Nonlinear Dynamics of Time Delay Systems: Methods and Applications

✍ Scribed by Jian Xu

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 492
• Edition: 1st ed. 2024
• Category: Library

✦ Synopsis

This book presents research advancements in the dynamics of systems with time delay conducted by the group led by Professor Jian Xu. Addressing the challenges arising from the joint impact of time delay and nonlinearity, novel theoretical approaches are developed to formulate the nonlinear response of the system. This facilitates the classification of complex nonlinear dynamics, especially the non-resonant and resonant double Hopf bifurcation. In contrast to systems without time delay, time delay systems require specific considerations when identifying system parameters, particularly the time delay. Consequently, inverse problems of systems with time delay are also explored in this book. Moreover, detailed investigations on vibration suppression methods and experimental prototypes based on time delay, such as time delay isolators with quasi-zero stiffness, are conducted. Simultaneously, this book is enriched with a large number of case studies ranging from manufacturing, network science, biology, and public transportation, illuminating the mechanisms of delay-induced nonlinear dynamics in practical applications. This book is suitable for graduate students and researchers who are eager to understand the delay-induced nonlinear dynamics, or technical personnel in whose projects small variations of time delay may cause significant changes in system responses.

✦ Table of Contents

Preface
Acknowledgments
Contents
About the Author
1 Introduction
1.1 Brief Review of Nonlinear Ordinary Differential Equations
1.2 Brief Review of Differential Equations with Time Delay
1.3 Main Challenge of Study on Nonlinear Dynamics of Systems with Time Delay
1.3.1 Nonlinear Dynamics of Time Delay Systems
1.3.2 Methods of Solving Nonlinear Systems with Time Delay
1.3.3 Identification of Nonlinear Systems with Time Delay
1.3.4 Utilization of Time Delay in Vibration Suppression
1.3.5 Other Applications in the Presence of Time Delay
1.4 Structure of the Book
References
2 Delay Induced Nonlinear Dynamics
2.1 Asymptotic Analytical Method for Periodic Solution of the Delay Induced VDP-Duffing System
2.1.1 Hopf Bifurcation and Btability Bwitching
2.1.2 Reduction on Center Manifold
2.1.3 The Phase-Shifting and Phase-Locked Periodic Solutions
2.2 1:3 Resonant Double Hopf Bifurcation in a Delayed Coupled VDP System
2.2.1 Multi-scale Analysis for 1:3 Resonance Double Hopf Bifurcation
2.2.2 1:3 Resonance Analysis of Time Delay Coupled VDP Oscillator System
2.2.3 Numerical Simulation
2.3 Multistability and Attraction Basin of VDP-Duffing System with Delayed Feedback
2.3.1 Mathematical Model
2.3.2 Approximated Analytical Solution of Periodic Solution
2.3.3 Basin of Attraction for Periodic Solutions of Multiple Steady States
2.4 Quasi-Periodic Solutions and Chaos in Time Delay Systems
References
3 Perturbation-Incremental Scheme and Integral Equation Method for Solving Time Delay Systems
3.1 Perturbation-Incremental Scheme for Studying Hopf Bi-Furcation in Delay Differential Systems
3.1.1 Introduction
3.1.2 Perturbation-Incremental Scheme
3.1.3 Synchronization Solution in a Network of Three Identical Neurons
3.2 Perturbation-Incremental Scheme for Studying Weak Resonant Double Hopf Bifurcation of Nonlinear Systems with Time Delay
3.2.1 Weak Resonant Double Hopf Bifurcation
3.2.2 Perturbation-Incremental Scheme
3.2.3 Weak Resonant Double Hopf Bifurcation in the Van Der Pol–Duffing System with Delayed Feedback
3.3 Integral Equation Method for Nonlinear Systems with Time Delay
3.3.1 The Integral Equation Method
3.3.2 The Primary Resonance and 1:1 Internal Resonance of a Two-Degrees-of-Freedom Model
Appendix
References
4 Inverse Problem of Systems with Time Delay
4.1 Characteristics of Time Delay in the Frequency Domain
4.2 Parameter Identification for Linear Time Delay Systems
4.2.1 Qualitative Identification for Large Time Delays
4.2.2 Quantitative Identification for Practical Applications
4.2.3 Uniqueness in Delay Identification and Its Solution
4.3 Parameter Identification for Nonlinear Time Delay Systems
4.3.1 Algorithm Construction in Frequency Domain: Quasi-Linear Method
4.3.2 Algorithm Construction in Frequency Domain: Harmonic Balance Method
4.4 Algorithm Modification for Noise-Correction Identification
4.4.1 Problem Representation with Noise Correction
4.4.2 Local Linearization and Regularization
4.4.3 Algorithm Construction
4.4.4 Convergence Analysis
4.5 Experiment Realization and Validations
4.5.1 Realization of the Time-Delayed Control via YASKAWA Hardware
4.5.2 Identification Experiment of a Linear Time Delay System
4.5.3 Identification Experiment of a Nonlinear Time Delay Nonlinear System
4.6 Conclusions
References
5 Time-Delayed Control of Vibration
5.1 Effect of Time Delay on Vibration Isolation
5.1.1 Stability Criteria
5.1.2 Real and Imaginary Parts for Different Time Delays
