Author | Bogaevski, V. N. author |
---|---|
Title | Algebraic Methods in Nonlinear Perturbation Theory [electronic resource] / by V. N. Bogaevski, A. Povzner |
Imprint | New York, NY : Springer New York, 1991 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4438-7 |
Descript | XII, 266 p. online resource |
1 Matrix Perturbation Theory -- 1.1 Perturbation Theory for a Linear Operator -- 1.2 Main Formulas -- 1.3 Diagonal Leading Operator -- 1.4 General Case. The Normal Form of the Matrix of the Operator M -- 1.5 Nilpotent Leading Operator. The Reconstruction Problem -- 2 Systems of Ordinary Differential Equations with a Small Parameter -- 2.1 Passage to the Linear Problem. Change of Variables Operator -- 2.2 General Formulation of the Perturbation Theory Problem -- 2.3 Canonical Form of First Order Operator X0 -- 2.4 An Algebraic Formulation of the Perturbation Theory Problem -- 2.5 The Normal Form of an Operator with Respect to a Nilpotent X0. The Reconstruction Problem -- 2.6 A Connection with N. N. Bogolyubovโs Ideas -- 2.7 The Motion Near the Stationary Manifold -- 2.8 Hamiltonian Systems -- 3 Examples -- 3.1 Example: The Pendulum of Variable Length -- 3.2 Example: A Second Order Linear Equation -- 3.3 Example: P. L. Kapitsaโs Problem: A Pendulum Suspended from an Oscillating Point -- 3.4 Example: Van der Pol Oscillator with Small Damping -- 3.5 Example: Duffing Oscillator -- 3.6 Example: Drift of a Charged Particle in an Electromagnetic Field -- 3.7 Example: Nonlinear System: Example of an Extension of an Operator -- 3.8 Example: Nonlinear Oscillator with Small Mass and Damping -- 3.9 Example: A Nonlinear Equation; Boundary-Layer-Type Problem -- 3.10 Example: Resonances. Particular Solutions -- 3.11 Example: The Mathieu Equation -- 3.12 Example: Oscillating Spring -- 3.13 Example: Periodic Solution (Hopfโs Theorem) -- 3.14 Example: Bifurcation -- 3.15 Example: Problem of a Periodic Solution of an Autonomous System -- 3.16 Example: One Problem on Eigenvalues -- 3.17 Example: A. M. Lyapunovโs Problem -- 3.18 Example: Illustration for Section 2.5 -- 3.19 Example: Fast Rotation of a Solid Body -- 3.20 Example: The Langer Problem ([28]) -- 4 Reconstruction -- 4.1 Introduction -- 4.2 New Leading Operators in the First Type Problems -- 4.3 The Second Type Problems. โAlgebraicโ Method of Reconstruction -- 4.4 โTrajectoryโ Method of Reconstruction -- 4.5 Matching -- 4.6 Example: Illustration for 4.5 -- 4.7 Example: Appearance of a New Singularity -- 4.8 Example: Passing Through a Resonance -- 4.9 Example: WKB-Type Problem -- 4.10 Example: Lighthillโs Problem [38] -- 4.11 Example: Singularity of Coefficients of an Operator -- 4.12 Example: A Second Order Linear Equation -- 4.13 Example: Van der Pol Oscillator (Relaxation Oscillations) -- 5 Equations in Partial Derivatives -- 5.1 Functional Derivatives -- 5.2 Equations with Partial Derivatives Whose Principal Part Is an Ordinary Differential Equation -- 5.3 Partial Derivatives. On Whitham Method -- 5.4 Geometric Optics and the Maslov Method -- 5.5 Problem (Whitham) -- 5.6 Problem. Diffraction of Short Waves on a Circle (Semishade) -- 5.7 One-Dimensional Shock Wave -- References