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Abstract/Syllabus:

## Ordinary Differential Equations and Dynamical Systems

#### Abstract.

This book provides an introduction to ordinary di erential equations
solvable equations. Then we prove the fundamental results concerning
the initial value problem: existence, uniqueness, extensibility, dependence
on initial conditions. Furthermore we consider linear equations, the Floquet
theorem, and the autonomous linear fow.

Then we establish the Frobenius method for linear equations in the complex
domain and investigate Sturm{Liouville type boundary value problems
including oscillation theory.

Next we introduce the concept of a dynamical system and discuss stability
including the stable manifold and the Hartman{Grobman theorem for
both continuous and discrete systems.

We prove the Poincare{Bendixson theorem and investigate several examples
of planar systems from classical mechanics, ecology, and electrical
engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem,
and periodic solutions are discussed as well.

Finally, there is an introduction to chaos. Beginning with the basics for
iterated interval maps and ending with the Smale{Birkho theorem and the
Melnikov method for homoclinic orbits.

Keywords and phrases. Ordinary di erential equations, dynamical systems,
Sturm-Liouville equations.

Typeset by AMS-LATEX and Makeindex.
Version: March 25, 2009
2000{2009 by Gerald Teschl

## Lecture Notes

### Chapter 1. Introduction 3

1.1. Newton's equations 3
1.2. Classi cation of di erential equations 6
1.3. First order autonomous equations 8
1.4. Finding eplicit solutions 13
1.5. Qualitative analysis of rst-order equations 19
1.6. Qualitative analysis of rst-order periodic equations 26

#### Chapter 2. Initial value problems 31

2.1. Fied point theorems 31
2.2. The basic eistence and uniqueness result 33
2.3. Some etensions 36
2.4. Dependence on the initial condition 39
2.5. Etensibility of solutions 44
2.6. Euler's method and the Peano theorem 47

#### Chapter 3. Linear equations 51

3.1. The matri eponential 51
3.2. Linear autonomous rst-order systems 56
3.3. Linear autonomous equations of order n 62
3.4. General linear rst-order systems 69
3.5. Periodic linear systems 75
3.6. Appendi: Jordan canonical form 80

#### Chapter 4. Di erential equations in the comple domain 87

4.1. The basic eistence and uniqueness result 87
4.2. The Frobenius method for second-order equations 90
4.3. Linear systems with singularities 101
4.4. The Frobenius method 105

#### Chapter 5. Boundary value problems 111

5.1. Introduction 111
5.2. Compact symmetric operators 114
5.3. Regular Sturm-Liouville problems 120
5.4. Oscillation theory 127
5.5. Periodic operators 133
Part 2. Dynamical systems

#### Chapter 6. Dynamical systems 145

6.1. Dynamical systems 145
6.2. Thef ow of an autonomous equation 146
6.3. Orbits and invariant sets 149
6.4. The Poincare map 154
6.5. Stability of ed points 155
6.6. Stability via Liapunov's method 156
6.7. Newton's equation in one dimension 158

#### Chapter 7. Local behavior near ed points 163

7.1. Stability of linear systems 163
7.2. Stable and unstable manifolds 165
7.3. The Hartman-Grobman theorem 172
7.4. Appendi: Integral equations 177

#### Chapter 8. Planar dynamical systems 185

8.1. The Poincare{Bendison theorem 185
8.2. Eamples from ecology 189
8.3. Eamples from electrical engineering 194

#### Chapter 9. Higher dimensional dynamical systems 199

9.1. Attracting sets 199
9.2. The Lorenz equation 203
9.3. Hamiltonian mechanics 208
9.4. Completely integrable Hamiltonian systems 212
9.5. The Kepler problem 216
9.6. The KAM theorem 218

#### Chapter 10. Discrete dynamical systems 225

10.1. The logistic equation 225
10.2. Fied and periodic points 228
10.3. Linear di erence equations 230
10.4. Local behavior near ed points 232

#### Chapter 11. Discrete dynamical systems in one dimension 235

11.1. Period doubling 235
11.2. Sarkovskii's theorem 238
11.3. On the de nition of chaos 239
11.4. Cantor sets and the tent map 242
11.5. Symbolic dynamics 245
11.6. Strange attractors/repellors and fractal sets 250
11.7. Homoclinic orbits as source for chaos 254

#### Chapter 12. Periodic solutions 259

12.1. Stability of periodic solutions 259
12.2. The Poincare map 260
12.3. Stable and unstable manifolds 262
12.4. Melnikov's method for autonomous perturbations 265
12.5. Melnikov's method for nonautonomous perturbations 270

#### Chapter 13. Chaos in higher dimensional systems 273

13.1. The Smale horseshoe 273
13.2. The Smale-Birkho homoclinic theorem 275
13.3. Melnikov's method for homoclinic orbits 276
Bibliography 281
Glossary of notation 283
Inde 285

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