Nonlinear Science and Complexity

1. Auflage 2010
ISBN: 978-90-481-9884-9
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 426 Seiten

This book contains selected papers of NSC08, the 2nd Conference on Nonlinear Science and Complexity, held 28-31 July, 2008, Porto, Portugal. It focuses on fundamental theories and principles, analytical and symbolic approaches, computational techniques in nonlinear physics and mathematics. Topics treated include • Chaotic Dynamics and Transport in Classic and Quantum Systems • Complexity and Nonlinearity in Molecular Dynamics and Nano-Science • Complexity and Fractals in Nonlinear Biological Physics and Social Systems • Lie Group Analysis and Applications in Nonlinear Science • Nonlinear Hydrodynamics and Turbulence • Bifurcation and Stability in Nonlinear Dynamic Systems • Nonlinear Oscillations and Control with Applications • Celestial Physics and Deep Space Exploration • Nonlinear Mechanics and Nonlinear Structural Dynamics • Non-smooth Systems and Hybrid Systems • Fractional dynamical systems

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;Nonlinear Dynamics of Continuous and Discontinuous Dynamical Systems;12
3.1;On Synchronization and Its Complexity of Multiple Dynamical Systems;13
3.1.1;Introduction;13
3.1.2;Basic Concepts;14
3.1.3;Discontinuous Descriptions;16
3.1.4;Complexity by System Synchronization;19
3.1.5;References;22
3.2;Periodic and Chaotic Motions in a Gear-pair Transmission System with Impacts;23
3.2.1;Introduction;23
3.2.2;Equations of Motion;24
3.2.3;Switching Sets and Mappings;26
3.2.4;Mapping Structures;29
3.2.5;Parameter Maps and Illustrations;31
3.2.6;Conclusions;33
3.2.7;References;33
3.3;Analytical Prediction of Interrupted Cutting Periodic Motions in a Machine Tool;35
3.3.1;Introduction;35
3.3.2;Mechanical Model;36
3.3.3;Motion Switchability Conditions;39
3.3.4;Mappings Structure;41
3.3.5;Numerical and Analytical Predictions;42
3.3.6;Summary;44
3.3.7;Appendix;44
3.3.8;References;46
3.4;Control of Hopf Bifurcation and Chaos as Applied to Multimachine System;47
3.4.1;Introduction;47
3.4.2;System Description;48
3.4.3;Mathematical Model;49
3.4.4;System Response without Controller;49
3.4.5;Control of Hopf Bifurcation and Chaos;51
3.4.5.1;Control of Critical Modes;51
3.4.5.2;Control of Hopf Bifurcation;52
3.4.5.3;Numerical Simulation Results;55
3.4.5.3.1;Linear Controller ;55
3.4.5.3.2;Nonlinear Controller;56
3.4.6;References;58
4;Lie Group Analysis and Applications in Nonlinear Sciences;59
4.1;Group-Invariant Solutions of Fractional Differential Equations;60
4.1.1;Introduction;60
4.1.2;Transformation Groups and Symmetries of FDE;61
4.1.3;Group Classification of Equations Dalphax y = f(x,y);63
4.1.4;Exact Solutions of Equation Dalpha y =f(x,y);65
4.1.5;References;67
4.2;Type-II Hidden Symmetries for Some Nonlinear Partial Differential Equations;69
4.2.1;Introduction;69
4.2.2;Weak Symmetries for the Model Equation ;70
4.2.3;Linear Three-Dimensional Wave Equation;73
4.2.4;References;74
4.3;Nonclassical and Potential Symmetries for a Boussinesq Equation with Nonlinear Dispersion;75
4.3.1;Introduction;75
4.3.2;Nonclassical Symmetries;76
4.3.3;Classical Potential Symmetries;78
4.3.4;Nonclassical Potential Symmetries;79
4.3.5;Concluding Remarks;79
4.3.6;References;80
4.4;Application of the Composite Variational Principle to Shallow Water Equations;81
