) andthe Project of National Natural Foundation of China (no51379013)

References

[1] C J Vorosmarty P Green J Salisbury and R B LammersldquoGlobal water resources vulnerability from climate change andpopulation growthrdquo Science vol 289 no 5477 pp 284ndash2882000

[2] D G Zeitoun ldquoA compartmental model for stable time-dependent modeling of surface water and groundwaterrdquo Envi-ronmental Modeling and Assessment vol 17 no 6 pp 673ndash6972012

[3] L Z Wang Cooperative Water Resources Allocation amongCompetingUsers [PhD Disserttion] theUniversity ofWaterlooWestern Ontario Canada 2005

[4] C Qi and N-B Chang ldquoSystem dynamics modeling formunicipal water demand estimation in an urban region underuncertain economic impactsrdquo Journal of Environmental Man-agement vol 92 no 6 pp 1628ndash1641 2011

[5] P M Senge The Fifth Discipline The Art and Practice of theLearning Organization Dell Publishing Group New York 1994

[6] M A Rayan B Djebedjian and I Khaled ldquoWater supply anddemand and a desalination option for Sinai Egyptrdquo Desalina-tion vol 136 no 1ndash3 pp 73ndash81 2001

[7] W Wang P H A J M V Gelder J K Vrijling and JMa ldquoForecasting daily streamflow using hybrid ANN modelsrdquoJournal of Hydrology vol 324 no 1ndash4 pp 383ndash399 2006

[8] H R Maier A Jain G C Dandy and K P Sudheer ldquoMethodsused for the development of neural networks for the predictionof water resource variables in river systems current status andfuture directionsrdquo Environmental Modelling and Software vol25 no 8 pp 891ndash909 2010

[9] W Huang B Xu and A Chan-Hilton ldquoForecasting flowsin Apalachicola River using neural networksrdquo HydrologicalProcesses vol 18 no 13 pp 2545ndash2564 2004

[10] B A George H M Malano A R Khan A Gaur and BDavidson ldquoUrban water supply strategies for Hyderabad Indiafuture scenariosrdquo Environmental Modeling and Assessment vol14 no 6 pp 691ndash704 2009

[11] X-B Ji E-S Kang R-S Chen W-Z Zhao S-C Xiao andB-W Jin ldquoAnalysis of water resources supply and demand and


security of water resources development in irrigation regions ofthe middle reaches of the Heihe River Basin Northwest ChinardquoAgricultural Sciences in China vol 5 no 2 pp 130ndash140 2006

[12] I Almutaz A Ajbar Y Khalid and E Ali ldquoA probabilistic fore-cast of water demand for a tourist and desalination dependentcity case ofMecca Saudi ArabiardquoDesalination vol 294 pp 53ndash59 2012

[13] J G Liu H H G Savenije and J X Xu ldquoForecast of waterdemand in Weinan City in China using WDF-ANN modelrdquoPhysics and Chemistry of the Earth vol 28 no 4-5 pp 219ndash2242003

[14] M A Yurdusev M Firat M Mermer and M E Turan ldquoWateruse prediction by radial and feed-forward neural netsrdquo Proceed-ings of the Institution of Civil EngineersWaterManagement vol162 no 3 pp 179ndash188 2009

[15] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[16] C L Wu KW Chau and Y S Li ldquoPredicting monthly stream-flow using data-drivenmodels coupledwith data-preprocessingtechniquesrdquoWater Resources Research vol 45 no 8 Article IDW08432 2009

[17] X H Yang D X She Z F Yang Q H Tang and J Q LildquoChaotic bayesian method based on multiple criteria decisionmaking (MCDM) for forecasting nonlinear hydrological timeseriesrdquo International Journal of Nonlinear Sciences and Numeri-cal Simulation vol 10 no 11-12 pp 1595ndash1610 2009

[18] L P Zhang J Xia X Y Song and X Cheng ldquoSimilarity modelof chaos phase space and its application in mid- and long-termhydrologic predictionrdquoKybernetes vol 38 no 10 pp 1835ndash18422009

[19] S Paul M K Verma P Wahi S K Reddy and K KumarldquoBifurcation analysis of the flow patterns in two-dimensionalrayleighbenard convectionrdquo International Journal of Bifurcationand Chaos vol 22 no 5 Article ID 1230018 2012

