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It turns out there are many such orbits, and there are various problems and trade-offs associated with their periods, the energy needed for "taxi" spacecraft to rendezvous with them as they pass by Earth or Mars, and other factors. In order to have a reasonable transit time in both directions, you would need to have two or more cyclers in similar but complementary orbits.
Damn Interesting did a nice story on Mars cyclers, and this 2005 article by Buzz himself is a very good intro. This is a fascinating concept for building some key infrastructure for a spacefaring civilization that extends beyond the "local neighborhood" of the Earth and Moon. I would like to experiment with these orbits in Orbiter if I could find the time.
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