Nonlinear Science and Complexity (eBook)

Preface 5
Contents 7
Nonlinear Dynamics of Continuous and Discontinuous Dynamical Systems 12
On Synchronization and Its Complexity of Multiple Dynamical Systems 13
Introduction 13
Basic Concepts 14
Discontinuous Descriptions 16
Complexity by System Synchronization 19
References 22
Periodic and Chaotic Motions in a Gear-pair Transmission System with Impacts 23
Introduction 23
Equations of Motion 24
Switching Sets and Mappings 26
Mapping Structures 29
Parameter Maps and Illustrations 31
Conclusions 33
References 33
Analytical Prediction of Interrupted Cutting Periodic Motions in a Machine Tool 35
Introduction 35
Mechanical Model 36
Motion Switchability Conditions 39
Mappings Structure 41
Numerical and Analytical Predictions 42
Summary 44
Appendix 44
References 46
Control of Hopf Bifurcation and Chaos as Applied to Multimachine System 47
Introduction 47
System Description 48
Mathematical Model 49
System Response without Controller 49
Control of Hopf Bifurcation and Chaos 51
Control of Critical Modes 51
Control of Hopf Bifurcation 52
Numerical Simulation Results 55
Linear Controller 55
Nonlinear Controller 56
References 58
Lie Group Analysis and Applications in Nonlinear Sciences 59
Group-Invariant Solutions of Fractional Differential Equations 60
Introduction 60
Transformation Groups and Symmetries of FDE 61
Group Classification of Equations Dalphax y = f(x,y) 63
Exact Solutions of Equation Dalpha y =f(x,y) 65
References 67
Type-II Hidden Symmetries for Some Nonlinear Partial Differential Equations 69
Introduction 69
Weak Symmetries for the Model Equation 70
Linear Three-Dimensional Wave Equation 73
References 74
Nonclassical and Potential Symmetries for a Boussinesq Equation with Nonlinear Dispersion 75
Introduction 75
Nonclassical Symmetries 76
Classical Potential Symmetries 78
Nonclassical Potential Symmetries 79
Concluding Remarks 79
References 80
Application of the Composite Variational Principle to Shallow Water Equations 81
Introduction 81
Symmetry Group Analysis of the Shallow Water Equations in the Plane Flow 83
Derivation of Conservation Laws 83
Conclusion 85
References 85
Conserved Forms of Second Order-Ordinary Differential Equations 87
Introduction 87
lambda-Symmetries Associated to Integrating Factors 88
Integrating Factors Associated to lambda-Symmetries 89
Algorithm to Calculate Integrating Factors Based on lambda-Symmetries 90
Conclusions 91
References 92
Analytical Investigation of a Two-Phase Model Describing a Three-Way-Catalytic Converter 93
Introduction 93
The Group Theoretical Approach 94
Reductions and Solutions 96
Conclusion 98
References 99
Celestial Mechanics and Dynamical Astronomy: Methods and Applications 101
The Role of Invariant Manifolds in the Formation of Spiral Arms and Rings in Barred Galaxies 102
Introduction 103
Results 103
References 104
Continuous and Discrete Concepts for Detecting Transport Barriers in the Planar Circular Restricted Three Body Problem 106
Introduction 106
Methods and Results 107
Finite-Time Lyapunov Exponents 107
Graph Partitioning Techniques 108
Graph Based Expansion 109
Discussion 111
References 111
Low-Energy Transfers in the Earth-Moon System 113
Introduction 113
The Model 114
Collinear Points Dynamics 115
Methodology 116
Rescue Orbits 117
Results 117
LEO Transfers 118
Results 119
Conclusions 119
References 120
Gravitational Potential of a Massive Disk. Dynamics Around an Annular Disk 121
The Massive Disk and Its Potential Function 121
Dynamics Around a Circular Annulus 123
References 127
An Accounting Device for Biasymptotic Solutions: The Scattering Map in the Restricted Three Body Problem 128
References 129
Optimal Capture Trajectories Using Multiple Gravity Assists 130
Introduction 130
The Keplerian Map 131
Control Problem Formulation 132
Optimal Feedback 133
Discretization 133
Low Energy Multiple Gravity Assists 134
Conclusion 135
References 135
New Periodic Orbits in the Solar Sail Three-Body Problem 136
Introduction 136
Equations of Motion in the Rotating Frame 137
Linearised System 139
High-Order Approximations to Periodic Orbits 139
The Solar Sail ERTBP 141
Stability of Periodic Orbits in the Solar Sail RTBP 142
Conclusion 143
References 143
A Review of Invariant Manifold Dynamics of the CRTBP and Some Applications 144
The Models 144
Lindstedt Poincaré and Normal Forms Techniques 146
Normal Forms 146
Applications to Classical Problems of Libration Point Mission Design 148
