APPLIED MATHEMATICAL SCIENCES
1 Introduction to Mathematical Modeling1 Introduction1 What Is Mathematical Modeling?1 Characteristics of Mathematical Models4 Classification of Mathematical Models5 Classification as per the Nature of Basic Equations6 Some Simple Examples of Optimization Problems8 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 Preliminaries20 Construction of the Lyapunov Function and Testing of Stability27 Krasovskii’s Method27 Lyapunov Function for Linear Systems with Constant Coefficients29 The Routh–Hurwitz Criterion for Stability 32 Stability Discussion Based on the Linearization Procedure33 Global Asymptotic Stability40 Limit Cycles42 Liénard’s Equation and Existence of a Limit Cycle48 Energy Balance Method for Limit Cycles49 Focus 51 Dynamic System and Its Mathematical Model56 Hamiltonian Systems 61 Numerical Tools and Software Used63 MATLAB®63 Dynamics: Numerical Explorations66 Exercise 1168 References73 2 Modeling of Systems from Natural Science75 Introduction75 Models with Single Population77 Continuous Time Models77 Malthusian Model77 Verhulst–Pearl Logistic Model 78 Gompertz Growth Model 81 Theta-Logistic Model83 Model with Allee Effect83 Limited Growth Model 87 Harvest Model88 Models with Delay96 Discrete Time Models99 Linear Map99 Model 1 100 Model 2 102 Model 3 103 Model 4 105 Model 5 107 Model 6 107 Model 7 113 Exercise 21 114 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 Exercise 22 178 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 Structures206 2D Parameter Scans209 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 Bifurcation222 Period-Doubling Bifurcation223 Andronov–Hopf Bifurcation (Cycle Birth Bifurcation) 224 Poincaré Map and Poincaré Section225 Lyapunov Exponents226 Exercise 31230 References234 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 Diffusion250 A Variant of DeAngelis Model with Diffusion and Spatial Heterogeneity258 Spatiotemporal Dynamics of a Predator–Prey Model Using Holling Type IV Functional Response266 Holling–Tanner Model with Diffusion 271 A Variant of Holling–Tanner Model with Diffusion 273 Chaos in Multi-Species Model Systems 279 Hastings and Powell Model280 Simple Prey–Specialist Predator–Generalist Predator Interaction285 Single Predator, Two Competing Prey Model290 Exercise 41 295 References 298 5 Modeling of Some Engineering Systems305 Introduction305 Models in Mechanical Systems305 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 Oscillator330 Sprott’s Chaotic Electrical Circuit 332 References333