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We investigate spatially flat isotropic cosmological models which contain a scalar field with an exponential potential and a perfect fluid with a linear equation of state. We include an interaction term, through which the energy of the scalar field is transferred to the matter fields, consistent with a term that arises from scalar-tensor theory under a conformal transformation and field redefinition. The governing ordinary differential equations reduce to a dynamical system when appropriate normalized variables are defined. We analyze the dynamical system and find that the interaction term can significantly affect the qualitative behavior of the models. The late-time behavior of these models may be of cosmological interest. In particular, for a specific range of values for the model parameters there are late-time attracting solutions, corresponding to a novel attracting equilibrium point, which are inflationary and in which the scalar field's energy-density remains a fixed fraction of the matter field's energy density. These scalar field models may be of interest as late-time cosmologies, particularly in view of the recent observations of the current accelerated cosmic expansion. For appropriate values of the interaction coupling parameter, this equilibrium point is an attracting focus, and hence as inflating solutions approach this late-time attractor the scalar field oscillates. Hence these models may also be of importance in the study of inflation in the early universe.
Billyard et al. (Mon,) studied this question.