This article is cited in 18 papers

Nonlinear Evolution ODEs Featuring Many Periodic Solutions

F. Calogero^ab, J.-P. Françoise<sup>c

a</sup> INFN — National Institute of Nuclear Physics
^b University of Rome "La Sapienza"
^c Université Pierre & Marie Curie, Paris VI

Abstract: We identify certain (classes of) single autonomous nonlinear evolution ODEs of arbitrarily high order that, by a simple explicit prescription, can be modified to generate a one-parameter family of deformed autonomous ODEs with the following properties: for all positive values of the deformation parameter $\omega$, these deformed ODEs have completely periodic solutions (with a fixed period $\widetilde T=R\pi/\omega$, where $R$ is an appropriate rational number) emerging–in the context of the initial-value problem–from open initial-data domains whose measure in the space of such initial data depends on the parameter $\omega$ but is generally positive (i.e., nonvanishing). Several examples are presented, including a one-parameter deformation of a well-known third-order ODE originally introduced by J. Chazy. We then discuss the deformation of the Chazy equation fully and find an explicit open semialgebraic set of periodic orbits.

Keywords: periodic solutions, nonlinear oscillators, Chazy equation.

DOI: 10.4213/tmf278

 English version: Theoretical and Mathematical Physics, 2003, 137:3, 1663–1675

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