Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Periodic Solutions of Non-autonomous Ordinary Differential Equations

  • Jean Mawhin
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_392-3

Definition of the Subject

Many phenomena in nature can be modeled by systems of ordinary differential equations which depend periodically upon time. For example, a linear or nonlinear oscillator can be forced by a periodic external force, and an important question is to know if the oscillator can exhibit a periodic response under this forcing. This question originated from problems in classical and celestial mechanics, before receiving important applications in radioelectricity and electronics. Nowadays, it also plays a great role in mathematical biology and population dynamics, as well as in mathematical economics, where the considered systems are often subject to seasonal variations. The general theory originated with Henri Poincaré’s work in celestial mechanics, at the end of the nineteenth century and has been constantly developed since.


To motivate the problem and its difficulties, let us start with the simple linear oscillator with forcing (or input) hL 2(0, 2 π)


Periodic Solution Fixed Point Theorem Implicit Function Theorem Topological Degree Positive Periodic Solution 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium