"Nonlinear Modes of Vibration," by Ivar Ekeland. Nature, 10 September 1998, pages 116-117.
In classical mechanics, a system with n degrees of freedom has n fundamental modes of vibration, and any motion of the system can be expressed as a superposition of these fundamental modes. This basic result applies only to linear systems; whether it also holds true for nonlinear systems is unknown. However, there are some partial results, as this article indicates. It discusses a recent paper by mathematicians Hofer, Wysocki, and Zehnder showing that nonlinear systems with two degrees of freedom must have either two periodic orbits for a given energy, or infinitely many.
--- Allyn Jackson