4.4.1;Introduction;81
4.4.2;Symmetry Group Analysis of the Shallow Water Equations in the Plane Flow;83
4.4.3;Derivation of Conservation Laws;83
4.4.4;Conclusion;85
4.4.5;References;85
4.5;Conserved Forms of Second Order-Ordinary Differential Equations;87
4.5.1;Introduction;87
4.5.2;lambda-Symmetries Associated to Integrating Factors;88
4.5.3;Integrating Factors Associated to lambda-Symmetries;89
4.5.4;Algorithm to Calculate Integrating Factors Based on lambda-Symmetries;90
4.5.5;Conclusions;91
4.5.6;References;92
4.6;Analytical Investigation of a Two-Phase Model Describing a Three-Way-Catalytic Converter;93
4.6.1;Introduction;93
4.6.2;The Group Theoretical Approach;94
4.6.3;Reductions and Solutions;96
4.6.4;Conclusion;98
4.6.5;References;99
5;Celestial Mechanics and Dynamical Astronomy: Methods and Applications;101
5.1;The Role of Invariant Manifolds in the Formation of Spiral Arms and Rings in Barred Galaxies;102
5.1.1;Introduction;103
5.1.2;Results;103
5.1.3;References;104
5.2;Continuous and Discrete Concepts for Detecting Transport Barriers in the Planar Circular Restricted Three Body Problem;106
5.2.1;Introduction;106
5.2.2;Methods and Results;107
5.2.2.1;Finite-Time Lyapunov Exponents;107
5.2.2.2;Graph Partitioning Techniques;108
5.2.2.3;Graph Based Expansion;109
5.2.3;Discussion;111
5.2.4;References;111
5.3;Low-Energy Transfers in the Earth-Moon System;113
5.3.1;Introduction;113
5.3.2;The Model;114
5.3.3;Collinear Points Dynamics;115
5.3.4;Methodology;116
5.3.5;Rescue Orbits;117
5.3.5.1;Results;117
5.3.6;LEO Transfers;118
5.3.6.1;Results;119
5.3.7;Conclusions;119
5.3.8;References;120
5.4;Gravitational Potential of a Massive Disk. Dynamics Around an Annular Disk;121
5.4.1;The Massive Disk and Its Potential Function;121
5.4.2;Dynamics Around a Circular Annulus;123
5.4.3;References;127
5.5;An Accounting Device for Biasymptotic Solutions: The Scattering Map in the Restricted Three Body Problem;128
5.5.1;References;129
5.6;Optimal Capture Trajectories Using Multiple Gravity Assists;130
5.6.1;Introduction;130
5.6.2;The Keplerian Map;131
5.6.3;Control Problem Formulation;132
5.6.3.1;Optimal Feedback;133
5.6.3.2;Discretization;133
5.6.4;Low Energy Multiple Gravity Assists;134
5.6.5;Conclusion;135
5.6.6;References;135
5.7;New Periodic Orbits in the Solar Sail Three-Body Problem;136
5.7.1;Introduction;136
5.7.2;Equations of Motion in the Rotating Frame;137
5.7.3;Linearised System;139
5.7.4;High-Order Approximations to Periodic Orbits;139
5.7.5;The Solar Sail ERTBP;141
5.7.6;Stability of Periodic Orbits in the Solar Sail RTBP;142
5.7.7;Conclusion;143
5.7.8;References;143
5.8;A Review of Invariant Manifold Dynamics of the CRTBP and Some Applications;144
5.8.1;The Models;144
5.8.2;Lindstedt Poincaré and Normal Forms Techniques;146
5.8.2.1;Normal Forms;146
5.8.3;Applications to Classical Problems of Libration Point Mission Design;148
5.8.4;References;150
5.9;Solar Sail Orbits at the Earth-Moon Libration Points;152
5.9.1;Introduction;152
5.9.2;Solar Sail in the Earth-Moon Restricted Three-Body Problem;153
5.9.2.1;Qualitative Approach;153
5.9.2.2;Equations of Motion in Presence of Solar Sail;154
5.9.3;Solution of the Linearized Equations of Motion;156
5.9.4;Numerical Integration of the Nonlinear Equations of Motion;157
5.9.5;Conclusion;159
5.9.6;References;159
6;Mathematical Modeling of Nonlinear Structures in Bose-Einstein Condensates;161
