Geodesic
Nonlinear Evolution ODEs Featuring Many Periodic Solutions
Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 3, pp. 358-374

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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. 
@article{TMF_2003_137_3_a4,
     author = {F. Calogero and J. Fran\c{c}oise},
     title = {Nonlinear {Evolution} {ODEs} {Featuring} {Many} {Periodic} {Solutions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {358--374},
     publisher = {mathdoc},
     volume = {137},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_137_3_a4/}
}
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F. Calogero; J. Françoise. Nonlinear Evolution ODEs Featuring Many Periodic Solutions. Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 3, pp. 358-374. http://geodesic.mathdoc.fr/item/TMF_2003_137_3_a4/