Nonlinear Evolution ODEs Featuring Many Periodic Solutions

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 ω, these deformed ODEs have completely periodic solutions (with a fixed period \(\tilde T=R{\pi /\omega}\), 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 ω 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.