6.1;Collisions of Discrete Breathers in Nonlinear Schrödinger and Klein-Gordon Lattices;162
6.1.1;Introduction;162
6.1.2;DNLS Lattices;163
6.1.3;Klein-Gordon Lattices;164
6.1.4;Interpretation;165
6.1.5;References;167
6.2;Stability of BEC Systems in Nonlinear Optical Lattices;168
6.2.1;Introduction;168
6.2.2;Conservative Systems (gamma2 = 0);170
6.2.2.1;Attractive Condensate (gamma0gamma1 + 1/2);170
6.2.2.2;Repulsive Condensate (gamma0gamma1 - 1/2);171
6.2.3;Dissipative Systems;174
6.2.4;Conclusions;174
6.2.5;References;175
6.3;Nonlinear Schrödinger Equations with a Four-Well Potential in Two Dimensions: Bifurcations and Stability Analysis;176
6.3.1;Introduction;176
6.3.2;The Analytical Approach;177
6.3.3;Numerical Results;178
6.3.4;Conclusions and Future Challenges;181
6.3.5;References;182
6.4;Bose-Einstein Condensates and Multi-Component NLS Models on Symmetric Spaces of BD.I-Type. Expansions over Squared Solutions;183
6.4.1;Introduction;184
6.4.2;MNLS Equations for BD.I. Series of Symmetric Spaces;185
6.4.3;The Inverse Scattering Problem;186
6.4.4;The Generalized Fourier Transforms for Non-regular J;188
6.4.5;Fundamental Properties of the MNLS Equations;189
6.4.6;References;190
7;Mathematical Models in Engineering;191
7.1;Impulsive Boundary Layer Flow Past a Permeable Quadratically Stretching Sheet;192
7.1.1;Introduction;192
7.1.2;Governing Equations;193
7.1.3;Results and Discussion;194
7.1.3.1;Case (i): Linear Injection (s <=1 and b >=0);194
7.1.3.2;Case (ii): Linear Suction (s >=1 and b <=0);196
7.1.3.3;Case (iii): Linear Suction for x < xc and Linear Injection for x > xc (s > 1 and b > 0);196
7.1.3.4;Case(iv): Linear Injection for x < xc and Linear Suction for x > xc (s < 1 and b < 0);196
7.1.3.5;Note on the Value of s;199
7.1.4;Conclusions;199
7.1.5;References;199
7.2;Complete Dynamic Modeling of a Stewart Platform Using the Generalized Momentum Approach;200
7.2.1;Introduction;200
7.2.2;Stewart Platform Kinematic Structure;201
7.2.3;Dynamic Modeling Using the Generalized Momentum;203
7.2.3.1;Moving Platform Modeling;203
7.2.3.2;Cylinder Modeling;204
7.2.3.3;Piston Modeling;208
7.2.3.4;Dynamic Model Gravitational Components;210
7.2.4;Conclusions;211
7.2.5;References;211
7.3;Numerical Solution of a PDE System with Non-Linear Steady State Conditions that Translates the Air Stripping Pollutants Removal;212
7.3.1;Background;212
7.3.2;The Differential Model;213
7.3.2.1;The Boundary Conditions;214
7.3.2.2;The Steady State;214
7.3.3;The Transient State;215
7.3.4;The Lanczos's Tau Method;216
7.3.4.1;Initial Conditions and Differential Equations;217
7.3.4.2;Algebraic Equations;218
7.3.5;Conclusions and Future Work;219
7.3.6;References;219
7.4;Three Behavioural Scenarios for Contingent Claims Valuation in Incomplete Markets;221
7.4.1;Introduction;221
7.4.2; Three Scenarios for Price Selection;222
7.4.2.1;Allocation of Wealth and Indifference Pricing;223
7.4.2.2;Market Games Approach;224
7.4.2.3;The Risk Sharing Approach ;225
7.4.2.4;Optimal Choice of the Agents Market Price of Risk;226
7.4.3;Conclusion;227
7.4.4;References;228
7.5;Undesired Oscillations in Pneumatic Systems;229
7.5.1;Introduction;229
7.5.2;Pneumatic System;230
7.5.2.1;Experimental Setup;230
7.5.2.2;Nonlinear Model;231
7.5.2.2.1;Pneumatic Chamber Model;232
7.5.2.2.2;Servovalve Model;232
7.5.2.2.3;Mechanical Model;233
7.5.2.3;Linearised Model;234
