The Duffing equation stands as one of the fundamental nonlinear models in applied mathematics, encapsulating the essential mechanisms of stiffness nonlinearity, resonance, instability, and transition to chaos. Beyond its classical role in nonlinear vibration theory, the Duffing framework provides a unifying mathematical structure underlying a broad class of mechanical, electrical, plasma, and wave phenomena.This book develops a rigorous and modern treatment of Duffing-type equations and their principal generalizations, including cubic–quintic oscillators, mixed-parity (Duffing–Helmholtz) models, linearly and nonlinearly coupled systems, parametrically driven oscillators, and fractional-order extensions. Analytical methods are presented in a systematic and self-contained manner, with particular emphasis on perturbation techniques, averaging and multiple-scale methods, nonlinear normal modes, and exact and semi-exact solutions expressed in terms of elliptic and special functions. These analytical developments are complemented by carefully designed numerical schemes suitable for strongly nonlinear and memory-dependent systems.A distinctive feature of the volume is the consistent interplay between mathematical structure and physical interpretation. Stability, bifurcation, internal resonance, and chaotic dynamics are analyzed within a unified framework, revealing deep connections between seemingly disparate models. By bridging abstract nonlinear analysis with concrete applications, this book provides a comprehensive reference for researchers and advanced graduate students in applied mathematics, physics, and engineering seeking a mathematically rigorous and application-driven understanding of nonlinear oscillatory systems.
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