If the Taurid Complex asteroids (plus 2P/Encke) are derived from the fragmentation of a giant comet over the past 20-30 kyr, as we have suggested elsewhere, then we need to identify some mechanism which could cause the observed dispersion in semimajor axes (1.8<<e1>a<</e1>2.6 au), eccentricities (0.64<<e1>e<</e1>0.85), and longitudes of perihelion (100 deg<omega<190 deg) the range in inclinations (i) is easily producible by secular perturbations. The problem devolves to one of identifying a source for the variations in a and e since, for a range of values of those parameters, differential precession will lead to a spreading in omega. Here we show that, considering only gravitational effects for variational orbits based on those of 2P/Encke, 5025 P-L and (6063) Jason, chaotic dynamical evolution over 30 kyr produces alterations in a and e which are insufficient to match the observed spread, suggesting that the asteroids concerned cannot have been derived from the splitting of a parent asteroid (i.e., inert object) on such a time-scale. However, plausible non-gravitational forces modelled quantitatively upon the observed motion of 2P/Encke can explain the spread, which may be interpreted as being supportive of our hypothesis of a cometary origin for the asteroids in the complex. Such forces also allow a connection with the Hephaistos group of objects to be made.