Table Of ContentWater Science Nollet T H I R D E D I T I O N
De Gelder
Third Edition
Handbook of Water Analysis
Extensively revised and updated, Handbook of Water Analysis, Third Edition
provides current analytical techniques for detecting various compounds in water
samples. Maintaining the detailed and accessible style of the previous editions,
this third edition demonstrates water sampling and preservation methods by
enumerating different ways to measure chemical and radiological characteris-
tics. It gives step-by-step descriptions of separation, residue determination,
and clean-up techniques for a variety of freshwater and saltwater.
See What’s New in the Third Edition
• Includes five new chapters covering ammonia, nitrates, nitrites,
and petroleum hydrocarbons, as well as organoleptical and algal
analysis methodology
• Compares older methods still frequently used with recently developed
protocols, and examines future trends
• Features a new section regarding organoleptical analysis of water
acknowledging that ultimately the consumers of drinking water have
the final vote over its quality with respect to odor, flavor, and color
The book covers the physical, chemical, and other relevant properties of
various substances found in water. It then describes the sampling, cleanup,
extraction, and derivatization procedures, and concludes with detection meth-
ods. Illustrated with procedure flow charts and schematics, the text includes
numerous tables categorizing methods according to type of component, origin
of the water sample, parameters and procedures used, and application range.
With contributions from international experts, the book guides you through
the entire scientific investigation starting with a sampling strategy designed
to capture the real-world situation as closely as possible, and ending with an
adequate chemometrical and statistical treatment of the acquired data.
By organizing data into more than 300 tables, graphs, and charts, and
supplementing the text with equations and illustrations, the editors distill
a wealth of knowledge into a single accessible reference.
E D I T E D B Y
L e o M . L . N o l l e t
L e e n S . P. D e G e l d e r
T H I R D
E D I T I O N
K14097
© 2008 Taylor & Francis Group, LLC
Introduction to
Mathematical
Modeling and
Chaotic Dynamics
© 2008 Taylor & Francis Group, LLC
Introduction to
Mathematical
Modeling and
Chaotic Dynamics
Ranjit Kumar Upadhyay
Indian School of Mines
Dhanbad, India
Satteluri R. K. Iyengar
Gokaraju Rangaraju Institute of Engineering & Technology
Hyderabad, India
© 2008 Taylor & Francis Group, LLC
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by the MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink®
software.
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Contents
Preface ......................................................................................................................ix
Authors ....................................................................................................................xi
1. Introduction to Mathematical Modeling ...................................................1
Introduction ......................................................................................................1
What Is Mathematical Modeling? .................................................................1
Characteristics of Mathematical Models .................................................4
Classification of Mathematical Models .........................................................5
Classification as per the Nature of Basic Equations ...............................6
Some Simple Examples of Optimization Problems ...............................8
Limitations Associated with Mathematical Modeling .............................10
Modeling Approaches ...................................................................................11
Empirical Approach ..................................................................................11
Theoretical Approach ...............................................................................12
Stochastic or Probabilistic Approach .....................................................14
Deterministic Approach ..........................................................................14
Statistical Approach ..................................................................................14
Simulation Approach ................................................................................15
Discrete and Continuous Approaches ...................................................15
Modeling/Cyclic Processes ..........................................................................16
A Modeling Diagram ....................................................................................16
Compartment Models ...................................................................................17
Mathematical Preliminaries .........................................................................20
Construction of the Lyapunov Function and Testing of Stability .....27
Krasovskii’s Method ............................................................................27
Lyapunov Function for Linear Systems with Constant
Coefficients ............................................................................................29
The Routh–Hurwitz Criterion for Stability ...........................................32
Stability Discussion Based on the Linearization Procedure...............33
Global Asymptotic Stability ....................................................................40
Limit Cycles ...............................................................................................42
Liénard’s Equation and Existence of a Limit Cycle .........................48
Energy Balance Method for Limit Cycles .........................................49
Focus ...........................................................................................................51
Dynamic System and Its Mathematical Model .........................................56
Hamiltonian Systems ...............................................................................61
Numerical Tools and Software Used ..........................................................63
MATLAB® ...................................................................................................63
Dynamics: Numerical Explorations .......................................................66
© 2008 Taylor & Francis Group, LLC v
vi Contents
Exercise 1.1 ......................................................................................................68
References .......................................................................................................73
2. Modeling of Systems from Natural Science ...........................................75
Introduction ....................................................................................................75
Models with Single Population ....................................................................77
Continuous Time Models .........................................................................77
Malthusian Model ................................................................................77
Verhulst–Pearl Logistic Model ...........................................................78
Gompertz Growth Model....................................................................81
Theta-Logistic Model ...........................................................................83
Model with Allee Effect .......................................................................83
Limited Growth Model .......................................................................87
Harvest Model ......................................................................................88
Models with Delay ...............................................................................96
Discrete Time Models ...............................................................................99
Linear Map ............................................................................................99
Model 1 ................................................................................................100
Model 2 ................................................................................................102
Model 3 ................................................................................................103
Model 4 ................................................................................................105
