Periodic differential equations in the plane

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Periodic differential equations are prevalent in various fields, including nonlinear oscillators, celestial mechanics, and population dynamics with seasonal effects. Traditional methods for studying these equations often involve introducing small parameters, but exploring nonlocal results necessitates topological tools like fixed point theorems, degree theory, and bifurcation theory. These techniques, applicable to equations of any dimension, are primarily used to establish the existence of periodic solutions. Building on Massera's approach, this book introduces more nuanced techniques specifically for two dimensions, revealing additional dynamical insights such as the instability of periodic solutions and the convergence of all solutions to periodic solutions. The qualitative analysis of periodic planar equations naturally leads to discrete dynamical systems formed by homeomorphisms or embeddings of the plane. Brouwer's concept of a translation arc serves as an analogue to orbits in continuous systems. The exploration of these translation arcs is rich with intuition and often results in "non-rigorous proofs." This book provides complete proofs inspired by Brown's ideas, culminating in the Arc Translation Lemma, which parallels the Poincaré–Bendixson theorem in discrete dynamics. Throughout the five chapters, applications to differential equations and discussions on plane topology are interwoven.