Asymptotic Approaches in Nonlinear Dynamics

1. Introduction: Some General Principles of Asymptotology..- 1.1 An Illustrative Example.- 1.2 Reducing the Dimensionality of a System.- 1.3 Continualization.- 1.4 Averaging.- 1.5 Renormalization.- 1.6 Localization.- 1.7 Linearization.- 1.8 Padé Approximants.- 1.9 Modern Computers and Asymptotic Methods.- 1.10 Asymptotic Methods and Teaching Physics.- 1.11 Problems and Perspectives.- 2. Discrete Systems.- 2.1 The Classical Perturbation Technique: an Introduction.- 2.2 Krylov-Bogolubov-Mitropolskij Method.- 2.3 Equivalent Linearization.- 2.4 Analysis of Nonconservative Nonautonomous Systems.- 2.5 Nonstationary Nonlinear Systems.- 2.6 Parametric and Self-Excited Oscillation in a Three-Degree-of-Freedom Mechanical System.- 2.7 Modified Poincaré Method.- 2.8 Hopf Bifurcation.- 2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom.- 2.11 Nontraditional Asymptotic Approaches.- 2.12 Padé Approximants.- 3. Continuous Systems.- 3.1 Continuous Approximation for a Nonlinear Chain.- 3.2 Homogenization Procedure in the Nonlinear Dynamics of Thin-Walled Structures.- 3.3 Averaging Procedure in the Nonlinear Dynamics of Thin-Walled Structures.- 3.4 Bolotin-Like Approach for Nonlinear Dynamics.- 3.5 Regular and Singular Asymptotics in the Nonlinear Dynamics of Thin-Walled Structures.- 3.6 One-Point Padé Approximants Using the Method of Boundary Condition Perturbation.- 3.7 Two-Point Padé Approximants: A Plate on Nonlinear Support.- 3.8 Solitons and Soliton-Like Approaches in the Case of Strong Nonlinearity.- 3.9 Nonlinear Analysis of Spatial Structures.- 4. Discrete-Continuous Systems.- 4.1 Periodic Oscillations of Discrete-Continuous Systems with a Time Delay.- 4.2 Simple Perturbation Technique.- 4.3 Nonlinear Behaviour of Electromechanical Systems.- GeneralReferences.- Detailed References (d).