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.
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"]