Nonlinear evolution odes featuring many periodic solutions

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 (T) over tilde = Rpi/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 W 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.