Self-similar static spherically symmetric scalar field models

Dynamical systems techniques are used to study the class of self-similar static spherically symmetric models with two non-interacting scalar fields with exponential potentials. The global dynamics depends on the scalar self-interaction potential parameters k(1) and k(2). For all values of k(1), k(2), there always exists (a subset of) expanding massless scalar field models that are early-time attractors and (a subset of) contracting massless scalar field models that are late-time attractors. When k(1) greater than or equal to1/root3 and k(2)greater than or equal to1/root3, in general the solutions evolve from an expanding massless scalar fields model and then recollapse to a contracting massless scalar fields model. When k(1) < 1/√3 or k(2) < 1/root3, the solutions generically evolve away from an expanding massless scalar fields model or an expanding single scalar field model and thereafter asymptote towards a contracting massless scalar fields model or a contracting single scalar field model. It is interesting that in this case a single scalar field model can represent the early-time or late-time asymptotic dynamical state of the models. The dynamics in the physical invariant set which constitutes a part of the boundary of the five-dimensional timelike self-similar physical region are discussed in more detail.