We consider certain autonomous three-dimensional dynamical systems that can arise in mechanical and fluid-dynamical contexts. Extending a previous study in Craik and Okamoto (2002), to include linear forcing and damping, we find that the four-leaf structure discovered in that paper, and unbounded orbits, persist, but may now be accompanied by three distinct period-doubling cascades to chaos, and by orbits that approach stable equilibrium points. This rich structure is investigated both analytically and numerically, distinguishing three main cases determined by the damping and forcing parameter values.
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