Dynamics of tachyon and phantom field beyond the inverse square potentials

We investigate the cosmological evolution of the tachyon and phantom-tachyon scalar field by considering the potential parameter \(\Gamma(=\frac{VV''}{V'^{2}}\)) as a function of another potential parameter \(\lambda(=\frac{V'}{\kappa V^{3/2}}\)), which correspondingly extends the analysis of the evolution of our universe from a two-dimensional autonomous dynamical system to the three-dimensional case. It allows us to investigate the more general situation where the potential is not restricted to an inverse square potential. One particular result is that, apart from the inverse square potential, there are a large number of potentials which can give the scaling and dominant solution when the function Γ(λ) equals 3/2 for one or more values of λ*, as well as that the parameter λ* satisfies certain conditions. We also find that for a class of different potentials the possibilities for the dynamical evolution of the universe are actually the same and therefore undistinguishable.