7.5.3;Friction Generated Oscillations;235
7.5.3.1;Describing Function Analysis;235
7.5.3.2;Limit Cycle Prediction;238
7.5.3.3;Experimental Results;240
7.5.4;Pressure Dynamics Generated Oscillations;241
7.5.5;Conclusions;242
7.5.6;References;242
7.6;A Study of Correlation and Entropy for Multiple Time Series;244
7.6.1;Introduction;245
7.6.1.1;Goals;245
7.6.2;Entropy;245
7.6.3;Covariance;246
7.6.3.1;Data;251
7.6.4;Conclusions;251
7.6.5;References;252
7.7;Characterization and Parameterization of the Singular Manifold of a Simple 6-6 Stewart Platform;254
7.7.1;Introduction;254
7.7.2;Forward Kinematics;256
7.7.2.1;Non-singular Case;257
7.7.2.2;Singular Case;260
7.7.3;Conclusions;260
7.7.4;References;261
8;Fractional Calculus Applications;262
8.1;Some Advances on Image Processing by Means of Fractional Calculus;263
8.1.1;Introduction;263
8.1.2;Time Discretization;265
8.1.3;Numerical Experiments;266
8.1.4;Conclusions;268
8.1.5;References;268
8.2;Application of Genetic Algorithms in the Design of an Electrical Potential of Fractional Order;270
8.2.1;Introduction;270
8.2.2;Integer and Fractional Electrical Potential;271
8.2.3;Conclusions;276
8.2.4;References;276
8.3;Mellin Transform for Fractional Differential Equations with Variable Potential;278
8.3.1;Introduction;278
8.3.2;Fractional Operators;279
8.3.2.1;The Mellin Transform and Its Properties;280
8.3.3;Fractional Linear Equation with Riemann-Liouville Derivative and tbeta-Potential;280
8.3.3.1;Example: Solution for Case beta=0;284
8.3.3.2;Example: Solution for Case beta= -alpha/2;285
8.3.4;Fractional Linear Equation with Caputo Derivative and tbeta-Potential;285
8.3.5;Nonhomogeneous Fractional Equations with tbeta-Potential;287
8.3.6;Final Remarks;287
8.3.7;References;288
8.4;Phase Plane Characteristics of Marginally Stable Fractional Order Systems;290
8.4.1;Introduction;290
8.4.2;Analysis of the Phase Plane;291
8.4.2.1;Analysis of Phase Plane in Marginally Stable LTI Integer Order Systems;291
8.4.2.2;Analysis of Phase Plane in Marginally Stable LTI Fractional Order Systems;292
8.4.3;Simulation Results;296
8.4.4;Conclusion;297
8.4.5;References;297
8.5;Application of Fractional Controllers for Quad Rotor;299
8.5.1;Introduction;299
8.5.2;Helicopter Model;300
8.5.3;Fractional Control;302
8.5.4;The Flight Simulator;303
8.5.5;Conclusions;304
8.5.6;References;304
8.6;Regularity of a Degenerated Convolution Semi-Group Without to Use the Poisson Process;306
8.6.1;Introduction;306
8.6.2;Proof of Bismut's Theorem Without to Use the Poisson Process;308
8.6.3;Proof of Léandre's Theorem Without to Use the Poisson Process;311
8.6.4;References;312
9;Computational Techniques for Engineering Sciences;314
9.1;Image Processing for the Estimation of Drop Distribution in Agitated Liquid-Liquid Dispersion;315
9.1.1;Introduction;315
9.1.2;Description of the Method;316
9.1.2.1;Edges Detection;316
9.1.2.2;Detection of Drops in the Contour Image;318
9.1.3;Results and Discussions;319
9.1.4;Conclusions and Future Work;321
9.1.5;References;321
9.2;Music and Evolutionary Computation;322
9.2.1;A Brief History of the Western Music;322
9.2.2;Music, Mathematics and Physics;323
9.2.3;Foundations versus Styles;324
9.2.4;Music and Art;325
9.2.5;Music and Algorithms;325
9.2.6;Evolutionary Computation;326
9.2.6.1;Genetic Algorithms (GAs);326
9.2.6.2;Genetic Programming (GPs);327
9.2.6.3;Particle Swarm Optimization (PSO);327