[20] B Sivakumar ldquoChaos theory in geophysics past present andfuturerdquo Chaos Solitons and Fractals vol 19 no 2 pp 441ndash4622004

[21] X Cui G Huang W Chen and A Morse ldquoThreatening ofclimate change on water resources and supply case study ofNorth ChinardquoDesalination vol 248 no 1ndash3 pp 476ndash478 2009

[22] J Xia H-L Feng C-S Zhan and G-W Niu ldquoDeterminationof a reasonable percentage for ecological water-use in the HaiheRiver Basin Chinardquo Pedosphere vol 16 no 1 pp 33ndash42 2006

[23] A Zilinskas and J Zilinskas ldquoInterval arithmetic based opti-mization in nonlinear regressionrdquo Informatica vol 21 no 1 pp149ndash158 2010

[24] W R Esposito and C A Floudas ldquoGlobal optimization forthe parameter estimation of differential-algebraic systemsrdquoIndustrial and Engineering Chemistry Research vol 39 no 5 pp1291ndash1310 2000

[25] K Wang and N Wang ldquoA novel RNA genetic algorithm forparameter estimation of dynamic systemsrdquo Chemical Engineer-ing Research and Design vol 88 no 11 pp 1485ndash1493 2010

[26] L Hamm B W Brorsen and M T Hagan ldquoComparisonof stochastic global optimization methods to estimate neuralnetwork weightsrdquo Neural Processing Letters vol 26 no 3 pp145ndash158 2007

[27] Q J Wang ldquoUsing genetic algorithms to optimize modelparametersrdquo Environmental Modeling and Software vol 12 no1 pp 27ndash34 1997

[28] X H Yang Z F Yang G-H Lu and J Q Li ldquoA gray-encodedhybrid-accelerated genetic algorithm for global optimizationsin dynamical systemsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 10 no 4 pp 355ndash363 2005

[29] X H Yang Z F Yang and Z Y Shen ldquoGHHAGA forenvironmental systems optimizationrdquo Journal of EnvironmentalInformatics vol 5 no 1 pp 36ndash41 2005

[30] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquoComputer Journal vol 7 no 4 pp 308ndash313 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


water resources development A probabilistic forecast modelof future water demand for the city of Mecca in Saudi Arabiawas proposed in [12] Liu et al [13] adopted WDF-ANNmodel to forecast water demand in Weinan city Yurdusevet al [14] predicted monthly water consumption by usingfeed-forward and radial-basis neural networks However thenonlinear dynamic behavior of the water resources supply-demand system is seldom considered by these methods

Any very complicated dynamical system can be solvedby a simple mathematical model [15] The dynamical systemmethod is one approach to nonlinear modeling It hasbeen applied increasingly in water environments and waterresources system such as hydrological time series prediction[16ndash18] and the dynamical analysis of runoff flows [19 20]In this paper according to the current situation of the waterresources supply and demand we establish a new dynamicalmodel to express the complex relationship between waterresources supply and demand by using dynamical systemtheory This model can be used to analyse the internaldynamic evolved mechanism of the water resources supply-demand system of a river basin or a city

The outline of this paper is organized as follows Section 2introduces the study area and data In Section 3 the waterresources supply demand model is established and themethod of parameter identification of the proposed modelis introduced Section 4 provides the results and discussionson the real water resources supply-demand system in HaiheRiver Basin The conclusions of the paper are summarized inSection 5

2 Study Area and Data

The HRB is the political economic and cultural center ofChina However water resources shortage in the HRB hasbecome a serious problem in recent years due to the rapideconomic development and climate change [21]The locationof the HRB is shown in Figure 1 The basin covers an areaof 318200 km2 with 60 being mountains in the westernand northern part and 40 being plains in the eastern andsouthern parts It belongs to continental monsoon climatezone and locates in a semihumid and semiarid region Theaverage annual precipitation is 52469mm and the totalamount of water resources is 37439times10

8m3 (1956ndash2008) thewater volume per capita is 272m3 merely 17 of the nationalaverage and 124 of the world average [22]

