Summary: The weakly nonlinear, resonant response of a damped, spherical pendulum (length l, damping ratio \(\delta\), natural frequency \(\omega_ 0)\) to the planar displacement \(\epsilon\) l \(\cos\omega t(\epsilon \ll 1)\) of its point of suspension is examined in a four-dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion. The loci of equilibrium points and the corresponding bifurcation points in this space are determined. The control parameters are \(\alpha =2\delta /\epsilon^{2/3}\) and \(v=2(\omega^ 2-\omega^ 2_ 0)/\epsilon^{2/3}\omega^ 2\). If \(\alpha <0.441\) there is a finite interval of v within which no stable equilibrium points exist. As v decreases through the upper bound (a Hopf-bifurcation point) of this interval the motion in the phase space becomes periodic and then, following a period-doubling cascade, chaotic. There may be alternating sub-intervals of chaotic and periodic motion. The chaotic trajectories in the phase space appear to lie on fractal attractors.