References 150
Solar Sail Orbits at the Earth-Moon Libration Points 152
Introduction 152
Solar Sail in the Earth-Moon Restricted Three-Body Problem 153
Qualitative Approach 153
Equations of Motion in Presence of Solar Sail 154
Solution of the Linearized Equations of Motion 156
Numerical Integration of the Nonlinear Equations of Motion 157
Conclusion 159
References 159
Mathematical Modeling of Nonlinear Structures in Bose-Einstein Condensates 161
Collisions of Discrete Breathers in Nonlinear Schrödinger and Klein-Gordon Lattices 162
Introduction 162
DNLS Lattices 163
Klein-Gordon Lattices 164
Interpretation 165
References 167
Stability of BEC Systems in Nonlinear Optical Lattices 168
Introduction 168
Conservative Systems (gamma2 = 0) 170
Attractive Condensate (gamma0gamma1 + 1/2) 170
Repulsive Condensate (gamma0gamma1 - 1/2) 171
Dissipative Systems 174
Conclusions 174
References 175
Nonlinear Schrödinger Equations with a Four-Well Potential in Two Dimensions: Bifurcations and Stability Analysis 176
Introduction 176
The Analytical Approach 177
Numerical Results 178
Conclusions and Future Challenges 181
References 182
Bose-Einstein Condensates and Multi-Component NLS Models on Symmetric Spaces of BD.I-Type. Expansions over Squared Solutions 183
Introduction 184
MNLS Equations for BD.I. Series of Symmetric Spaces 185
The Inverse Scattering Problem 186
The Generalized Fourier Transforms for Non-regular J 188
Fundamental Properties of the MNLS Equations 189
References 190
Mathematical Models in Engineering 191
Impulsive Boundary Layer Flow Past a Permeable Quadratically Stretching Sheet 192
Introduction 192
Governing Equations 193
Results and Discussion 194
Case (i): Linear Injection (s < =1 and b >
Case (ii): Linear Suction (s > =1 and b <
Case (iii): Linear Suction for x < xc and Linear Injection for x >
Case(iv): Linear Injection for x < xc and Linear Suction for x >
Note on the Value of s 199
Conclusions 199
References 199
Complete Dynamic Modeling of a Stewart Platform Using the Generalized Momentum Approach 200
Introduction 200
Stewart Platform Kinematic Structure 201
Dynamic Modeling Using the Generalized Momentum 203
Moving Platform Modeling 203
Cylinder Modeling 204
Piston Modeling 208
Dynamic Model Gravitational Components 210
Conclusions 211
References 211
Numerical Solution of a PDE System with Non-Linear Steady State Conditions that Translates the Air Stripping Pollutants Removal 212
Background 212
The Differential Model 213
The Boundary Conditions 214
The Steady State 214
The Transient State 215
The Lanczos's Tau Method 216
Initial Conditions and Differential Equations 217
Algebraic Equations 218
Conclusions and Future Work 219
References 219
Three Behavioural Scenarios for Contingent Claims Valuation in Incomplete Markets 221
Introduction 221
Three Scenarios for Price Selection 222
Allocation of Wealth and Indifference Pricing 223
Market Games Approach 224
The Risk Sharing Approach 225
Optimal Choice of the Agents Market Price of Risk 226
Conclusion 227
References 228
Undesired Oscillations in Pneumatic Systems 229
Introduction 229
Pneumatic System 230
Experimental Setup 230
Nonlinear Model 231
Pneumatic Chamber Model 232
Servovalve Model 232
Mechanical Model 233
Linearised Model 234
Friction Generated Oscillations 235
Describing Function Analysis 235
Limit Cycle Prediction 238
Experimental Results 240
Pressure Dynamics Generated Oscillations 241
Conclusions 242
References 242
A Study of Correlation and Entropy for Multiple Time Series 244
Introduction 245
Goals 245
Entropy 245
Covariance 246
Data 251
Conclusions 251
References 252
Characterization and Parameterization of the Singular Manifold of a Simple 6-6 Stewart Platform 254
Introduction 254
Forward Kinematics 256
Non-singular Case 257
Singular Case 260
Conclusions 260
References 261
Fractional Calculus Applications 262
Some Advances on Image Processing by Means of Fractional Calculus 263
Introduction 263
Time Discretization 265
Numerical Experiments 266
Conclusions 268
References 268
Application of Genetic Algorithms in the Design of an Electrical Potential of Fractional Order 270
Introduction 270
Integer and Fractional Electrical Potential 271
Conclusions 276
References 276
Mellin Transform for Fractional Differential Equations with Variable Potential 278
Introduction 278
Fractional Operators 279
The Mellin Transform and Its Properties 280
Fractional Linear Equation with Riemann-Liouville Derivative and tbeta-Potential 280
