Nonlinear Differential Equations and Dynamical Systems

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...

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Bibliographic Details
Online Access: Full text (MCPHS users only)
Main Author: Verhulst, Ferdinand
Format: Electronic eBook
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 1996
Edition:Second, rev. and Expanded edition.
Series:Universitext.
Subjects:
Mathematics.
Differentiable dynamical systems.
Engineering mathematics.
Mathematics
Local Note:ProQuest Ebook Central
Table of Contents:
  • 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.

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