Averaging for nonlinear dynamics with applications and numerical bifurcations : parametric and autoparametric systems, Hamiltonian systems, FPU systems, coupled oscillators and chaos / Ferdinand Verhulst.

Format:

Book

Author/Creator:

Verhulst, F. (Ferdinand), 1939- Author.

Series:

Applied mathematical sciences (Springer-Verlag New York Inc.) ; 2196-968X volume 223.

Applied mathematical sciences, 0066-5452 ; volume 223

Language:

English

Subjects (All):

Averaging method (Differential equations).

Nonlinear systems.

Physical Description:

xiii, 276 pages : illustrations (some color) ; 25 cm.

Place of Publication:

Cham, Switzerland : Springer, [2026]

Summary:

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.

Contents:

First order periodic averaging

Periodic solutions

Second order periodic averaging

First order general averaging

Approximations on timescales longer than 1/ε

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.

Notes:

Includes bibliographical references (pages 265-271) and index.

ISBN:

3032127440

9783032127440

OCLC:

1547287149