The exact solution for the simplest type of this system contains a complete elliptic integral of the  rst kind.
Perturbative expansion of one type of this system in terms of two coupling constants with a  xed ratio produces an equation of motion where the second time derivative of the more slowly changing variable is proportional to the partial derivative of an e1⁄2fective potential with respect to that variable. David Acheson derived the conditions under which another type of this system may be coupled to itself an arbitrary number of times while remaining stable. The Furuta type of this system is a benchmark for testing nonlinear control strategies for under-actuated mechanical devices. The exact solution for the simplest type of this system contains a complete elliptic integral of the  rst kind. The equation "three cosine theta-one" plus "sine theta-two" equals "two" de nes the onset of chaotic behavior in the "double" type of this system. For 10 points, name this oscillatory system in which a mass is suspended from a pivot.
Perturbative expansion of one type of this system in terms of two coupling constants with a  xed ratio produces an equation of motion where the second time derivative of the more slowly changing variable is proportional to the partial derivative of an e1⁄2fective potential with respect to that variable.
pendulums [accept Kapitza pendulum, inverted pendulum, Furuta pendulum, or double pendulum; prompt on parametric oscillators or driven harmonic oscillators; do not accept "simple harmonic oscillators"]