5.1.3 Optimal Control for Time-Delayed Control
5.2 Time-Delayed Vibration Isolator
5.2.1 Asymmetrical Isolation System
5.2.2 Adjustable Time Delay
5.3 Experimental Investigation on the Time-Delayed Vibration Absorber
5.3.1 Time-Delayed Vibration Absorber
5.3.2 Time-Delayed Absorber for Nonlinear System
5.3.3 Time-Delayed Absorber for M-DOF System
5.3.4 Time-Delayed Absorber for Continuous Structure
References
6 Effects of Time Delay on Manufacturing
6.1 Modeling of Cutting Dynamics by Delay Differential Equation
6.1.1 Cutting Dynamics with Single Time Delay—Turning
6.1.2 Cutting Dynamics with Two Time Delays—Grinding
6.1.3 Cutting Dynamics with Three Time Delays—Parallel Grinding
6.2 Analysis of Nonlinear Cutting Chatter Based on Time Delay Model
6.2.1 Grinding Chatter with External Excitation—Workpiece Imbalance
6.2.2 Grinding Chatter with Time-Varying Parameters—Transverse Grinding
6.3 Estimate of Cutting Safety by Time Delay
6.3.1 Unsafe Cutting and Unsafe Zones
6.3.2 Statistical Basin of Attraction (SBoA)
References
7 Effect of Time Delay on Network Dynamics
7.1 Chaotic Oscillation of Time Delay Coupled Wilson-Cowan Neural Oscillator System
7.1.1 Delay-Coupled Wilson-Cowan Neural Oscillator System
7.1.2 Periodic Oscillation Under the Effect of Time Delay
7.1.3 Torus and Chaotic Oscillations
7.2 Fold-Hopf Bifurcation and Approximated Computation of the Synchronous Periodic Solutions in the BAM Network System
7.2.1 System Model
7.2.2 Fold-Hopf Bifurcation
7.2.3 Delay-Induced Approximated Computation of Synchronous Periodic Solutions
7.3 Hyperchaos and Synchronous Control of Neural Network System with Time Delay
7.3.1 Hyperchaotic Synchronization in Neural Networks with Time Delay
7.3.2 Adaptive Synchronization with Uncertain Parameters
7.3.3 Projective Synchronization Based on Sliding Mode Variable Structure Control
7.4 Nonlinear Dynamics of Internet Congestion Control Model with Time Delay
7.4.1 Time-Varying Delayed Feedback Control of an Internet Congestion Control Model
7.4.2 Oscillation Suppression Through Periodic Delay
7.4.3 Analytical Solution of Time-Varying Delayed System (7.56)
7.4.4 Discussion on the Oscillation Control of System (7.54)
7.4.5 Discussions on Cases for Different Positive Gain Parameter k in System (7.54)
7.4.6 Utilization of the Bursting-Like Phenomenon
7.4.7 Generalization to the Case with n Users
7.5 Oscillatory Dynamics Induced by Time Delay in an Internet Congestion Control Model with a Ring Topology
7.5.1 Model of Congestion Control for a Ring Network
7.5.2 Analysis on the Stability of the Equilibrium
7.5.3 Study on the Periodic Motion Induced by the Delay Through the Method of Multiple Scales
7.5.4 Effects of M: Long Transmission Distance Will Induce Oscillation
7.6 Desynchronization-Based Congestion Suppression for a Star-Type Internet System
7.6.1 Model Setup
7.6.2 Critical Conditions for Stability Switch and Oscillatory Patterns
7.6.3 Method of Multiple Scales
7.6.4 Examples
References
8 Delay Effect in Biology
8.1 Dynamic Analysis of a Coupled FitzHugh-Nagumo Neural System with Time Delay
8.1.1 A Single FHN Neuron Model and Delay Coupled System
8.1.2 Analysis on Eigenvalues
8.1.3 Numerical Simulations
8.2 Effects of the Technological Delay and Physiological Delay on the Insulin and Blood Glucose System
8.2.1 Mathematical Model
8.2.2 Double Hopf Bifurcation Analysis
8.3 Effects of Time Delay and Noise on Asymptotic Stability in Human Quiet Standing Model
8.3.1 Model Formulation
8.3.2 Asymptotically Stable Analysis
8.4 Pattern Dynamics of Population Reaction–Diffusion Models with Spatiotemporal Delay
8.4.1 Single Species Reaction–Diffusion Model
8.4.2 Predator–Prey Reaction–Diffusion Model
8.4.3 Three-Species Food Chain System and Chaotic Behaviour
References
9 Impact of Time Delay on Traffic Flow
9.1 Full Velocity Difference Model and Traffic Patterns
9.1.1 Time Delay Full Velocity Difference Model
9.1.2 Criteria for Traffic Jams and Traffic Patterns
9.1.3 Example
9.2 Bistable Traffic Patterns Induced by Reaction Time Delay
9.2.1 Analysis on Stability of Uniform Flow
9.2.2 Bistable Phenomenon Induced by Subcritical Hopf Bifurcation
9.2.3 Examples
9.3 Control of Traffic Jam by Time-Varying Delay
9.3.1 Model Setup
9.3.2 Traffic Jam Mode Under Constant Delay
9.3.3 Suppress Traffic Jam Through Time-Varying Delay
Appendix
References
10 Nonlinear Dynamics of Car-Following Model Induced by Time Delay and Other Parameters
10.1 Traffic Modes Induced by Reaction Delay and Road Length
10.1.1 Stability in the Parameter Plane Defined by the Average Inter-Vehicle Distance and Driver’s Response Time Delay
10.1.2 Classification of Nonlinear Dynamics in Two-Parameter Plane
10.1.3 Explanations from the Perspective of Physics
10.1.4 Conclusion
10.2 Traffic Modes Induced by Reaction Time Delay and Sensitivity Coefficients
10.2.1 Stability in the Plane of Sensitivity Coefficient and Reaction Delay
10.2.2 Dynamics Classification
10.2.3 Modes of Traffic Flow Corresponding to the Solutions in Different Regions
Appendix
References