Thewater resources demand and supply data from 1998 to2009 are obtained from the water resources bulletin of HRBTable 1 is the water resources statistic data (1998ndash2009) ofHRB where 119883 is water demand 119884 is surface water supply 119885is groundwater supply and119882 is other kinds of water supply

3 Modeling and Methods

31 Establishment of the Water Resources Supply-DemandModel Let 119883(119905) be the demand for water resources 119884(119905)

the surface water supply (including diversion water supply)119885(119905) the groundwater supply and 119882(119905) other kinds of watersupply (mainly including wastewater reuse direct seawater

China

RiverHaihe River Basin

Haihe River Basin

0 80 160 320

(km)

42∘N

40∘N

38∘N

36∘N

112∘E 114

∘E 116∘E 118

∘E 120∘E

N

Figure 1 The location of the Haihe River Basin

utilization and collection of rainwater) The model for thewater resource system is established as follows

119889119883

119889119905= 1198861119883(1 minus

119883

119872) minus 1198862119884 minus 1198863119885 minus 1198864119882

119889119884

119889119905= minus 119887

1119884 minus 1198872119885 + 1198873119883 [119873 minus (119883 minus 119885 minus 119882)]

119889119885

119889119905= 1198881119883 minus 1198882119884 minus 1198883119885 minus 1198884119882

119889119882

119889119905= 1198891119882(119883 minus 119889

2) minus 1198893119884

(1)

where 119886119894 119887119894 119888119894 119889119894119872119873119870 are constants and119872 gt 0119873 gt 0

Additionally 1198861is the elastic coefficient of the water

resources demand 1198862 1198863 and 119886

4are respectively the influ-

ence coefficients of the surface water supply the groundwatersupply and other kinds of water supply on thewater resourcesdemand 119887

1is the influence coefficient of the surface water

supply on the change rate of the surface water supply 1198872is the

influence coefficient of the groundwater supply on the surfacewater supply 119887

3is the influence coefficient of water resources

demand on the surface water supply 1198881 1198882 1198883 and 119888

4are

respectively the influence coefficients of the water resourcesdemand the surface water supply the groundwater supplyand other kinds of water supply on the groundwater supply1198891is the velocity constant of the recycled water supply

1198892is the cost of recycled water supply 119889

3is the influence

coefficient of the surface water supply on the other kinds ofwater supply 119872 is the maximum value of water resourcesdemand119873 is the valve value of diversion water supply

In system (1) 1198861119883(1 minus (119883119872)) denotes that the rate of

water resources demand 119889119883119889119905 is in direct ratio to waterresources demand when the current water resources demand



Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009119883 42400 43150 40060 39200 39983 37705 36801 38046 39285 38452 37339 37002119884 16070 16268 13660 12270 12794 11364 11823 12335 13415 12864 12310 12550119885 26190 26750 26279 26774 27029 26138 24697 25294 25185 25001 24060 23588119882 140 129 121 156 160 203 283 417 685 583 969 864


2119884

minus1198863119885 and minus119886










2119884 minus 119888

3119885 minus 119888



1119882(119883 minus 119889




1=

006 1198862

= 02 1198863

= 009 1198864

= 015 1198871

= 022 1198872

= 021198873

= 02 1198881

= 013 1198882

= 014 1198883

= 012 1198884

= 021198891= 016 119889

2= 02 119889

3= 0001 119872 = 11 and 119873 = 06 and


1= minus00156 119871

2= minus00371

1198713

= minus00853 and 1198714


1= 006 119886

2= 008

1198863= 03 119886

4= 02 119887

1= 005 119887

2= 004 119887

3= 004 119888

1= 01

1198882= 02 119888

3= 005 119888

4= 004 119889

1= 02 119889

2= 03 119889

3= 0001


1= minus00020 119871

2= minus00134 119871

3= minus00169 and



2= 013 119886

3= 006

1198864

= 008 1198871

= 015 1198872

= 015 1198873

= 014 1198881

= 0131198882

= 015 1198883

= 019 1198884

= 018 1198891

= 016 1198892



1= 00421 119871

2= minus00104 119871

3= minus01144 and






minus4minus2

02

minus6minus4

minus20

2minus2

minus1

0

1

2

3

XY

Z

(a)

minus4minus2

02

minus6minus4

minus20

20

005

01

015

02

XY

W

(b)

