This book covers those topics necessary for a clear understanding of the 
qualitative theory of ordinary differential equations and the concept of 
a dynamical system. It is written for advanced undergraduates and for 
beginning graduate students. It begins with a study of linear systems of 
ordinary differential equations, a topic already familiar to the student who 
has completed a first course in differential equations. An efficient method 
for solving any linear system of ordinary differential equations is presented 
in Chapter 1. 
The major part of this book is devoted to a study of nonlinear sys- 
tems of ordinary differential equations and dynamical systems. Since most 
nonlinear differential equations cannot be solved, this book focuses on the 
qualitative or geometrical theory of nonlinear systems of differential equa- 
tions originated by Henri Poincarc in his work on differential equations at 
the end of the nineteenth century as well as on the functional properties 
inherent in the solution set of a system of nonlinear differential equations 
embodied in the more recent concept of a dynamical system. Our primary 
goal is to describe the qualitative behavior of the solution set of a given 
system of differential equations including the invariant sets and limiting 
behavior of the dynamical system or flow defined by the system of dif- 
ferential equations. In order to achieve this goal, it is first necessary to 
develop the local theory for nonlinear systems. This is done in Chapter 2 
which includes the fundamental local existence-uniqueness theorem, the 
Hartman-Grobman Theorem and the Stable Manifold Theorem. These lat- 
ter two theorems establish that the qualitative behavior of the solution set 
of a nonlinear system of ordinary differential equations near an equilibrium 
point is typically the same as the qualitative behavior of the solution set 
of the corresponding linearized system near the equilibrium point. 
After developing the local theory, we turn to the global theory in Chap- 
ter 3. This includes a study of limit sets of trajectories and the behavior of 
trajectories at infinity. Some unresolved problems of current research inter- 
est are also presented in Chapter 3. For example, the Poincare-Bendixson 
Theorem, established in Chapter 3, describes the limit sets of trajectories 
of two-dimensional systems; however, the limit sets of trajectories of three- 
dimensional (and higher dimensional) systems can be much more compli- 
cated and establishing the nature of these limit sets is a topic of current

Differential Equations and Dynamical Systems PDF

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