Nonlinear Science and Complexity

Preface

Contents

Nonlinear Dynamics of Continuous and Discontinuous Dynamical Systems

On Synchronization and Its Complexity of Multiple Dynamical Systems

Introduction

Basic Concepts

Discontinuous Descriptions

Complexity by System Synchronization

References

Periodic and Chaotic Motions in a Gear-pair Transmission System with Impacts

Introduction

Equations of Motion

Switching Sets and Mappings

Mapping Structures

Parameter Maps and Illustrations

Conclusions

References

Analytical Prediction of Interrupted Cutting Periodic Motions in a Machine Tool

Introduction

Mechanical Model

Motion Switchability Conditions

Mappings Structure

Numerical and Analytical Predictions

Summary

Appendix

References

Control of Hopf Bifurcation and Chaos as Applied to Multimachine System

Introduction

System Description

Mathematical Model

System Response without Controller

Control of Hopf Bifurcation and Chaos

Control of Critical Modes

Control of Hopf Bifurcation

Numerical Simulation Results

Linear Controller

Nonlinear Controller

References

Lie Group Analysis and Applications in Nonlinear Sciences

Group-Invariant Solutions of Fractional Differential Equations

Introduction

Transformation Groups and Symmetries of FDE

Group Classification of Equations Dalphax y = f(x,y)

Exact Solutions of Equation Dalpha y =f(x,y)

References

Type-II Hidden Symmetries for Some Nonlinear Partial Differential Equations

Introduction

Weak Symmetries for the Model Equation

Linear Three-Dimensional Wave Equation

References

Nonclassical and Potential Symmetries for a Boussinesq Equation with Nonlinear Dispersion

Introduction

Nonclassical Symmetries

Classical Potential Symmetries

Nonclassical Potential Symmetries

Concluding Remarks

References

Application of the Composite Variational Principle to Shallow Water Equations

Introduction

Symmetry Group Analysis of the Shallow Water Equations in the Plane Flow

Derivation of Conservation Laws

Conclusion

References

Conserved Forms of Second Order-Ordinary Differential Equations

Introduction

lambda-Symmetries Associated to Integrating Factors

Integrating Factors Associated to lambda-Symmetries

Algorithm to Calculate Integrating Factors Based on lambda-Symmetries

Conclusions

References

Analytical Investigation of a Two-Phase Model Describing a Three-Way-Catalytic Converter

Introduction

The Group Theoretical Approach

Reductions and Solutions

Conclusion

References

Celestial Mechanics and Dynamical Astronomy: Methods and Applications

The Role of Invariant Manifolds in the Formation of Spiral Arms and Rings in Barred Galaxies

Introduction

Results

References

Continuous and Discrete Concepts for Detecting Transport Barriers in the Planar Circular Restricted Three Body Problem

Introduction

Methods and Results

Finite-Time Lyapunov Exponents

Graph Partitioning Techniques

Graph Based Expansion

Discussion

References

Low-Energy Transfers in the Earth-Moon System

Introduction

The Model

Collinear Points Dynamics

Methodology

Rescue Orbits

Results

LEO Transfers

Results

Conclusions

References

Gravitational Potential of a Massive Disk. Dynamics Around an Annular Disk

The Massive Disk and Its Potential Function

Dynamics Around a Circular Annulus

References

An Accounting Device for Biasymptotic Solutions: The Scattering Map in the Restricted Three Body Problem

References

Optimal Capture Trajectories Using Multiple Gravity Assists

Introduction

The Keplerian Map

Control Problem Formulation

Optimal Feedback

Discretization

Low Energy Multiple Gravity Assists

Conclusion

References

New Periodic Orbits in the Solar Sail Three-Body Problem

Introduction

Equations of Motion in the Rotating Frame

Linearised System

High-Order Approximations to Periodic Orbits

The Solar Sail ERTBP

Stability of Periodic Orbits in the Solar Sail RTBP

Conclusion

References

A Review of Invariant Manifold Dynamics of the CRTBP and Some Applications

The Models

Lindstedt Poincaré and Normal Forms Techniques

Normal Forms

Applications to Classical Problems of Libration Point Mission Design

References

Solar Sail Orbits at the Earth-Moon Libration Points

Introduction

Solar Sail in the Earth-Moon Restricted Three-Body Problem

Qualitative Approach

Equations of Motion in Presence of Solar Sail

Solution of the Linearized Equations of Motion

Numerical Integration of the Nonlinear Equations of Motion

Conclusion

References

Mathematical Modeling of Nonlinear Structures in Bose-Einstein Condensates

Collisions of Discrete Breathers in Nonlinear Schrödinger and Klein-Gordon Lattices