Example: Solution for Case beta=0 284
Example: Solution for Case beta= -alpha/2 285
Fractional Linear Equation with Caputo Derivative and tbeta-Potential 285
Nonhomogeneous Fractional Equations with tbeta-Potential 287
Final Remarks 287
References 288
Phase Plane Characteristics of Marginally Stable Fractional Order Systems 290
Introduction 290
Analysis of the Phase Plane 291
Analysis of Phase Plane in Marginally Stable LTI Integer Order Systems 291
Analysis of Phase Plane in Marginally Stable LTI Fractional Order Systems 292
Simulation Results 296
Conclusion 297
References 297
Application of Fractional Controllers for Quad Rotor 299
Introduction 299
Helicopter Model 300
Fractional Control 302
The Flight Simulator 303
Conclusions 304
References 304
Regularity of a Degenerated Convolution Semi-Group Without to Use the Poisson Process 306
Introduction 306
Proof of Bismut's Theorem Without to Use the Poisson Process 308
Proof of Léandre's Theorem Without to Use the Poisson Process 311
References 312
Computational Techniques for Engineering Sciences 314
Image Processing for the Estimation of Drop Distribution in Agitated Liquid-Liquid Dispersion 315
Introduction 315
Description of the Method 316
Edges Detection 316
Detection of Drops in the Contour Image 318
Results and Discussions 319
Conclusions and Future Work 321
References 321
Music and Evolutionary Computation 322
A Brief History of the Western Music 322
Music, Mathematics and Physics 323
Foundations versus Styles 324
Music and Art 325
Music and Algorithms 325
Evolutionary Computation 326
Genetic Algorithms (GAs) 326
Genetic Programming (GPs) 327
Particle Swarm Optimization (PSO) 327
Applications of EAs to Computer Music 328
Conclusions and Future Work 329
References 329
Application of Computational Intelligence to Engineering 330
Introduction 330
Identification of Material Constants with NNs 331
Problem Definition and NN Model 332
Results 333
Equipment Fault Diagnosis with Fuzzy Systems 333
Problem Definition and Fuzzy Model 334
Results 335
Digital Circuit Synthesis with EC 335
Problem Definition and the Genetic Algorithm 336
Results 337
Conclusion 337
References 338
Evolutionary Trajectory Optimization for Redundant Robots 339
Introduction 339
Kinematics of Redundant Manipulators 340
Robot Trajectory Control 341
The CLGA Formulation 341
Representation and Operators in the CLGA 342
Optimization Criteria 343
Simulation Results 343
Conclusions 345
References 345
Nonlinear Systems 346
Robust Communication-Masking via a Synchronized Chaotic Lorenz Transmission System 347
Introduction 347
The Lorenz System 348
Master-Slave Synchronization Using a Luenberger-Type Observer 349
Robustness Study 350
Modified Lorenz-Based Observer 352
Conclusions 355
References 355
A Boundary Layer Problem in Power Law Fluids through a Moving Flat Plate 356
Introduction 356
Proofs of Two Theorems 357
Proof of Theorem 1 362
Proof of Theorem 2 362
References 362
An Overview of the Behaviour of a Scattering Map for the Dynamics of Two Interacting Particles in a Uniform Magnetic Field 364
Introduction 364
The Planar Problem 365
The Spatial Problem 365
The Scattering Map for the Spatial Problem 366
Possible Applications 367
References 368
A Generalised Entropy of Curves Approach for the Analysis of Dynamical Systems 369
Introduction 369
The Generalised Entropy of Curves and its Application 370
Properties of the Generalised Entropy 372
A Comparison with Other Chaotic Indicators 372
Simulation Examples 373
Conclusions 375
References 376
Uncertainty on a Bertrand Duopoly with Product Differentiation 377
Introduction 377
The Model and the Equilibrium 378
Conclusions 382
References 382
Price-Setting Dynamical Duopoly with Incomplete Information 384
Introduction 384
The Model and the Equilibrium 385
Conclusions 389
References 390
Inductor-Free Version for Chua's Oscillator Based in Electronic Analogy 391
Introduction 391
Electronic Analogy 393
Design of an Analogous Chua's Oscillator 394
Experimental Implementation 396
Conclusions 396
References 397
Model Reduction of Nonlinear Continuous Dynamic Systems on Inertial Manifolds with Delay 398
Introduction 398
Inertial Manifolds with Time Delay 399
Governing Equations of Shallow Arch under Impact 400
Numerical Examples 401
Conclusions 402
References 403
A Fuzzy Crisis in a Duffing-Van der Pol System 404
Introduction 404
A Fuzzy Crisis in a Duffing-Van der Pol System 405
Concluding Remarks 407
References 408
Name Index 410