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Mathematical Methods
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·
Chuanwen Luo Chuncheng Wang
Mathematical Theory
of Uniformity and its
Applications in Ecology
and Chaos
ChuanwenLuo ChunchengWang
CenterofUniformityTheory DepartmentofMathematics
andApplication HarbinInstituteofTechnology
NortheastForestryUniversity Harbin,Heilongjiang,China
Harbin,Heilongjiang,China
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefsinAppliedSciencesandTechnology
ISSN 2365-0826 ISSN 2365-0834 (electronic)
SpringerBriefsinMathematicalMethods
ISBN 978-981-19-5511-2 ISBN 978-981-19-5512-9 (eBook)
https://doi.org/10.1007/978-981-19-5512-9
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Preface
Thedistributionofafinitenumberofpointsinapolyhedronisreferredtoasapattern
inthisbook.Thedegreeofuniformityofplantsdistributedinapolygonalareahas
been studied for more than a century. The mathematical problem behind this is to
measuretherandomnessofasetofdiscretepoints.Theuniformdegree,introduced
inthisbook,summarizesthecorrelatedresultsintheliterature,unifiestheirmethods,
andsimplifiestheprocessofsampling,computationandtesting.Moreimportantly,
the extent of randomness and chaos is also put together within the framework of
uniformdegree.
The randomness called the external stochasticity and chaos called the internal
stochasticityhavelongbeenseparated,andconsideredtobelongtodifferentsubjects.
The reason behind this is the lack of theory and method, that can unify these two
concepts.Thestandpointinthisbook,basedonthemeasurementofuniformdegree
forthechaotictimeseries,isinclinedtoadmitthatthereisnosubstantialdifference
betweenrandomnessandchaos.Auniformdegreeisanindexthatcanmeasureboth
stochasticandchaotictimeseries.Thetheoryofuniformityisestablishedbasedon
theconceptofuniformdegree.
The uniform degree of the pattern generated by a distribution function F in a
polyhedronisabbreviatedastheuniformdegreeofadistributionfunction,denoted
byL ,andtheuniformdegreeofauniformdistributionfunctionisdenotedbyL .
F U
TheuniformdegreeisanalogoustotheconceptofShannon’sentropy.Itisconjectured
that the uniform degree of a uniform distribution function is greater than those of
all the other distribution functions, that is, “a uniform distribution function is the
mostuniform”,or L ≤ L .Thisconjecturewillexplainwhytheuniformdegree
F U
of a chaotic orbit must lie in [0, L ], since the uniform degree L is the critical
U U
situationthatachaoticorbitcanneverattain.Manynumericalillustrationsshowthat
theuniformdegreeexhibitsbetterresultsthantheotherindexesformeasuringchaos.
BasedontheMonteCarlomethod,anypatternobeyingadistributionfunction F
inapolyhedroncanbeviewedasthetransformationF−1ofthepatterngeneratedby
auniformdistributionfunction.RecallthatthenonlinearityofFisinlinewiththatof
F−1.NumericalillustrationsshowthatthenonlinearityofFisinverselyproportional
toL ,thatis,thenonlineartransformationofapatternwillreduceitsuniformdegree,
F
v
vi Preface
andtheuniformdegreeofapatternwillnotchangeunderatransformationonlyif
F islinear.Recallthatonlytheuniformdistributionfunctionislinear.Toacertain
extent,thisimpliestheconjectureistrue.Thisalsoseemstobeconnectedwiththe
processofformationofadissipativestructure.
Inaword,itseemsthatorderedandunordered,periodicandchaotic,deterministic
and stochastic, aggregate and uniform patterns can be unified in the framework of
uniformdegree.
As illustrations, k-step chaometry, which is equivalent to a uniform degree, is
usedtoexaminetheinternalfaultsintransformers,basedonthetimeseriescaused
byvibration.Otherexamplesinvolvethetestofelectrocardiogramsandelectroen-
cephalograms, and the theory of uniformity may also have wide applications in
practice.
Turbulenceisanintricatepatterninaboundedarea(suchasagivenpolyhedron),
whichcanbedescribedbythemulti-dimensionaluniformdegree.Itisworthtrying
forscholars,whohavethedataonturbulence.
Inamaterialcomposition,theparticleswillbecomeuniforminthemediumafter
agitating,andauniformdegreecanmeasuretheuniformityofparticlesinthisprocess.