Introduction

DNLS Lattices

Klein-Gordon Lattices

Interpretation

References

Stability of BEC Systems in Nonlinear Optical Lattices

Introduction

Conservative Systems (gamma2 = 0)

Attractive Condensate (gamma0gamma1 + 1/2)

Repulsive Condensate (gamma0gamma1 - 1/2)

Dissipative Systems

Conclusions

References

Nonlinear Schrödinger Equations with a Four-Well Potential in Two Dimensions: Bifurcations and Stability Analysis

Introduction

The Analytical Approach

Numerical Results

Conclusions and Future Challenges

References

Bose-Einstein Condensates and Multi-Component NLS Models on Symmetric Spaces of BD.I-Type. Expansions over Squared Solutions

Introduction

MNLS Equations for BD.I. Series of Symmetric Spaces

The Inverse Scattering Problem

The Generalized Fourier Transforms for Non-regular J

Fundamental Properties of the MNLS Equations

References

Mathematical Models in Engineering

Impulsive Boundary Layer Flow Past a Permeable Quadratically Stretching Sheet

Introduction

Governing Equations

Results and Discussion

Case (i): Linear Injection (s <=1 and b >=0)

Case (ii): Linear Suction (s >=1 and b <=0)

Case (iii): Linear Suction for x < xc and Linear Injection for x > xc (s > 1 and b > 0)

Case(iv): Linear Injection for x < xc and Linear Suction for x > xc (s < 1 and b < 0)

Note on the Value of s

Conclusions

References

Complete Dynamic Modeling of a Stewart Platform Using the Generalized Momentum Approach

Introduction

Stewart Platform Kinematic Structure

Dynamic Modeling Using the Generalized Momentum

Moving Platform Modeling

Cylinder Modeling

Piston Modeling

Dynamic Model Gravitational Components

Conclusions

References

Numerical Solution of a PDE System with Non-Linear Steady State Conditions that Translates the Air Stripping Pollutants Removal

Background

The Differential Model

The Boundary Conditions

The Steady State

The Transient State

The Lanczos's Tau Method

Initial Conditions and Differential Equations

Algebraic Equations

Conclusions and Future Work

References

Three Behavioural Scenarios for Contingent Claims Valuation in Incomplete Markets

Introduction

Three Scenarios for Price Selection

Allocation of Wealth and Indifference Pricing

Market Games Approach

The Risk Sharing Approach

Optimal Choice of the Agents Market Price of Risk

Conclusion

References

Undesired Oscillations in Pneumatic Systems

Introduction

Pneumatic System

Experimental Setup

Nonlinear Model

Pneumatic Chamber Model

Servovalve Model

Mechanical Model

Linearised Model

Friction Generated Oscillations

Describing Function Analysis

Limit Cycle Prediction

Experimental Results

Pressure Dynamics Generated Oscillations

Conclusions

References

A Study of Correlation and Entropy for Multiple Time Series

Introduction

Goals

Entropy

Covariance

Data

Conclusions

References

Characterization and Parameterization of the Singular Manifold of a Simple 6-6 Stewart Platform

Introduction

Forward Kinematics

Non-singular Case

Singular Case

Conclusions

References

Fractional Calculus Applications

Some Advances on Image Processing by Means of Fractional Calculus

Introduction

Time Discretization

Numerical Experiments

Conclusions

References

Application of Genetic Algorithms in the Design of an Electrical Potential of Fractional Order

