Abstract
The weakly nonlinear, resonant response of a damped, spherical pendulum (length l, damping ratio δ, natural frequency ω 0) to the planar displacement ε l cos ω t (ε ⪡ 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 α= 2δ/ε 2 3 and v= 2(ω 2 - ω 2 0/ ε 2 3 ω 2 . If α < 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.
Published Version
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