0 200 400 600 800 1000minus5

05

0 200 400 600 800 1000minus10

010

0 200 400 600 800 1000minus5

05

0 200 400 600 800 10000

01

02

X

Y

Z

t

t

t

t

W

(c)



119883 (119896 + 1)

= 119883 (119896) + 119905 [1198861119883 (119896) (1 minus

119883 (119896)

119872) minus 1198862119884 (119896)

minus1198863119885 (119896) minus 119886

4119882(119896) ]

119884 (119896 + 1)

= 119884 (119896) + 119905 [ minus 1198871119884 (119896) minus 119887

2119885 (119896)


119885 (119896 + 1)

= 119885 (119896) + 119905 [1198881119883 (119896) minus 119888

2119884 (119896) minus 119888

3119885 (119896) minus 119888

4119882(119896)]

119882 (119896 + 1) = 119882 (119896) + 119905 [1198891119882(119896) (119883 (119896) minus 119889

2) minus 1198893119884 (119896)]

(2)



min 119891 (1199091 1199092 119909

119899)

st 119886119894le 119909119894le 119887119894 for 119894 = 1 2 119899

(3)

where 119909 = 119909119894 119894 = 1 2 119899 119909






minus020

0204

0608

minus010

0102

03minus02

minus01

0

01

02

XY

Z

(a)

minus020

0204

0608

minus02minus01

001

02minus005

0

005

01

015

XY

Z

(b)

0 200 400 600 800 1000minus1

01

0 200 400 600 800 1000minus02

002

0 200 400 600 800 1000minus02

002

0 200 400 600 800 10000

01

02

X

Y

t

t

t

t

Z

W

(c)




tervals

119909119894= 119886119894+ 119868119894sdot 119888119894 (4)


=


119894is a



119868119894=

119890

sum

119895=1

(

119890

⨁

119896=119895

119889 (119894 119896)) sdot 2119895minus1

(5)




119868119894(119895) = int (119906 (119894 119895) sdot 2

119890)

119894 = 1 2 119899 119895 = 1 2 119901

(6)





minus20minus10

010

minus100minus50

050

minus10

0

10

20

30

X

Y

Z

(a)

minus20minus10

010

minus100minus50

0500

1

2

3

XY

W

(b)

0 500 1000 1500 2000 2500 3000minus20

020

t

t

t

t

0 500 1000 1500 2000 2500 3000minus100

0100

0 500 1000 1500 2000 2500 3000minus50

050

0 500 1000 1500 2000 2500 30000

2

4

X

Y

Z

W

(c)





119865 (119895) =1

[119891 (119895)]2

+ 01

(7)


119904(119895)

119875119904(119895) = 119865(119895)sum

119899






int(1198801sdot (119890 + 1)) and 119868

2= int(119880





1198891015840

1(119894 119896 119895) =

1198892(119894 119896 119895) 119896 isin [119868

1 1198682]

1198891(119894 119896 119895) 119896 notin [119868

1 1198682]

(8)

1198891015840

2(119894 119896 119895) =

1198891(119894 119896 119895) 119896 isin [119868

1 1198682]

1198892(119894 119896 119895) 119896 notin [119868

1 1198682]

(9)


119888ge 05


1 1198812 1198813 1198814 isin [0 1]

If 1198811






No


Gray encoding


Decoding


Simplex searching




Yes

Stop


be 1198691= int(119881

2sdot (119890 + 1)) 119869

2= int(119881



1198893(119894 119896 119895)

(119894119896119895)

=

0 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

0 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

1198891015840(119894 119896 119895)

(119894119896119895)otherwise

(10)








119904= 10 the



119898= 05 and



1(09093 07669

08118 03969) 1198782(08856 04171 36269 minus09523) and 119878

3


= 06921 gt 0 1205822

= 00320 gt 0 12058223

= minus01155 gt 0and 120582




1= minus00022 lt 0

1205822= minus05721 lt 0 and 120582





1(09093 07669 08118 03969)


2and 1198783




P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)




0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)






