Breaking of Symmetrical Periodic Solutions in a Singularly Perturbed KDV Model

There are several recent developments in the well-known problem of breaking of homoclinic orbits (splitting of separatrices) of a system that undergoes a singular perturbation. First, survival of a homoclinic orbit is an exceptional situation that can be linked to triviality of the Stokes phenomenon of the underlying “truncated” equation. Second, homoclinic connections to exponentially small periodic orbits survive the perturbation in the generic case. In this paper we consider a different problem: we study deformations of “genuine” periodic orbits of the second order equation $y”=y+y^2$ that undergoes the singular perturbation $\varepsilon^2y””+(1-\varepsilon^2)y”=y+y^2$, where $\varepsilon>0$ is a small parameter. We prove that if the period and the constant of motion do not change too rapidly (in $\varepsilon$), a genuine (nontrivial) periodic solution does not survive the perturbation.