Nonlinear Differential Equations and Dynamical Systems /
On the subject of differential equations many elementary books have been written. This book bridges the gap between elementary courses and research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invar...
Clasificación: | Libro Electrónico |
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Autor principal: | |
Formato: | Electrónico eBook |
Idioma: | Inglés |
Publicado: |
Berlin, Heidelberg :
Springer Berlin Heidelberg,
1996.
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Edición: | Second, rev. and Expanded edition. |
Colección: | Universitext.
|
Temas: | |
Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 1 Introduction
- 1.1 Definitions and notation
- 1.2 Existence and uniqueness
- 1.3 Gronwall's inequality
- 2 Autonomous equations
- 2.1 Phase-space, orbits
- 2.2 Critical points and linearisation
- 2.3 Periodic solutions
- 2.4 First integrals and integral manifolds
- 2.5 Evolution of a volume element, Liouville's theorem
- 2.6 Exercises
- 3 Critical points
- 3.1 Two-dimensional linear systems
- 3.2 Remarks on three-dimensional linear systems
- 3.3 Critical points of nonlinear equations
- 3.4 Exercises
- 4 Periodic solutions
- 4.1 Bendixson's criterion
- 4.2 Geometric auxiliaries, preparation for the Poincaré-Bendixson theorem
- 4.3 The Poincaré-Bendixson theorem
- 4.4 Applications of the Poincaré-Bendixson theorem
- 4.5 Periodic solutions in?n
- 4.6 Exercises
- 5 Introduction to the theory of stability
- 5.1 Simple examples
- 5.2 Stability of equilibrium solutions
- 5.3 Stability of periodic solutions
- 5.4 Linearisation
- 5.5 Exercises
- 6 Linear Equations
- 6.1 Equations with constant coefficients
- 6.2 Equations with coefficients which have a limit
- 6.3 Equations with periodic coefficients
- 6.4 Exercises
- 7 Stability by linearisation
- 7.1 Asymptotic stability of the trivial solution
- 7.2 Instability of the trivial solution
- 7.3 Stability of periodic solutions of autonomous equations
- 7.4 Exercises
- 8 Stability analysis by the direct method
- 8.1 Introduction
- 8.2 Lyapunov functions
- 8.3 Hamiltonian systems and systems with first integrals
- 8.4 Applications and examples
- 8.5 Exercises
- 9 Introduction to perturbation theory
- 9.1 Background and elementary examples
- 9.2 Basic material
- 9.3 Naïve expansion
- 9.4 The Poincaré expansion theorem
- 9.5 Exercises
- 10 The Poincaré-Lindstedt method
- 10.1 Periodic solutions of autonomous second-order equations
- 10.2 Approximation of periodic solutions on arbitrary long time-scales
- 10.3 Periodic solutions of equations with forcing terms
- 10.4 The existence of periodic solutions
- 10.5 Exercises
- 11 The method of averaging
- 11.1 Introduction
- 11.2 The Lagrange standard form
- 11.3 Averaging in the periodic case
- 11.4 Averaging in the general case
- 11.5 Adiabatic invariants
- 11.6 Averaging over one angle, resonance manifolds
- 11.7 Averaging over more than one angle, an introduction
- 11.8 Periodic solutions
- 11.9 Exercises
- 12 Relaxation Oscillations
- 12.1 Introduction
- 12.2 Mechanical systems with large friction
- 12.3 The van der Pol-equation
- 12.4 The Volterra-Lotka equations
- 12.5 Exercises
- 13 Bifurcation Theory
- 13.1 Introduction
- 13.2 Normalisation
- 13.3 Averaging and normalisation
- 13.4 Centre manifolds
- 13.5 Bifurcation of equilibrium solutions and Hopf bifurcation
- 13.6 Exercises
- 14 Chaos
- 14.1 Introduction and historical context
- 14.2 The Lorenz-equations
- 14.3 Maps associated with the Lorenz-equations
- 14.4 One-dimensional dynamics
- 14.5 One-dimensional chaos: the quadratic map
- 14.6 One-dimensional chaos: the tent map
- 14.7 Fractal sets
- 14.8 Dynamical characterisations of fractal sets
- 14.9 Lyapunov exponents
- 14.10 Ideas and references to the literature
- 15 Hamiltonian systems
- 15.1 Introduction
- 15.2 A nonlinear example with two degrees of freedom
- 15.3 Birkhoff-normalisation
- 15.4 The phenomenon of recurrence
- 15.5 Periodic solutions
- 15.6 Invariant tori and chaos
- 15.7 The KAM theorem
- 15.8 Exercises
- Appendix 1: The Morse lemma
- Appendix 2: Linear periodic equations with a small parameter
- Appendix 3: Trigonometric formulas and averages
- Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3
- Appendix 5: Bifurcations of self-excited oscillations
- Appendix 6: Normal forms of Hamiltonian systems near equilibria
- Answers and hints to the exercises
- References.