0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009119883 42400 43150 40060 39200 39983 37705 36801 38046 39285 38452 37339 37002119884 16070 16268 13660 12270 12794 11364 11823 12335 13415 12864 12310 12550119885 26190 26750 26279 26774 27029 26138 24697 25294 25185 25001 24060 23588119882 140 129 121 156 160 203 283 417 685 583 969 864


2119884

minus1198863119885 and minus119886










2119884 minus 119888

3119885 minus 119888



1119882(119883 minus 119889




1=

006 1198862

= 02 1198863

= 009 1198864

= 015 1198871

= 022 1198872

= 021198873

= 02 1198881

= 013 1198882

= 014 1198883

= 012 1198884

= 021198891= 016 119889

2= 02 119889

3= 0001 119872 = 11 and 119873 = 06 and


1= minus00156 119871

2= minus00371

1198713

= minus00853 and 1198714


1= 006 119886

2= 008

1198863= 03 119886

4= 02 119887

1= 005 119887

2= 004 119887

3= 004 119888

1= 01

1198882= 02 119888

3= 005 119888

4= 004 119889

1= 02 119889

2= 03 119889

3= 0001


1= minus00020 119871

2= minus00134 119871

3= minus00169 and



2= 013 119886

3= 006

1198864

= 008 1198871

= 015 1198872

= 015 1198873

= 014 1198881

= 0131198882

= 015 1198883

= 019 1198884

= 018 1198891

= 016 1198892



1= 00421 119871

2= minus00104 119871

3= minus01144 and






minus4minus2

02

minus6minus4

minus20

2minus2

minus1

0

1

2

3

XY

Z

(a)

minus4minus2

02

minus6minus4

minus20

20

005

01

015

02

XY

W

(b)

0 200 400 600 800 1000minus5

05

0 200 400 600 800 1000minus10

010

0 200 400 600 800 1000minus5

05

0 200 400 600 800 10000

01

02

X

Y

Z

t

t

t

t

W

(c)



119883 (119896 + 1)

= 119883 (119896) + 119905 [1198861119883 (119896) (1 minus

119883 (119896)

119872) minus 1198862119884 (119896)

minus1198863119885 (119896) minus 119886

4119882(119896) ]

119884 (119896 + 1)

= 119884 (119896) + 119905 [ minus 1198871119884 (119896) minus 119887

2119885 (119896)


119885 (119896 + 1)

= 119885 (119896) + 119905 [1198881119883 (119896) minus 119888

2119884 (119896) minus 119888

3119885 (119896) minus 119888

4119882(119896)]

119882 (119896 + 1) = 119882 (119896) + 119905 [1198891119882(119896) (119883 (119896) minus 119889

2) minus 1198893119884 (119896)]

(2)



min 119891 (1199091 1199092 119909

119899)

st 119886119894le 119909119894le 119887119894 for 119894 = 1 2 119899

(3)

where 119909 = 119909119894 119894 = 1 2 119899 119909






minus020

0204

0608

minus010

0102

03minus02

minus01

0

01

02

XY

Z

(a)

minus020

0204

0608

minus02minus01

001

02minus005

0

005

01

015

XY

Z

(b)

0 200 400 600 800 1000minus1

01

0 200 400 600 800 1000minus02

002

0 200 400 600 800 1000minus02

002

0 200 400 600 800 10000

01

02

X

Y

t

t

t

t

Z

W

(c)




tervals

119909119894= 119886119894+ 119868119894sdot 119888119894 (4)


=


119894is a



119868119894=

119890

sum

119895=1

(

119890

⨁

119896=119895

119889 (119894 119896)) sdot 2119895minus1

(5)




119868119894(119895) = int (119906 (119894 119895) sdot 2

119890)

119894 = 1 2 119899 119895 = 1 2 119901

(6)





minus20minus10

010

minus100minus50

050

minus10

0

10

20

30

X

Y

Z

(a)

minus20minus10

010

minus100minus50

0500

1

2

3

XY

W

(b)

0 500 1000 1500 2000 2500 3000minus20

020

t

t

t

t

0 500 1000 1500 2000 2500 3000minus100

0100

0 500 1000 1500 2000 2500 3000minus50

050

0 500 1000 1500 2000 2500 30000

2

4

X

Y

Z

W

(c)