Model 5 ................................................................................................107
Model 6 ................................................................................................107
Model 7 ................................................................................................113
Exercise 2.1 ....................................................................................................114
Two-Dimensional (2D) Continuous Models (Modeling
of Population Dynamics of Two Interacting Species) .............................117
Analytical Tool ........................................................................................123
Kolmogorov Theorem ........................................................................123
Local Stability Analysis .....................................................................125
Lotka–Volterra Model .............................................................................126
Variation of the Classical LV Model ................................................135
Leslie–Gower Model ...............................................................................136
Rosenzweig–MacArthur Model ............................................................140
Variations of the RM Model ..............................................................142
Prey–Generalist Predator Model ..........................................................153
Holling–Tanner Model ...........................................................................156
Modified HT Model ................................................................................163
Competition Model .................................................................................166
Model 1: Gause Model .......................................................................167
Model 2 ................................................................................................168
2D Discrete Models......................................................................................170
Nicholson–Bailey Model ........................................................................171
Modified NB Model ................................................................................173
Aihara Model ...........................................................................................177
© 2008 Taylor & Francis Group, LLC
Contents vii
Exercise 2.2 ....................................................................................................178
References .....................................................................................................180
3. Introduction to Chaotic Dynamics .........................................................189
Introduction ..................................................................................................189
Chaos and Chaotic Dynamics ....................................................................189
Basin of Attraction ..................................................................................193
Primary Routes to Study Chaos ................................................................193
Types of Chaos, Transients, and Attractors..............................................196
Methods of Investigation for Detecting Chaos ........................................201
Method for Selection of Parameter Values ..........................................201
Calculation of the Basin Boundary Structures ...................................206
2D Parameter Scans ................................................................................209
Bifurcation Diagrams .............................................................................211
Hopf Bifurcation Analysis ................................................................213
Time-Series Analysis and Phase-Space Diagram ...............................217
Types of Bifurcations ..............................................................................219
Saddle-Node Bifurcation or Tangent Bifurcation ..........................219
Transcritical Bifurcation ....................................................................221
Pitchfork Bifurcation ..........................................................................222
Period-Doubling Bifurcation ............................................................223
Andronov–Hopf Bifurcation (Cycle Birth Bifurcation) .................224
Poincaré Map and Poincaré Section ..........................................................225
Lyapunov Exponents ...................................................................................226
Exercise 3.1 ....................................................................................................230
References .....................................................................................................234
4. Chaotic Dynamics in Model Systems from Natural Science ............241
Introduction ..................................................................................................241
Chaos in Single Species Model Systems ...................................................242
Model 1 .....................................................................................................242
Model 2 .....................................................................................................243
Model with Predator Saturation ......................................................244
Model with Mating Limitation .........................................................244
Chaos in Two Species Model Systems ......................................................246
Chaos in Two Species Model Systems with Diffusion ...........................247
Rosenzweig–MacArthur Model with Diffusion.................................249
A Variant of Rosenzweig–MacArthur Model with Diffusion ..........250
A Variant of DeAngelis Model with Diffusion and Spatial
Heterogeneity ..........................................................................................258
Spatiotemporal Dynamics of a Predator–Prey Model Using
Holling Type IV Functional Response .................................................266
Holling–Tanner Model with Diffusion ................................................271
A Variant of Holling–Tanner Model with Diffusion..........................273
Chaos in Multi-Species Model Systems ....................................................279
© 2008 Taylor & Francis Group, LLC
viii Contents
Hastings and Powell Model ..................................................................280
Simple Prey–Specialist Predator–Generalist Predator Interaction ..285
Single Predator, Two Competing Prey Model .....................................290
Exercise 4.1 ....................................................................................................295
References .....................................................................................................298
5. Modeling of Some Engineering Systems ..............................................305
Introduction ..................................................................................................305
Models in Mechanical Systems ..................................................................305
Nonlinear Oscillators .............................................................................312
Chaos in Mass–Spring System ..............................................................317
Chaotic Oscillations in Duffing–Holmes Oscillator ..........................318
Chaos in Jerk Systems .............................................................................320
Models in Electronic Circuits .....................................................................321
Nonlinear Circuits .......................................................................................324
Chua’s Diode and Chua’s Circuit ..........................................................324
Murali–Lakshmanan–Chua Circuit .....................................................327
A Third-Order Autonomous Chaotic Oscillator ................................330
Sprott’s Chaotic Electrical Circuit .........................................................332
References .....................................................................................................333
Solutions to Odd-Numbered Problems .........................................................339
© 2008 Taylor & Francis Group, LLC
