  We analyze a tilt instability of the orbit of an outer planet in a two planet
circumbinary system that we recently reported. The binary is on an eccentric
orbit and the inner circumbinary planet is on a circular polar orbit that
causes the the binary to undergo apsidal precession. The outer circumbinary
planet is initially on a circular or eccentric orbit that is coplanar with
respect to the binary. We apply a Hamiltonian in quadrupole order of the binary
potential to show that the tilt instability is the result of a secular
resonance in which the apsidal precession rate of the binary matches the nodal
precession rate of the outer planet. Resonance is possible because the polar
inner planet causes the apsidal precession of the binary to be retrograde. The
outer planet periodically undergoes large tilt oscillations for which we
analytically determine the initial evolution and maximum inclination. Following
a typically relatively short adjustment phase, the tilt grows exponentially in
time at a characteristic rate that is of order the absolute value of the binary
apsidal precession rate. The analytic results agree well with numerical
simulations. This instability is analogous to the Kozai-Lidov instability, but
applied to a circumbinary object. The instability fails to operate if the
binary mass ratio is too extreme. The instability occurs even if the outer
planet is instead an object of stellar mass and involves tilt oscillations of
the inner binary.
