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Description
This book presents a comprehensive and practical survey of averaging methods for differential equations. Combining rigorous theory with applied perspectives, this book serves as both a study text and a reference for mathematicians and scientists in fields such as engineering, physics, and biology.
Divided into two complementary parts, the book begins with Part I, the Toolbox of Averaging Theorems, providing clear definitions, theorem formulations, and foundational results. While mathematicians may be content with existence proofs and qualitative analyses, applied scientists require tools that link theory to real-world problems an essential motivation for Part II.
Part II explores applications in physics and engineering, blending theory with practice and incorporating numerical bifurcation analysis using tools such as AUTO, Mathematica, and MatCont. Interspersed theoretical interludes provide the background necessary for understanding and applying these methods.
Highlights include:
Hamiltonian systems (Ch. 9), examining resonance phenomena in physics and engineering.
Fermi-Pasta-Ulam chains (Ch. 10), extending fundamental theory.
Parametric excitation (Ch. 11) and dissipation-induced instability (Ch. 13), showcasing classical but lesser-known engineering results.
Coupled oscillators and chaos (Ch. 12), a detailed exploration of complex nonlinear dynamics.
Diffusion and waves (Ch. 14), providing essential guidance while pointing to broader material for further study.
Whether as a reference, teaching aid, or bridge between theory and application, Averaging for Nonlinear Dynamics equips readers with the tools to analyze, approximate, and apply nonlinear systems across a wide range of scientific disciplines.
Introduction.- First order periodic averaging.- Periodic solutions.- Second order periodic averaging.- First order general averaging.- Approximations on timescales longer than 1/ .- Averaging over angles.- Averaging for partial differential equations.- Hamiltonian systems.- Fermi-Pasta-Ulam chains.- Parametric and autoparametric oscillations.- Interactions, bifurcations and chaos.- Instability induced by dissipation.- Diffusion and waves.
Ferdinand Verhulst graduated in theoretical astrophysics in 1966 at the University of Amsterdam, and he got his Ph.D. in mathematics at Utrecht University in 1973. His main interest and research are on approximation theory of solutions of differential equations and applications in physics and engineering. The author wrote a number of books on these topics, partly co-authored; also he edited a number of lecture notes in mathematics and conference proceedings. He is now still active in these fields as Emeritus Professor of nonlinear dynamics at Utrecht University.