Thetheoryofuniformityisamathematicaltheoryontheuniformdegree,andit
hasmanyapplicationsinvariousfields.Nevertheless,thetheoryisstilldeveloping
andisfarfromcomplete.
Harbin,China ChuanwenLuo
ChunchengWang
Contents
1 UniformDegree ................................................. 1
1.1 Introduction ................................................ 1
1.2 UniformDegree ............................................. 2
1.3 ContainedUniformDegreeTheorem ........................... 4
1.4 UniformDegreeTheoremforn-dimRandomPattern ............. 11
1.5 UniformDegreeandEntropy .................................. 17
1.6 NumericalTestfortheConjecture ............................. 21
1.7 ApplicationsofUniformDegreeTheoremsinPlantPattern
TypeTest ................................................... 31
1.8 UniversalityofUniformityMeasurement ....................... 33
References ...................................................... 52
2 AnInterpretationofChaosbyUniformDegree .................... 55
2.1 Introduction ................................................ 55
2.2 InstantaneousChaometryandk-StepChaometry ................. 55
2.3 InstantaneousChaometryandUniformDegree .................. 61
2.4 MoreApplicationsofk-StepChaometry ........................ 63
2.5 Applicationof250-StepChaometryinHeartRateProblem ........ 65
2.6 K-StepChaometryinVibrationFaultDetection .................. 67
References ...................................................... 69
3 Simulationsonk-StepChaometry ................................ 71
3.1 Introduction ................................................ 71
3.2 LorenzSystem .............................................. 72
3.3 ApplicationofInstantaneousChaometry ........................ 78
3.4 ICM forLargek ............................................ 81
4 ApplicationsofUniformDegreeinForestryandEcology ............ 85
4.1 EcologicalPattern ........................................... 85
4.2 MonopolizedDiskandUniformDegree ........................ 87
vii
viii Contents
√
4.3 ApplicationsofUniformDegree—TheRuleof 2 ............... 88
4.3.1 MonopolizedDiskandUniformDegree .................. 90
4.3.2 DiversityofDistance .................................. 92
References ...................................................... 93
Chapter 1
Uniform Degree
1.1 Introduction
Thedistributionofafinitenumberofpointsinapolygonisreferredtoasapattern
in this book. Aggregation and uniformity are two typical opposite features of a
pattern.Basedontheuniformdegreedefinedinthiscontext,apatternwithuniform
degree zero, like a periodic orbit, is the most aggregate pattern and the uniform
degree of any chaotic orbit must be greater than 0. For any segment of a chaotic
orbit,itmaybehaveineitheranaggregateorauniformway.However,itsaverage
uniform degree is invariant, which is the reason for measuring chaotic behaviour
by the uniform degree. Based on this, the word “chaometry” has been invented,
meaning the measurement of chaos. The uniform degree of the pattern generated
byadistributionfunction F inapolygonisabbreviatedastheuniformdegreeofa
distributionfunction,denotedby L .
F
TheuniformdegreeofauniformdistributionfunctionisdenotedbyL ,andthe
U
uniformdegreeofthepatterngeneratedbyachaoticorbitvariesfrom0to L .
U
TheuniformdegreeisanalogoustotheconceptofShannon’sentropy.Itiscon-
jectured that the uniform degree of a uniform distribution function is greater than
thoseofalltheotherdistributionfunctions,thatis,theuniformdistributionfunction
isthemostuniform,orL ≤ L .Thisconjecturewillexplainthatwhytheuniform
F U
degree of a chaotic orbit must lie in [0, L ], since the uniform degree L is the
U U
criticalsituationthatachaoticorbitcanneverattain.Manynumericalillustrations
showthatuniformdegreeperformbetterthantheotherindexesformeasuringchaos.
Based on the Monte Carlo method, any complex pattern obeying a distribution
function F in a bounded area can be viewed as the nonlinear transformation F−1
ofthepatterngeneratedbyauniformdistributionfunction,anditsfeaturescanbe
characterizedbytheuniformdegree.
Turbulenceisanintricatepatterninaboundedarea(suchasagivenpolyhedron),
whichcanbedescribedbythemulti-dimensionaluniformdegree.Itisworthtrying
forscholars,whohavedataonturbulence.
©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2022 1
C.LuoandC.Wang,MathematicalTheoryofUniformityanditsApplicationsinEcology
andChaos,SpringerBriefsinMathematicalMethods,
https://doi.org/10.1007/978-981-19-5512-9_1