119865 (119895) =1

[119891 (119895)]2

+ 01

(7)


119904(119895)

119875119904(119895) = 119865(119895)sum

119899






int(1198801sdot (119890 + 1)) and 119868

2= int(119880





1198891015840

1(119894 119896 119895) =

1198892(119894 119896 119895) 119896 isin [119868

1 1198682]

1198891(119894 119896 119895) 119896 notin [119868

1 1198682]

(8)

1198891015840

2(119894 119896 119895) =

1198891(119894 119896 119895) 119896 isin [119868

1 1198682]

1198892(119894 119896 119895) 119896 notin [119868

1 1198682]

(9)


119888ge 05


1 1198812 1198813 1198814 isin [0 1]

If 1198811






No


Gray encoding


Decoding


Simplex searching




Yes

Stop


be 1198691= int(119881

2sdot (119890 + 1)) 119869

2= int(119881



1198893(119894 119896 119895)

(119894119896119895)

=

0 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

0 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

1198891015840(119894 119896 119895)

(119894119896119895)otherwise

(10)








119904= 10 the



119898= 05 and



1(09093 07669

08118 03969) 1198782(08856 04171 36269 minus09523) and 119878

3


= 06921 gt 0 1205822

= 00320 gt 0 12058223

= minus01155 gt 0and 120582




1= minus00022 lt 0

1205822= minus05721 lt 0 and 120582





1(09093 07669 08118 03969)


2and 1198783




P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)




0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)






0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus4minus2

02

minus6minus4

minus20

2minus2

minus1

0

1

2

3

XY

Z

(a)

minus4minus2

02

minus6minus4

minus20

20

005

01

015

02

XY

W

(b)

0 200 400 600 800 1000minus5

05

0 200 400 600 800 1000minus10

010

0 200 400 600 800 1000minus5

05

0 200 400 600 800 10000

01

02

X

Y

Z

t

t

t

t

W

(c)



119883 (119896 + 1)

= 119883 (119896) + 119905 [1198861119883 (119896) (1 minus

119883 (119896)

119872) minus 1198862119884 (119896)

minus1198863119885 (119896) minus 119886

4119882(119896) ]

119884 (119896 + 1)

= 119884 (119896) + 119905 [ minus 1198871119884 (119896) minus 119887

2119885 (119896)


119885 (119896 + 1)

= 119885 (119896) + 119905 [1198881119883 (119896) minus 119888

2119884 (119896) minus 119888

3119885 (119896) minus 119888

4119882(119896)]

119882 (119896 + 1) = 119882 (119896) + 119905 [1198891119882(119896) (119883 (119896) minus 119889

2) minus 1198893119884 (119896)]

(2)



min 119891 (1199091 1199092 119909

119899)

st 119886119894le 119909119894le 119887119894 for 119894 = 1 2 119899

(3)

where 119909 = 119909119894 119894 = 1 2 119899 119909






minus020

0204

0608

minus010

0102

03minus02

minus01

0

01

02

XY

Z

(a)

minus020

0204

0608

minus02minus01

001

02minus005

0

005

01

015

XY

Z

(b)

0 200 400 600 800 1000minus1

01

0 200 400 600 800 1000minus02

002

0 200 400 600 800 1000minus02

002

0 200 400 600 800 10000

01

02

X

Y

t

t

t

t

Z

W

(c)




tervals

119909119894= 119886119894+ 119868119894sdot 119888119894 (4)


=


119894is a



119868119894=

119890

sum

119895=1

(

119890

⨁

119896=119895

119889 (119894 119896)) sdot 2119895minus1

(5)




119868119894(119895) = int (119906 (119894 119895) sdot 2

119890)

119894 = 1 2 119899 119895 = 1 2 119901

(6)





minus20minus10

010

minus100minus50

050

minus10

0

10

20

30

X

Y

Z

(a)

minus20minus10

010

minus100minus50

0500

1

2

3

XY

W

(b)

0 500 1000 1500 2000 2500 3000minus20

020

t

t

t

t

0 500 1000 1500 2000 2500 3000minus100

0100

0 500 1000 1500 2000 2500 3000minus50

050

0 500 1000 1500 2000 2500 30000

2

4

X

Y

Z

W

(c)





