非线性动力学入门（影印版）

　　片断：Finite-DifferenceEquations1.1AMYTHICALFIELDImaginethatagraduatestudentgoestoameadowonthefirstdayofMay，walksthroughthemeadowwavingaflynet，andcountsthenumberoffliescaughtinthenet.Sherepeatsthisritualforseveralyears，followingupontheworkofpreviousgraduatestudents.TheresultingmeasurementsmightlooklikethegraphshowninFigure1.1.Thegraduatestudentnotesthevariabilityinhermeasurementsandwantstofindoutiftheycontainanyimportantbiologicalinformation.Severaldifferentapproachescouldbetakentostudythedata.Thestudentcoulddostatisticalanalysesofthedatatocalculatethemeanvalueortodetectlong-termtrends.Shecouldalsotrytodevelopadetailedandrealisticmodeloftheecosystem，takingintoaccountsuchfactorsasweather，predators，andtheflypopulationsinpreviousyears.Orshecouldconstructasimplifiedtheoreticalmodelforflypopulationdensity.Stickingtowhatsheknows，thestudentdecidestomodelthepopulationvariabilityintermsofactualmeasurements.Thenumberoffliesinonesummerdependsonthenumberofeggslaidthepreviousyear.Thenumberofeggslaiddependsonthenurnberoffliesaliveduringthatsummer.Thus，thenumberoffliesinonesummerdependsonthenumberoffliesintheprevioussummer.Innaathematicaltenns，thisisarelationship，orfunction，Thisequationsayssimplythatthenumberoffliesinthet 1summerisde-terminedby（orisafunctionof）thenumberoffliesinsummert，whichistheprevioussummer.Equationsofthisform，whichrelatevaluesatdiscretetimes（e.g.，eachMay），arecalledfinite-differenceequations.N，iscalledthestateofthesystemattimet.Weareinterestedinhowthestatechangesintime：thedynamicsofthesystem.Sincethereal-worldecosystemiscomplicatedandsincethemeasurementsareimperfect，wedonotexpectamodellikeEq.1.1tobeabletoduplicateexactlytheactualflypopulationmeasurements.Forexample，birdseatflies，sothepopulationoffliesisinfluencedbythebirdpopulation，whichitselfdependsonacomplicatedarrayoffactors.TheassumptionbehindEq.1.1isthatthenumberoffliesinyeart 1dependssolelyonthenumberoffliesinyeart.Whilethisisnotstrictlytrue，itmayserveasaworkingapproximation.Theproblemnowistofigureoutanappropriatefonnforthisdependencethatisconsistentwiththedataandthatencapsulatestheimportantaspectsofflypopulationbiology.

　　Daniel Kaplan specializes in the analysis of data using tachniques motivated by nonlinear dynamics. His primary interest is in the interpretation of irregular physiological rhythms， but the methods he has developed have been used in geophysics， economics， marine ecology， and other fields. He joined McGill in 1991，after receiving his Ph.D from Harvard University and working at MIT. His undergraduate studies were completed at Swarthomore College. He has worked with several intrumentation companies to develop novel types of medical monitors.Leon Glass is one of the pioneers of what has come to be callled chaos theory， specializing in applications to medicine and biology. He has worked in areas as diverse as physical chemistry， visual perception， and cardiology， and is one of the originators of the concept of "dynamical disease."

PREFACE
ABOUT THE AUTHORS
1
FINITE-DIFFERENCE EQUATIONS
1.1 A Mythical Field
1.2 The Linear Finite-Difference Equation
1.3 Methods of Iteration
1.4 Nonlinear Finite-Difference Equations
1.5 Steady States and Their Stability
1.6 Cycles and Their Stability
1.7 Chaos
1.8 Quasiperiodicity
1 Chaos in Periodically Stimulated Heart Cells
Sources and Notes
Exercises
Computer Projects
2 BOOLEAN NETWORKS AND CELLULAR
AUTOMATA
2.1 Elements and Networks
2.2 Boolean Variables, Functions, and Networks
2 A Lambda Bacteriophage Model
3 Locomotion in Salamanders
2.3 Boolean Functions and Biochemistry
2.4 Random Boolean Networks
2.5 Cellular Automata
4 Spiral Waves in Chemistry and Biology
2.6 Advanced Topic: Evolution and Computation
Sources and Notes
Exercises
Computer Projects
3 SELF-SIMILARITY AND FRACTAL GEOMETRY
3.1 Describing a Tree
3.2 Fractals
3.3 Dimension
5 The Box-Counting Dimension
3.4 Statistical Self-Similarity
6 Self-Similarity in Time
3.5 Fractals and Dynamics
7 Random Walks and Levy Walks
8 Fractal Growth
Sources and Notes
Exercises
Computer Projects
4 ONE-DIMENSIONAL DIFFERENTIAL
EQUATIONS
4.1 Basic Definitions
4.2 Growth and Decay
9 Traffic on the Intemet
10 Open ''nme Histograms in Patch Clamp Experiments
11 Gompertz Growth of Tumors
4.3 Multiple Fixed Points
4.4 Geometrical Analysis of One-Dimensional Nonlinear Ordinary
Differential Equations
4.5 Algebraic Analysis of Fixed Points
4.6 Differential Equations versus Finite-Difference Equations
4.7 Differential Equations with Inputs
12 Heart Rate Response to Sinusoid Inputs
4.8 Advanced Topic: Time Delays and Chaos
13 Nicholson''s Blowflies
Sources and Notes
Exercises
Computer Projects
5
TWO-DIMENSIONAL DIFFERENTIAL
EQUATIONS
5.1 The Harmonic Oscillator
5.2 Solutions, Trajectories, and Flows
5.3 The Two-Dimensional Linear Ordinary Differential Equation
5.4 Coupled First-Order Linear Equations
14 Metastasis of Malignant Tumors
5.5 The Phase Plane
5.6 Local Stability Analysis of Two-Dimensional, Nonlinear
Differential Equations
5.7 Limit Cycles and the van der Poi Oscillator
5,8 Finding Solutions to Nonlinear Differential Equations
15 Action Potentials in Nerve Cells
5.9 Advanced Topic: Dynamics in Three or More Dimensions
5.10 Advanced Topic: Poincare Index Theorem
Sources and Notes
Exercises
Computer Projects
6
TIME-SERIES ANALYSIS
6.1 Starting with Data
6.2 Dynamics, Measurements, and Noise
16 Fluctuations in Marine Populations
6.3 The Mean and Standard Deviation
6.4 Linear Correlations
6.5 PoWer Spectrum Analysis
17 Daily Oscillations in Zooplankton
6.6 Nonlinear Dynamics and Data Analysis
18 Reconstructing Nerve Cell Dynamics
6.7 Characterizing Chaos
19 Predicting the Next Ice Age
6.8 Detecting Chaos and Nonlinearity
6.9 Algorithms and Answers
Sources and Notes
Exercises
Computer Projects
APPENDIX A A MULTI-FUNCTIONAL APPENDIX
A.1 The Straight Line
A.2 The Quadratic Function
A.3 The Cubic and Higher-Order Polynomials
A.4 The Exponential Function
A.5 Sigmoidal Functions
A.6 The Sine and Cosine Functions
A.7 The Gaussian or Normal Distribution
A.8 The Ellipse
A.9 The Hyperbola
Exercises
APPENDIX B
A NOTE ON COMPUTER NOTATION
SOLUTIONS TO SELECTED EXERCISES
BIBLIOGRAPHY
INDEX