Given the pervasive character of nonlinearity throughout the physical universe, a case is made for
introducing undergraduate students to its consequences and signatures earlier rather than later. The
dynamics of two well-known systems—a spring and a pendulum—are reviewed when the standard textbook
linearising assumptions are relaxed. Some qualitative effects of nonlinearity can be anticipated
from symmetry (e.g., inspection of potential energy functions), and further physical insight gained
by applying a simple successive-approximation method that might be taught in parallel with courses
on classical mechanics, ordinary differential equations, and computational physics. We conclude with
a survey of how these ideas have been deployed on programmes at a UK university.