Introduction

Integer and Fractional Electrical Potential

Conclusions

References

Mellin Transform for Fractional Differential Equations with Variable Potential

Introduction

Fractional Operators

The Mellin Transform and Its Properties

Fractional Linear Equation with Riemann-Liouville Derivative and tbeta-Potential

Example: Solution for Case beta=0

Example: Solution for Case beta= -alpha/2

Fractional Linear Equation with Caputo Derivative and tbeta-Potential

Nonhomogeneous Fractional Equations with tbeta-Potential

Final Remarks

References

Phase Plane Characteristics of Marginally Stable Fractional Order Systems

Introduction

Analysis of the Phase Plane

Analysis of Phase Plane in Marginally Stable LTI Integer Order Systems

Analysis of Phase Plane in Marginally Stable LTI Fractional Order Systems

Simulation Results

Conclusion

References

Application of Fractional Controllers for Quad Rotor

Introduction

Helicopter Model

Fractional Control

The Flight Simulator

Conclusions

References

Regularity of a Degenerated Convolution Semi-Group Without to Use the Poisson Process

Introduction

Proof of Bismut's Theorem Without to Use the Poisson Process

Proof of Léandre's Theorem Without to Use the Poisson Process

References

Computational Techniques for Engineering Sciences

Image Processing for the Estimation of Drop Distribution in Agitated Liquid-Liquid Dispersion

Introduction

Description of the Method

Edges Detection

Detection of Drops in the Contour Image

Results and Discussions

Conclusions and Future Work

References

Music and Evolutionary Computation

A Brief History of the Western Music

Music, Mathematics and Physics

Foundations versus Styles

Music and Art

Music and Algorithms

Evolutionary Computation

Genetic Algorithms (GAs)

Genetic Programming (GPs)

Particle Swarm Optimization (PSO)

Applications of EAs to Computer Music

Conclusions and Future Work

References

Application of Computational Intelligence to Engineering

Introduction

Identification of Material Constants with NNs

Problem Definition and NN Model

Results

Equipment Fault Diagnosis with Fuzzy Systems

Problem Definition and Fuzzy Model

Results

Digital Circuit Synthesis with EC

Problem Definition and the Genetic Algorithm

Results

Conclusion

References

Evolutionary Trajectory Optimization for Redundant Robots

Introduction

Kinematics of Redundant Manipulators

Robot Trajectory Control

The CLGA Formulation

Representation and Operators in the CLGA

Optimization Criteria

Simulation Results

Conclusions

References

Nonlinear Systems

Robust Communication-Masking via a Synchronized Chaotic Lorenz Transmission System

Introduction

The Lorenz System

Master-Slave Synchronization Using a Luenberger-Type Observer

Robustness Study

Modified Lorenz-Based Observer

Conclusions

References

A Boundary Layer Problem in Power Law Fluids through a Moving Flat Plate

Introduction

Proofs of Two Theorems

Proof of Theorem 1

Proof of Theorem 2

References

An Overview of the Behaviour of a Scattering Map for the Dynamics of Two Interacting Particles in a Uniform Magnetic Field

Introduction

The Planar Problem

The Spatial Problem

The Scattering Map for the Spatial Problem

Possible Applications

References

A Generalised Entropy of Curves Approach for the Analysis of Dynamical Systems

Introduction

The Generalised Entropy of Curves and its Application

Properties of the Generalised Entropy

A Comparison with Other Chaotic Indicators

Simulation Examples

Conclusions

References

Uncertainty on a Bertrand Duopoly with Product Differentiation

Introduction

The Model and the Equilibrium

Conclusions

References

Price-Setting Dynamical Duopoly with Incomplete Information

Introduction

The Model and the Equilibrium

Conclusions

References

Inductor-Free Version for Chua's Oscillator Based in Electronic Analogy

Introduction

Electronic Analogy

Design of an Analogous Chua's Oscillator

Experimental Implementation

Conclusions

References

Model Reduction of Nonlinear Continuous Dynamic Systems on Inertial Manifolds with Delay

Introduction

Inertial Manifolds with Time Delay

Governing Equations of Shallow Arch under Impact

Numerical Examples

Conclusions

References

A Fuzzy Crisis in a Duffing-Van der Pol System

Introduction

A Fuzzy Crisis in a Duffing-Van der Pol System

Concluding Remarks

References

Name Index