Periodic differential equations appear in many contexts such as in
the theory of nonlinear oscillators, in celestial mechanics, or in
population dynamics with seasonal effects. The most traditional
approach to study these equations is based on the introduction of
small parameters, but the search of nonlocal results leads to the
application of several topological tools. Examples are fixed point
theorems, degree theory, or bifurcation theory. These well-known
methods are valid for equations of arbitrary dimension and they are
mainly employed to prove the existence of periodic solutions.
Following the approach initiated by Massera, this book presents
some more delicate techniques whose validity is restricted to two
dimensions. These typically produce additional dynamical
information such as the instability of periodic solutions, the
convergence of all solutions to periodic solutions, or connections
between the number of harmonic and subharmonic solutions. The
qualitative study of periodic planar equations leads naturally to a
class of discrete dynamical systems generated by homeomorphisms or
embeddings of the plane. To study these maps, Brouwer introduced
the notion of a translation arc, somehow mimicking the notion of an
orbit in continuous dynamical systems. The study of the properties
of these translation arcs is full of intuition and often leads to
"non-rigorous proofs". In the book, complete proofs following ideas
developed by Brown are presented and the final conclusion is the
Arc Translation Lemma, a counterpart of the Poincare-Bendixson
theorem for discrete dynamical systems. Applications to
differential equations and discussions on the topology of the plane
are the two themes that alternate throughout the five chapters of
the book.
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