Nonlinear Differential Equations and Dynamical Systems by Ferdinand Verhulst.

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Bibliographic Details
Main Author:	Verhulst, Ferdinand (Author)
Corporate Author:	SpringerLink (Online service)
Format:	eBook
Language:	English
Published:	Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1990.
Edition:	1st ed. 1990.
Series:	Universitext, Springer eBook Collection.
Subjects:	Mathematical analysis. Analysis (Mathematics). Physics. Statistical physics. Dynamical systems. Applied mathematics. Engineering mathematics. Electronic resources (E-books)
Online Access:	Click to view e-book
Holy Cross Note:	Loaded electronically. Electronic access restricted to members of the Holy Cross Community.

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 Rn.
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 The van der Pol-equation
12.3 The Volterra-Lotka equations
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 The Lorenz-equations
14.2 A mapping associated with the Lorenz-equations
14.3 A mapping of R into R as a dynamical system
14.4 Results for the quadratic mapping
15 Hamiltonian systems
15.1 Summary of results obtained earlier
15.2 A nonlinear example with two degrees of freedom
15.3 The phenomenon of recurrence
15.4 Periodic solutions
15.5 Invariant tori and chaos
15.6 The KAM theorem
15.7 Exercises
Appendix 1: The Morse lemma
Appendix 2: Linear periodic equations with a small parameter
Appendix 3: Trigonometric formulas and averages
Answers and hints to the exercises
References.

Nonlinear Differential Equations and Dynamical Systems by Ferdinand Verhulst.

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