The dynamics of a universe dominated by a self-interacting nonminimally
coupled scalar field are considered. The structure of the phase space and
complete phase portraits are given. New dynamical behaviors include
superinflation ($\dot{H}>0$), avoidance of big bang singularities through
classical birth of the universe, and spontaneous entry into and exit from
inflation. This model is promising for describing quintessence as a
nonminimally coupled scalar field.Comment: 4 pages, 2 figure
A spatially flat FLRW universe (motivated by inflation) is studied; by a dimensional reduction of the dynamical equations of scalar field cosmology, it is demonstrated that a spatially flat universe cannot exhibit chaotic behaviour. The result holds when the source of gravity is a non-minimally coupled scalar field, for any self-interaction potential and for arbitrary values of the coupling constant with the Ricci curvature. The phase space of the dynamical system is studied, and regions inaccessible to the evolution are found.
The topology of the forbidden regions, their dependence on the parameters, the fixed points and their stability character, and the asymptotic behaviour of the solutions are studied. New attractors are found, in addition to those known from the minimal coupling case, certain exact solutions are presented and the implications for inflation are discussed. The equation of state is not prescribed a priori
, but rather is deduced self-consistently from the field equations.
A natural embedding for most time-continuous systems is presented. A set of nonlinear transformations is shown to split the whole family of dynamical systems into equivalence classes. The degree of nonlinearity is not a relevant characteristic of these classes of ordinary differential equations (ODE's). In each class there exist two particular simple canonical forms into which any such ODE can be cast. We show on an example how to use these canonical forms to find nontrivial integrability conditions.
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