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Delay Differential Equations (DDEs) are a crucial class of mathematical models used to describe systems where the rate of change is influenced by past states, not solely the current state. These time-delay systems are vital for accurately capturing phenomena across engineering, biology, and control theory, particularly when analyzing stability analysis DDE and understanding the complex dynamical systems with delay. Investigating nonlinear delay equations allows for a deeper understanding of behaviors that might involve amplification or oscillations, providing insights into system responses over time.

This document explores the fundamental principles of oscillation theory as applied to neutral differential equations with delay. It provides a comprehensive analysis of the oscillatory behavior of solutions, crucial for understanding the stability and long-term dynamics of complex systems modeled by functional differential equations, offering vital insights for mathematical analysis in various scientific and engineering applications.