9.2.7;Applications of EAs to Computer Music;328
9.2.8;Conclusions and Future Work;329
9.2.9;References;329
9.3;Application of Computational Intelligence to Engineering;330
9.3.1;Introduction;330
9.3.2;Identification of Material Constants with NNs;331
9.3.2.1;Problem Definition and NN Model;332
9.3.2.2;Results;333
9.3.3;Equipment Fault Diagnosis with Fuzzy Systems;333
9.3.3.1;Problem Definition and Fuzzy Model;334
9.3.3.2;Results;335
9.3.4;Digital Circuit Synthesis with EC;335
9.3.4.1;Problem Definition and the Genetic Algorithm;336
9.3.4.2;Results;337
9.3.5;Conclusion;337
9.3.6;References;338
9.4;Evolutionary Trajectory Optimization for Redundant Robots;339
9.4.1;Introduction;339
9.4.2;Kinematics of Redundant Manipulators;340
9.4.3;Robot Trajectory Control;341
9.4.3.1;The CLGA Formulation;341
9.4.3.2;Representation and Operators in the CLGA;342
9.4.3.3;Optimization Criteria;343
9.4.4;Simulation Results;343
9.4.5;Conclusions;345
9.4.6;References;345
10;Nonlinear Systems;346
10.1;Robust Communication-Masking via a Synchronized Chaotic Lorenz Transmission System;347
10.1.1;Introduction;347
10.1.2;The Lorenz System;348
10.1.3;Master-Slave Synchronization Using a Luenberger-Type Observer;349
10.1.3.1;Robustness Study;350
10.1.4;Modified Lorenz-Based Observer;352
10.1.5;Conclusions;355
10.1.6;References;355
10.2;A Boundary Layer Problem in Power Law Fluids through a Moving Flat Plate;356
10.2.1;Introduction;356
10.2.2;Proofs of Two Theorems;357
10.2.2.1;Proof of Theorem 1;362
10.2.2.2;Proof of Theorem 2;362
10.2.3;References;362
10.3;An Overview of the Behaviour of a Scattering Map for the Dynamics of Two Interacting Particles in a Uniform Magnetic Field;364
10.3.1;Introduction;364
10.3.2;The Planar Problem;365
10.3.3;The Spatial Problem;365
10.3.4;The Scattering Map for the Spatial Problem;366
10.3.5;Possible Applications;367
10.3.6;References;368
10.4;A Generalised Entropy of Curves Approach for the Analysis of Dynamical Systems;369
10.4.1;Introduction;369
10.4.2;The Generalised Entropy of Curves and its Application;370
10.4.2.1;Properties of the Generalised Entropy;372
10.4.3;A Comparison with Other Chaotic Indicators;372
10.4.4;Simulation Examples;373
10.4.5;Conclusions;375
10.4.6;References;376
10.5;Uncertainty on a Bertrand Duopoly with Product Differentiation;377
10.5.1;Introduction;377
10.5.2;The Model and the Equilibrium;378
10.5.3;Conclusions;382
10.5.4;References;382
10.6;Price-Setting Dynamical Duopoly with Incomplete Information;384
10.6.1;Introduction;384
10.6.2;The Model and the Equilibrium;385
10.6.3;Conclusions;389
10.6.4;References;390
10.7;Inductor-Free Version for Chua's Oscillator Based in Electronic Analogy;391
10.7.1;Introduction;391
10.7.2;Electronic Analogy;393
10.7.3;Design of an Analogous Chua's Oscillator;394
10.7.4;Experimental Implementation;396
10.7.5;Conclusions;396
10.7.6;References;397
10.8;Model Reduction of Nonlinear Continuous Dynamic Systems on Inertial Manifolds with Delay;398
10.8.1;Introduction;398
10.8.2;Inertial Manifolds with Time Delay;399
10.8.3;Governing Equations of Shallow Arch under Impact;400
10.8.4;Numerical Examples;401
10.8.5;Conclusions;402
10.8.6;References;403
10.9;A Fuzzy Crisis in a Duffing-Van der Pol System;404
10.9.1;Introduction;404
10.9.2;A Fuzzy Crisis in a Duffing-Van der Pol System;405
10.9.3;Concluding Remarks;407
10.9.4;References;408
11;Name Index;410