The differential equation describing the dynamics of a nonlinear one of these systems contains a term proportional to the sine of the angle, which means that displacement of such a system is proportional to an elliptic integral of the first kind.
Holmes and Marsden used the Melnikov formalism to show that coupling perturbations in this system render it non-integrable due to the introductions of horseshoes, which are intersections between stable and unstable manifolds. The differential equation describing the dynamics of a nonlinear one of these systems contains a term proportional to the sine of the angle, which means that displacement of such a system is proportional to an elliptic integral of the first kind. A linear variant of this system avoids that complication by Taylor expanding around the equilibrium point and assuming small-angle oscillations, which gives a period of two pi times the square root of the length over the gravitational acceleration. Also exhibiting chaos when two of these are connected in a "double" variant, for ten points, identify this dynamical system, a harmonic oscillator consisting of a weight on a massless rod.
Holmes and Marsden used the Melnikov formalism to show that coupling perturbations in this system render it non-integrable due to the introductions of horseshoes, which are intersections between stable and unstable manifolds.
pendulum [accept harmonic oscillator before "nonlinear"]