Numerical integrations are performed on test-particle orbits similar to those of 2P/Encke and (6063) Jason. For purely gravitational integrations, or integrations with non-gravitational forces applied of a magnitude similar to those currently observed for 2P/Encke, the orbits do not alter substantially, all remaining decoupled from Jupiter. By using non-gravitational forces about four times as strong, which are plausible given 2P/Encke's known history as a prolific producer of meteoroids and dust, it is found that some orbits become Jupiter-crossers. It is not necessary for the non-gravitational forces to be exceptionally strong, nor to act consistently in one direction for an extended period, to produce such a result. Rather, the importance of the non-gravitational forces lies in them causing the semimajor axes of the test orbits to scan across the complex of jovian and saturnian mean motion resonances at 2.3<<e1>a<</e1>2.9 au, residence in strong resonances leading to enhanced secular orbital changes. By appealing to a reversal of the arrow of time one can then hypothesize that the sub-jovian orbit of 2P/Encke was achieved through the opposite process. This result also has implications for the origin of Apollo-Amor asteroids derived from de-volatilized cometary nuclei.