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This graduate-level introduction to ordinary differential equations
combines both qualitative and numerical analysis of solutions, in
line with Poincare's vision for the field over a century ago.
Taking into account the remarkable development of dynamical systems
since then, the authors present the core topics that every young
mathematician of our time--pure and applied alike--ought to learn.
The book features a dynamical perspective that drives the
motivating questions, the style of exposition, and the arguments
and proof techniques. The text is organized in six cycles. The
first cycle deals with the foundational questions of existence and
uniqueness of solutions. The second introduces the basic tools,
both theoretical and practical, for treating concrete problems. The
third cycle presents autonomous and non-autonomous linear theory.
Lyapunov stability theory forms the fourth cycle. The fifth one
deals with the local theory, including the Grobman-Hartman theorem
and the stable manifold theorem. The last cycle discusses global
issues in the broader setting of differential equations on
manifolds, culminating in the Poincare-Hopf index theorem. The book
is appropriate for use in a course or for self-study. The reader is
assumed to have a basic knowledge of general topology, linear
algebra, and analysis at the undergraduate level. Each chapter ends
with a computational experiment, a diverse list of exercises, and
detailed historical, biographical, and bibliographic notes seeking
to help the reader form a clearer view of how the ideas in this
field unfolded over time.
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