Introduction

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.