119865 (119895) =1

[119891 (119895)]2

+ 01

(7)


119904(119895)

119875119904(119895) = 119865(119895)sum

119899






int(1198801sdot (119890 + 1)) and 119868

2= int(119880





1198891015840

1(119894 119896 119895) =

1198892(119894 119896 119895) 119896 isin [119868

1 1198682]

1198891(119894 119896 119895) 119896 notin [119868

1 1198682]

(8)

1198891015840

2(119894 119896 119895) =

1198891(119894 119896 119895) 119896 isin [119868

1 1198682]

1198892(119894 119896 119895) 119896 notin [119868

1 1198682]

(9)


119888ge 05


1 1198812 1198813 1198814 isin [0 1]

If 1198811






No


Gray encoding


Decoding


Simplex searching




Yes

Stop


be 1198691= int(119881

2sdot (119890 + 1)) 119869

2= int(119881



1198893(119894 119896 119895)

(119894119896119895)

=

0 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

0 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

1198891015840(119894 119896 119895)

(119894119896119895)otherwise

(10)








119904= 10 the



119898= 05 and



1(09093 07669

08118 03969) 1198782(08856 04171 36269 minus09523) and 119878

3


= 06921 gt 0 1205822

= 00320 gt 0 12058223

= minus01155 gt 0and 120582




1= minus00022 lt 0

1205822= minus05721 lt 0 and 120582





1(09093 07669 08118 03969)


2and 1198783




P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)




0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)






0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus020

0204

0608

minus010

0102

03minus02

minus01

0

01

02

XY

Z

(a)

minus020

0204

0608

minus02minus01

001

02minus005

0

005

01

015

XY

Z

(b)

0 200 400 600 800 1000minus1

01

0 200 400 600 800 1000minus02

002

0 200 400 600 800 1000minus02

002

0 200 400 600 800 10000

01

02

X

Y

t

t

t

t

Z

W

(c)




tervals

119909119894= 119886119894+ 119868119894sdot 119888119894 (4)


=


119894is a



119868119894=

119890

sum

119895=1

(

119890

⨁

119896=119895

119889 (119894 119896)) sdot 2119895minus1

(5)




119868119894(119895) = int (119906 (119894 119895) sdot 2

119890)

119894 = 1 2 119899 119895 = 1 2 119901

(6)





minus20minus10

010

minus100minus50

050

minus10

0

10

20

30

X

Y

Z

(a)

minus20minus10

010

minus100minus50

0500

1

2

3

XY

W

(b)

0 500 1000 1500 2000 2500 3000minus20

020

t

t

t

t

0 500 1000 1500 2000 2500 3000minus100

0100

0 500 1000 1500 2000 2500 3000minus50

050

0 500 1000 1500 2000 2500 30000

2

4

X

Y

Z

W

(c)





119865 (119895) =1

[119891 (119895)]2

+ 01

(7)


119904(119895)

119875119904(119895) = 119865(119895)sum

119899






int(1198801sdot (119890 + 1)) and 119868

2= int(119880





1198891015840

1(119894 119896 119895) =

1198892(119894 119896 119895) 119896 isin [119868

1 1198682]

1198891(119894 119896 119895) 119896 notin [119868

1 1198682]

(8)

1198891015840

2(119894 119896 119895) =

1198891(119894 119896 119895) 119896 isin [119868

1 1198682]

1198892(119894 119896 119895) 119896 notin [119868

1 1198682]

(9)


119888ge 05


1 1198812 1198813 1198814 isin [0 1]

If 1198811






No


Gray encoding


Decoding


Simplex searching




Yes

Stop


be 1198691= int(119881

2sdot (119890 + 1)) 119869

2= int(119881



1198893(119894 119896 119895)

(119894119896119895)

=

0 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

0 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

1198891015840(119894 119896 119895)

(119894119896119895)otherwise

(10)








119904= 10 the



119898= 05 and



1(09093 07669

08118 03969) 1198782(08856 04171 36269 minus09523) and 119878

3


= 06921 gt 0 1205822

= 00320 gt 0 12058223

= minus01155 gt 0and 120582




1= minus00022 lt 0

1205822= minus05721 lt 0 and 120582





1(09093 07669 08118 03969)


