Coupled pendulums and normal modes

What you are seeing: two identical pendulums of length $L$ and mass $m$, coupled by a spring of constant $k$ attached at distance $d$ from the pivot. The small-angle EOM has two normal modes: symmetric (both swing in phase) at $\omega_+ = \sqrt{g/L}$, and antisymmetric (opposite phase) at $\omega_- = \sqrt{g/L + 2 k d^2 / (m L^2)}$.

Start with only pendulum 1 displaced ($\theta_2 = 0$) and watch the energy slosh back and forth: pendulum 1 slows while pendulum 2 grows, until after half a beat period almost all the energy lives on pendulum 2. Beat period $T_\text{beat} = 2 \pi / (\omega_- - \omega_+)$. Symmetric or antisymmetric initial conditions stay locked in their respective modes forever.