Two pendulums that start 1e-6 rad apart. Watch the butterfly effect, then try the tiny swing.
The science
Equations. From the Lagrangian of two point masses on rigid rods the angular accelerations follow (standard closed form); they are integrated with fixed-step RK4 (dt up to 2 ms). Energy E = T + V with V = -(m1+m2) g l1 cos(th1) - m2 g l2 cos(th2) is conserved when damping is zero.
Sensitive dependence. Two copies start a tiny distance apart. In the chaotic regime their separation grows roughly as exp(lambda t) with lambda of order 1 per second, so a 1e-6 rad difference becomes visible after about 10-20 s. The log-separation chart shows that rising line until it saturates at the size of the attractor.
Small angles. For small amplitudes the motion is a sum of two normal modes with omega^2 from det(K - omega^2 M) = 0, M = [[(m1+m2)l1^2, m2 l1 l2],[m2 l1 l2, m2 l2^2]], K = diag((m1+m2) g l1, m2 g l2). The checks compare these periods with the nonlinear simulation.
Simplifications. Massless rods, frictionless pivots unless damping is set (linear viscous term), no air drag. Beyond roughly 20 s of chaotic motion no numerical solution matches a real pendulum.