Nonlinear Differential Equations and Dynamical Systems 2nd rev. and expanded ed., 3rd printing 2006 [Pehme köide]

1 Introduction		1	(6)
1.1 Definitions and notation		1	(2)
1.2 Existence and uniqueness		3	(1)
1.3 Gronwall's inequality		4	(3)
2 Autonomous equations		7	(18)
2.1 Phase-space, orbits		7	(3)
2.2 Critical points and linearisation		10	(4)
2.3 Periodic solutions		14	(2)
2.4 First integrals and integral manifolds		16	(5)
2.5 Evolution of a volume element, Liouville's theorem		21	(2)
2.6 Exercises		23	(2)
3 Critical points		25	(13)
3.1 Two-dimensional linear systems		25	(4)
3.2 Remarks on three-dimensional linear systems		29	(2)
3.3 Critical points of nonlinear equations		31	(5)
3.4 Exercises		36	(2)
4 Periodic solutions		38	(21)
4.1 Bendixson's criterion		38	(2)
4.2 Geometric auxiliaries, preparation for the Poincare-Bendixson theorem		40	(3)
4.3 The Poincare-Bendixson theorem		43	(4)
4.4 Applications of the Poincare-Bendixson theorem		47	(6)
4.5 Periodic solutions in Rn		53	(4)
4.6 Exercises		57	(2)
5 Introduction to the theory of stability		59	(10)
5.1 Simple examples		59	(2)
5.2 Stability of equilibrium solutions		61	(1)
5.3 Stability of periodic solutions		62	(4)
5.4 Linearisation		66	(1)
5.5 Exercises		67	(2)
6 Linear Equations		69	(14)
6.1 Equations with constant coefficients		69	(2)
6.2 Equations with coefficients which have a limit		71	(4)
6.3 Equations with periodic coefficients		75	(5)
6.4 Exercises		80	(3)
7 Stability by linearisation		83	(13)
7.1 Asymptotic stability of the trivial solution		83	(5)
7.2 Instability of the trivial solution		88	(3)
7.3 Stability of periodic solutions of autonomous equations		91	(2)
7.4 Exercises		93	(3)
8 Stability analysis by the direct method		96	(14)
8.1 Introduction		96	(2)
8.2 Lyapunov functions		98	(5)
8.3 Hamiltonian systems and systems with first integrals		103	(4)
8.4 Applications and examples		107	(1)
8.5 Exercises		108	(2)
9 Introduction to perturbation theory		110	(12)
9.1 Background and elementary examples		110	(3)
9.2 Basic material		113	(3)
9.3 Naive expansion		116	(3)
9.4 The Poincare expansion theorem		119	(1)
9.5 Exercises		120	(2)
10 The Poincare-Lindstedt method		122	(14)
10.1 Periodic solutions of autonomous second-order equations		122	(5)
10.2 Approximation of periodic solutions on arbitary long time-scales		127	(2)
10.3 Periodic solutions of equations with forcing terms		129	(2)
10.4 The existence of periodic solutions		131	(4)
10.5 Exercises		135	(1)
11 The method of averaging		136	(30)
11.1 Introduction		136	(2)
11.2 The Lagrange standard form		138	(2)
11.3 Averaging in the periodic case		140	(4)
11.4 Averaging in the general case		144	(3)
11.5 Adiabatic invariants		147	(3)
11.6 Averaging over one angle, resonance manifolds		150	(4)
11.7 Averaging over more than one angle, an introduction		154	(3)
11.8 Periodic solutions		157	(5)
11.9 Exercises		162	(4)
12 Relaxation Oscillations		166	(7)
12.1 Introduction		166	(1)
12.2 Mechanical systems with large friction		167	(1)
12.3 The van der Pol-equation		168	(2)
12.4 The Volterra-Lotka equations		170	(2)
12.5 Exercises		172	(1)
13 Bifurcation Theory		173	(20)
13.1 Introduction		173	(2)
13.2 Normalisation		175	(5)
13.3 Averaging and normalisation		180	(2)
13.4 Centre manifolds		182	(4)
13.5 Bifurcation of equilibrium solutions and Hopf bifurcation		186	(4)
13.6 Exercises		190	(3)
14 Chaos		193	(31)
14.1 Introduction and historical context		193	(1)
14.2 The Lorenz-equations		194	(3)
14.3 Maps associated with the Lorenz-equations		197	(2)
14.4 One-dimensional dynamics		199	(4)
14.5 One-dimensional chaos: the quadratic map		203	(4)
14.6 One-dimensional chaos: the tent map		207	(1)
14.7 Fractal sets		208	(5)
14.8 Dynamical characterisations of fractal sets		213	(3)
14.9 Lyapunov exponents		216	(2)
14.10 Ideas and references to the literature		218	(6)
15 Hamiltonian systems		224	(24)
15.1 Introduction		224	(2)
15.2 A nonlinear example with two degrees of freedom		226	(4)
15.3 Birkhoff-normalisation		230	(3)
15.4 The phenomenon of recurrence		233	(3)
15.5 Periodic solutions		236	(2)
15.6 Invariant tori and chaos		238	(4)
15.7 The KAM theorem		242	(4)
15.8 Exercises		246	(2)
Appendix 1: The Morse lemma		248	(2)
Appendix 2: Linear periodic equations with a small parameter		250	(2)
Appendix 3: Trigonometric formulas and averages		252	(1)
Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3		253	(2)
Appendix 5: Bifurcations of self-excited oscillations		255	(5)
Appendix 6: Normal forms of Hamiltonian systems near equilibria		260	(7)
Answers and hints to the exercises		267	(28)
References		295	(6)
Index		301

Ferdinand Verhulst was born in Amsterdam, The Netherlands, in 1939.

He graduated at the University of Amsterdam in Astrophysics and Mathematics. A period of five years at the Technological University of Delft, started his interest in technological problems, resulting in various cooperations with engineers. His other interests include the methods and applications of asymptotic analysis, nonlinear oscillations and wave theory.

He holds a chair of dynamical systems at the department of mathematics at the University of Utrecht.

Among his other interests are a publishing company, Epsilon Uitgaven, that he founded in 1985, and the relation between dynamical systems and psychoanalysis.

For more information see www.math.uu.nl/people/verhulst