本书为低年级研究生提供了一个关于常微分方程和动力系统的自封式的导引。
第一部分从一些显式可解方程的简单例子和对定性方法的初步了解开始；然后证明了有关初值问题的基本结果：存在性，唯一性，可延拓性，对初始条件的依赖性；此外，还考虑了线性方程组，包括Floquet定理和一些摄动结果；作为有些独立的主题，本部分还建立了复数域中线性方程组的Frobenius方法，研究了Sturm-Liouville边值问题（包括振动理论）。
第二部分介绍了动力系统的概念，证明了Poincaré-Bendixson定理，并研究了来自经典力学、生态学和电气工程的平面系统的几个例子；此外，还讨论了吸引子、Hamilton系统、KAM定理和周期解；最后，研究了稳定性，包括连续系统和离散系统的稳定流形和Hartman-Grobman定理。
第三部分介绍了混沌，从迭代区间映射的基础知识开始，以Smale-Birkhoff定理和同宿轨道的Melnikov方法结束。
本书包含近300道习题。此外，数学软件系统的使用贯穿始终，展示了如何使用软件帮助读者研究微分方程。

Preface
Part 1. Classical theory
Chapter 1. Introduction
§1.1. Newton's equations
§1.2. Classification of differential equations
§1.3. First-order autonomous equations
§1.4. Finding explicit solutions
§1.5. Qualitative analysis of first-order equations
§1.6. Qualitative analysis of first-order periodic equations
Chapter 2. Initial value problems
§2.1. Fixed point theorems
§2.2. The basic existence and uniqueness result
§2.3. Some extensions
§2.4. Dependence on the initial condition
§2.5. Regular perturbation theory
§2.6. Extensibility of solutions
§2.7. Euler's method and the Peano theorem
Chapter 3. Linear equations
§3.1. The matrix exponential
§3.2. Linear autonomous first-order systems
§3.3. Linear autonomous equations of order n
§3.4. General linear first-order systems
§3.5. Linear equations of order n
§3.6. Periodic linear systems
§3.7. Perturbed linear first-order systems
§3.8. Appendix:Jordan canonical form
Chapter 4. Differential equations in the complex domain
§4.1. The basic existence and uniqueness result
§4.2. The Frobenius method for second-order equations
§4.3. Linear systems with singularities
§4.4. The Frobenius method
Chapter 5. Boundary value problems
§5.1. Introduction
§5.2. Compact symmetric operators
§5.3. Sturm-Liouville equations
§5.4. Regular Sturm-Liouville problems
§5.5. Oscillation theory
§5.6. Periodic Sturm-Liouville equations
Part 2. Dynamical systems
Chapter 6. Dynamical systems
§6.1. Dynamical systems
§6.2. The flow of an autonomous equation
§6.3. Orbits and invariant sets
§6.4. The Poincaré map
§6.5. Stability of fixed points
§6.6. Stability via Liapunov's method
§6.7. Newton's equation in one dimension
Chapter 7. Planar dynamical systems
§7.1. Examples from ecology
§7.2. Examples from electrical engineering
§7.3. The Poincaré-Bendixson theorem
Chapter 8. Higher dimensional dynamical systems
§8.1. Attracting sets
§8.2. The Lorenz equation
§8.3. Hamiltonian mechanics
§8.4. Completely integrable Hamiltonian systems
§8.5. The Kepler problem
§8.6. The KAM theorem
Chapter 9. Local behavior near fixed points
§9.1. Stability of linear systems
§9.2. Stable and unstable manifolds
§9.3. The Hartman-Grobman theorem
§9.4. Appendix: Integral equations
Part 3. Chaos
Chapter 10. Discrete dynamical systems
§10.1. The logistic equation
§10.2. Fixed and periodic points
§10.3. Linear difference equations
§10.4. Local behavior near fixed points
Chapter 11. Discrete dynamical systems in one dimension
§11.1. Period doubling
§11.2. Sarkovskii's theorem
§11.3. On the definition of chaos
§11.4. Cantor sets and the tent map
§11.5. Symbolic dynamics
§11.6. Strange attractors/repellers and fractal sets
§11.7. Homoclinic orbits as source for chaos
Chapter 12. Periodic solutions
§12.1. Stability of periodic solutions
§12.2. The Poincaré map
§12.3. Stable and unstable manifolds
§12.4. Melnikov's method for autonomous perturbations
§12.5. Melnikov's method for nonautonomous perturbations
Chapter 13. Chaos in higher dimensional systems
§13.1. The Smale horseshoe
§13.2. The Smale-Birkhoff homoclinic theorem
§13.3. Melnikov's method for homoclinic orbits
Bibliographical notes
Bibliography
Glossary of notation
Index