Two identical frictionless pendulums coupled together by a torsion spring oscillate with equal displacements and velocities. Unstable synchronized states lead to chaotic behavior.

where is the spring constant, proportional to the stiffness of the spring. The total momentum of the system is conserved because the forces are equal and opposite, so the system satisfies Newton’s third law.

References

[1] J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, Singapore: World Scientific, 2010 pp. 121–123.

[2] E. J. Banning and J. P. van der Weele, "Mode Competition in a System of Two Parametrically Driven Pendulums; the Hamiltonian Case," Physica A220(3–4), 1995 pp. 485–533. doi:10.1016/0378-4371(95)00153-X.