2and 1198783




P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)




0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)






0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus20minus10

010

minus100minus50

050

minus10

0

10

20

30

X

Y

Z

(a)

minus20minus10

010

minus100minus50

0500

1

2

3

XY

W

(b)

0 500 1000 1500 2000 2500 3000minus20

020

t

t

t

t

0 500 1000 1500 2000 2500 3000minus100

0100

0 500 1000 1500 2000 2500 3000minus50

050

0 500 1000 1500 2000 2500 30000

2

4

X

Y

Z

W

(c)





119865 (119895) =1

[119891 (119895)]2

+ 01

(7)


119904(119895)

119875119904(119895) = 119865(119895)sum

119899






int(1198801sdot (119890 + 1)) and 119868

2= int(119880





1198891015840

1(119894 119896 119895) =

1198892(119894 119896 119895) 119896 isin [119868

1 1198682]

1198891(119894 119896 119895) 119896 notin [119868

1 1198682]

(8)

1198891015840

2(119894 119896 119895) =

1198891(119894 119896 119895) 119896 isin [119868

1 1198682]

1198892(119894 119896 119895) 119896 notin [119868

1 1198682]

(9)


119888ge 05


1 1198812 1198813 1198814 isin [0 1]

If 1198811






No


Gray encoding


Decoding


Simplex searching




Yes

Stop


be 1198691= int(119881

2sdot (119890 + 1)) 119869

2= int(119881



1198893(119894 119896 119895)

(119894119896119895)

=

0 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

0 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

1198891015840(119894 119896 119895)

(119894119896119895)otherwise

(10)








119904= 10 the



119898= 05 and



1(09093 07669

08118 03969) 1198782(08856 04171 36269 minus09523) and 119878

3


= 06921 gt 0 1205822

= 00320 gt 0 12058223

= minus01155 gt 0and 120582




1= minus00022 lt 0

1205822= minus05721 lt 0 and 120582





1(09093 07669 08118 03969)


2and 1198783




P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)




0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)






0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










No


Gray encoding


Decoding


Simplex searching




Yes

Stop


be 1198691= int(119881

2sdot (119890 + 1)) 119869

2= int(119881



1198893(119894 119896 119895)

(119894119896119895)

=

0 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198691 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

0 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=1

1 119896=1198692 1198814le119901119898 1198891015840(119894 119896 119895)

(119894119896119895)=0

1198891015840(119894 119896 119895)

(119894119896119895)otherwise

(10)








119904= 10 the



119898= 05 and



1(09093 07669

08118 03969) 1198782(08856 04171 36269 minus09523) and 119878

3


= 06921 gt 0 1205822

= 00320 gt 0 12058223

= minus01155 gt 0and 120582




1= minus00022 lt 0

1205822= minus05721 lt 0 and 120582





1(09093 07669 08118 03969)


2and 1198783




P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)




0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)






0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











P 1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198881

1198882

1198883

1198884

1198891

1198892

1198893

119872 119873

II [001100]

[001020]

[002020]

[005020]

[001015]

[004015]

[005020]

[010040]

[005030]

[001010]

[001020]

[001010]

[001010]

[001010]

[010200]

[010200]

IP 070 001 003 005 003 005 010 019 011 006 010 010 089 001 099 040

0 10 20 30 40 50 60 70 80 90 1000709

t

X

(a)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Y

(b)

t

0 10 20 30 40 50 60 70 80 90 1000

051

Z

(c)

t

0 10 20 30 40 50 60 70 80 90 100045

05055

W

(d)




0 50 100 150 200 250 300 350 40005

115

t

X

(a)

0 50 100 150 200 250 300 350 4000

5001000

t

Y

(b)

0 50 100 150 200 250 300 350 400minus5000

05000

t

Z

(c)

0 50 100 150 200 250 300 350 4000

20004000

t

W

(d)






0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 10009

095

t

X

(a)

t

0 20 40 60 80 100040608

Y

(b)

t

0 20 40 60 80 100012

Z

(c)

t

0 20 40 60 80 100minus05

005

W

(d)



5 Conclusions









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article modeling and dynamical analysis of the...

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