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2024 | Book

Nonlinear Dynamics of Time Delay Systems

Methods and Applications

Author: Jian Xu

Publisher: Springer Nature Singapore

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About this book

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

Frontmatter
Chapter 1. Introduction
Abstracts
Due to the ubiquity of time delay in real systems, it has been an important topic of concern to scientists in many fields such as mathematics, mechanics, high-precision mechanical manufacturing engineering, biology, etc. Investigations of systems with time delays are essential due to their significant impact on the stability, performance, and dynamical behaviour across various domains. Time delays are inherent in numerous real-world processes ranging from natural sciences to engineering applications, and understanding the effect induced by time delays is crucial for understating underlying mechanisms or designing effective control strategies to ensure the reliability of complicated artificial systems.
Jian Xu
Chapter 2. Delay Induced Nonlinear Dynamics
Abstract
Since the 1970s, with the development of computer technology and numerical methods, researchers have gradually recognized that parameter perturbations can lead to complex dynamic behaviors in nonlinear dynamic systems. Based on the typical nonlinear examples commonly found in mechanics and physics, numerous new nonlinear phenomena, including chaos and fractals, have been discovered. The corresponding fundamental theories and methods have been proposed and established.
Jian Xu
Chapter 3. Perturbation-Incremental Scheme and Integral Equation Method for Solving Time Delay Systems
Abstract
This chapter presents two new methods for analyzing dynamics of nonlinear systems with time delay. The first method is called the perturbation-incremental scheme (PIS). The perturbation step and incremental step, which are the two key components of the PIS, allow us to track the bifurcated solution arising from Hopf or double Hopf bifurcation with high accuracy, without the tedious calculations needed for center manifold reduction or normal form computation. In doing so, the PIS avoides the shortcomings of the traditional incremental harmonic balance method in addition to inheriting the advantages of the method of multiple scales.
Jian Xu
Chapter 4. Inverse Problem of Systems with Time Delay
Abstract
As mentioned in the preface, time delay identification in the time domain (Chai et al. in Appl Math Comput 219:9543–9560, 2013; Hartung in Discrete Contin Dyn Syst Ser B 18:1611–1631, 2013; Lin et al. in Automatica 60:48–56, 2015; Abooshahab et al. in J Optim Theory Appl 180:1036–1064, 2019; Zhang et al. in Nonlinear Dyn 104:4045–4061, 2021 [15]) may be unstable when the initial estimation is improperly set.
Jian Xu
Chapter 5. Time-Delayed Control of Vibration
Abstract
Active control can effectively inhibit vibration responses in aerospace and vehicle engineering applications (Xu and Pei in Advances in Mechanics 36:17–30, [1]). Notably, the control loop’s time lag cannot be overlooked as it significantly affects the system’s stability and vibration characteristics, which can alter fluidly with the adjustment of variables like time delay and control strength.
Jian Xu
Chapter 6. Effects of Time Delay on Manufacturing
Abstract
Manufacturing effectiveness has the potential for significant growth, yet its progress is impeded by the dynamics of the manufacturing processes. Specifically, the material removal rate (MRR) in machining processes experiences a reduction to avoid chatter. Consequently, the anticipation and management of chatter have emerged as prominent subjects over the last two decades. Cutting instabilities have frictional, regenerative, mode-coupling, and thermo-mechanical types, where the primary (frictional) and secondary (regenerative) are the most significant ones (Wiercigroch and Budak in Philosoph Trans R Soc Lond Ser A Math Phys Eng Sci 359(1781):663–93, [1]). These groups have extensively investigated self-excited chatter within turning (Fu et al. in Int J Mech Sci 88:145–153, [2]) and other manufacturing processes.
Jian Xu
Chapter 7. Effect of Time Delay on Network Dynamics
Abstract
Network systems widely exist in human and nature society, such as the internet, electric power, transportation, and interpersonal relationships (Bullo in Lectures on network systems. Kindle Direct Publishing Seattle, DC, USA, 2020). It can be said that any system composed of the same or different individuals can be represented by a network. When the individuals and the interactions between them are respectively abstracted as nodes and edges, these individuals can be represented by a network system.
Jian Xu
Chapter 8. Delay Effect in Biology
Abstract
In fact, life phenomena are far more complicated than physical and mechanical phenomena. In the study of life science, dynamical systems have been developed and successfully used in a variety of subjects, including physics, mechanics, and life science (Sriram K, Rodriguez-Fernandez and Doyle, LoS ONE 7:e42958 (2012); Parmar et al., Comput Math Methods Med 2015:347273 (2015); Bonella et al., Phys Chem 22:10775–10785 (2020) [13]). Especially since the late 1970s, the discipline has developed rapidly, and it has penetrated almost every branch of life science, which promotes the development of life science from micro to macro.
Jian Xu
Chapter 9. Impact of Time Delay on Traffic Flow
Abstract
Since the 1930s, there has been a consistent increase in human’s demand for transportation. A substantial number of vehicles have been manufactured and integrated into road transportation operations. However, due to limited transportation resources, an imbalance between transportation supply and demand has arisen, leading to the emergence of issues in traffic flow. Among these issues, traffic jam causes concern.
Jian Xu
Chapter 10. Nonlinear Dynamics of Car-Following Model Induced by Time Delay and Other Parameters
Abstract
Chapter 9 explores the bistable solution induced by bifurcation and presents the corresponding bistable traffic mode by studying the subcritical Hopf bifurcation when the driver’s reaction delay is near its critical value. The research is conducted when other parameters such as sensitivity coefficient and inter-vehicle distance are fixed, with only driver’s reaction delay being allowed to vary near the critical value. The study finds that the mechanisms of stability switch of uniform flow are dependent on the road length as well as the sensitivity coefficient of the driver. Therefore, this chapter proposes to analyze the dynamic behaviour induced by the bifurcation and the corresponding traffic modes under the joint influence of driver’s reaction time delay and road length/sensitivity coefficient.
Jian Xu
Metadata
Title
Nonlinear Dynamics of Time Delay Systems
Author
Jian Xu
Copyright Year
2024
Publisher
Springer Nature Singapore
Electronic ISBN
978-981-9999-07-1
Print ISBN
978-981-9999-06-4
DOI
https://doi.org/10.1007/978-981-99-9907-1

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